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Table of Contents
RSICC Abstract
Section 1: README4C2.txt
Section 2: LANL Memo X-5:RN
Section 3: MCNP - A General Monte Carlo N-Particle...Version 4C

ccc-70 1

MCNP4C2

OAK RIDGE NATIONAL LABORATORY

managed by

UT-BATTELLE, LLC

for the

U.S. DEPARTMENT OF ENERGY

RSICC COMPUTER CODE COLLECTION

MCNP4C2

Monte Carlo N-Particle Transport Code System

Contributed by:

Los Alamos National Laboratory

Los Alamos, New Mexico

RADIATION SAFETY INFORMATION COMPUTATIONAL CENTER

Legal Notice: This material was prepared as an account of Government sponsored work and describes a code

system or data library which is one of a series collected by the Radiation Safety Information Computational

Center (RSICC). These codes/data were developed by various Government and private organizations who

contributed them to RSICC for distribution; they did not normally originate at RSICC. RSICC is informed that

each code system has been tested by the contributor, and, if practical, sample problems have been run by

RSICC. Neither the United States Government, nor the Department of Energy, nor UT-BATTELLE, LLC,

nor any person acting on behalf of the Department of Energy or UT-BATTELLE, LLC, makes any warranty,

expressed or implied, or assumes any legal liability or responsibility for the accuracy, completeness, usefulness

or functioning of any information code/data and related material, or represents that its use would not infringe

privately owned rights. Reference herein to any specific commercial product, process, or service by trade name,

trademark, manufacturer, or otherwise, does not necessarily constitute or imply its endorsement,

recommendation, or favoring by the United States Government, the Department of Energy, UT-BATTELLE,

LLC, nor any person acting on behalf of the Department of Energy or UT-BATTELLE, LLC.

Distribution Notice: This code/data package is a part of the collections of the Radiation Safety Information

Computational Center (RSICC) developed by various government and private organizations and contributed

to RSICC for distribution. Any further distribution by any holder, unless otherwise specifically provided for

is prohibited by the U.S. Department of Energy without the approval of RSICC, P.O. Box 2008, Oak Ridge,

TN 37831-6362.

Documentation for CCC-701/MCNP4C2 Code Package

PAGE

RSICC Computer Code Abstract ...................................................... iii

E. Selcow, LANL, “README4C2.txt” (June 6, 2001) .................................. Section 1

J. S. Hendricks, “MCNP4C2,” LANL Memo X-5:RN (U)-JSH-01-01 (30 January, 2001) ......... Section 2

J. F. Briesmeister, Ed., “MCNP - A General Monte Carlo N-Particle Transport Code, Version 4C,”

LA-13709-M (April 2000) ...................................................... Section 3

(June 2001)

iii

RSICC CODE PACKAGE CCC-701

1. NAME AND TITLE

MCNP4C2: Monte Carlo N-Particle Transport Code System.

AUXILIARY PROGRAMS

PRPR: Pre-processor for Extracting the Various Hardware Versions of MCNP and other

codes.

MAKXSF: Preparer of MCNP Cross-Section Libraries.

RELATED DATA LIBRARY

MCNP4C2 includes a test library of cross sections for running the sample problems. The DLC-

200/MCNPDATA code package includes data for use with MCNP and is distributed with the code for

the convenience of users. A new LA150U photonuclear library of particle emission data for nuclear

events from incident neutrons, protons and photons with energies up to 150 MeV is included in the

MCNP4C2 package. The following twelve isotopes have photonuclear evaluations in LA150U: C-12,

O-16, Al-27, Si-28, Ca-40, Fe-56, Cu-63, Ta-181, W-184, Pb-206, Pb-207, and Pb-208.

2. CONTRIBUTOR

Diagnostics Applications Group, Los Alamos National Laboratory, Los Alamos, New Mexico.

3. CODING LANGUAGE AND COMPUTERS

Fortran 77 or 90 and C; Unix workstations, Intel-based PCs, and Cray (C00701/ALLCP/00).

4. NATURE OF PROBLEM SOLVED

MCNP is a general-purpose, continuous-energy, generalized geometry, time-dependent, coupled

neutron-photon-electron Monte Carlo transport code system. MCNP4C2 is an interim release of

MCNP4C with distribution restricted to the Criticality Safety community and attendees of the LANL

MCNP workshops. The major new features of MCNP4C2 include:

* Photonuclear physics.

* Interactive plotting.

* Plot superimposed weight window mesh.

* Implement remaining macrobody surfaces.

* Upgrade macrobodies to surface sources and other capabilities.

* Revised summary tables.

* Weight window improvements

See the MCNP home page more information http://www-xdiv.lanl.gov/XCI/PROJECTS/MCNP

with a link to the MCNP Forum. See the Electronic Notebook at http://www-rsicc.ornl.gov/rsic.html

for information on user experiences with MCNP.

5. METHOD OF SOLUTION

MCNP treats an arbitrary three-dimensional configuration of materials in geometric cells bounded

by first- and second-degree surfaces and some special fourth-degree surfaces. Pointwise continuous-

energy cross section data are used, although multigroup data may also be used. Fixed-source adjoint

calculations may be made with the multigroup data option. For neutrons, all reactions in a particular

cross-section evaluation are accounted for. Both free gas and S(alpha, beta) thermal treatments are

used. Criticality sources as well as fixed and surface sources are available. For photons, the code takes

account of incoherent and coherent scattering with and without electron binding effects, the possibility

of fluorescent emission following photoelectric absorption, and absorption in pair production with local

emission of annihilation radiation. A very general source and tally structure is available. The tallies

have extensive statistical analysis of convergence. Rapid convergence is enabled by a wide variety of

variance reduction methods. Energy ranges are 0-60 MeV for neutrons (data generally only available up

to 20 MeV) and 1 keV - 1 GeV for photons and electrons.

6. RESTRICTIONS OR LIMITATIONS

None noted.

7. TYPICAL RUNNING TIME

The 32 test cases ran in ~4 minutes on a Pentium III 550 MHz in a DOS window of WindowsNT

and in ~6 minutes on an IBM 43P-260.

8. COMPUTER HARDWARE REQUIREMENTS

MCNP is operable on Cray computers under UNICOS, workstations or PC’s running Unix or

Linux, and Windows-based PC’s. Executable files for Windows-based PC’s are provided for running

on Pentium computers. Expanding the code system requires 50 MB, and expanding the ASCII cross

sections require 880 MB of hard disk space.

9. COMPUTER SOFTWARE REQUIREMENTS

Compilation of MCNP requires both FORTRAN and ANSI C standard compilers for Unix and

under Windows for the dynamic memory option (pointer) with DVF. Executables are included for

Windows users. PVM is required for multiprocessing on a cluster of workstations and can be

downloaded from www.netlib.org. Scripts are provided for installation on both PC and Unix systems.

The PC Windows distribution includes MCNP and MAKXSF executables. For the PC Windows

systems, the supported operating systems are Windows NT/9x. The included executables also run under

Windows 2000. Both DVF and LF95 compilers are supported. The Lahey Fortran 95 5.50h LF95 PRO

v5.5 Professional Edition compiler was used to create an executable with MDAS=4,000,000. The

Digital Visual Fortran 6.0 Professional Edition and Microsoft Visual C++ 6.0 Professional Edition

compilers were used to create MCNP executables with the dynamic memory option (pointer). PC

executables linked with the standard DVF and Lahey graphics are included, and PC executables linked

with X11 graphics routines are also included. To use the later, X11 must be installed on your PC. An

X-windows server is required to display the X11 graphics. Suggested servers include ReflectionX,

Exceed, and X-Deep/32. RSICC tested this release on the following systems:

1. AIX 4.3.3 (IBM 43P-260) with XL C/C++ 4.4; XL Fortran 6.1

2. Redhat Linux Version 6.1 on 450 MHz Pentium III (9 nodes) with g77 0.5.24

(Case 14 fails; runs correctly with g77 0.5.25.)

3. Sun Solaris 2.6 on UltraSparc 60 using F77 Version 5.0 and C/C++ Version 5.0

4. HP B1000 (PA-8500) under HP-UX 10.20 with FORTRAN 77 V0.20 and HP C V10.32.00

5. DEC 500 AU under Digital Unix 4.0D with DEC Fortran 5.1-8 and DEC C 5.6-075

6. SGI MIPS R10000 (225MHz) under IRIX 6.5.5 with MIPS Fortran 77 Version 7.3

7. Pentium III 550MHz in a DOS window of Windows NT4 with Digital Visual Fortran

professional Edition 6.0 Fortran 90 compiler with QuickWin graphics

8. Pentium III 550MHz in a DOS window of Windows NT4 with Lahey/Fujitsu Fortran 95 --

LF95 Version 5.50h Fortran compiler with Winteracter graphics.

10. REFERENCES

The Adobe Acrobat Reader freeware is available from http://www.adobe.com to read and print the

electronic documentation.

a. included documentation in electronic format on the CD in DOC/C701DOC.PDF:

E. Selcow, LANL, “README4C2.txt” (June 6, 2001).

J. S. Hendricks, “MCNP4C2,” LANL Memo X-5:RN (U)-JSH-01-01 (30 January, 2001).

J. F. Briesmeister, Ed., “MCNP - A General Monte Carlo N-Particle Transport Code, Version 4C,”

LA-13709-M (April 2000).

b. background information:

D. J. Whalen, D. A. Cardon, J. L. Uhle, J. S. Hendricks, “MCNP: Neutron Benchmark Problems,”

LA-12212 (November 1993).

C. D. Harmon, II, R. D. Busch, J. F. Briesmeister, R. A. Forster, “Criticality Calculations with

MCNP: A Primer,” LA-12827-M (August 1994).

R. C. Little and R. E. Seamon, “Dosimetry/Activation Cross Sections for MCNP,” LANL Memo

(March 13, 1984).

11. CONTENTS OF CODE PACKAGE

Included are the referenced electronic documents in (10.a) and the source codes, test problems, PC

executables, and installation scripts transmitted on CD in Windows and UNIX format. The ASCII

DLC-200/MCNPDATA data library is included on the distribution media. See the README files for

details on package contents and installation.

12. DATE OF ABSTRACT

June 2001.

KEYWORDS: COMPLEX GEOMETRY; COUPLED; CROSS SECTIONS; ELECTRON;

GAMMA-RAY; MICROCOMPUTER; MONTE CARLO; NEUTRON;

WORKSTATION

___________________________________________________________________________

MCNP4C2 Notes LODDAT: 01/20/01

___________________________________________________________________________

___________________

1.0 Copyright

___________________

MCNP was prepared by the Regents of the University of

California at Los Alamos National Laboratory (the University) under

Contract number W-7405-ENG-36 with the U. S. Department of Energy

(DOE). The University has certain rights in the program pursuant to

the contract and the program should not be copied or distributed

outside your organization. All rights in the program are reserved by

the DOE and the University. Neither the U. S. government nor the

University makes any warranty, express or implied, or assumes any

liability or responsibility for the use of this software.

___________________

2.0 MCNP4C2

___________________

The major new features of MCNP4C2 include:

* Photonuclear physics;

* Interactive plotting;

* Plot superimposed weight window mesh;

* Implement remaining macrobody surfaces;

* Upgrade macrobodies to surface sources and other capabilities;

* Revised summary tables;

* Weight window improvements:

(a) Add weight window scaling factor;

(b) Allow 1 wwg coarse mesh per direction;

(d) Write out normalization constant for mesh windows.

In addition, there are 9 minor new features and 35 corrections.

___________________

3.0 User Support

___________________

A LIMITED amount of free user support is available from

Larry Cox, mcnp@lanl.gov. Users are encouraged to

communicate with other users via the list server,

mcnp-forum@lanl.gov. Our WWW Web site is:

http://www-xdiv.lanl.gov/XCI/PROJECTS/MCNP

________________________

4.0 DISTRIBUTION FILES

________________________

The following files should be present with the MCNP 4C2 distribution:

FILE DESCRIPTION

----------------------------------------------------------------------

Readme This file.

INSTALL Installation controller.

Named INSTALL.BAT for PC Windows systems.

INSTALL.FIX Installation fix file.

MCSETUP.ID Setup FORTRAN code.

PRPR.ID FORTRAN preprocessor code.

MAKXS.ID Cross-section processor source code.

MCNPC.ID MCNP C source code.

MCNPF.ID MCNP FORTRAN source code.

RUNPROB Script file for MCNP verification.

Named RUNPROB.BAT for PC Windows systems.

TESTINP.TAR Compressed input files for MCNP verification.

Named TESTINP.ZIP for PC Windows systems.

TESTMCTL.SYS Compressed tally output files for MCNP verification.

Named TESTMCTL.ZIP for PC Windows systems.

TESTOUTP.SYS Compressed MCNP output files for MCNP verification.

Named TESTOUTP.ZIP for PC Windows systems.

TESTDIR Cross-section directory for MCNP verification.

TESTLIB1 Cross-section data for MCNP verification.

Substitute the appropriate system identifier from the following table

for the "SYS" suffix.

SYSTEM IDENTIFIER SYSTEM IDENTIFIER

----------------------------------------------------------------------

Cray UNICOS ucos DEC ALPHA dec

PC DVF Windows n/a PC Lahey Windows n/a

IBM RS/6000 AIX aix Sun Solaris sun

HP-9000 HPUX hp SGI IRIX sgi

PC LINUX linux

The INSTALL.FIX file is used to implement corrections to either the MCNP

source or the MAKEMCNP script. The latter is important for future

changes/bugs in compilers and/or operating systems. The format of this

file is provided within INSTALL.FIX, and more details can be found in

Appendix C of the MCNP manual. The MCSETUP utility is a user-friendly

interface for creating system-dependent files. The remaining files in

the first group are MCNP related source code, and the second group of

files are used for MCNP verification (i.e. running the 32 MCNP test

problems).

For PC Windows systems, one additional utility has been included: the

archive utility PKUNZIP.EXE.

________________________

5.0 SYSTEM REQUIREMENTS

________________________

Software Requirements:

(1) A FORTRAN 77 compiler. The supported compiler for each system is

listed in the 1.1 MCSETUP menu (see below). The PC DVF compiler is

FORTRAN 90 and the PC Lahey compiler is FORTRAN 95.

(2) A C compiler with an ANSI C library is required for UNIX system timing,

as well as the X-Window graphics and dynamic memory allocation options.

On PC Windows systems, the Microsoft Visual C++ compiler is required

to implement the X-Window graphics and dynamic memory allocation options.

A Bourne-shell command interpreter is needed to execute the installation

Script on UNIX systems.

Hardware Requirements:

Minimum Recommended

RAM 2 Mbytes 16 Mbytes

Disk Space 50 Mbytes 100 Mbytes

________________________

6.0 GETTING STARTED

________________________

Before proceeding, read the "IMPORTANT ADDITIONAL INFORMATION" section below.

On all systems, initiate the installation controller with the following

commands:

COMMANDS COMMENT

---------------------------------------------------------------------

chmod a+x install UNIX systems - SYS keyword

./install SYS mcnp given in the table above.

---------------------------------------------------------------------

install mcnp PC Windows systems

The MCSETUP utility is initiated first. Simply alter the main menu

according to the MCNP options you desire. Note the following:

(1) Section 1.1 of the main menu SHOULD BE ALTERED FIRST.

This sets the appropriate computer system which in turn selects

suitable defaults for the remaining options.

(2) Default responses are indicated, and these will be activated

by typing a <CR>. Additional options are also included,

from which the user can select the desired configuration.

Several user-specific parameters, such as the cross section

data path, graphics library path, library name, and

include path may be also entered.

(3) If the dynamic memory option is turned "off", an appropriate value

for the MDAS parameter should be set (default is mdas=4000000).

In general MDAS should be greater than 100000 and less than

(R-2)/4 * 1000000, where R is your available RAM in Mbytes.

(4) More information on the setup options is available in the

MCNP manual. If you are unsure as to the graphics libraries

available on your system or their location, contact your system

administrator. Default library names and directory paths are

supplied by the MCSETUP utility; however these may not be

applicable to your system. An error message is displayed

if needed libraries could not be located. Included in

this error message is the expected library name and path.

When done altering the main menu, use the PROCESS command to continue

the installation. The MCSETUP utility creates three system dependent

files: the PRPR C patch file (PATCHC), the PRPR FORTRAN patch file (PATCHF),

and the MAKEMCNP script. PATCHF and PATCHC include the *define preprocessor

directives that reflect the options chosen in the execution of the MCSETUP

code. MCSETUP also creates an ANSWER file which contains the MCSETUP input

for future installations. This file reflects all options chosen during the

initial installation and can be used in future installations by

COMMAND(S) COMMENT

---------------------------------------------------------------------

./install mcnp SYS < answer UNIX systems

---------------------------------------------------------------------

install mcnp < answer PC Windows systems

Next, the installation controller initiates the MAKEMCNP script which

creates the MCNP executable. System differences can result in

compilation errors (e.g., unsatisfied externals). If this occurs,

contact MCNP@LANL.GOV regarding a fix. In most cases a two line fix

can be added to your INSTALL.FIX file to rectify the situation (the

INSTALL.FIX file included with the distribution contains examples of

such fixes).

The last section of the installation controller performs MCNP

verification by running the 32 MCNP test problems. If this step is

to be omitted, rename the RUNPROB file with some other name (e.g.,

RUNPROB.ORG).

On most dedicated systems, compilation time is roughly 15-30 minutes

and verification an additional 20-40 minutes.

___________________

7.0 UPON COMPLETION

___________________

A successful compilation generates an MCNP executable, called mcnp on

UNIX systems and mcnp.exe on PC Windows systems. The MCNP FORTRAN

source is split into subroutines, called subroutine.f on UNIX and

subroutine.for on PC Windows, and is placed in the flib directory.

The object code for individual subroutines is placed in the olib directory.

A normal completion results in the following message:

Installation complete - see Readme file.

A log of the installation process is written to the INSTALL.LOG file.

An abnormal completion results in one of the following messages:

SETUP ERROR OR USER ABORT.

COMPILATION ERROR - see INSTALL.LOG file.

VERIFICATION ERROR - see INSTALL.LOG file.

The cause of the error can be found in the INSTALL.LOG file.

Upon completion of MCNP verification, 32 difm?? files will exist

containing the MCNP tally differences between your runs and the

standard. Similarly, the 32 difo?? files will contain the MCNP output

file differences between your runs and the standard. Exact tracking

is required for MCNP verification, thus significant differences

(i.e. other than round-off in the last digit) may prove to be serious

(e.g. compiler bugs, etc.). In such cases the INSTALL.LOG file should

be reviewed to ensure that the 32 test problems ran successfully.

On all systems, EXACT tracking of ALL the test problems is required

to verify proper code installation. If you do not track exactly, or the code

crashes while running the test problems, try again using a lower optimization,

and eventually completely turn off all optimization. If verification errors

persist without optimization, try compiling without graphics.

Approximately 99% of installation problems are due to compiler

optimization bugs, compiler bugs, bad graphics libraries, or bad operating

system environments.

It should be noted that the results for a 32-bit compilation differ from

those for a 64-bit compilation.

_____________________________________

8.0 IMPORTANT ADDITIONAL INFORMATION

_______________________________________

The install.fix file contains directives to generate debuggable versions of

the code for all the supported systems. In order to activate this capability,

uncomment the specified lines for the system of interest. In particular,

delete the leading "c" plus one blank space for the indicated number of lines.

________________________

8.1 PC DVF Windows

________________________

For the PC Windows systems, the supported operating systems are

Windows NT/9x. The code can be installed and run from a DOS command

line prompt.

The following combination of software packages are required to achieve

full functionality with MCNP on the PC DVF Windows system:

___________________________________________________________________________

PACKAGE VERSION

------- -------

Digital Visual Fortran 6.0

Professional Edition

http://www5.compaq.com/fortran

This product is now known as

Compaq Visual Fortran.

Microsoft Visual C++ 6.0

Professional Edition

http://msdn.microsoft.com/visualc

___________________________________________________________________________

Two graphics systems are supported: X-windows graphics and DVF QuickWin.

It is important that your Path, Include, and Lib environment variables are

set accordingly. See the DVF and Microsoft Visual C++ manuals for appropriate

settings.

The X-windows library, X11, release 6.4, X11R6.4, can be downloaded

free-of-charge from the web-site "http://www.x.org".

This site contains the code needed to generate the X-windows libraries

to display MCNP geometry, cross section and tally plots. In addition, an

X-windows server is required to display the graphics. Suggested servers

include ReflectionX, Exceed, and X-Deep/32. It should be noted that the

development versions of the X-servers, which may be more expensive than the

standard versions, also include the additional software necessary to generate

the X11R6 development libraries. For this application, a custom installation

of the X-servers is recommended.

The following are guidelines for installing the X-Windows graphics

from the www.x.org download.

It is first necessary to unpack the X11R6.4 source code release

distribution (use WinZip), compile it, and then install it. The distribution

includes imake files, library files, fonts, language support files, auxiliary

programs, as well as detailed documentation. The imake utility, included in

the distribution, creates system-specific Makefiles from system-independent

Imakefiles. The system-dependent configuration parameters are defined in the

file site.def. There is a sample site.def (called site.sample) included in

the distribution. Copy this file to site.def and add the following as the

second line in the file:

#define RmTreeCmd del /q /s

When installing X11R6.4, is it necessary to create the following

subdirectories a priori:

\exports\include

\exports\lib

Follow the directions in the documentation to build the libraries, and type

the following line in your local directory:

nmake World.Win32 > world.log

After the build has had a successful completion, install the software

by typing:

nmake install > install.log

The generated files will include X11.lib and Xlib.h, which are required for

the X-Windows graphics version on PC Windows systems.

The MCSETUP utility will query the user on the graphics library path, library

filename, and include path only for the X-windows graphics option for the PC

Windows systems. There are default graphics paths, libraries, and include

paths which can be changed upon installation.

In addition, on all PC Windows systems, the graphics plots can be saved to

a postscript file using the FILE command at the PLOT or MCPLOT prompt.

These postscript files can be sent to any postscript-ready printer

for printing in color or black and white.

The archive utility PKUNZIP.EXE can also be downloaded free-of-charge as a

Shareware version:

http://www.pkware.com

________________________

8.2 PC LINUX

________________________

The dynamic memory option (pointer) is not currently available with

the LINUX system with the supported operating system and compiler.

For the LINUX system, using Redhat 6.0, there is a known bug

with the g77 compiler, version 05.24. Installation and execution

with this compiler version results in a verification error; the

code fails to execute test problem 14, which uses the

like-but construct. This bug has been rectified in version 05.25,

which we support.

For the LINUX system, the fsplit utility is available to be downloaded

free-of-charge from the following web-site.

http://imsb.au.dk/~mok/linux/dist/fsplit-5.5-1.i386.html

In order to download the fsplit utility from this site, simply click on

the title text: "fsplit-5.5-1 RPM for i386", and specify the desired path

for storage on your local computer system. This is a RPM (Red Hat Package

Manager) software tool that must subsequently be installed on your

local linux system. You must have rpm on your system, in addition to the

following files:

ld-linux.so.2

libc.so.6

Later versions of these shared object files will also be compatible with this

installation.

________________________

8.3 PC Lahey Windows

________________________

The following combination of software packages are required to achieve

full functionality with MCNP on PC Lahey Windows system:

___________________________________________________________________________

PACKAGE VERSION

------- -------

Lahey Fortran 95 5.50h

LF95 PRO v5.5

Professional Edition

http://www.lahey.com

This product is now known as

Lahey/Fujitsu Fortran 95.

Microsoft Visual C++ 6.0

Professional Edition

http://msdn.microsoft.com/visualc

___________________________________________________________________________

Two graphics systems are supported: X-windows graphics and Lahey Winteracter.

Please see the PC DVF Windows section for additional applicability to

the Lahey Fortran system.

For the Lahey Winteracter graphics, it is necessary to move all open windows

to the periphery of the windows screen in order to be enable visualization

of the plot. In addition, when executing the Lahey Winteracter version, it

is recommended to minimize the number of additional open windows in your

system.

The Lahey Fortran system does not include the fsplit utility.

For LF95, the Fortran 77 source code for the fsplit utility can

be downloaded free-of-charge from the following web-site:

http://members.aol.com/~Draine3/fsplit.html

After downloading the source, compile the source under the Lahey

Fortran 95 compiler, and specify name the executable as fsplit.exe.

Place this file in your local directory file-space when installing the code.

The dynamic memory option (pointer) is not currently available with

the PC Lahey Fortran system with the supported operating systems.

Los Alamos

NATIONAL LABORATORY

memorandum

Applied Physics Division

X-5: Diagnostics Applications

TO/MS: Distribution

From/MS:

John S. Hendricks/X-5 F663

Phone/FAX:

(505)667-6997

Symbol:

X-5:RN(U)-JSH-01-01

Date:

30 January, 2001

Subject: MCNP4C2

MCNP4C2TM1 ’

IS finished. The load date is Zoddat = 01/20/01. MCNP4C2 2 will be released to

RSICC for sponsors, such as the criticality safety community, and others whom we designate.

This MCNP4C2 documentation supersedes the preliminary version 3 released December 22, 2000.

The code has changed since then as required 4 by the MCNP Board of Directors (BoD) at their

January 9, 2001, meeting:

1. Revise interactive geometry plotting to make the “ROTATE”, “COLOR”, and ‘SCALES”

(both options 1 and 2) buttons into toggles rather than immediately redrawing. (JSH)

2. Implement “NoLines” option in interactive plotter so geometry plots can have any combina-

tion of lines for cell boundaries or the weight window mesh. (JSH)

3. Lee Carter’s patch 5 to extend macrobodies to MCTAL files, SSW and SSR surface sources,

event logs and PTRAK was integrated. (LLC)

Summary of New MCNP4C2 Features

Major New Features:

1. Photonuclear physics. (MCW)

2. Interactive plotting. (JSH)

3. Plot superimposed weight window mesh. (JSH)

4. Implement remaining macrobody surfaces. (LLC)

5. Upgrade macrobodies to surface sources and other capabilities. (LLC)

6. Revised summary tables. (MCW/JSH)

7. Weight window improvements:

(a) Add weight window scaling factor. (JSH)

‘MCNP is a trademark of the Regents of the University of California, Los Alamos National Laboratory

‘5. F. Briesmeister, Ed., “MCNP - A General Monte Carlo N-Particle Transport Code, Version 4C,” LA-13709-M,

Los Alamos National Laboratory (April 2000)

‘John S. Hendricks, “MCNP4C2,” X-S:RN(U)-JSH-00-48 (December 22, 2000)

4John S. Hendricks and Gregg C. Giesler, “Jan 9, 2001 MCNP BoD,” X-5:JSH-w-02 (January 9, 2001)

‘John S. Hendricks, “Macrobody Upgrade,” X-5:RN(U)-JSH-01-03 (January 31, ‘2001)

To Distribution

X-S:RN(U)-JSH-01-01

-2-

(b) Allow 1 wwg coarse mesh per direction. (JAF)

(d) Write out normalization constant for mesh windows. (JSH)

30 January, 2001

Minor

New Features:

1. Remove 4B tracking fixes. (JSH)

2. Save particle attributes in stack. (JSH)

3. Shortcut for electrons below cutoff. (KJA)

4. Include bremsstrahlung produced below energy cutoff in photon summary table. Make elec-

tron summary balance. (AS)

5. Warn of unavailable delayed neutrons. (JSH)

6. Print random number index. (JSH)

7. Fatal error for CTME time cutoff and PVM. (JSH)

8. Fatal error if analog capture with alpha. (JSH)

9. Eliminate a DVF Qwin prompt inconvenience. (GWM)

Summary of MCNP4C2 Corrections

Significant Bugs:

1. Wrong record size causes PVM/SSW, SSR combination crash. (LJC)

2. KCODE source overwrites common in PVM mode. (JAF)

3. $20 PVM hangs with positive number of PVM tasks. (JSH)

4. $20 Bad pointers for unresolved resonance treatment. (JSH)

5. $20 Interrupts crash Lahey Fortran executables. (ECS)

6. $20 Bad energies with law 61 scatter and detectors. (JSH)

7. $20 Identical surfaces with reflection or white boundary fail. (LLC)

8. $4 Cannot read datapath on newer PC compilers. (JFB/GWM)

9. $4 Crash if inadequate space for FG:n,p tallies. (CJW/JSH)

10. $4 Torus will not translate. (LLC)

Lesser Bugs and corrections:

1. Corrected net multiplication. (REP)

2. Correct exponential transform. (JSH/TEB)

3. Perturbations wrong with P-group xsecs. (JAF)

4. Better diagnostics for failed source position sampling. (AS)

5. Faulty surface transformation initiation causes crash on tray. (JSH)

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6. Multigroup adjoint puts upper weight cutoff in wrong place in summary table. (JSH)

7. Correct setting of DBCN(8). (REP)

8. Correct error messages (write hangs multitasking). (JSH)

9. Avoid infinite loop (unicos roundoff) if 1 azimuth bin of mesh-based weight window. (TEB)

10. Protect from floating to integer roundoff errors. (JSH)

11. Fix numerical weight window mesh tracking problems. (JAF)

12. Consistency between rectangular and cylindrical mesh tracking. (JAF)

13. Cleanup: unpack IEX in BANKIT. (JSH)

14. Wrong PVM line count. (GWM)

15. More precise error message (KPRINT). (JAF)

16. Solaris F90 bug workaround. (REP)

17. Solaris F90 problems with JSOURC ERPRNT. (REP)

18. Correct harmless 4B plot logic error. (JSH)

19. Remove unused variables. (JAF/TEB/JSH)

20. Typos in comments. (JSH/JAF)

21. Workarounds for Sun F90 compiler. (REP)

22. Correct weight window theta mesh indexing. (TEB/JAF/JSH)

23. Warn of missing material on BBREM (Bremsstrahlung biasing) card. (AS)

24. Print reaction number in event log and PTRAK. (GWM)

25. Eliminate overwrite in MCPLOT. (TBK/JSH)

Major New MCNP4C2 Features

1. Photonuclear Physics.

Morgan White’s Doctoral Dissertation 6 has been integrated into MCNP. 7 Morgan has pre-

pared a detailed description of the photonuclear interface ’ and a brief primer for simulating

photonuclear interactions. ’ Also available are the MCNP Manual Appendix F (data for-

mats) lo and Appendix G (data libraries). rp The photonuclear capability produces both

photoneutrons and photonuclear photons from photon collisions.

6M. C. White, “Development and Implementation of Photonuclear Cross-Section Data for Mutually Coupled Neutron-

Photon Transport Calculations in the Monte Carlo N-Particle (MCNP) Radiation Transport Code,” Los Alamos

National Laboratory report LA-13744-T (July 2000).

‘John S. Hendricks, “MCNP Photonuclear Physics,” X-5:RN(U)-JSH-00-19 (November 13, 2000)

‘Morgan C. White, “User Interface for Photonuclear Physics in MCNP(X),” X-5:MCW-00-88(U) (July 26, 2000)

‘Morgan C. White, A Brief Primer for Simulating Photonuclear Interactions with MCNP(X),” X-5:MCW-00-89(U)

(July 26,200O)

“Morgan C. White, “Class ‘u’ ACE Format - Photonuclear Data,” X-5:MCW-OO-86U (July 26, 2000)

‘lMorgan C. White, “Release of the LA150U Photonuclear Data Library,” X-S:MCW-00-87 (July 26, 2099)

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User Interface Changes:

Mm card:

PNLIB = bd changes the default photonuclear table identifier to id.

Nevl MPNm Photon&ear material card:

MPNm ZApl~i ZApl~z . . .

The MPNm card allows different photonuclear ZAIDs than specified on the Mn card. For

example,

M23 1001.6OC 2 8016.60~ .9 8017.60~ .l

MPN23 0 8016 8016

PH YS: P cad:

Form: PHYS:P EMCPF IDES NOCOH PNB

PNB = -1 Analog photonuclear particle production

= 0 No photonuclear particle production

= 1 Biased photonuclear particle production

The user interface changes are described in more detail in References 2, 3 and 4.

2. Interactive Plotting.

MCNP4C2 introduces interactive point-and-click geometry plotting I2 for all systems with

XLIB graphics (basically, everything.) Figure 1 displays 3-cell macrobody geometry with

interactive geometry plot legends and buttons. The legend for the plot is in the upper left

hand corner and is unchanged from MCNP4C. All the other (red) markings in the margin

are commands for manipulating the plot.

On the top horizontal legend, UP, RT, DN, LF move the plot frame to the right, left, or up

or down. The origin (center) of the plot can be moved by clicking “Origin” and then clicking

the new location of the origin within the picture. “.l .2 Zoom 5. 10.” enables zooming in and

out. For example, if you click “5.” and then any point within the picture, the plot zooms in

to that point by a factor of 5.

The “Edit” command in the left legend provides information for the current plot cell quantity

at the cursor point. It is followed by black lettering identifying the present cell and coordinates

of wherever the last click was in the picture. The commands “CURSOR” and “SCALES”

are the same as MCNP4C, namely form a cursor to zoom into a part of the picture’ or add

scales showing the dimensions of the plot. “WW MESH” is described in the next section.

“ROTATE” rotates the picture 90”. “PostScript” creates a PostScript publication quality

picture in the file plotm.ps (“FILE” command in MCNP4C.) “COLOR” is a toggle to turn

off colors and produce a line drawing only. “XY YZ ZX” can be clicked to get MCNP4C PX,

PZ, or PY plots. “LABEL” controls surface and cell labels.

laJohn S. Hendricks, “Point-and-Click Plotting with MCNP,” Radiation Protection for Our National Priorities,

Spokane, Washington, p. 313-315 (September 17-21, 2000)

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The right legend lists plot cell quantities. If “ccl” is clicked, then the cell labels (“LABEL”)

will be cell numbers, If “imp” is clicked then the cell labels will be importances. The particle

type is controlled by “PAR” in the right margin, and “N” in the right margin controls the

number on the cell quantity. For example, “wwn3:p” would provide photon weight windows

in the 3rd energy group and be clicked in using the “wwn”, “P”, and “N” in the right margin.

The lower legend controls the plots. ‘<Redraw” redraws the picture in case part of it got

cropped or otherwise needs to be refreshed. “Plot>” returns control to the command window

so that plot commands can be entered in the old MCNP4C command style. “End” terminates

the plot session. Command style commands can also be entered in the Plot Window by click-

ing in the lower left hand corner where it says “Click here or picture or menu.” The lower

left legend also suggests what further action is needed. For example, if you click “Zoom”

the lower left legend will change to tell you to either double click or make your next click

somewhere within the picture.

User Interface Change:

“Interact” is a new plot command to return from the command window mode to the point-

and-click mode.

3. Plot Superimposed Weight Window Meeh.

Figure 1 also shows the new plotting of the superimposed weight window mesh. In problems

where the weight window mesh is input from the WWINP file the point-and-click button

“MESH off” appears. It can be toggled to “WW MESH” to get the lines of the mesh-based

weight window boundaries. l3 i4 Both the XYZ rectangular and the RZ8 cylindrical meshes

can be plotted in any arbitrary combination of mesh and plot orientations. In the plot com-

mand window mode the PLOT) command is meshpl N where N = O/1/2/3 = No Lines /

CellLine / WW MESH/ WWSCell.

To plot the values of the mesh windows, click wwn in the right margin, toggle par and N in

the lower right margin to get the weight window particle type and number, and then click

the cell label entry (LABEL 2nd parameter, lower left).

User Interface Change:

“Meshpl N” is a new plot command for problems where a WWINP file is input. N = -l/O/l =

No Lines / MESH off / WW MESH. The interactive plotting buttons are No Lines / MESH

off / WW MESH which appear only if a WWINP file is read in.

4. Implement Remaining Macrobody Surfaces.

MCNP4C introduced five macrobodies: SPH, BOX, RPP, RCC, RHP/HEX. Lee Carter has

added five more I5 to MCNP4C2:

13John S. Hendricks, “Plotting Superimposed Meshes in MCNP,” X-5:RN(U)-JSH-01-04 (December 21, 2000)

l*John S. Hendricks, “Mathematics for Plotting Superimposed Meshes in MCNP,” X-5:RN(U)-JSH-01-04 (February

5, 2000)

“John S. Hendricks, “Extended Macrobodies,” X-5:RN(U)-JSH-00-32 (September 6, 2000)

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REC Right Elliptical Cylinder

TRC Truncated Right-angle Cone

ELL ELLipsoid

WED WEDge

ARB ARBitrary polyhedron

User Interface Change:

REC vx vy vz

Hx Hy Hz Vlx Vly VIZ v2x v2y v2z

30 January, 2001

where Vx Vy Vz = x,y,z coordinates of bottom cylinder

Hx Hy Hz = cylinder axis height vector

Vlx Vly Viz = ellipse major axis vector (normal to Hx Hy Hz)

v2x v2y v2z = ellipse minor axis vector (orthogonal to H and Vl)

If there are IO entries instead of 12, the 10th entry is the minor

axis radius, where the direction is determined from the cross product

of H and vi.

Example: REC 0 -5 0 0 10 0 4 0 0 2

a IO-cm high elliptical cylinder about the y-axis with

the center of the base at x,y,z=O,-5,0 and with

major radius 4 in the x-direction and minor radius 2

in the z-direction.

TRC: Truncated Right-angle Cone

TRC Vx Vy Vz Hx By Hz RI R2

where Vx Vy Vz = x,y,z coordinates of botto? of truncated cone

Hx Hy Hz = cone axis height vector

Rl = radius of lower cone base

R2 = radius of upper cone base

Example: TRC -500 1000 42

a IO-cm high truncated cone about the x-axis with the

center of the 4 cm radius base at x,y,z = -5,O,O and with

the 2 cm radius top at x,y,z = 5,0,0

ELL: ELLipsoid

ELL Vlx Vly Viz v2x v2y v2z Rm

If Rm > 0:

Vlx Vly VIZ = 1st foci coordinate

v2x v2y v2z = 2nd foci coordinate

Rm = length of major axis

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If Rm < 0:

-7-

Vlx Vly Viz = center of ellipsoid

V2x V2y V2z = major axis vector (length = major radius)

Rm = minor radius length

30Januar~y 2001

Examples: ELL 00-2 002 006

ELL 0 0 0 003 2

an ellipsoid at the origin with major axis of length 6

in the z-direction and minor axis radius of length 4

normal to the z-axis

WED: Wedge

WED vx vy vz Vlx Vly VIZ

v2x v2y v2z v3x v3y v3z

Vx Vy Vz = vertex.

Vlx Vly Viz = vector of 1st side of triangular base

V2x V2y V2z = vector of 2nd side of triangular base

V3x V3y V3z = height vector

A right-angle wedge has a right triangle for a base defined

by VI and V2 and a height of V3.

The vectors Vl, V2, and V3 are orthogonal to each other.

Example: WED 00-6 400 030 0012

a 12 cm high wedge with vertex at x,y,z = O,O,-6.

The triangular base and top are a right triangle

with sides of length 4 (x-direction) and 3 (y-direction)

and hypotenuse of length 5.

ARB: ARBitrary polyhedron

ARB ax ay az bx by bz cx cy cz . . . hx hy hz Nl N2 N3 N4 N5 N6

There must be 8 triplets of entries input for the ARB to describe the (x,y,z) of the corners,

although some may not be used (just use zero triplets of entries). These are followed by six

more entries, N, which follow the prescription: each entry is a 4 digit integer that defines

a side of the ARB in terms of the corners for the side. For example, the entry 1278 would

define this plane surface to be bounded by the lst, 2nd, 7th, and 8th above triplets (corners).

Since three points are sufficient to determine the plane, only the lst, 2nd, and 7th corners

would be used in this example to determine the plane. The distance from the plane to the

fourth corner (corner 8 in the example) is determined by MCNP. If the absolute value of this

distance is greater than P.e-6, an error message is given and the distance is printed in the outp

file along with the (x,y,z) that would lie on the plane. If the 4th digit is zero, the fourth point

is ignored. For a four sided ARB, 4 non-zero 4-digit integers (last digit is zero for four sided

since there are only 3 corners for each side) are required to define the sides. For a five sided

ARB, 5 non-zero 4-digit integers are required, and 6 non-zero 4-digit integers are required for

a six sided ARB. Since there must be 30 entries altogether for an ARB (or MCNP gives an

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error message), the last two integers are zero for the four sided ARB and the last integer is

zero for a five sided ARB.

Example: ARB -5 -10 -5 -5 -IQ 5

5 -10 -5 5

-10 5

0 12 0 000

00 0 0 0 0 1234 1250 1350 2450 3450 0

a 5-sided polyhedron with corners at x,y,z = (-5,-IO,-59,

(-5,-10,5),(5,-IO,-5),(5,-10~5),(0,12,0) and planar facets

constructed from corners 1234, etc.

Facet numbering:

REC:

TRC:

ELL:

WED:

ARB:

I Elliptical cylinder

2 Plane normal to end of Hx Hy Hz

3 Plane normal to beginning of Hx Hy Hz

1 Conical surface

2 Plane normal to end of Hx Hy Hz

3 Plane normal to beginning of Hx Hy Hz

Treated as regular surface, so no facet

I Slant plane including top and bottom hypotenuses

2 Plane including vectors V2 and V3

3 Plane including vectors Vl and V3

4 Plane includng vectors VI and V2 at end of V3

(top triangle)

5 Plane includng vectors VI and V2 at beginning of V3

(bottom triangle, including vertex point)

I plane defined by corners Nl

2 plane defined by corners N2

3 plane defined by corners N3

4 plane defined by corners N4

5 plane defined by corners N5

6 plane defined by corners N6

5. Upgrade macrobodies to surface sources and other capabilities.

Lee Carter upgraded5 MCNP macrobody capability to

e Allow macrobody facets on SSW surface source writes and SSR surface source reads;

Allow surface source facets on SF (surface

flagging)

tally cards;

Print surface facets in the event log output and PTRAK files.

Print surface facets in the MCTAL file.

6. Revised Summary Tables.

Morgan White proposed (and the 7/25/00 MCNP Board of Directors meeting approved)

sweeping changes in the summary tables and provided a good first-cut rewrite. I have further

rewritten much ofthe summary table arrays and output asillustratedin Figure 2. The main

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changes are Print Table 130 which has a new horizontal format for cells so that the increasing

number of events and reactions can be vertical. Print table 140 separates photonuclear and

photoatomic events. The problem summary also regroups events and adds photonuclear

interactions.

7. Weight Window Improvements.

The following improvements have been made for the weight window and weight window

generator variance reduction methods.

(a) Add weight window scaling factor. Now input windows may be multiplied by a user-

specified constant (7th entry on WWP card); l6

(b) Allow 1 superimposed mesh weight window coarse mesh per direction and make the

default 1 fine mesh in each direction; I7

(d) Write out normalization constant used in generating weight windows (usually half the

average source weight) for mesh windows.

User Interface Changes:

WWP:n card, new 7th entry is multiplicative constant for all lower weight bounds on WWNim

cards or WWINP file mesh-based windows of particle type n.

WWG card 9th entry flags undocumented developmental recursive Monte Carlo feature.

MESH card defaults are now 1 fine mesh per coarse mesh and now 1 coarse mesh per direction

is allowed.

Description of Minor New Features

1. Remove 4B tracking fixes. The 20th entry on the DBCN card now causes MCNP4C2 to track

MCNP4C. (JSH)

2. Save particle attributes in stack. Morgan White in his photoneutron patch proposed a subrou-

tine to put particle descriptors (GPBLCM, JPBLCM and sometimes UDT arrays) in a stack

while photonuclear events took place. This functionality has been generalized and applied

wherever it is needed. (JSH)

3. Shortcut for electrons below cutoff. If electrons are below the electron energy cutoff they

do not produce bremsstrahlung photons as in MCNP4C. This speeds the code but affects

tracking of MCNP test problem 23. (KJA)

4. Include bremsstrahlung produced below energy cutoff in the photon summary table and make

electron summary balance. Ken Adams’ MCNP4C electron enhancements deliberately let

the electron summary table be out of balance in order to show energy lost to bremsstrahlung

production below the photon energy cutoff. (AS) l8 has put the electron table back in

balance and shows the bremsstrahlung photons not produced below the photon energy cutoff

‘sThomas E. Booth, “Theoretical and Practical Mesh-Based Weight Window Generator Suggestions for MCNP,”

X-5:RN(U)-TEB-00-40 (September 27, 2000)

17.Jef?rey A. Favorite, “Four Enhancements for the MCNP Mesh-Based Weight Window Generator,” X-5:RN(U)-JAF-

00-13

(May 25, 2000)

‘sAvneet Sood, “Electron Summary Table Balance,” X-5:AS-00-153 (U) (December 11, 2000)

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as produced and captured in the photon summary table. (AS)

5. Warn of unavailable delayed neutrons. If delayed neutrons are requested and a fissionable

nuclide does not have delayed neutron data available a warning is issued. Approved at 2/10/00

MCNP BoD. (JSH)

6. Print random number index. In ERRPRN messages (warnings and fatal errors during the

transport of particles) and for large histories at point detectors the random number index

rather than the octal random number itself is printed. Approved at 7/25/00 MCNP BoD.

CJ w

7. Fatal error for CTME time cutoff and PVM. This caused wrong answers because of incomplete

accumulation of task data. Approved at 7/25/00 MCNP BoD. (JSH)

8. Fatal error if analog capture with alpha. With analog capture it was possible for alpha

time absorption to cause very low particle weights which, unchecked by weight cutoff, caused

underflow. Approved at 7/25/00 MCNP BoD. (JSH)

9. Eliminate a DVF Qwin prompt inconvenience that caused the code to wait for a user prompt

on PCs with DVF Qwin. (GWM)

Summary of MCNP4C2 Corrections

Significant Bugs:

1. Wrong record size causes PVM/SSW, SSR combination crash. Surface source reads and

writes simply do not work with PVM multiprocessing. (LJC)

2. KCODE source overwrites common in PVM mode. (JAF)

3. PVM hangs with positive number of PVM tasks. $20 to Neil1 Taylor (UKAEA Fusion,

Abingdon, UK) ls (JSH)

4. Bad pointers for unresolved resonance treatment. $20 to Alfred Hogenbirk, NRG, Petten,

Netherlands. 2o (JSH)

5. Interrupts crash Lahey Fortran executables. $20 to David Seagraves (ESH-4, LANL) 21 (ECS)

6. Bad energies with law 61 scatter and detectors. $20 to Chikara Konno (JAERI, Japan). 22

(JSH)

7. Identical surfaces with reflection or white boundary fail. $20 to Bruce Wilkin (AECL Re-

search, Chalk River, Ontario, Canada) 23 (LLC)

8. Cannot read datapath on newer PC compilers. $4 to Nick Savin (Westinghouse Savannah

River, Aiken, SC) 24 (JFB/GWM)

“John S. Hendricks, “MCNP Cash Award,” X-5:JSH-00-155 (December 20, 2000)

2030hn S. Hendricks, “MCNP Cash Award,,, X-5:JSH-00-53 (April 24, 2000)

‘lElizabeth C. Selcow, “MCNP Cash Award,” X-5:ECS-00-101 (August 10, 2000)

“John S. Hendicks, “MCNP Cash Award,,, X-5:JSH-00-127 (October 30, 2000)

23John S. Hendricks, “MCNP Cash Award,” X-5:JSH-00-152 (December 6, 2000)

24John S. Hendricks, “MCNP Cash Award,,’ X-5:JSH-00-150 (November 20, 2000)

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9. Crash if inadequate space for FG:n,p tallies. $4 to Frej Wasastjerna (VTT, Finland) 25

(CJW/JSH)

10. Torus will not translate. (LLC) $4 to Dennis Allen (BNFL, UK) 26 (LLC)

Lesser Bugs and corrections:

1. Correct the net multiplication in the problem summary table 27 (REP)

2. Correct exponential transform. 28 The following are wrong when the exponential transform

(EXP card) is used in MCNP4C: generated mesh-based weight windows, track length h,ff

estimate, track length cy perturbation estimates, summary accounts for the exponential trans-

form, multigroup weight window generation, and the DXTRAN weight cutoffs. Fortunately,

the exponential transform is seldom used for these applications. (JSH/TEB)

3. Perturbations are wrong with one-group multigroup cross section data. 2g (JAF)

4. Better diagnostics for failed source position sampling, namely, print the source distribution

number and the coordinates of the source point. so (AS)

5. Faulty surface transformation initiation causes crash on tray (subroutine TRFMAT). (JSH)

6. Multigroup adjoint puts upper weight cutoff in wrong summary table array. (subroutine

MGACOL) (JSH)

7. Correct setting of random number index (8th entry on DBCN card.) 31 (REP)

8. Error message corrections. Write statements during multitasking cause the code to hang

without proper multitasking lock settings. (JSH)

9. Avoid a UNICOS roundoff error which causes the code to hang in an infinite loop if there is

1 azimuthal bin in the mesh-based weight window. (TEB)

10. Protect from floating to integer roundoff errors by adding nint functions in appropriate places.

(Jfw

11. Fix numerical weight window mesh tracking problems.r7 (JAF)

12. Consistency between rectangular and cylindrical mesh tracking.17 (JAF)

13. Cleanup the unpacking of variable IEX in BANKIT for later use in PTRAK” (JSH)

14. Wrong PVM line count if *if de f,pvm compiler directives. (GWM)

15. More precise error message (subroutine KPRINT).2Q (JAF)

26Christopher J. Werner, “MCNP Cash Award,” X-5:CJW-00-93 (August 3, 2000)

26John S. Hendricks, “MCNP Cash Award,” X-5:JSH-00-128 (October 30, 2000)

27Richard E. Prael, “Reformulation of the New Multiplication Calculation,” X-5:REP-00-14 (January 26, 2000)

28Thomas E. Booth, “Correcting the Exponential Transform in MCNP4C,” X-5:RN(U)-TEB-00-42 (October 17,200O)

2gJefiey A. Favorite, LLAn Error in the MCNP4C Perturbation Capability for Eigenvalue Problems,” X-5:RH(U)-JAF-

00-39 (September 25, 2000)

“Avneet Sood, Ymproved Source Distribution Efficiency Message,” X-5:AS-00-104 (August 15,200O)

31Richard E. Prael, “Inconsistency in Setting Initial Conditions for Random Number Generator,” X-5:REP-00-117

(September 14, 2000)

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16. Solaris F90 bug workaround (block data: n*’ ’ fails). (REP)

17. Solaris F90 problems with ERPRNT call in JSOURC. (REP)

18. Correct harmless 4B plot logic error (subroutine PTOST). (JSH)

19. Remove unused variables (subroutines AVRWGI, KSKCYC, etc.) (JAF/TEB/JSH)

20. Typos in comments (subroutine IPBC, ACALC, EXORDP, etc.). (JSH,JAF)

21. Workarounds for the Sun Solaris F90 compiler. (subroutines MAIN, GXAXIS) (REP)

22. Correct mesh-based weight window theta mesh indexing. (TEB/JAF/JSH)

23. Warn of missing material on BBREM (B remsstrahlung biasing) card. The 1st 49 entries are

energy bins, and the 50th entry onward is materials. If the count is off or the material(s)

omitted, MCNP4C would assume the 1st problem material, sometimes giving wrong answers

without warning. (AS)

24. Print reaction number (MTP) rather than type (NTYN) in event log and PTRAK. (GWM)

25. Eliminate overwrite in MCPLOT. If more than 100 Million histories were run then stars would

partially overwrite the legend NPS print field. (TBK/JSH)

File Location

The MCNP4C2 installation, test, and executable files are located on both open and closed systems

in directories install, test, exe under the following nodes:

cfs get dir=/x5code/mcnp4c2/. . .

hpss get /hpss/mcnp/mcnp4c2/...

Acknowledgement

MCNP4C2 is the collaborative effort of the X-5 Eolus Monte Carlo code development team:

Gregg W. McKinney (Team Leader), Thomas E. Booth, Judith F. Briesmeister, Leland L. Carter,

Lawrence J. Cox, R. Arthur Forster, William B. Hamilton, John S. Hendricks, Russell D. Mosteller,

Richard E. Prael, Elizabeth C. Selcow, Avneet Sood, Stephen White.

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Figure B

MCNP4C2 Interactive Plotter

Plot shows the MCNP4C2 interactive geometry plot

with superimposed weight window mesh and mesh values.

30 January, 2001

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Fignse 2

New MCNP4C2 Output

prd.hn m.mury

rmn terminated when

10000 particle histories Pera doao.

UC1 1.0000H-10

=52 1.0000E-21

might energy

(per smr~e particle)

1.6891E-01 6.94693+00

0. 0.

0. *.20,8E+oo

1.76523+00 i..%o~E-o*

8.61823-01 *.6432E+Ol

2.0186E-03 1.833JE-01

2.*0*0E+00 1.0091E+02

cntot*r

(CO 1.0000E+3*

ece 1.0000E-03

PC1 1.ooooz-10

I752 S.OOOOE-21

30 January, 2001

rango of *ampled 10nrce weights = 1.0000E+OO to 1,0000E+OO

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neutron weight balance in each cell

30January, 2001

print table 130

cell index 1 2

cell number I 2 total

external events:

entering O.OOOOE+OO 3.5482E-03 3.5482E-03

source O.OOOOE+OO O.OOOOE+OO O.OOOOE+OO

energy cutoff O.OOOOE+OO O.OOOOE+OO O.OOOOE+OO

time cutoff O.OOOOE+OO O.OOOOE+OO O.OOOOE+OO

exiting -3.5482E-03 -3.5482E-03 -7.0963E-03

---_----_- ---------- ---_----__

total -3.5482E-03 O.OOOOE+OO -3.5482E-03

variance reduction events:

weight window O.OOOOE+OO

cell importance O.OOOOE+OO

weight cutoff O.OOOOE+OO

energy importance O.OOOOE+OO

dxtran O.OOOOE+OO

forced collisions O.OOOOE+OO

exp. transform O.OOOOE+OO

O.OOOOE+OO

0.0000E+00

O.OOOOE+OO

total O.OOOOE+OO

physical events:

capture -2.4858E-04

(n,xn) 5.979OE-04

loss to (n,xn) -2.9895E-04

fission O.OOOOE+OO

loss to fission O.OOOOE+OO

photonuclear 3.4978E-03

----------

total 3.5482E-03

O.OOOOE+OO

O.OOOOE+OO -2.4858E-04

O.OOOOE+OO 5.9790E-04

O.OOOOE+OO -2.9895E-04

O.OOOOE+OO O.OOOOE+OO

O.OOOOE+OO 3.4978E-03

-------VW- ---------_

O.OOOOE+OO 3.5482E-03

To Distribution

X-s:RN(U)-JSH-01-01

photon weight balance in each cell

-16-

cell index 1 2

cell number f 2 total

external events:

entering O.OOOOE+OO 1.6891E-01 1.689lE-01

source l.OOOOE+OO O.OOOOE+OO 1.0000E+OO

energy cutoff O.OOOOE+OO O.OOOOE+OO O.OOOOE+OO

time cutoff O.OOOOE+OO O.OOOOE+OO O.OOOOE+OO

exiting -1.6891E-01 -1.689lE-01 -3.378fE-01

----__---- ---------- --------__

total 8.3109E-01 O.OOOOE+OO 8.3109E-01

variance reduction events:

weight window O.OOOOE+OO

cell importance O.OOOOE+OO

weight cutoff O.OOOOE+OO

energy importance O.OOOOE+OO

dxtran O.OOOOE+OO

forced collisions O.OOOOE+OO

exp. transform O.OOOOE+OO

O.OOOOE+OO

total O.OOOOE+OO

physical events:

from neutrons 5.7000E-03

bremsstrahlung O.OOOOE+OO

capture -1.7652E+OO

p-annihilation 1.7356E+OO

pair production -8.6782E-01

photonuclear 2.6750E-03

photonuclear abs -2.0186E-03

electron x-rays O.OOOOE+OO

flourescence 5.9956E-02

----------

total -8.3109E-01

O.OOOOE+OO

O.OOOOE+OO 5.7000E-03

O.OOOOE+OO O.OOOOE+OO

O.OOOOE+OO -1.7652E+OO

O.OOOOE+OO 1.7356E+OO

O.OOOOE+OO -8.6782E-01

O.OOOOE+OO 2.6750E-03

O.OOOOE+OO -2.0186E-03

O.OOOOE+OO O.OOOOE+OO

O.OOOOE+OO 5.9956E-02

---------- ---_----__

O.OOOOE+OO -8.3109E-01

30Januar35 2001

print table 130

To Distribution

X-C-C:RN(U)-ASH-01-01

3OJanuary, 2001

To Distribution

X-5:RN(U)-JSH-01-01

-18-

photmmlear activity e* each mclide in each 0011, par smrso particle

onorgy

interval

20.000

16.000

10.000

9.000

8.000

7.000

6.000

5.000

4.000

3.000

2.000

1.000

0.600

0.100

0.010

0.000

total

weight per onorgy par

soxr'co mlllt son?xe noat

6.700003-03 1.30116E-02

0.00000E+00 0.00000E+00

0.00000E+00 0.00000H+00

5.70000E-03 1.30ilSE-02

30January, 2001

1.1827iE+oo 1.797603-08 T.*0816E-01 i.78138B-01

0.00000B+00 0.00000E+00 0.00000E+00 0.00000E+00

0.00000E+00 0.00000E+00 0.00000E+00 0.00000E+00

1.2*2T1E+oo

cm might

dirtribation

To Distribution

X-C:RiV(U)-JSH-ol-01

JSH:jsh

Distribution:

X-5 File

A. R. Heath, X-5, MS F663

T. J. Seed, X-5, MS F663

G. W. McKinney, X-5, MS F663

T. E. Booth, X-5, MS F663

J. F. Briesmeister, X-5, MS F663

L. L. Carter, X-5, MS F663

L. J. Cox, X-5, MS F663

J. D. Court, X-5, MS F663

G. P. Estes, X-5, MS F663

J. A. Favorite, X-5, MS F663

S. C. Frankle, X-5, MS F663

R. A. Forster, X-5, MS F663

W. B. Hamilton, X-5, MS F663

J. S. Hendricks, X-5, MS F663

R. C. Little, X-5, MS F663

R. D. Mosteller, X-5, MS F663

R. E. Prael, X-5, MS F663

C. E. Ragan, X-5, MS F663

R. R. Roberts, X-5, MS F663

E. C. Selcow, X-5, MS F663

A. Sood, X-5, MS F663

C. J. Werner, X-5, MS F663

M. C. White, X-5, MS F663

S. W. White, X-5, MS F663

H. G. Hughes, CCS-4, MS D409

H. Lichtenstein, CCS-4, MS D409

G. C. Giesler, CIC-12, MS B295

D. A. Rutherford, NIS-8, MS B230

-19- 30 January, 2001

18 December 2000 i

LA–13709–M

Manual

MCNPTM–A General Monte Carlo

N–Particle Transport Code

Version 4C

Judith F. Briesmeister, Editor

UC abc

and

UC 700

Issued: March 2000

ii 18 December 2000

An Afﬁrmative Action/Equal Opportunity Employer

DISCLAIMER

This report was prepared as an account of work sponsored by an agency of the United States

Government. Neither the United States Government nor any agency thereof, nor any of their

employees, makes any warranty, express or implied, or assumes any legal liability or

responsibility for the accuracy, completeness, or usefulness of any information, apparatus,

product, or process disclosed, or represents that its use would not infringe privately owned

rights. Reference herein to any speciﬁc commercial product, process, or service by trade name,

trademark, manufacturer, or otherwise, does not necessarily constitute or imply its

endorsement, recommendation, or favoring by the United States Government or any agency

thereof. The views and opinions of authors expressed herein do not necessarily state or reﬂect

those of the United States government or any agency thereof.

18 December 2000 iii

FOREWORD

This manual is a practical guide for the use of our general-purpose Monte Carlo code MCNP. The

ﬁrst chapter is a primer for the novice user. The second chapter describes the mathematics, data,

physics, and Monte Carlo simulation found in MCNP. This discussion is not meant to be

exhaustive---details of the particular techniques and of the Monte Carlo method itself will have to

be found elsewhere. The third chapter shows the user how to prepare input for the code. The fourth

chapter contains several examples, and the ﬁfth chapter explains the output. The appendices show

how to use MCNP on various computer systems and also give details about some of the code

internals.

The Monte Carlo method emerged from work done at Los Alamos duringWorld War II. The

invention is generally attributed to Fermi,von Neumann, Ulam, Metropolis, and Richtmyer. MCNP

is the successor to their work and represents over 450 person-years of development.

Neither the code nor the manual is static. The code is changed as the need arises and the manual

is changed to reﬂect the latest version of the code. This particular manual refers to Version 4C.

MCNP and this manual are the product of the combined effort of many people in the Diagnostics

Applications Group (X-5) in the Applied Physics Division (X Division) at the Los Alamos National

Laboratory.

The code and manual can be obtained from the Radiation Safety InformationComputational Center

(RSICC), P. O. Box 2008, Oak Ridge, TN, 37831-6362

J. F. Briesmeister

Editor

505-667-7277

email: mcnp@lanl.gov

iv 18 December 2000

Unless otherwise indicated, this information has been authored by anemployee or employees of the

University of California, operator of the Los Alamos National Laboratory under Contract No. W-

-7405--ENG--36 with the U.S. Department of Energy. The U.S. Government has rights to use,

reproduce, and distribute this information. The public maycopy and use this information without

charge, provided that this Notice and any statement of authorship are reproduced on all copies.

Neither the government nor the University makes any warranty, express or implied, or assumes any

liability or responsibility for the use of this information.

18 December 2000 v

TABLE OF CONTENTS

CHAPTER 1 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1

I. MCNP AND THE MONTE CARLO METHOD. . . . . . . . . . . . . . . . . . . . . . . . . . . 1

A. Monte Carlo Method vs Deterministic Method . . . . . . . . . . . . . . . . . . . . . . . . 2

B. The Monte Carlo Method . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3

II. INTRODUCTION TO MCNP FEATURES . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4

A. Nuclear Data and Reactions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4

B. Source Specification . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5

C. Tallies and Output. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6

D. Estimation of Monte Carlo Errors. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7

E. Variance Reduction. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9

III. MCNP GEOMETRY . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 13

A. Cells . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 14

B. Surface Type Specification. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 19

C. Surface Parameter Specification. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 19

IV. MCNP INPUT FOR SAMPLE PROBLEM. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 20

A. INP File. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 22

B. Cell Cards . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 23

C. Surface Cards . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 24

D. Data Cards. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 25

V. HOW TO RUN MCNP. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31

A. Execution Line . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 32

B. Interrupts. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 35

C. Running MCNP . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 35

VI. TIPS FOR CORRECT AND EFFICIENT PROBLEMS . . . . . . . . . . . . . . . . . . . . 36

A. Problem Setup. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 36

B. Preproduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 36

C. Production. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 37

VII. REFERENCES . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 38

CHAPTER 2 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1

I. INTRODUCTION . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1

A. History. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2

B. MCNP Structure . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5

C. History Flow . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7

II. GEOMETRY . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9

A. Complement Operator. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9

B. Repeated Structure Geometry. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11

vi 18 December 2000

C. Surfaces. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11

III. CROSS SECTIONS . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 16

A. Neutron Interaction Data: Continuous-Energy and Discrete-Reaction . . . . . 18

B. Photon Interaction Data . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 21

C. Electron Interaction Data . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 23

D. Neutron Dosimetry Cross Sections. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 23

E. Neutron Thermal S(α,β) Tables. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 24

F. Multigroup Tables. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 25

IV. PHYSICS . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 25

A. Particle Weight . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 26

B. Particle Tracks . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 27

C. Neutron Interactions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 27

D. Photon Interactions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 55

E. Electron Interactions. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 63

V. TALLIES . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 76

A. Surface Current Tally . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 78

B. Flux Tallies . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 78

C. Track Length Cell Energy Deposition Tallies . . . . . . . . . . . . . . . . . . . . . . . . 80

D. Pulse Height Tallies . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 83

E. Flux at a Detector . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 85

F. Additional Tally Features . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 95

VI. ESTIMATION OF THE MONTE CARLO PRECISION . . . . . . . . . . . . . . . . . . . 99

A. Monte Carlo Means, Variances, and Standard Deviations . . . . . . . . . . . . . . . 99

B. Precision and Accuracy. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 101

C. The Central Limit Theorem and Monte Carlo Confidence Intervals . . . . . . 103

D. Estimated Relative Errors in MCNP. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 104

E. MCNP Figure of Merit . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 108

F. Separation of Relative Error into Two Components. . . . . . . . . . . . . . . . . . . 109

G. Variance of the Variance . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 111

H. Empirical History Score Probability Density Function f(x) . . . . . . . . . . . . . 113

I. Forming Statistically Valid Confidence Intervals. . . . . . . . . . . . . . . . . . . . . 119

J. A Statistically Pathological Output Example . . . . . . . . . . . . . . . . . . . . . . . . 123

VII. VARIANCE REDUCTION . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 127

A. General Considerations. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 127

B. Variance Reduction Techniques . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 132

VIII. CRITICALITY CALCULATIONS . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 159

A. Criticality Program Flow . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 159

B. Estimation of keff Confidence Intervals and Prompt Neutron Lifetimes . . . 162

C. Recommendations for Making a Good Criticality Calculation . . . . . . . . . . 178

IX. VOLUMES AND AREAS114. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 181

A. Rotationally Symmetric Volumes and Areas . . . . . . . . . . . . . . . . . . . . . . . . 181

B. Polyhedron Volumes and Areas . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 182

18 December 2000 vii

C. Stochastic Volume and Area Calculation . . . . . . . . . . . . . . . . . . . . . . . . . . . 183

X. PLOTTER. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 183

XI. PSEUDORANDOM NUMBERS. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 187

XII. PERTURBATIONS . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 188

A. Derivation of the Operator . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 189

B. Limitations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 195

C. Accuracy . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 196

XIII. REFERENCES . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 197

CHAPTER 3 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1

I. INP FILE. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1

A. Message Block . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1

B. Initiate-Run . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2

C. Continue−Run . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2

D. Card Format . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4

E. Particle Designators . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7

F. Default Values . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8

G. Input Error Messages . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8

H. Geometry Errors . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8

II. CELL CARDS . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10

A. Shorthand Cell Specification . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 12

III. SURFACE CARDS . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 12

A. Surfaces Defined by Equations. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 12

B. Axisymmetric Surfaces Defined by Points. . . . . . . . . . . . . . . . . . . . . . . . . . . 16

C. General Plane Defined by Three Points . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 19

D. Surfaces Defined by Macrobodies . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 19

IV. DATA CARDS . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 22

A. Problem Type (MODE) Card . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 23

B. Geometry Cards . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 23

C. Variance Reduction. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 32

D. Source Specification . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 49

E. Tally Specification . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 73

F. Material Specification Cards. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 107

G. Energy and Thermal Treatment Specification . . . . . . . . . . . . . . . . . . . . . . . 116

H. Problem Cutoff Cards . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 123

I. User Data Arrays. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 126

J. Peripheral Cards . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 127

V. SUMMARY OF MCNP INPUT FILE . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 146

A. Input Cards . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 146

B. Storage Limitations. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 150

viii 18 December 2000

CHAPTER 4 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1

I. GEOMETRY SPECIFICATION . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1

II. COORDINATE TRANSFORMATIONS . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 15

A. TR1 and M = 1 Case: . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 18

B. TR2 and M= −1 Case: . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 19

III. REPEATED STRUCTURE AND LATTICE EXAMPLES . . . . . . . . . . . . . . . . . 20

IV. TALLY EXAMPLES . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 37

A. FMn Examples (Simple Form) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 37

B. FMn Examples (General Form) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 39

C. FSn Examples . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 40

D. FTn Examples . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 42

E. Repeated Structure/Lattice Tally Example. . . . . . . . . . . . . . . . . . . . . . . . . . . 44

F. TALLYX Subroutine Examples . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 47

V. SOURCE EXAMPLES. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 50

VI. SOURCE SUBROUTINE . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 53

VII. SRCDX SUBROUTINE. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 55

CHAPTER 5 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1

I. DEMO PROBLEM AND OUTPUT . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1

II. TEST1 PROBLEM AND OUTPUT. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9

III. CONC PROBLEM AND OUTPUT. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 48

IV. KCODE. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 61

V. EVENT LOG AND GEOMETRY ERRORS. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 99

A. Event Log . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 99

B. Debug Print . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 102

APPENDIX B. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1

I. SYSTEM GRAPHICS INFORMATION. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1

A. X–Windows . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2

II. THE PLOT GEOMETRY PLOTTER . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3

A. PLOT Input and Execute Line Options . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3

B. Plot Commands Grouped by Function . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5

C. Geometry Debugging and Plot Orientation . . . . . . . . . . . . . . . . . . . . . . . . . . . 9

III. THE MCPLOT TALLY AND CROSS SECTION PLOTTER . . . . . . . . . . . . . . . 10

A. Input for MCPLOT and Execution Line Options . . . . . . . . . . . . . . . . . . . . . . 11

B. Plot Conventions and Command Syntax . . . . . . . . . . . . . . . . . . . . . . . . . . . . 12

C. Plot Commands Grouped by Function . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 13

D. MCTAL Files . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 20

E. Example of Use of COPLOT . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 22

APPENDIX C. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1

I. INSTALLING MCNP . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1

18 December 2000 ix

A. On Supported Systems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2

B. VMS System . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4

C. On Other Systems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5

II. MODIFYING MCNP . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8

A. Creating a PRPR Patch File . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8

B. Creating a New MCNP Executable . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11

III. MCNP VERIFICATION . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 12

A. On Supported Systems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 12

B. On VMS System . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 13

IV. CONVERTING CROSS-SECTION FILES WITH MAKXSF . . . . . . . . . . . . . . . 14

APPENDIX D. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1

I. PREPROCESSORS . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1

II. PROGRAMMING LANGUAGE. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1

III. SYMBOLIC NAMES. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2

IV. SYSTEM DEPENDENCE . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3

V. COMMON BLOCKS . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4

VI. DYNAMICALLY ALLOCATED STORAGE . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5

VII. THE RUNTPE FILE. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6

VIII. C FUNCTIONS . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7

IX. REFERENCES . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7

APPENDIX E . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1

I. DICTIONARY OF SYMBOLIC NAMES. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1

II. SOME IMPORTANT COMPLICATED ARRAYS . . . . . . . . . . . . . . . . . . . . . . . 32

A. Source Arrays . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 32

B. Transport Arrays . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 33

C. Tally Arrays . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 35

D. Accounting Arrays . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 41

E. KCODE Arrays. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 46

F. Alpha Arrays. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 48

G. Universe Map/ Lattice Activity Arrays for Table 128 . . . . . . . . . . . . . . . . . . 48

H. Weight Window Mesh Parameters . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 49

I. Perturbation Parameters . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 49

J. Macrobody and Identical Surface Arrays . . . . . . . . . . . . . . . . . . . . . . . . . . . . 51

APPENDIX F . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1

I. Data Types and Classes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1

II. XSDIR— Data Directory File . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2

III. Data Tables . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4

A. Locating Data on a Type 1 Table . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5

B. Locating Data on a Type 2 Table . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10

x 18 December 2000

C. Locating Data Tables in MCNP . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11

D. Individual Data Blocks . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11

IV. Data Blocks for Continuous–Energy and Discrete Neutron Transport Tables. . . . 12

V. Data Blocks for Dosimetry Tables . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 34

VI. Data Blocks for Thermal S(α,β) Tables . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 35

VII. Data Blocks for Photon Transport Tables. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 38

VIII. Format for Multigroup Transport Tables . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 40

IX. Format for Electron Transport Tables. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 52

Appendix G. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1

I. ENDF/B REACTION TYPES . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1

II. S(a,b) DATA FOR USE WITH THE MTm CARD . . . . . . . . . . . . . . . . . . . . . . . . 5

III. MCNP NEUTRON CROSS–SECTION LIBRARIES. . . . . . . . . . . . . . . . . . . . . . . 6

IV. MULTIGROUP DATA FOR MCNP . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 29

V. DOSIMETRY DATA FOR MCNP . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 32

VI. REFERENCES . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 46

Appendix H. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1

I. CONSTANTS FOR FISSION SPECTRA . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1

A. Constants for the Maxwell fission spectrum (neutron-induced). . . . . . . . . . . . 1

B. Constants for the Watt Fission Spectrum . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3

II. FlUX-TO-DOSE CONVERSION FACTORS . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3

A. Biological Dose Equivalent Rate Factors . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4

B. Silicon Displacement Kerma Factors . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5

III. REFERENCES . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7

Appendix I . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1

Appendix J . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1

18 December 2000 xi

MCNP–A General Monte Carlo N–Particle Transport Code

Version 4C

Diagnostics Applications Group

Los Alamos National Laboratory

ABSTRACT

MCNP is a general-purpose Monte Carlo N–Particle code that can be used for neutron, photon,

electron, or coupled neutron/photon/electron transport, including the capability to calculate

eigenvalues for critical systems. The code treats an arbitrary three-dimensional conﬁguration of

materials in geometric cells bounded by ﬁrst- and second-degree surfaces and fourth-degree

elliptical tori.

Pointwise cross-section data are used. For neutrons, all reactions given in a particular cross-section

evaluation (such as ENDF/B-VI) are accounted for. Thermal neutrons are described by both the

free gas and S(α,β) models. For photons, the code takes account of incoherent and coherent

scattering, the possibility of ﬂuorescent emission after photoelectric absorption, absorption in pair

production with local emission of annihilation radiation, and bremsstrahlung. A continuous-

slowing-down model is used for electron transport that includes positrons, k x-rays, and

bremsstrahlung but does not include external or self-induced ﬁelds.

Important standard features that make MCNP very versatile and easy to use include a powerful

general source, criticality source, and surface source; both geometry and output tally plotters; a rich

collection of variance reduction techniques; a ﬂexible tally structure; and an extensive collection

of cross-section data.

xii 18 December 2000

CHAPTER 2

INP File

NOTES:

April 10, 2000 1-1

CHAPTER 1

MCNP AND THE MONTE CARLO METHOD

CHAPTER 1

PRIMER

WHAT IS COVERED IN CHAPTER 1

Brief explanation of the Monte Carlo method.

Summary of MCNP features.

Introduction to geometry.

Description of MCNP data input illustrated by a sample problem.

How to run MCNP.

Tips on problem setup.

Chapter 1 will enable the novice to start using MCNP, assuming very little knowledge of the Monte

Carlo method and no experience with MCNP. The primer begins with a short discussion of the

Monte Carlo method. Five features of MCNP are introduced: (1) nuclear data and reactions, (2)

source speciﬁcations, (3) tallies and output, (4) estimation of errors, and (5) variance reduction.

The third section explains MCNP geometry setup, including the concept of cells and surfaces. A

general description of an input deck is followed by a sample problem and a detailed description of

the input cards used in the sample problem. Section V tells how to run MCNP, VI lists tips for

setting up correct problems and running them efﬁciently, and VII is the references for Chapter 1.

The word “card” is used throughout this document to describe a single line of input up to 80

characters.

I. MCNP AND THE MONTE CARLO METHOD

MCNP is a general-purpose, continuous-energy, generalized-geometry, time-dependent, coupled

neutron/photon/electron Monte Carlo transport code. It can be used in several transport modes:

neutron only, photon only, electron only, combined neutron/photon transport where the photons are

produced by neutron interactions, neutron/photon/electron, photon/electron, or electron/photon.

The neutron energy regime is from 10-11 MeV to 20 MeV, and the photon and electron energy

regimes are from 1 keV to 1000 MeV. The capability to calculate keff eigenvalues for ﬁssile

systems is also a standard feature.

The user creates an input ﬁle that is subsequently read by MCNP. This ﬁle contains information

about the problem in areas such as:

the geometry specification,

the description of materials and selection of cross-section evaluations,the location and

characteristics of the neutron, photon, or electron source,

the type of answers or tallies desired, and

any variance reduction techniques used to improve efficiency.

1-2 April 10, 2000

CHAPTER 1

MCNP AND THE MONTE CARLO METHOD

Each area will be discussed in the primer by use of a sample problem. Remember ﬁve “rules’’ when

running a Monte Carlo calculation. They will be more meaningful as you read this manual and

gain experience with MCNP, but no matter how sophisticated a user you may become, never forget

the following ﬁve points:

1. Define and sample the geometry and source well;

2. You cannot recover lost information;

3. Question the stability and reliability of results;

4. Be conservative and cautious with variance reduction biasing; and

5. The number of histories run is not indicative of the quality of the answer.

The following sections compare Monte Carlo and deterministic methods and provide a simple

description of the Monte Carlo method.

A. Monte Carlo Method vs Deterministic Method

Monte Carlo methods are very different from deterministic transport methods. Deterministic

methods, the most common of which is the discrete ordinates method, solve the transport equation

for the average particle behavior. By contrast, Monte Carlo does not solve an explicit equation, but

rather obtains answers by simulating individual particles and recording some aspects (tallies) of

their average behavior. The average behavior of particles in the physical system is then inferred

(using the central limit theorem) from the average behavior of the simulated particles. Not only are

Monte Carlo and deterministic methods very different ways of solving a problem, even what

constitutes a solution is different. Deterministic methods typically give fairly complete information

(for example, ﬂux) throughout the phase space of the problem. Monte Carlo supplies information

only about speciﬁc tallies requested by the user.

When Monte Carlo and discrete ordinates methods are compared, it is often said that Monte Carlo

solves the integral transport equation, whereas discrete ordinates solves the integro-differential

transport equation. Two things are misleading about this statement. First, the integral and integro-

differential transport equations are two different forms of the same equation; if one is solved, the

other is solved. Second, Monte Carlo “solves” a transport problem by simulating particle histories

rather than by solving an equation. No transport equation need ever be written to solve a transport

problem by Monte Carlo. Nonetheless, one can derive an equation that describes the probability

density of particles in phase space; this equation turns out to be the same as the integral transport

equation.

Without deriving the integral transport equation, it is instructive to investigate why the discrete

ordinates method is associated with the integro-differential equation and Monte Carlo with the

integral equation. The discrete ordinates method visualizes the phase space to be divided into many

small boxes, and the particles move from one box to another. In the limit as the boxes get

April 10, 2000 1-3

CHAPTER 1

MCNP AND THE MONTE CARLO METHOD

progressively smaller, particles moving from box to box take a differential amount of time to move

a differential distance in space. In the limit this approaches the integro-differential transport

equation, which has derivatives in space and time. By contrast, Monte Carlo transports particles

between events (for example, collisions) that are separated in space and time. Neither differential

space nor time are inherent parameters of Monte Carlo transport. The integral equation does not

have time or space derivatives.

Monte Carlo is well suited to solving complicated three-dimensional, time-dependent problems.

Because the Monte Carlo method does not use phase space boxes, there are no averaging

approximations required in space, energy, and time. This is especially important in allowing

detailed representation of all aspects of physical data.

B. The Monte Carlo Method

Monte Carlo can be used to duplicate theoretically a statistical process (such as the interaction of

nuclear particles with materials) and is particularly useful for complex problems that cannot be

modeled by computer codes that use deterministic methods. The individual probabilistic events

that comprise a process are simulated sequentially. The probability distributions governing these

events are statistically sampled to describe the total phenomenon. In general, the simulation is

performed on a digital computer because the number of trials necessary to adequately describe the

phenomenon is usually quite large. The statistical sampling process is based on the selection of

random numbers—analogous to throwing dice in a gambling casino—hence the name “Monte

Carlo.” In particle transport, the Monte Carlo technique is pre-eminently realistic (a theoretical

experiment). It consists of actually following each of many particles from a source throughout its

life to its death in some terminal category (absorption, escape, etc.). Probability distributions are

randomly sampled using transport data to determine the outcome at each step of its life.

Figure 1-1.

Event Log

1. Neutron scatter

Photon Production

2. Fission

Photon Production

3. Neutron Capture

4. Neutron Leakage

5. Photon Scatter

6. Photon Leakage

7. Photon Capture

Incident

Neutron

Void

Fissionable

Material Void

1-4 April 10, 2000

CHAPTER 1

INTRODUCTION TO MCNP FEATURES

Figure 1.1 represents the random history of a neutron incident on a slab of material that can

undergo ﬁssion. Numbers between 0 and 1 are selected randomly to determine what (if any) and

where interaction takes place, based on the rules (physics) and probabilities (transport data)

governing the processes and materials involved. In this particular example, a neutron collision

occurs at event 1. The neutron is scattered in the direction shown, which is selected randomly from

the physical scattering distribution. A photon is also produced and is temporarily stored, or banked,

for later analysis. At event 2, ﬁssion occurs, resulting in the termination of the incoming neutron

and the birth of two outgoing neutrons and one photon. One neutron and the photon are banked for

later analysis. The ﬁrst ﬁssion neutron is captured at event 3 and terminated. The banked neutron

is now retrieved and, by random sampling, leaks out of the slab at event 4. The ﬁssion-produced

photon has a collision at event 5 and leaks out at event 6. The remaining photon generated at

event 1 is now followed with a capture at event 7. Note that MCNP retrieves banked particles such

that the last particle stored in the bank is the ﬁrst particle taken out.

This neutron history is now complete. As more and more such histories are followed, the neutron

and photon distributions become better known. The quantities of interest (whatever the user

requests) are tallied, along with estimates of the statistical precision (uncertainty) of the results.

II. INTRODUCTION TO MCNP FEATURES

Various features, concepts, and capabilities of MCNP are summarized in this section. More detail

concerning each topic is available in later chapters or appendices.

A. Nuclear Data and Reactions

MCNP uses continuous-energy nuclear and atomic data libraries. The primary sources of nuclear

data are evaluations from the Evaluated Nuclear Data File (ENDF)1system, the Evaluated Nuclear

Data Library (ENDL)2 and the Activation Library (ACTL)3 compilations from Livermore, and

evaluations from the Applied Nuclear Science (T–2) Group4,5,6 at Los Alamos. Evaluated data are

processed into a format appropriate for MCNP by codes such as NJOY.7 The processed nuclear

data libraries retain as much detail from the original evaluations as is feasible to faithfully

reproduce the evaluator’s intent.

Nuclear data tables exist for neutron interactions, neutron-induced photons, photon interactions,

neutron dosimetry or activation, and thermal particle scattering S(α,β). Photon and electron data

are atomic rather than nuclear in nature. Each data table available to MCNP is listed on a directory

ﬁle, XSDIR. Users may select speciﬁc data tables through unique identiﬁers for each table, called

ZAIDs. These identiﬁers generally contain the atomic number Z, mass number A, and library

speciﬁer ID.

April 10, 2000 1-5

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INTRODUCTION TO MCNP FEATURES

Over 500 neutron interaction tables are available for approximately 100 different isotopes and

elements. Multiple tables for a single isotope are provided primarily because data have been

derived from different evaluations, but also because of different temperature regimes and different

processing tolerances. More neutron interaction tables are constantly being added as new and

revised evaluations become available. Neutron−induced photon production data are given as part

of the neutron interaction tables when such data are included in the evaluations.

Photon interaction tables exist for all elements from Z = 1 through Z = 94. The data in the photon

interaction tables allow MCNP to account for coherent and incoherent scattering, photoelectric

absorption with the possibility of ﬂuorescent emission, and pair production. Scattering angular

distributions are modiﬁed by atomic form factors and incoherent scattering functions.

Cross sections for nearly 2000 dosimetry or activation reactions involving over 400 target nuclei in

ground and excited states are part of the MCNP data package. These cross sections can be used as

energy-dependent response functions in MCNP to determine reaction rates but cannot be used as

transport cross sections.

Thermal data tables are appropriate for use with the S(α,β)scattering treatment in MCNP. The data

include chemical (molecular) binding and crystalline effects that become important as the

neutron’s energy becomes sufﬁciently low. Data at various temperatures are available for light and

heavy water, beryllium metal, beryllium oxide, benzene, graphite, polyethylene, and zirconium and

hydrogen in zirconium hydride.

B. Source Speciﬁcation

MCNP’s generalized user-input source capability allows the user to specify a wide variety of

source conditions without having to make a code modiﬁcation. Independent probability

distributions may be speciﬁed for the source variables of energy, time, position, and direction, and

for other parameters such as starting cell(s) or surface(s). Information about the geometrical extent

of the source can also be given. In addition, source variables may depend on other source variables

(for example, energy as a function of angle) thus extending the built-in source capabilities of the

code. The user can bias all input distributions.

In addition to input probability distributions for source variables, certain built-in functions are

available. These include various analytic functions for ﬁssion and fusion energy spectra such as

Watt, Maxwellian, and Gaussian spectra; Gaussian for time; and isotropic, cosine, and

monodirectional for direction. Biasing may also be accomplished by special built−in functions.

A surface source allows particles crossing a surface in one problem to be used as the source for a

subsequent problem. The decoupling of a calculation into several parts allows detailed design or

analysis of certain geometrical regions without having to rerun the entire problem from the

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INTRODUCTION TO MCNP FEATURES

beginning each time. The surface source has a ﬁssion volume source option that starts particles

from ﬁssion sites where they were written in a previous run.

MCNP provides the user three methods to deﬁne an initial criticality source to estimate keff, the

ratio of neutrons produced in successive generations in ﬁssile systems.

C. Tallies and Output

The user can instruct MCNP to make various tallies related to particle current, particle ﬂux, and

energy deposition. MCNP tallies are normalized to be per starting particle except for a few special

cases with criticality sources. Currents can be tallied as a function of direction across any set of

surfaces, surface segments, or sum of surfaces in the problem. Charge can be tallied for electrons

and positrons. Fluxes across any set of surfaces, surface segments, sum of surfaces, and in cells,

cell segments, or sum of cells are also available. Similarly, the ﬂuxes at designated detectors (points

or rings) are standard tallies. Heating and ﬁssion tallies give the energy deposition in speciﬁed

cells. A pulse height tally provides the energy distribution of pulses created in a detector by

radiation. In addition, particles may be ﬂagged when they cross speciﬁed surfaces or enter

designated cells, and the contributions of these ﬂagged particles to the tallies are listed separately.

Tallies such as the number of ﬁssions, the number of absorptions, the total helium production, or

any product of the ﬂux times the approximately 100 standard ENDF reactions plus several

nonstandard ones may be calculated with any of the MCNP tallies. In fact, any quantity of the form

can be tallied, where is the energy-dependent ﬂuence, and f(E) is any product or summation

of the quantities in the cross-section libraries or a response function provided by the user. The

tallies may also be reduced by line-of-sight attenuation. Tallies may be made for segments of cells

and surfaces without having to build the desired segments into the actual problem geometry. All

tallies are functions of time and energy as speciﬁed by the user and are normalized to be per starting

particle.

In addition to the tally information, the output ﬁle contains tables of standard summary information

to give the user a better idea of how the problem ran. This information can give insight into the

physics of the problem and the adequacy of the Monte Carlo simulation. If errors occur during the

running of a problem, detailed diagnostic prints for debugging are given. Printed with each tally is

also its statistical relative error corresponding to one standard deviation. Following the tally is a

detailed analysis to aid in determining conﬁdence in the results. Ten pass/no pass checks are made

for the user-selectable tally ﬂuctuation chart (TFC) bin of each tally. The quality of the conﬁdence

interval still cannot be guaranteed because portions of the problem phase space possibly still have

not been sampled. Tally ﬂuctuation charts, described in the following section, are also

CφE()fE()Ed

∫

φE()

April 10, 2000 1-7

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INTRODUCTION TO MCNP FEATURES

automatically printed to show how a tally mean, error, variance of the variance, and slope of the

largest history scores ﬂuctuate as a function of the number of histories run.

Tally results can be displayed graphically, either while the code is running or in a separate

postprocessing mode.

D. Estimation of Monte Carlo Errors

MCNP tallies are normalized to be per starting particle and are printed in the output accompanied

by a second number R, which is the estimated relative error deﬁned to be one estimated standard

deviation of the mean divided by the estimated mean . In MCNP, the quantities required for

this error estimate−−the tally and its second moment−−are computed after each complete Monte

Carlo history, which accounts for the fact that the various contributions to a tally from the same

history are correlated. For a well-behaved tally, R will be proportional to where N is the

number of histories. Thus, to halve R, we must increase the total number of histories fourfold. For

a poorly behaved tally, R may increase as the number of histories increases.

The estimated relative error can be used to form conﬁdence intervals about the estimated mean,

allowing one to make a statement about what the true result is. The Central Limit Theorem states

that as N approaches inﬁnity there is a 68% chance that the true result will be in the range

and a 95% chance in the range . It is extremely important to note that these

conﬁdence statements refer only to the precision of the Monte Carlo calculation itself and not to

the accuracy of the result compared to the true physical value. A statement regarding accuracy

requires a detailed analysis of the uncertainties in the physical data, modeling, sampling

techniques, and approximations, etc., used in a calculation.

The guidelines for interpreting the quality of the conﬁdence interval for various values of R are

listed in Table 1.1.

TABLE 1.1:

Guidelines for Interpreting the Relative Error R*

* and represents the estimated relative error at the 1σlevel.

These interpretations of Rassume that all portions of the problem phase

space are being sampled well by the Monte Carlo process.

Range of R Quality of the Tally

0.5 to 1.0 Not meaningful

0.2 to 0.5 Factor of a few

0.1 to 0.2 Questionable

< 0.10 Generally reliable

< 0.05 Generally reliable for point detectors

Sx x

1N⁄

x1R±() x12R±()

xx⁄=

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INTRODUCTION TO MCNP FEATURES

For all tallies except next-event estimators, hereafter referred to as point detector tallies, the

quantity R should be less than 0.10 to produce generally reliable conﬁdence intervals. Point

detector results tend to have larger third and fourth moments of the individual tally distributions,

so a smaller value of R, < 0.05, is required to produce generally reliable conﬁdence intervals. The

estimated uncertainty in the Monte Carlo result must be presented with the tally so that all are

aware of the estimated precision of the results.

Keep in mind the footnote to Table 1.1. For example, if an important but highly unlikely particle

path in phase space has not been sampled in a problem, the Monte Carlo results will not have the

correct expected values and the conﬁdence interval statements may not be correct. The user can

guard against this situation by setting up the problem so as not to exclude any regions of phase

space and by trying to sample all regions of the problem adequately.

Despite one’s best effort, an important path may not be sampled often enough, causing conﬁdence

interval statements to be incorrect. To try to inform the user about this behavior, MCNP calculates

a ﬁgure of merit (FOM) for one tally bin of each tally as a function of the number of histories and

prints the results in the tally ﬂuctuation charts at the end of the output. The FOM is deﬁned as

where Tis the computer time in minutes. The more efﬁcient a Monte Carlo calculation is, the larger

the FOM will be because less computer time is required to reach a given value of R.

The FOM should be approximately constant as Nincreases because R2is proportional to 1/N and

Tis proportional to N.Always examine the tally ﬂuctuation charts to be sure that the tally appears

well behaved, as evidenced by a fairly constant FOM. A sharp decrease in the FOM indicates that

a seldom-sampled particle path has signiﬁcantly affected the tally result and relative error estimate.

In this case, the conﬁdence intervals may not be correct for the fraction of the time that statistical

theory would indicate. Examine the problem to determine what path is causing the large scores and

try to redeﬁne the problem to sample that path much more frequently.

After each tally, an analysis is done and additional useful information is printed about the TFC tally

bin result. The nonzero scoring efﬁciency, the zero and nonzero score components of the relative

error, the number and magnitude of negative history scores, if any, and the effect on the result if the

largest observed history score in the TFC were to occur again on the very next history are given. A

table just before the TFCs summarizes the results of these checks for all tallies in the problem. Ten

statistical checks are made and summarized in table 160 after each tally, with a pass yes/no

criterion. The empirical history score probability density function (PDF) for the TFC bin of each

tally is calculated and displayed in printed plots.

FOM 1R2T()⁄≡

April 10, 2000 1-9

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INTRODUCTION TO MCNP FEATURES

The TFCs at the end of the problem include the variance of the variance (an estimate of the error

of the relative error), and the slope (the estimated exponent of the PDF large score behavior) as a

function of the number of particles started.

All this information provides the user with statistical information to aid in forming valid conﬁdence

intervals for Monte Carlo results. There is no GUARANTEE, however. The possibility always

exists that some as yet unsampled portion of the problem may change the conﬁdence interval if

more histories were calculated. Chapter 2 contains more information about estimation of Monte

Carlo precision.

E. Variance Reduction

As noted in the previous section, R(the estimated relative error) is proportional to , where

Nis the number of histories. For a given MCNP run, the computer time Tconsumed is proportional

to N. Thus , where Cis a positive constant. There are two ways to reduce R: (1)

increase Tand/or (2) decrease C. Computer budgets often limit the utility of the ﬁrst approach. For

example, if it has taken 2 hours to obtain R=0.10, then 200 hours will be required to obtain R=0.01.

For this reason MCNP has special variance reduction techniques for decreasing C. (Variance is the

square of the standard deviation.) The constant Cdepends on the tally choice and/or the sampling

choices.

1. Tally Choice

As an example of the tally choice, note that the ﬂuence in a cell can be estimated either by a

collision estimate or a track length estimate. The collision estimate is obtained by tallying 1/Σt

(Σt=macroscopic total cross section) at each collision in the cell and the track length estimate is

obtained by tallying the distance the particle moves while inside the cell. Note that as Σtgets very

small, very few particles collide but give enormous tallies when they do, a high variance situation

(see page 2–109). In contrast, the track length estimate gets a tally from every particle that enters

the cell. For this reason MCNP has track length tallies as standard tallies, whereas the collision

tally is not standard in MCNP, except for estimating keff.

2. Nonanalog Monte Carlo

Explaining how sampling affects C requires understanding of the nonanalog Monte Carlo model.

The simplest Monte Carlo model for particle transport problems is the analog model that uses the

natural probabilities that various events occur (for example, collision, ﬁssion, capture, etc.).

Particles are followed from event to event by a computer, and the next event is always sampled

(using the random number generator) from a number of possible next events according to the

natural event probabilities. This is called the analog Monte Carlo model because it is directly

analogous to the naturally occurring transport.

1N⁄

RCT⁄=

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INTRODUCTION TO MCNP FEATURES

The analog Monte Carlo model works well when a signiﬁcant fraction of the particles contribute

to the tally estimate and can be compared to detecting a signiﬁcant fraction of the particles in the

physical situation. There are many cases for which the fraction of particles detected is very small,

less than 10-6. For these problems analog Monte Carlo fails because few, if any, of the particles

tally, and the statistical uncertainty in the answer is unacceptable.

Although the analog Monte Carlo model is the simplest conceptual probability model, there are

other probability models for particle transport. They estimate the same average value as the analog

Monte Carlo model, while often making the variance (uncertainty) of the estimate much smaller

than the variance for the analog estimate. Practically, this means that problems that would be

impossible to solve in days of computer time can be solved in minutes of computer time.

A nonanalog Monte Carlo model attempts to follow “interesting” particles more often than

“uninteresting” ones. An “interesting” particle is one that contributes a large amount to the

quantity (or quantities) that needs to be estimated. There are many nonanalog techniques, and they

all are meant to increase the odds that a particle scores (contributes). To ensure that the average

score is the same in the nonanalog model as in the analog model, the score is modiﬁed to remove

the effect of biasing (changing) the natural odds. Thus, if a particle is artiﬁcially made q times as

likely to execute a given random walk, then the particle’s score is weighted by (multiplied by) .

The average score is thus preserved because the average score is the sum, over all random walks,

of the probability of a random walk multiplied by the score resulting from that random walk.

A nonanalog Monte Carlo technique will have the same expected tallies as an analog technique if

the expected weight executing any given random walk is preserved. For example, a particle can be

split into two identical pieces and the tallies of each piece are weighted by 1/2 of what the tallies

would have been without the split. Such nonanalog, or variance reduction, techniques can often

decrease the relative error by sampling naturally rare events with an unnaturally high frequency and

weighting the tallies appropriately.

3. Variance Reduction Tools in MCNP

There are four classes of variance reduction techniques8that range from the trivial to the esoteric.

Truncation Methods are the simplest of variance reduction methods. They speed up calculations

by truncating parts of phase space that do not contribute signiﬁcantly to the solution. The simplest

example is geometry truncation in which unimportant parts of the geometry are simply not

modeled. Speciﬁc truncation methods available in MCNP are energy cutoff and time cutoff.

Population Control Methods use particle splitting and Russian roulette to control the number of

samples taken in various regions of phase space. In important regions many samples of low weight

are tracked, while in unimportant regions few samples of high weight are tracked. A weight

adjustment is made to ensure that the problem solution remains unbiased. Speciﬁc population

1q⁄

April 10, 2000 1-11

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INTRODUCTION TO MCNP FEATURES

control methods available in MCNP are geometry splitting and Russian roulette, energy splitting/

roulette, weight cutoff, and weight windows.

Modiﬁed Sampling Methods alter the statistical sampling of a problem to increase the number of

tallies per particle. For any Monte Carlo event it is possible to sample from any arbitrary

distribution rather than the physical probability as long as the particle weights are then adjusted to

compensate. Thus, with modiﬁed sampling methods, sampling is done from distributions that send

particles in desired directions or into other desired regions of phase space such as time or energy,

or change the location or type of collisions. Modiﬁed sampling methods in MCNP include the

exponential transform, implicit capture, forced collisions, source biasing, and neutron-induced

photon production biasing.

Partially-Deterministic Methods are the most complicated class of variance reduction methods.

They circumvent the normal random walk process by using deterministic-like techniques, such as

next event estimators, or by controlling the random number sequence. In MCNP these methods

include point detectors, DXTRAN, and correlated sampling.

Variance reduction techniques, used correctly, can greatly help the user produce a more efﬁcient

calculation. Used poorly, they can result in a wrong answer with good statistics and few clues that

anything is amiss. Some variance reduction methods have general application and are not easily

misused. Others are more specialized and attempts to use them carry high risk. The use of weight

windows tends to be more powerful than the use of importances but typically requires more input

data and more insight into the problem. The exponential transform for thick shields is not

recommended for the inexperienced user; rather, use many cells with increasing importances (or

decreasing weight windows) through the shield. Forced collisions are used to increase the

frequency of random walk collisions within optically thin cells but should be used only by an

experienced user. The point detector estimator should be used with caution, as should DXTRAN.

For many problems, variance reduction is not just a way to speed up the problem but is absolutely

necessary to get any answer at all. Deep penetration problems and pipe detector problems, for

example, will run too slowly by factors of trillions without adequate variance reduction.

Consequently, users have to become skilled in using the variance reduction techniques in MCNP.

Most of the following techniques cannot be used with the pulse height tally.

The following summarizes brieﬂy the main MCNP variance reduction techniques. Detailed

discussion is in Chapter 2, page 2–127.

1. Energy cutoff: Particles whose energy is out of the range of interest are terminated so

that computation time is not spent following them.

2. Time cutoff: Like the energy cutoff but based on time.

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INTRODUCTION TO MCNP FEATURES

3. Geometry splitting with Russian roulette: Particles transported from a region of higher

importance to a region of lower importance (where they will probably contribute little to

the desired problem result) undergo Russian roulette; that is, some of those particles will

be killed a certain fraction of the time, but survivors will be counted more by increasing

their weight the remaining fraction of the time. In this way, unimportant particles are

followed less often, yet the problem solution remains undistorted. On the other hand, if

a particle is transported to a region of higher importance (where it will likely contribute

to the desired problem result), it may be split into two or more particles (or tracks), each

with less weight and therefore counting less. In this way, important particles are followed

more often, yet the solution is undistorted because on average total weight is conserved.

4. Energy splitting/Russian roulette: Particles can be split or rouletted upon entering

various user−supplied energy ranges. Thus important energy ranges can be sampled

more frequently by splitting the weight among several particles and less important

energy ranges can be sampled less frequently by rouletting particles.

5. Weight cutoff/Russian roulette: If a particle weight becomes so low that the particle

becomes insignificant, it undergoes Russian roulette. Most particles are killed, and some

particles survive with increased weight. The solution is unbiased because total weight is

conserved, but computer time is not wasted on insignificant particles.

6. Weight window: As a function of energy, geometrical location, or both, low−weighted

particles are eliminated by Russian roulette and high−weighted particles are split. This

technique helps keep the weight dispersion within reasonable bounds throughout the

problem. An importance generator is available that estimates the optimal limits for a

weight window.

7. Exponential transformation: To transport particles long distances, the distance between

collisions in a preferred direction is artificially increased and the weight is

correspondingly artifically decreased. Because large weight fluctuations often result, it

is highly recommended that the weight window be used with the exponential transform.

8. Implicit capture: When a particle collides, there is a probability that it is captured by the

nucleus. In analog capture, the particle is killed with that probability. In implicit capture,

also known as survival biasing, the particle is never killed by capture; instead, its weight

is reduced by the capture probability at each collision. Important particles are permitted

to survive by not being lost to capture. On the other hand, if particles are no longer

considered useful after undergoing a few collisions, analog capture efficiently gets rid of

them.

9. Forced collisions: A particle can be forced to undergo a collision each time it enters a

designated cell that is almost transparent to it. The particle and its weight are

appropriately split into a collided and uncollided part. Forced collisions are often used to

generate contributions to point detectors, ring detectors, or DXTRAN spheres.

10. Source variable biasing: Source particles with phase space variables of more

importance are emitted with a higher frequency but with a compensating lower weight

April 10, 2000 1-13

CHAPTER 1

MCNP GEOMETRY

than are less important source particles. This technique can be used with pulse height

tallies.

11. Point and ring detectors: When the user wishes to tally a flux−related quantity at a point

in space, the probability of transporting a particle precisely to that point is vanishingly

small. Therefore, pseudoparticles are directed to the point instead. Every time a particle

history is born in the source or undergoes a collision, the user may require that a

pseudoparticle be tallied at a specified point in space. In this way, many pseudoparticles

of low weight reach the detector, which is the point of interest, even though no particle

histories could ever reach the detector. For problems with rotational symmetry, the point

may be represented by a ring to enhance the efficiency of the calculation.

12. DXTRAN: DXTRAN, which stands for deterministic transport, improves sampling in

the vicinity of detectors or other tallies. It involves deterministically transporting

particles on collision to some arbitrary, user−defined sphere in the neighborhood of a

tally and then calculating contributions to the tally from these particles. Contributions to

the detectors or to the DXTRAN spheres can be controlled as a function of geometric

cell or as a function of the relative magnitude of the contribution to the detector or

DXTRAN sphere.

The DXTRAN method is a way of obtaining large numbers of particles on user–speciﬁed

“DXTRAN spheres.” DXTRAN makes it possible to obtain many particles in a small

region of interest that would otherwise be difﬁcult to sample. Upon sampling a collision

or source density function, DXTRAN estimates the correct weight fraction that should

scatter toward, and arrive without collision at, the surface of the sphere. The DXTRAN

method then puts this correct weight on the sphere. The source or collision event is

sampled in the usual manner, except that the particle is killed if it tries to enter the sphere

because all particles entering the sphere have already been accounted for

deterministically.

13. Correlated sampling: The sequence of random numbers in the Monte Carlo process is

chosen so that statistical fluctuations in the problem solution will not mask small

variations in that solution resulting from slight changes in the problem specification. The

ith history will always start at the same point in the random number sequence no matter

what the previous i−1 particles did in their random walks.

III. MCNP GEOMETRY

We will present here only basic information about geometry setup, surface speciﬁcation, and cell

and surface card input. Areas of further interest would be the complement operator, use of

parentheses, and repeated structure and lattice deﬁnitions, found in Chapter 2. Chapter 4 contains

geometry examples and is recommended as a next step. Chapter 3 has detailed information about

the format and entries on cell and surface cards and discusses macrobodies.

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MCNP GEOMETRY

The geometry of MCNP treats an arbitrary three-dimensional conﬁguration of user-deﬁned

materials in geometric cells bounded by ﬁrst- and second-degree surfaces and fourth-degree

elliptical tori. The cells are deﬁned by the intersections, unions, and complements of the regions

bounded by the surfaces. Surfaces are deﬁned by supplying coefﬁcients to the analytic surface

equations or, for certain types of surfaces, known points on the surfaces.

MCNP has a more general geometry than is available in most combinatorial geometry codes.

Rather than combining several predeﬁned geometrical bodies, as in a combinatorial geometry

scheme, MCNP gives the user the added ﬂexibility of deﬁning geometrical regions from all the ﬁrst

and second degree surfaces of analytical geometry and elliptical tori and then of combining them

with Boolean operators. The code does extensive internal checking to ﬁnd input errors. In addition,

the geometry-plotting capability in MCNP helps the user check for geometry errors.

MCNP treats geometric cells in a Cartesian coordinate system. The surface equations recognized

by MCNP are listed in Table 3.1 on page 3–14. The particular Cartesian coordinate system used is

arbitrary and user deﬁned, but the right−handed system shown in Figure 1.2 is often chosen.

Figure 1-2.

Using the bounding surfaces speciﬁed on cell cards, MCNP tracks particles through the geometry,

calculates the intersection of a track’s trajectory with each bounding surface, and ﬁnds the

minimum positive distance to an intersection. If the distance to the next collision is greater than

this minimum distance and there are no DXTRAN spheres along the track, the particle leaves the

current cell. At the appropriate surface intersection, MCNP ﬁnds the correct cell that the particle

will enter by checking the sense of the intersection point for each surface listed for the cell. When

a complete match is found, MCNP has found the correct cell on the other side and the transport

continues.

A. Cells

When cells are deﬁned, an important concept is that of the sense of all points in a cell with respect

to a bounding surface. Suppose that is the equation of a surface in the

sfxyz),,()0==

April 10, 2000 1-15

CHAPTER 1

MCNP GEOMETRY

problem. For any set of points (x,y,z), if s= 0 the points are on the surface. However, for points

not on the surface, if sis negative, the points are said to have a negative sense with respect to that

surface and, conversely, a positive sense if sis positive. For example, a point at x= 3 has a positive

sense with respect to the plane . That is, the equation is

positive for x= 3 (where D = constant).

Cells are deﬁned on cells cards. Each cell is described by a cell number, material number, and

material density followed by a list of operators and signed surfaces that bound the cell. If the sense

is positive, the sign can be omitted. The material number and material density can be replaced by

a single zero to indicate a void cell. The cell number must begin in columns 1−5. The remaining

entries follow, separated by blanks. A more complete description of the cell card format can be

found on page 1–23. Each surface divides all space into two regions, one with positive sense with

respect to the surface and the other with negative sense. The geometry description deﬁnes the cell

to be the intersection, union, and/or complement of the listed regions.

The subdivision of the physical space into cells is not necessarily governed only by the different

material regions, but may be affected by problems of sampling and variance reduction techniques

(such as splitting and Russian roulette), the need to specify an unambiguous geometry, and the tally

requirements. The tally segmentation feature may eliminate most of the tally requirements.

Be cautious about making any one cell very complicated. With the union operator and disjointed

regions, a very large geometry can be set up with just one cell. The problem is that for each track

ﬂight between collisions in a cell, the intersection of the track with each bounding surface of the

cell is calculated, a calculation that can be costly if a cell has many surfaces. As an example,

consider Figure 1.3a. It is just a lot of parallel cylinders and is easy to set up. However, the cell

containing all the little cylinders is bounded by fourteen surfaces (counting a top and bottom). A

much more efﬁcient geometry is seen in Figure 1.3b, where the large cell has been broken up into

a number of smaller cells.

Figure 1-3.

x2–0=xD–32–s1===

a b

1-16 April 10, 2000

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MCNP GEOMETRY

1. Cells Defined by Intersections of Regions of Space

The intersection operator in MCNP is implicit; it is simply the blank space between two surface

numbers on the cell card.

If a cell is speciﬁed using only intersections,all points in the cell must have the same sense with

respect to a given bounding surface. This means that, for each bounding surface of a cell, all points

in the cell must remain on only one side of any particular surface. Thus, there can be no concave

corners in a cell speciﬁed only by intersections. Figure 1.4, a cell formed by the intersection of ﬁve

surfaces (ignore surface 6 for the time being), illustrates the problem of concave corners by

allowing a particle (or point) to be on two sides of a surface in one cell. Surfaces 3 and 4 form a

concave corner in the cell such that points p1and p2are on the same side of surface 4 (that is, have

the same sense with respect to 4) but point p3 is on the other side of surface 4 (opposite sense).

Points p2and p3have the same sense with respect to surface 3, but p1has the opposite sense. One

way to remedy this dilemma (and there are others) is to add surface 6 between the 3/4 corner and

surface 1 to divide the original cell into two cells.

Figure 1-4.

With surface 6 added to Figure 1.4, the cell to the right of surface 6 is number~1 (cells indicated

by circled numbers); to the left number 2; and the outside cell number 3. The cell cards (in two

dimensions, all cells void) are

101–2–36

201–6–45

Cell 1 is a void and is formed by the intersection of the region above (positive sense) surface 1 with

the region to the left (negative sense) of surface 2 intersected with the region below (negative sense)

surface 3 and ﬁnally intersected with the region to the right (positive sense) of surface 6. Cell 2 is

described similarly.

Cell 3 cannot be speciﬁed with the intersection operator. The following section about the union

operator is needed to describe cell 3.

April 10, 2000 1-17

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MCNP GEOMETRY

2. Cells Defined by Unions of Regions of Space

The union operator, signiﬁed by a colon on the cell cards, allows concave corners in cells and also

cells that are completely disjoint. The intersection and union operators are binary Boolean

operators, so their use follows Boolean algebra methodology; unions and intersections can be used

in combination in any cell description.

Spaces on either side of the union operator are irrelevant, but remember that a space without the

colon signiﬁes an intersection. In the hierarchy of operations, intersections are performed ﬁrst and

then unions. There is no left to right ordering. Parentheses can be used to clarify operations and in

some cases are required to force a certain order of operations. Innermost parentheses are cleared

ﬁrst. Spaces are optional on either side of a parenthesis. A parenthesis is equivalent to a space and

signiﬁes an intersection.

For example, let A and B be two regions of space. The region containing points that belong to both

A and B is called the intersection of A and B. The region containing points that belong to A alone

or to B alone or to both A and B is called the union of A and B. The lined area in Figure 1.5a

represents the union of A and B (or A : B), and the lined area in Figure 1.5b represents the

intersection of A and B (or A B). The only way regions of space can be added is with the union

operator. An intersection of two spaces always results in a region no larger than either of the two

spaces. Conversely, the union of two spaces always results in a region no smaller than either of the

two spaces.

Figure 1-5.

A simple example will further illustrate the concept of Figure 1.5 and the union operator to solidify

the concept of adding and intersecting regions of space to deﬁne a cell. See also the second example

in Chapter 4. In Figure 1.6 we have two inﬁnite planes that meet to form two cells. Cell 1 is easy

to deﬁne; it is everything in the universe to the right of surface 1 (that is, a positive sense) that is

also in common with (or intersected with) everything in the universe below surface 2 (that is, a

negative sense). Therefore, the surface relation of cell 1 is 1 –2.

1-18 April 10, 2000

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MCNP GEOMETRY

Figure 1-6.

Cell 2 is everything in the universe to the left (negative sense) of surface 1 plus everything in the

universe above (positive sense) surface 2, or –1 : 2, illustrated in Figure 1.6b by all the shaded

regions of space. If cell 2 were speciﬁed as –1 2, that would represent the region of space common

to –1 and 2, which is only the cross-hatched region in the ﬁgure and is obviously an improper

speciﬁcation for cell 2.

Returning to Figure 1.4 on page 1–16, if cell 1 is inside the solid black line and cell 2 is the entire

region outside the solid line, then the MCNP cell cards in two dimensions are (assuming both cells

are voids)

101–2(–3:–4)5

2 0 –5:–1:2:3 4

Cell 1 is deﬁned as the region above surface 1 intersected with the region to the left of surface 2,

intersected with the union of regions below surfaces 3 and 4, and ﬁnally intersected with the region

to the right of surface 5. Cell 2 contains four concave corners (all but between surfaces 3 and 4),

and its speciﬁcation is just the converse (or complement) of cell 1. Cell 2 is the space deﬁned by

the region to the left of surface 5 plus the region below 1 plus the region to the right of 2 plus the

space deﬁned by the intersections of the regions above surfaces 3 and 4.

A simple consistency check can be noted with the twocell cards above. All intersections for cell 1

become unions for cell 2 and vice versa. The senses are also reversed.

Note that in this example, all corners less than 180 degrees in a cell are handled by intersections

and all corners greater than 180 degrees are handled by unions.

To illustrate some of the concepts about parentheses, assume an intersection is thought of

mathematically as multiplication and a union is thought of mathematically as addition.

Parentheses are removed ﬁrst, with multiplication being performed before addition. The cell cards

for the example cards above from Figure 1.4 may be written in the form

(a)

(b)

April 10, 2000 1-19

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MCNP GEOMETRY

Note that parentheses are required for the ﬁrst cell but not for the second, although the second could

have been written as , etc.

Several more examples using the union operator are given in Chapter 4. Study them to get a better

understanding of this powerful operator that can greatly simplify geometry setups.

B. Surface Type Speciﬁcation

The ﬁrst- and second-degree surfaces plus the fourth-degree elliptical and degenerate tori of

analytical geometry are all available in MCNP. The surfaces are designated by mnemonics such as

C/Z for a cylinder parallel to the z-axis. A cylinder at an arbitrary orientation is designated by the

general quadratic GQ mnemonic. A paraboloid parallel to a coordinate axis is designated by the

special quadratic SQ mnemonic. The 29 mnemonics representing various types of surfaces are

listed in Table 3.1 on page 3–14.

C. Surface Parameter Speciﬁcation

There are two ways to specify surface parameters in MCNP: (1) by supplying the appropriate

coefﬁcients needed to satisfy the surface equation, and (2) by specifying known geometrical points

on a surface that is rotationally symmetric about a coordinate axis.

1. Coefficients for the Surface Equations

The ﬁrst way to deﬁne a surface is to use one of the surface-type mnemonics from Table 3.1 on

page 3–14 and to calculate the appropriate coefﬁcients needed to satisfy the surface equation. For

example, a sphere of radius 3.62-cm with the center located at the point (4,1,–3) is speciﬁed by

S41–33.62

An ellipsoid whose axes are not parallel to the coordinate axes is deﬁned by the GQ mnemonic plus

up to 10 coefﬁcients of the general quadratic equation. Calculating the coefﬁcients can be (and

frequently is) nontrivial, but the task is greatly simpliﬁed by deﬁning an auxiliary coordinate

system whose axes coincide with the axes of the ellipsoid. The ellipsoid is easily deﬁned in terms

of the auxiliary coordinate system, and the relationship between the auxiliary coordinate system

and the main coordinate system is speciﬁed on a TRn card, described on page 3–30.

The use of the SQ (special quadratic) and GQ (general quadratic) surfaces is determined by the

orientation of the axes. One should always use the simplest possible surface in describing

ab c d+()e⋅⋅ ⋅

eabcd⋅+++

eab cd⋅()eab++()cd⋅()e() a() b() cd⋅()+++,+,+++

1-20 April 10, 2000

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MCNP INPUT FOR SAMPLE PROBLEM

geometries; for example, using a GQ surface instead of an S to specify a sphere will require more

computational effort for MCNP.

2. Points that Define a Surface

The second way to deﬁne a surface is to supply known points on the surface. This method is

convenient if you are setting up a geometry from something like a blueprint where you know the

coordinates of intersections of surfaces or points on the surfaces. When three or more surfaces

intersect at a point, this second method also produces a more nearly perfect point of intersection if

the common point is used in the surface speciﬁcation. It is frequently difﬁcult to get complicated

surfaces to meet at one point if the surfaces are speciﬁed by the equation coefﬁcients. Failure to

achieve such a meeting can result in the unwanted loss of particles.

There are, however, restrictions that must be observed when specifying surfaces by points that do

not exist when specifying surfaces by coefﬁcients. Surfaces described by points must be either

skew planes or surfaces rotationally symmetric about the x, y, or z axes. They must be unique, real,

and continuous. For example, points speciﬁed on both sheets of a hyperboloid are not allowed

because the surface is not continuous. However, it is valid to specify points that are all on one sheet

of the hyperboloid. (See the X,Y,Z, and P input cards description on page 3–16 for additional

explanation.)

IV. MCNP INPUT FOR SAMPLE PROBLEM

The main input ﬁle for the user is the INP (the default name) ﬁle that contains the input information

to describe the problem. We will present here only the subset of cards required to run the simple

ﬁxed source demonstration problem. All input cards are discussed in Chapter 3 and summarized in

Table 3.8 starting on page 3–148.

MCNP does extensive input checking but is not foolproof. A geometry should be checked by

looking at several different views with the geometry plotting option. You should also surround the

entire geometry with a sphere and ﬂood the geometry with particles from a source with an inward

cosine distribution on the spherical surface, using a VOID card to remove all materials speciﬁed in

the problem. If there are any incorrectly speciﬁed places in your geometry, this procedure will

usually ﬁnd them. Make sure the importance of the cell just inside the source sphere is not zero.

Then run a short job and study the output to see if you are calculating what you think you are

calculating.

The basic constants used in MCNP are printed in optional print table 98 in the output ﬁle. The units

used are:

1. lengths in centimeters,

2. energies in MeV,

3. times in shakes (10-8 sec),

April 10, 2000 1-21

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MCNP INPUT FOR SAMPLE PROBLEM

4. temperatures in MeV (kT),

5. atomic densities in units of atoms/barn-cm,

6. mass densities in g/cm3,

7. cross sections in barns (10-24 cm2),

8. heating numbers in MeV/collision, and

9. atomic weight ratio based on a neutron mass of 1.008664967. In these units, Avogadro’s

number is 0.59703109 x 10-24.

A simple sample problem illustrated in Figure 1.7 is referred to throughout the remainder of this

chapter. We wish to start 14-MeV neutrons at a point isotropic source in the center of a small sphere

of oxygen that is embedded in a cube of carbon. A small sphere of iron is also embedded in the

carbon. The carbon is a cube 10 cm on each side; the spheres have a 0.5-cm radius and are centered

between the front and back faces of the cube. We wish to calculate the total and energy-dependent

ﬂux in increments of 1 MeV from 1 to 14 MeV, where bin 1 will be the tally from 0 to 1 MeV

1. on the surface of the iron sphere and

2. averaged in the iron sphere volume.

This geometry has four cells, indicated by circled numbers, and eight surfaces—six planes and two

spheres. Surface numbers are written next to the appropriate surfaces. Surface 5 comes out from

the page in the +x direction and surface 6 goes back into the page in the –x direction.

Figure 1-7.

With knowledge of the cell card format, the sense of a surface, and the union and intersection

operators, we can set up the cell cards for the geometry of our example problem. To simplify this

step, assume the cells are void, for now. Cells 1 and 2 are described by the following cards:

10–7

20–8

where the negative signs denote the regions inside (negative sense) surfaces 7 and 8. Cell 3 is

everything in the universe above surface 1 intersected with everything below surface 2 intersected

with everything to the left of surface 3 and so forth for the remaining three surfaces. The region in

1-22 April 10, 2000

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MCNP INPUT FOR SAMPLE PROBLEM

common to all six surfaces is the cube, but we need to exclude the two spheres by intersecting

everything outside surface 7 and outside surface 8. The card for cell 3 is

30 1–2–34–5678

Cell 4 requires the use of the union operator and is similar to the idea illustrated in Figure 1.6. Cell

4 is the outside world, has zero importance, and is deﬁned as everything in the universe below

surface 1 plus everything above surface 2 plus everything to the right of surface 3 and so forth. The

cell card for cell 4 is

40–1:2:3:–4:5:–6

A. INP File

An input ﬁle has the following form:

All input lines are limited to 80 columns. Alphabetic characters can be upper, lower, or mixed case.

A $ (dollar sign) terminates data entry. Anything that follows the $ is interpreted as a comment.

Blank lines are used as delimiters and as an optional terminator. Data entries are separated by one

or more blanks.

Comment cards can be used anywhere in the INP ﬁle after the problem title card and before the

optional blank terminator card. Comment lines must have a C somewhere in columns 1-5 followed

by at least one blank and can be a total of 80 columns long.

Cell, surface, and data cards must all begin within the ﬁrst ﬁve columns. Entries are separated by

one or more blanks. Numbers can be integer or ﬂoating point. MCNP makes the appropriate

conversion. A data entry item, e.g., IMP:N or 1.1e2, must be completed on one line.

Message Block

Blank Line Delimiter

One Line Problem Title Card

Cell Cards

Blank Line Delimiter

Surface Cards

Blank Line Delimiter

Data Cards

Blank Line Terminator (optional)

}Optional

April 10, 2000 1-23

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MCNP INPUT FOR SAMPLE PROBLEM

Blanks ﬁlling the ﬁrst ﬁve columns indicate a continuation of the data from the last named card.

An & (ampersand) ending a line indicates data will continue on the following card, where data on

the continuation card can be in columns 1-80.

The optional message block, discussed in detail on page 3–1, is used to change ﬁle names and

specify running options such as a continuation run. On most systems these options and ﬁles may

alternatively be speciﬁed with an execution line message (see page 1–32). Message block entries

supersede execution line entries. The blank line delimiter signals the end of the message block.

The ﬁrst card in the ﬁle after the optional message block is the required problem title card. If there

is no message block, this must be the ﬁrst card in the INP ﬁle. It is limited to one 80-column line

and is used as a title in various places in the MCNP output. It can contain any information you

desire but usually contains information describing the particular problem.

MCNP makes extensive checks of the input ﬁle for user errors. A FATAL error occurs if a basic

constraint of the input speciﬁcation is violated, and MCNP will terminate before running any

particles. The ﬁrst fatal error is real; subsequent error messages may or may not be real because

of the nature of the ﬁrst fatal message.

B. Cell Cards

The cell number is the ﬁrst entry and must begin in the ﬁrst ﬁve columns.

The next entry is the cell material number, which is arbitrarily assigned by the user. The material

is described on a material card (Mn) that has the same material number (see page 1–29). If the cell

is a void, a zero is entered for the material number. The cell and material numbers cannot exceed

5 digits.

Next is the cell material density. A positive entry is interpreted as atom density in units of 1024

atoms/cm3. A negative entry is interpreted as mass density in units of g/cm3. No density is entered

for a void cell.

A complete speciﬁcation of the geometry of the cell follows. This speciﬁcation includes a list of

the signed surfaces bounding the cell where the sign denotes the sense of the regions deﬁned by the

surfaces. The regions are combined with the Boolean intersection and union operators. A space

indicates an intersection and a colon indicates a union.

Optionally, after the geometry description, cell parameters can be entered. The form is

keyword=value. The following line illustrates the cell card format:

11–0.0014 –7 IMP:N=1

1-24 April 10, 2000

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MCNP INPUT FOR SAMPLE PROBLEM

Cell 1 contains material 1 with density 0.0014 g/cm3, is bounded by only one surface (7), and has

an importance of 1. If cell 1 were a void, the cell card would be

10–7 IMP:N=1

The complete cell card input for this problem (with 2 comment cards) is

c cell cards for sample problem

–0.0014 –7

2 2 –7.86 –8

3 3 –1.60 1 –2 –34–5678

4 0 –1:2:3:–4:5:–6

c end of cell cards for sample problem

blank line delimiter

The blank line terminates the cell card section of the INP ﬁle. We strongly suggest that the cells be

numbered sequentially starting with one. A complete explanation of the cell card input is found in

Chapter 3, page 3–9.

C. Surface Cards

The surface number is the ﬁrst entry. It must begin in columns 1-5 and not exceed 5 digits. The next

entry is an alphabetic mnemonic indicating the surface type. Following the surface mnemonic are

the numerical coefﬁcients of the equation of the surface in the proper order. This simpliﬁed

description enables us to proceed with the example problem. For a full description of the surface

card see page 3–12.

Our problem uses planes normal to the x, y, and z axes and two general spheres. The respective

mnemonics are PX, PY, PZ, and S. Table 1.2 shows the equations that determine the sense of the

surface for the cell cards and the entries required for the surface cards. A complete list of available

surface equations is contained in Table 3.1 on page 3–14.

TABLE 1.2:

Surface Equations

Mnemonic Equation Card Entries

PX x - D = 0 D

PY y - D = 0 D

PZ x - D = 0 D

Sxx–()

2xy–()

2zz–()

2R2

–++ 0=xyzR

April 10, 2000 1-25

CHAPTER 1

MCNP INPUT FOR SAMPLE PROBLEM

For the planes, D is the point where the plane intersects the axis. If we place the origin in the center

of the 10-cm cube shown in Figure 1.7, the planes will be at x = –5, x = 5, etc. The two spheres are

not centered at the origin or on an axis, so we must give the x,y,z of their center as well as their

radii. The complete surface card input for this problem is shown below. A blank line terminates the

surface card portion of the input.

C Beginning of surfaces for cube

1 PZ −5

2 PZ 5

3 PY 5

4 PY −5

5 PX 5

6 PX −5

C End of cube surfaces

7 S 0 -4-2.5 .5 $ oxygen sphere

8 S 0 4 4 .5 $ iron sphere

blank line delimiter

D. Data Cards

The remaining data input for MCNP follows the second blank card delimiter, or third blank card if

there is a message block. The card name is the ﬁrst entry and must begin in the ﬁrst ﬁve columns.

The required entries follow, separated by one or more blanks.

Several of the data cards require a particle designator to distinguish between input data for

neutrons, data for photons, and data for electrons. The particle designator consists of the symbol :

(colon) and the letter N or P or E immediately following the name of the card. For example, to enter

neutron importances, use an IMP:N card; enter photon importances on an IMP:P card; enter

electron importances on an IMP:E card. No data card can be used more than once with the same

mnemonic, that is, M1 and M2 are acceptable, but two M1 cards are not allowed. Defaults have

been set for cards in some categories. A summary starting on page 3–147 shows which cards are

required, which are optional, and whether defaults exist and if so, what they are. The sample

problem will use cards in the following categories:

MCNP card name

1. mode, MODE

2. cell and surface parameters, IMP:N

3. source speciﬁcation, SDEF

4. tally speciﬁcation, Fn, En

5. material speciﬁcation, and Mn

6. problem cutoffs. NPS

1-26 April 10, 2000

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MCNP INPUT FOR SAMPLE PROBLEM

A complete description of the data cards is found on page 3–22 in Chapter 3.

1. MODE Card

MCNP can be run in several different modes:

Mode N — neutron transport only (default)

N P — neutron and neutron-induced photon transport

P — photon transport only

E — electron transport only

P E — photon and electron transport

N P E — neutron, neutron-induced photon and electron transport

The MODE card consists of the mnemonic MODE followed by the choices shown above. If the

MODE card is omitted, mode N is assumed.

Mode N P does not account for photo-neutrons but only neutron-induced photons. Photon-

production cross sections do not exist for all nuclides. If they are not available for a Mode N P

problem, MCNP will print out warning messages. To ﬁnd out whether a particular table for a

nuclide has photon-production cross sections available, check the Appendix G cross-section list.

Mode P or mode N P problems generate bremsstrahlung photons with a computationally expensive

thick-target bremsstrahlung approximation. This approximation can be turned off with the PHYS:E

card.

The sample problem is a neutron-only problem, so the MODE card can be omitted because MODE

N is the default.

2. Cell and Surface Parameter Cards

Most of these cards deﬁne values of cell parameters. Entries correspond in order to the cell or

surface cards that appear earlier in the INP ﬁle. A listing of all available cell and surface parameter

cards is found on page 3–32. A few examples are neutron and photon importance cards

(IMP:N,IMP:P), weight window cards (WWE:N, WWE:P, WWNi:N, WWNi:P), etc. Some

method of specifying relative cell importances is required; the majority of the other cell parameter

cards are for optional variance reduction techniques. The number of entries on a cell or surface

parameter card must equal the number of cells or surfaces in the problem or MCNP prints out a

WARNING or FATAL error message. In the case of a WARNING, MCNP assumes zeros.

The IMP:N card is used to specify relative cell importances in the sample problem. There are four

cells in the problem, so the IMP:N card will have four entries. The IMP:N card is used (a) for

terminating the particle’s history if the importance is zero and (b) for geometry splitting and

April 10, 2000 1-27

CHAPTER 1

MCNP INPUT FOR SAMPLE PROBLEM

Russian roulette to help particles move more easily to important regions of the geometry. An

IMP:N card for the sample problem is

IMP:N 1110

Cell parameters also can be deﬁned on cell cards using the keyword=value format. If a cell

parameter is speciﬁed on any cell card, it must be speciﬁed only on cell cards and not at all in the

data card section.

3. Source Specification Cards

A source deﬁnition card SDEF is one of four available methods of deﬁning starting particles.

Chapter 3 has a complete discussion of source speciﬁcation. The SDEF card deﬁnes the basic

source parameters, some of which are

POS = x y z default is 0 0 0;

CEL = starting cell number

ERG = starting energy default is 14 MeV;

WGT = starting weight default is 1;

TME = time default is 0;

PAR = source particle type 1 for N, N P, N P E; 2 for P, P E; 3 for E.

MCNP will determine the starting cell number for a point isotropic source, so the CEL entry is not

always required. The default starting direction for source particles is isotropic.

For the example problem, a fully speciﬁed source card is

SDEF POS = 0 –4–2.5 CEL = 1 ERG = 14 WGT = 1 TME = 0 PAR = 1

Neutron particles will start at the center of the oxygen sphere (0 –4 –2.5), in cell 1, with an energy

of 14 MeV, and with weight 1 at time 0. All these source parameters except the starting position

are the default values, so the most concise source card is

SDEF POS = 0 –4 –2.5

If all the default conditions applied to the problem, only the mnemonic SDEF would be required.

4. Tally Specification Cards

The tally cards are used to specify what you want to learn from the Monte Carlo calculation,

perhaps current across a surface, ﬂux at a point, etc. You request this information with one or more

tally cards. Tally speciﬁcation cards are not required, but if none is supplied, no tallies will be

1-28 April 10, 2000

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MCNP INPUT FOR SAMPLE PROBLEM

printed when the problem is run and a warning message is issued. Many of the tally speciﬁcation

cards describe tally “bins.” A few examples are energy (En), time (Tn), and cosine (Cn) cards.

MCNP provides six standard neutron, six standard photon, and four standard electron tallies, all

normalized to be per starting particle. Some tallies in criticality calculations are normalized

differently. Chapter 2, page 2–76, discusses tallies more completely, and Chapter 3, page 3–73, lists

all the tally cards and fully describes each one.

Tally Mnemonic Description

F1:N or F1:P or F1:E Surface current

F2:N or F2:P or F2:E Surface ﬂux

F4:N or F4:P or F4:E Track length estimate of cell ﬂux

F5a:N or F5a:P Flux at a point (point detector)

F6:N or F6:N,P Track length estimate of energy deposition

or F6:P

F7:N Track length estimate of ﬁssion energy deposition

F8:P or F8:E Energy distribution of pulses created

or F8:P,E in a detector

The tallies are identiﬁed by tally type and particle type. Tallies are given the numbers 1, 2, 4, 5, 6,

7, 8, or increments of 10 thereof, and are given the particle designator :N or :P or :E (or :N,P only

in the case of tally type 6 or P,E only for tally type 8). Thus you may have as many of any basic

tally as you need, each with different energy bins or ﬂagging or anything else. F4:N, F14:N,

F104:N, and F234:N are all legitimate neutron cell ﬂux tallies; they could all be for the same cell(s)

but with different energy or multiplier bins, for example. Similarly F5:P, F15:P, and F305:P are all

photon point detector tallies. Having both an F1:N card and an F1:P card in the same INP ﬁle is not

allowed. The tally number may not exceed three digits.

For our sample problem we will use Fn cards (Tally type) and En cards (Tally energy).

a. Tally (Fn) Cards: The sample problem has a surface ﬂux tally and a track length cell ﬂux

tally. Thus, the tally cards for the sample problem shown in Figure 1.7 are

F2:N 8 $ ﬂux across surface 8

F4:N 2 $ track length in cell 2

Printed out with each tally bin is the relative error of the tally corresponding to one estimated

standard deviation. Read page 1−6 for an explanation of the relative error. Results are not reliable

until they become stable as a function of the number of histories run. Much information is provided

for one bin of each tally in the tally ﬂuctuation charts at the end of the output ﬁle to help determine

tally stability. The user is strongly encouraged to look at this information carefully.

April 10, 2000 1-29

CHAPTER 1

MCNP INPUT FOR SAMPLE PROBLEM

b. Tally Energy (En) Card: We wish to calculate ﬂux in increments of 1 MeV from 14 to 1

MeV. Another tally speciﬁcation card in the sample input deck establishes these energy bins.

The entries on the En card are the upper bounds in MeV of the energy bins for tally n. The entries

must be given in order of increasing magnitude. If a particle has an energy greater than the last

entry, it will not be tallied, and a warning is issued. MCNP automatically provides the total over all

speciﬁed energy bins unless inhibited by putting the symbol NT as the last entry on the selected En

card.

The following cards will create energy bins for the sample problem:

E2 1234567891011121314

E4 1 12I 14

If no En card exists for tally n, a single bin over all energy will be used. To change this default, an

E0 (zero) card can be used to set up a default energy bin structure for all tallies. A speciﬁc En card

will override the default structure for tally n. We could replace the E2 and E4 cards with one E0

card for the sample problem, thus setting up identical bins for both tallies.

5. Materials Specification

The cards in this section specify both the isotopic composition of the materials and the cross-

section evaluations to be used in the cells. For a comprehensive discussion of materials

speciﬁcation, see page 3–108.

a. Material (Mm) Card: The following card is used to specify a material for all cells

containing material m, where m cannot exceed 5 digits:

Mm ZAID1fraction1ZAID2 fraction2

The m on a material card corresponds to the material number on the cell card (see page 1–23). The

consecutive pairs of entries on the material card consist of the identiﬁcation number (ZAID) of the

constituent element or nuclide followed by the atomic fraction (or weight fraction if entered as a

negative number) of that element or nuclide, until all the elements and nuclides needed to deﬁne

the material have been listed.

i. Nuclide Identiﬁcation Number (ZAID). This number is used to identify the

element or nuclide desired. The form of the number is ZZZAAA.nnX, where

ZZZ is the atomic number of the element or nuclide,

AAA is the mass number of the nuclide, ignored for photons and electrons,

nn is the cross-section evaluation identiﬁer; if blank or zero, a default

cross-section evaluation will be used, and

…

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MCNP INPUT FOR SAMPLE PROBLEM

X is the class of data: C is continuous energy; D is discrete reaction; T

is thermal; Y is dosimetry; P is photon; E is electron; and M is

multigroup.

For naturally occurring elements, AAA=000. Thus ZAID=74182 represents

the isotope W, and ZAID=74000 represents the element tungsten.

ii. Nuclide Fraction. The nuclide fractions may be normalized to 1 or left

unnormalized. For example, if the material is H2O, the fractions can be entered

as .667 and .333, or as 2 and 1 for H and O respectively. If the fractions are

entered with negative signs, they are weight fractions; otherwise they are

atomic fractions. Weight fractions and atomic fractions cannot be mixed on the

same Mm card.

The material cards for the sample problem are

M1 8016 1 $ oxygen 16

M2 26000 1 $ natural iron

M3 6000 1 $ carbon

b. VOID Card: The VOID card removes all materials and cross sections in a problem and

sets all nonzero importances to unity. It is very effective for ﬁnding errors in the geometry

description because many particles can be run in a short time. Flooding the geometry with many

particles increases the chance of particles going to most parts of the geometry—in particular, to an

incorrectly speciﬁed part of the geometry—and getting lost. The history of a lost particle often

helps locate the geometry error. The other actions of and uses for the VOID card are discussed on

page 3–113.

The sample input deck could have a VOID card while testing the geometry for errors. When you

are satisﬁed that the geometry is error-free, remove the VOID card.

6. Problem Cutoffs

Problem cutoff cards are used to specify parameters for some of the ways to terminate execution

of MCNP. The full list of available cards and a complete discussion of problem cutoffs is found on

page 3–124. For our problem we will use only the history cutoff (NPS) card. The mnemonic NPS

is followed by a single entry that speciﬁes the number of histories to transport. MCNP will

terminate after NPS histories unless it has terminated earlier for some other reason.

182

April 10, 2000 1-31

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HOW TO RUN MCNP

7. Sample Problem Summary

The entire input deck for the sample problem follows. Recall that the input can be upper, lower, or

mixed case.

Sample Problem Input Deck

c cell cards for sample problem

1 1 -0.0014 -7

2 2 -7.86 -8

3 3 -1.60 1 -2 -3 4 -5 6 7 8

4 0 -1:2:3:-4:5:-6

c end of cell cards for sample problem

C Beginning of surfaces for cube

1PZ-5

2PZ5

3PY 5

4PY-5

5PX 5

6PX-5

C End of cube surfaces

7 S 0 -4 -2.5 .5 $ oxygen sphere

8 S 0 4 4.5 $ iron sphere

blank line delimiter

IMP:N1110

SDEF POS=0 -4 -2.5

F2:N 8 $ flux across surface 8

F4:N 2 $ track length in cell 2

E0 1 12I 14

M1 8016 1 $ oxygen 16

M2 26000 1 $ natural iron

M3 6000 1 $ carbon

NPS 100000

blank line delimiter (optional)

V. HOW TO RUN MCNP

This section assumes a basic knowledge of UNIX. Lines the user will type are shown in lower

case typewriter style type. Press the RETURN key after each input line. MCNP is the

executable binary ﬁle and XSDIR is the cross-section directory. If XSDIR is not in your current

directory, you may need to set the environmental variable:

setenv DATAPATH /ab/cd

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HOW TO RUN MCNP

where /ab/cd is the directory containing both XSDIR and the data libraries.

A. Execution Line

The MCNP execution line has the following form:

mcnp

Files Options

Files

and

Options

are described below. Their order on the execution line is irrelevant. If there

are no changes in default ﬁle names, nothing need be entered for

Files

and

Options

1. Files

MCNP uses several ﬁles for input and output. The ﬁle names cannot be longer than eight

characters. The ﬁles pertinent to the sample problem are shown in Table 1.3. File INP must be

present as a local ﬁle. MCNP will create OUTP and RUNTPE.

The default name of any of the ﬁles in Table 1.3 can be changed on the MCNP execution line by

entering

default_file_name=newname

For example, if you have an input ﬁle called MCIN and want the output ﬁle to be MCOUT and the

runtpe to be MCRUNTPE, the execution line is

mcnp inp=mcin outp=mcout runtpe=mcruntpe

Only enough letters of the default name are required to uniquely identify it. For example,

mcnp i=mcin o=mcout ru=mcrntpe

also works. If a ﬁle in your local ﬁle space has the same name as a ﬁle MCNP needs to create, the

ﬁle is created with a different unique name by changing the last letter of the name of the new ﬁle

TABLE 1.3:

MCNP Files

Default File Name Description

INP Problem input specification

OUTP BCD output for printing

RUNTPE Binary start-restart data

XSDIR Cross-section directory

April 10, 2000 1-33

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HOW TO RUN MCNP

to the next letter in the alphabet. For example, if you already have an OUTP, MCNP will create

OUTQ.

Sometimes it is useful for all ﬁles from one run to have similar names. If your input ﬁle is called

JOB1, the following line

mcnp name=job1

will create an OUTP ﬁle called JOB1O and a RUNTPE ﬁle called JOB1R. If these ﬁles already

exist, MCNP will NOT overwrite them, but will issue a message that JOB1O already exists and

then will terminate.

2. Options

There are two kinds of options: program module execution options and other options. Execution

options are discussed next.

MCNP consists of ﬁve distinct execution operations, each given a module name. These operations,

their corresponding module names, and a one-letter mnemonic for each operation are listed in

Table 1.4.

When

Options

are omitted, the default is ixr. The execution of the modules is controlled by

entering the proper mnemonic on the execution line. If more than one operation is desired, combine

the single characters (in any order) to form a string. Examples of use are as follows: i to look for

input errors, ip to debug a geometry by plotting, ixz to plot cross-section data, and z to plot tally

results from the RUNTPE ﬁle.

After a job has been run, the BCD print ﬁle OUTP can be examined with an editor on the computer

and/or sent to a printer. Numerous messages about the problem execution and statistical quality of

the results are displayed at the terminal.

TABLE 1.4:

Execution Options

Mnemonic Module Operation

i IMCN Process problem input file

p PLOT Plot geometry

x XACT Process cross sections

r MCRUN Particle transport

z MCPLOT Plot tally results or cross section data

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HOW TO RUN MCNP

The “other” options add more ﬂexibility when running MCNP and are shown in Table 1.5.

The TASK option must be used to invoke multiprocessing on common or distributed memory

computer systems and is followed by the number of tasks or CPUs to be used for particle tracking.

The multiprocessing capability must be invoked at the time of compilation to create a compatible

executable. Two compilation options exist: common memory systems (UNICOS, etc.) and

distributed memory systems (workstation clusters, Cray T3D, etc.) While multiprocessing on

common memory systems is invoked and handled by the compiler with compiler directives, on

distributed memory systems it is performed by the software communications package Parallel

Virtual Machine9(PVM). Thus, using this capability on distributed memory systems requires the

installation and execution of PVM.10 On such systems, a negative entry following the TASKS

option will maximize efﬁciency for homogeneous dedicated systems (e.g., workstation with

multiple CPUs). For heterogeneous or multiuser systems, a positive entry should be used, in which

case load balancing and fault tolerance are enabled.11 In either case, the absolute value of this entry

indicates the number of hosts (or CPUs) available for use during particle tracking. On both

common and distributed memory systems, a table is provided in the output ﬁle that lists the number

of particles tracked by each host.

mcnp i=input o=output tasks 8

Indicates eight processors are to be used for particle tracking. On a common memory system, eight

tasks are initiated (if fewer processors are actually available, multiple tasks are run on each

processor.) On a distributed memory system, the master task and one subtask are initiated on the

primary host (i.e., machine from which the execution is initiated), and a subtask is initiated on each

of the seven secondary hosts.

TABLE 1.5:

Other Options

Mnemonic Operation

C m Continue a run starting with mth dump. If m is omitted, last dump is used.

See page 3–2

CN Like C, but dumps are written immediately after the ﬁxed part of the

RUNTPE, rather than at the end. See page 3–2

DBUG n Write debug information every n particles. See DBCN card, page 3–130

NOTEK Indicates that your terminal has no graphics capability. PLOT output is in

PLOTM.PS. Equivalent to TERM=0. See

FATAL Transport particles and calculate volumes even if fatal errors are found.

PRINT Create the full output file; equivalent to PRINT card. See page 3–134

TASKS n Invokes multiprocessing on common or distributed memory systems.

n=number of processors to be used.

–n is allowed only on distributed memory systems to disable load

balancing and fault tolerance, increasing system efficiency.

April 10, 2000 1-35

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HOW TO RUN MCNP

mcnp name=inp tasks -4

A negative entry following the TASKS option is allowed only on a distributed memory system and

is recommended for homogeneous dedicated systems. As in the previous example, the master task

and one subtask are initiated on the primary host, and a subtask is initiated on each of the three

secondary hosts. The negative entry disables load balancing and fault tolerance, increasing system

efﬁciency.

B. Interrupts

MCNP allows four interactive interrupts while it is running:

(ctrl c)<cr> (default) MCNP status

(ctrl c)s MCNP status

(ctrl c)m Make interactive plots of tallies

(ctrl c)q Terminate MCNP normally after current history

(ctrl c)k Kill MCNP immediately

The (ctrl c)s interrupt prints the computer time used so far, the number of particles run so far, and

the number of collisions. In the IMCN module, it prints the input line being processed. In the

XACT module, it prints the cross section being processed.

The (ctrl c)q interrupt has no effect until MCRUN is executed. (Ctrl c)q causes the code to stop

after the current particle history, to terminate “gracefully,” and to produce a ﬁnal print output ﬁle

and RUNTPE ﬁle.

The (ctrl c)k interrupt kills MCNP immediately, without normal termination. If (ctrl c)k fails, enter

(ctrl c) three or more times in a row.

C. Running MCNP

To run the example problem, have the input ﬁle in your current directory. For illustration, assume

the ﬁle is called SAMPLE. Type

mcnp n=sample

where n uniquely identiﬁes NAME. MCNP will produce an output ﬁle SAMPLEO that you can

examine at your terminal, send to a printer, or both. To look at the geometry with the PLOT module

using an interactive graphics terminal, type in

mcnp ip n=sample

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TIPS FOR CORRECT AND EFFICIENT PROBLEMS

After the plot prompt plot > appears, type in

px=0 ex=20

This plot will show an intersection of the surfaces of the problem by the planeX=0with an extent

in the x-direction of 20 cm on either side of the origin. If you want to do more with PLOT, see the

instructions on page B-1. Otherwise type “end” after the next prompt to terminate the session.

VI. TIPS FOR CORRECT AND EFFICIENT PROBLEMS

This section has a brief checklist of helpful hints that apply to three phases of your calculation:

deﬁning and setting up the problem, preparing for the long computer runs that you may require,

and making the runs that will give you results. Not everything mentioned in the checklist has been

covered in this chapter, but the list can serve as a springboard for further reading in preparation for

tackling more difﬁcult problems.

A. Problem Setup

1. Model the geometry and source distribution accurately.

2. Use the best problem cutoffs.

3. Use zero (default) for the neutron energy cutoff (MODE N P).

4. Do not use too many variance reduction techniques.

5. Use the most conservative variance reduction techniques.

6. Do not use cells with many mean free paths.

7. Use simple cells.

8. Use the simplest surfaces.

9. Study warning messages.

10. Always plot the geometry.

11. Use the VOID card when checking geometry.

12. Use separate tallies for the fluctuation chart.

13. Generate the best output (consider PRINT card).

14. RECHECK the INP file (materials, densities, masses, sources, etc.).

15. GARBAGE into code = GARBAGE out of code.

B. Preproduction

1. Run some short jobs.

2. Examine the outputs carefully.

3. Study the summary tables.

4. Study the statistical checks on tally quality and the sources of variance.

5. Compare the figures of merit and variance of the variance.

April 10, 2000 1-37

CHAPTER 1

TIPS FOR CORRECT AND EFFICIENT PROBLEMS

6. Consider the collisions per source particle.

7. Examine the track populations by cell.

8. Scan the mean free path column.

9. Check detector diagnostic tables.

10. Understand large detector contributions.

11. Strive to eliminate unimportant tracks.

12. Check MODE N P photon production.

13. Do a back-of-the-envelope check of the results.

14. DO NOT USE MCNP AS A BLACK BOX.

C. Production

1. Save RUNTPE for expanded output printing, continue run, tally plotting.

2. Look at figure of merit stability.

3. Make sure answers seem reasonable.

4. Make continue runs if necessary.

5. See if stable errors decrease by (that is, be careful of the brute force approach).

6. Remember, accuracy is only as good as the nuclear data, modeling, MCNP sampling

approximations, etc.

1N⁄

1-38 April 10, 2000

CHAPTER 1

REFERENCES

VII. REFERENCES

1. R. Kinsey, “Data Formats and Procedures for the Evaluated Nuclear Data File, ENDF,”

Brookhaven National Laboratory report BNL-NCS-50496 (ENDF 102) 2nd Edition (ENDF/

B-V) (October 1979).

2. R. J. Howerton, D. E. Cullen, R. C. Haight, M. H. MacGregor, S. T. Perkins, and E. F.

Plechaty, “The LLL Evaluated Nuclear Data Library (ENDL): Evaluation Techniques,

Reaction Index, and Descriptions of Individual Reactions,” Lawrence Livermore National

Laboratory report UCRL-50400, Vol. 15, Part A (September 1975).

3. M. A. Gardner and R. J. Howerton, “ACTL: Evaluated Neutron Activation Cross–Section

Library-Evaluation Techniques and Reaction Index,” Lawrence Livermore National

Laboratory report UCRL-50400, Vol. 18 (October 1978).

4. E. D. Arthur and P. G. Young, “Evaluated Neutron-Induced Cross Sections for 54,56Fe to 40

MeV,” Los Alamos Scientific Laboratory report LA-8626-MS (ENDF-304) (December

1980).

5. D. G. Foster, Jr. and E. D. Arthur, “Average Neutronic Properties of “Prompt” Fission

Products,” Los Alamos National Laboratory report LA-9168-MS (February 1982).

6. E. D. Arthur, P. G. Young, A. B. Smith, and C. A. Philis, “New Tungsten Isotope Evaluations

for Neutron Energies Between 0.1 and 20 MeV,” Trans. Am. Nucl. Soc.39, 793 (1981).

7. R. E. MacFarlane, D. W. Muir, and R. M. Boicourt, “The NJOY Nuclear Data Processing

System, Volume I: User’s Manual,” Los Alamos National Laboratory report LA-9303-M,

Vol. I (ENDF-324) (May 1982).

R. E. MacFarlane, D. W. Muir, and R. M. Boicourt, “The NJOY Nuclear Data Processing

System, Volume II: The NJOY, RECONR, BROADR, HEATR, and THERMR Modules,”

Los Alamos National Laboratory report LA-9303-M, Vol. II (ENDF-324) (May 1982).

8. R. A. Forster, R. C. Little, J. F. Briesmeister, and J. S. Hendricks, “MCNP Capabilities For

Nuclear Well Logging Calculations,” IEEE Transactions on Nuclear Science,37 (3), 1378

(June 1990)

9. A. Geist et al, “PVM 3 User’s Guide and Reference Manual,” ORNL/TM-12187, Oak Ridge

National Laboratory (1993).

10. G. McKinney, “A Practical Guide to Using MCNP with PVM,” Trans. Am. Nucl. Soc. 71,

397 (1994).

11. G. McKinney, “MCNP4B Multiprocessing Enhancements Using PVM,” LANL memo

X-6:GWM-95-212 (1995).

April 10, 2000 2-1

CHAPTER 2

INTRODUCTION

CHAPTER 2

GEOMETRY, DATA, PHYSICS, AND MATHEMATICS

I. INTRODUCTION

Chapter 2 discusses the mathematics and physics of MCNP, including geometry, cross−section

libraries, sources, variance reduction schemes, Monte Carlo simulation of neutron and photon

transport, and tallies. This discussion is not meant to be exhaustive; many details of the particular

techniques and of the Monte Carlo method itself will be found elsewhere. Carter and Cashwell's

book Particle-Transport Simulation with the Monte Carlo Method,1a good general reference on

radiation transport by Monte Carlo, is based upon what is in MCNP. A more recent reference is

Lux and Koblinger's book, Monte Carlo Particle Transport Methods: Neutron and Photon

Calculations.2 Methods of sampling from standard probability densities are discussed in the

Monte Carlo samplers by Everett and Cashwell.3

MCNP was originally developed by the Monte Carlo Group, currently the Diagnostic

Applications Group, (Group X-5) in the Applied Physics Division (X Division) at the Los

Alamos National Laboratory. Group X-5 improves MCNP (releasing a new version every two

to three years), maintains it at Los Alamos and at other laboratories where we have collaborators

or sponsors, and provides limited free consulting and support for MCNP users. MCNP is

distributed to other users through the Radiation Safety Information Computational Center

(RSICC) at Oak Ridge, Tennessee, and the OECD/NEA data bank in Paris, France.

MCNP has approximately 48,000 lines of FORTRAN and 1000 lines of C source coding,

including comments, and with the COMMON blocks listed only once and not in every

subroutine. There are about 385 subroutines. There is only one source code; it is used for all

systems. At Los Alamos, there are about 250 active users. Worldwide, there are about 3000

active users at about 200 installations.

MCNP takes advantage of parallel computer architectures. It is supported in multitasking mode

on some mainframes and in multiprocessing mode on a cluster of workstations where the

distributed processing uses the Parallel Virtual Machine (PVM) software from Oak Ridge.

MCNP has not been successfully vectorized because the overhead required to set up and break

apart vector queues at random decision points is greater than the savings from vectorizing the

simple arithmetic between the decision points. MCNP (and any general Monte Carlo code) is

little more than a collection of random decision points with some simple arithmetic in between.

Because MCNP does not take advantage of vectorization, it is fairly inefﬁcient on vectorized

computers. In particular, many workstations and PCs run MCNP as fast or faster than

mainframes. MCNP has been made as system independent as possible to enhance its portability,

2-2 April 10, 2000

CHAPTER 2

INTRODUCTION

and has been written to comply with the ANSI FORTRAN 77 standard. With one source code,

MCNP is maintained on many platforms.

A. History

The Monte Carlo method is generally attributed to scientists working on the development of

nuclear weapons in Los Alamos during the 1940s. However, its roots go back much farther.

Perhaps the earliest documented use of random sampling to solve a mathematical problem was

that of Compte de Buffon in 1772.4A century later people performed experiments in which they

threw a needle in a haphazard manner onto a board ruled with parallel straight lines and inferred

the value of π from observations of the number of intersections between needle and lines.5,6

Laplace suggested in 1786 that πcould be evaluated by random sampling.7Lord Kelvin appears

to have used random sampling to aid in evaluating some time integrals of the kinetic energy that

appear in the kinetic theory of gasses8 and acknowledged his secretary for performing

calculations for more than 5000 collisions.9

According to Emilio Segrè, Enrico Fermi's student and collaborator, Fermi invented a form of

the Monte Carlo method when he was studying the moderation of neutrons in Rome.9,10 Though

Fermi did not publish anything, he amazed his colleagues with his predictions of experimental

results. After indulging himself, he would reveal that his “guesses” were really derived from the

statistical sampling techniques that he performed in his head when he couldn't fall asleep.

During World War II at Los Alamos, Fermi joined many other eminent scientists to develop the

ﬁrst atomic bomb. It was here that Stan Ulam became impressed with electromechanical

computers used for implosion studies. Ulam realized that statistical sampling techniques were

considered impractical because they were long and tedious, but with the development of

computers they could become practical. Ulam discussed his ideas with others like

John von Neumann and Nicholas Metropolis. Statistical sampling techniques reminded

everyone of games of chance, where randomness would statistically become resolved in

predictable probabilities. It was Nicholas Metropolis who noted that Stan had an uncle who

would borrow money from relatives because he “just had to go to Monte Carlo” and thus named

the mathematical method “Monte Carlo.”10

Meanwhile, a team of wartime scientists headed by John Mauchly was working to develop the

ﬁrst electronic computer at the University of Pennsylvania in Philadelphia. Mauchly realized

that if Geiger counters in physics laboratories could count, then they could also do arithmetic

and solve mathematical problems. When he saw a seemingly limitless array of women cranking

out ﬁring tables with desk calculators at the Ballistic Research Laboratory at Aberdeen, he

proposed10 that an electronic computer be built to deal with these calculations. The result was

ENIAC (Electronic Numerical Integrator and Computer), the world’s ﬁrst computer, built for

April 10, 2000 2-3

CHAPTER 2

INTRODUCTION

Aberdeen at the University of Pennsylvania. It had 18,000 double triode vacuum tubes in a

system with 500,000 solder joints.10

John von Neumann was a consultant to both Aberdeen and Los Alamos. When he heard about

ENIAC, he convinced the authorities at Aberdeen that he could provide a more exhaustive test

of the computer than mere ﬁring-table computations. In 1945 John von Neumann, Stan Frankel,

and Nicholas Metropolis visited the Moore School of Electrical Engineering at the University of

Pennsylvania to explore using ENIAC for thermonuclear weapon calculations with Edward

Teller at Los Alamos.10 After the successful testing and dropping of the ﬁrst atomic bombs a few

months later, work began in earnest to calculate a thermonuclear weapon. On March 11, 1947,

John von Neumann sent a letter to Robert Richtmyer, leader of the Theoretical Division at Los

Alamos, proposing use of the statistical method to solve neutron diffusion and multiplication

problems in ﬁssion devices.10 His letter was the ﬁrst formulation of a Monte Carlo computation

for an electronic computing machine. In 1947, while in Los Alamos, Fermi invented a

mechanical device called FERMIAC11 to trace neutron movements through ﬁssionable materials

by the Monte Carlo Method.

By 1948 Stan Ulam was able to report to the Atomic Energy Commission that not only was the

Monte Carlo method being successfully used on problems pertaining to thermonuclear as well

as ﬁssion devices, but also it was being applied to cosmic ray showers and the study of partial

differential equations.10 In the late 1940s and early 1950s, there was a surge of papers describing

the Monte Carlo method and how it could solve problems in radiation or particle transport and

other areas.12,13,14 Many of the methods described in these papers are still used in Monte Carlo

today, including the method of generating random numbers15 used in MCNP. Much of the

interest was based on continued development of computers such as the Los Alamos MANIAC

(Mechanical Analyzer, Numerical Integrator, and Computer) in March, 1952.

The Atomic Energy Act of 1946 created the Atomic Energy Commission to succeed the

Manhattan Project. In 1953 the United States embarked upon the “Atoms for Peace” program

with the intent of developing nuclear energy for peaceful applications such as nuclear power

generation. Meanwhile, computers were advancing rapidly. These factors led to greater interest

in the Monte Carlo method. In 1954 the ﬁrst comprehensive review of the Monte Carlo method

was published by Herman Kahn16 and the ﬁrst book was published by Cashwell and Everett17

in 1959.

At Los Alamos, Monte Carlo computer codes developed along with computers. The ﬁrst Monte

Carlo code was the simple 19−step computing sheet in John von Neumann's letter to Richtmyer.

But as computers became more sophisticated, so did the codes. At ﬁrst the codes were written

in machine language and each code would solve a speciﬁc problem. In the early 1960s, better

computers and the standardization of programming languages such as FORTRAN made possible

more general codes. The ﬁrst Los Alamos general−purpose particle transport Monte Carlo code

was MCS,18 written in 1963. Scientists who were not necessarily experts in computers and

2-4 April 10, 2000

CHAPTER 2

INTRODUCTION

Monte Carlo mathematical techniques now could take advantage of the Monte Carlo method for

radiation transport. They could run the MCS code to solve modest problems without having to

do either the programming or the mathematical analysis themselves. MCS was followed by

MCN19 in 1965. MCN could solve the problem of neutrons interacting with matter in a three−

dimensional geometry and used physics data stored in separate, highly−developed libraries.

In 1973 MCN was merged with MCG,20 a Monte Carlo gamma code that treated higher energy

photons, to form MCNG, a coupled neutron−gamma code. In 1977 MCNG was merged with

MCP,20 a Monte Carlo Photon code with detailed physics treatment down to 1 keV, to accurately

model neutron-photon interactions. The code has been known as MCNP ever since. Though at

ﬁrst MCNP stood for Monte Carlo Neutron Photon, now it stands for Monte Carlo N−Particle.

Other major advances in the 70s included the present generalized tally structure, automatic

calculation of volumes, and a Monte Carlo eigenvalue algorithm to determine for nuclear

criticality (KCODE).

In 1983 MCNP3 was released, entirely rewritten in ANSI standard FORTRAN 77. MCNP3 was

the ﬁrst MCNP version internationally distributed through the Radiation Shielding and

Information Center at Oak Ridge, Tennessee. Other 1980s versions of MCNP were MCNP3A

(1986) and MCNP3B (1988), that included tally plotting graphics (MCPLOT), the present

generalized source, surface sources, repeated structures/lattice geometries, and multigroup/

adjoint transport.

MCNP4 was released in 1990 and was the ﬁrst UNIX version of the code. It accommodated N−

particle transport and multitasking on parallel computer architectures. MCNP4 added electron

transport (patterned after the Integrated TIGER Series (ITS) continuous−slowing−down

approximation physics),21 the pulse height tally (F8), a thick−target bremsstrahlung

approximation for photon transport, enabled detectors and DXTRAN with the S(α,β) thermal

treatment, provided greater random number control, and allowed plotting of tally results while

the code was running.

MCNP4A, released in 1993, featured enhanced statistical analysis, distributed processor

multitasking for running in parallel on a cluster of scientiﬁc workstations, new photon libraries,

ENDF/B−VI capabilities, color X−Windows graphics, dynamic memory allocation, expanded

criticality output, periodic boundaries, plotting of particle tracks via SABRINA, improved tallies

in repeated structures, and many smaller improvements.

MCNP4B, released in 1997, featured differential operator perturbations, enhanced photon

physics equivalent to ITS3.0, PVM load balance and fault tolerance, cross section plotting,

postscript ﬁle plotting, 64−bit workstation upgrades, PC X−windows, inclusion of LAHET

HMCNP, lattice universe mapping, enhanced neutron lifetimes, coincident−surface lattice

capability, and many smaller features and improvements.

keff

April 10, 2000 2-5

CHAPTER 2

INTRODUCTION

MCNP4C, released in 2000 features an unresolved resonance treatment, macrobodies,

superimposed importance mesh, perturbation enhancements, electron physics enhancements, an

alpha eigenvalue search, plotter upgrades, cumulative tallies, parallel enhancements and other

small features and improvements.

Large production codes such as MCNP have revolutionized science −− not only in the way it is

done, but also by becoming the repositories for physics knowledge. MCNP represents about 500

person-years of sustained effort. The knowledge and expertise contained in MCNP is

formidable.

Current MCNP development is characterized by a strong emphasis on quality control,

documentation, and research. New features continue to be added to the code to reﬂect new

advances in computer architecture, improvements in Monte Carlo methodology, and better

physics models. MCNP has a proud history and a promising future.

B. MCNP Structure

MCNP is written in the style of Dr. Thomas N. K. Godfrey, the principal MCNP programmer

from 1975−1989. Variable dimensions for arrays are achieved by massive use of

EQUIVALENCE statements and offset indexing. All variables local to a routine are no more

than two characters in length, and all COMMON variables are between three and six characters

in length. The code strictly complies with the ANSI FORTRAN 77 standard. The principal

characteristic of Tom Godfrey’s style is its terseness. Everything is accomplished in as few lines

of code as possible. Thus MCNP does more than some other codes that are more than ten times

larger. It was Godfrey’s philosophy that anyone can understand code at the highest level by

making a ﬂow chart and anyone can understand code at the lowest level (one FORTRAN line);

it is the intermediate level that is most difﬁcult. Consequently, by using a terse programming

style, subroutines could ﬁt within a few pages and be most easily understood. Tom Godfrey’s

style is clearly counter to modern computer science programming philosophies, but it has served

MCNP well and is preserved to provide stylistic consistency throughout.

The general structure of MCNP is as follows:

Initiation (IMCN):

• Read input ﬁle (INP) to get dimensions (PASS1);

• Set up variable dimensions or dynamically allocated storage (SETDAS);

• Re-read input ﬁle (INP) to load input (RDPROB);

• Process source (ISOURC);

• Process tallies (ITALLY);

• Process materials speciﬁcations (STUFF) including masses without loadingthe data ﬁles;

• Calculate cell volumes and surface areas (VOLUME).

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INTRODUCTION

Interactive Geometry Plot (PLOT).

Cross Section Processing (XACT):

• Load libraries (GETXST);

• Eliminate excess neutron data outside problem energy range (EXPUNG);

• Doppler broaden elastic and total cross sections to the proper temperature if the problem

temperature is higher than the library temperature (BROADN);

• Process multigroup libraries (MGXSPT);

• Process electron libraries (XSGEN) including calculation of range tables, straggling tables,

scattering angle distributions, and bremsstrahlung.

MCRUN sets up multitasking and multiprocessing, runs histories (by calling TRNSPT, which

calls HSTORY), and returns to OUTPUT to print, write RUNTPE dumps, or process another

criticality (KCODE) cycle.

Under MCRUN, MCNP runs neutron, photon, or electron histories (HSTORY), calling

ELECTR for electron tracks:

• Start a source particle (STARTP);

• Find the distance to the next boundary (TRACK), cross the surface (SURFAC) and enter

the next cell (NEWCEL);

• Find the total neutron cross section (ACETOT) and process neutron collisions (COLIDN)

producing photons as appropriate (ACEGAM);

• Find the total photon cross section (PHOTOT) and process photon collisions (COLIDP)

producing electrons as appropriate (EMAKER);

• Use the optional thick−target bremsstrahlung approximation if no electron transport

(TTBR);

• Follow electron tracks (ELECTR);

• Process optional multigroup collisions (MGCOLN, MGCOLP, MGACOL);

• Process detector tallies (TALLYD) or DXTRAN;

• Process surface, cell, and pulse height tallies (TALLY).

Periodically write output ﬁle, restart dumps, update to next criticality (KCODE) cycle,

rendezvous for multitasking and updating detector and DXTRAN Russian roulette criteria, etc.

(OUTPUT):

• Go to the next criticality cycle (KCALC);

• Print output ﬁle summary tables (SUMARY, ACTION);

• Print tallies (TALLYP);

• Generate weight windows (OUTWWG).

Plot tallies, cross sections, and other data (MCPLOT).

GKS graphics simulation routines.

PVM distributed processor multiprocessing routines.

Random number generator and control (RANDOM).

Mathematics, character manipulation, and other slave routines.

April 10, 2000 2-7

CHAPTER 2

INTRODUCTION

C. History Flow

The basic ﬂow of a particle history for a coupled neutron/photon/electron problem is handled in

subroutine HSTORY. HSTORY is called from TRNSPT after the random number sequence is

set up and the number of the history, NPS, is incremented. The ﬂow of HSTORY is then as

follows.

First, STARTP is called. The ﬂag IPT is set for the type of particle being run: 1 for a neutron, 2

for a photon, and 3 for an electron. Some arrays and variables (such as NBNK, the number of

particles in the bank) are initialized to zero. The starting random number is saved (RANB,

RANS, RNRTC0), and the branch of the history, NODE, is set to 1.

Next, the appropriate source routine is called. Source options are the standard ﬁxed sources

(SOURCB), the surface source (SURSRC), the KCODE criticality source (SOURCK), or a user-

provided source (SOURCE). All of the parameters describing the particle are set in these source

routines, including position, direction of ﬂight, energy, weight, time, and starting cell (and

possibly surface), by sampling the various distributions described on the source input control

cards. Several checks are made at this time to verify that the particle is in the correct cell or on

the correct surface, and directed toward the correct cell; then control is returned to STARTP.

Next in STARTP, the initial parameters of the ﬁrst ﬁfty particle histories are printed. Then some

of the summary information is incremented (see Appendix E for an explanation of these arrays).

Energy, time, and weight are checked against cutoffs. A number of error checks are made.

TALLYD is called to score any detector contributions, and then DXTRAN is called (if used in

the problem) to create particles on the spheres. The particles are saved with BANKIT for later

tracking. TALPH is called to start the bookkeeping for the pulse height cell tally energy balance.

The weight window game is played, with any additional particles from splitting put into the bank

and any losses to Russian roulette terminated. Control is returned to HSTORY.

Back in HSTORY, the actual particle transport is started. For an electron source, ELECTR is

called and electrons are run separately. For a neutron or photon source, TRACK is called to

calculate the intersection of the particle trajectory with each bounding surface of the cell. The

minimum positive distance DLS to the cell boundary indicates the next surface JSU the particle

is heading toward. The distance to the nearest DXTRAN sphere DXL is calculated, as is the

distance to time cutoff DTC, and energy boundary for multigroup charged particles DEB. The

cross sections for cell ICL are calculated using a binary table lookup in ACETOT for neutrons

and in PHOTOT for photons. The total cross section is modiﬁed in EXTRAN by the exponential

transformation if necessary. The distance PMF to the next collision is determined (if a forced

collision is required, FORCOL is called and the uncollided part is banked). The track length D

of the particle in the cell is found as the minimum of the distance PMF to collision, the distance

DLS to the surface JSU, the distance DXL to a DXTRAN sphere, the distance DTC to time

cutoff, or the distance DEB to energy boundary. TALLY then is called to increment any track

2-8 April 10, 2000

CHAPTER 2

INTRODUCTION

length cell tallies. Some summary information is incremented. The particle’s parameters (time,

position, and energy) are then updated. If the particle's distance DXL to a DXTRAN sphere (of

the same type as the current particle) is equal to the minimum track length D, the particle is

terminated because particles reaching the DXTRAN sphere are already accounted for by the

DXTRAN particles from each collision. If the particle exceeds the time cutoff, the track is

terminated. If the particle was detected leaving a DXTRAN sphere, the DXTRAN ﬂag IDX is

set to zero and the weight cutoff game is played. The particle is either terminated to weight cutoff

or survives with an increased weight. Weight adjustments then are made for the exponential

transformation.

If the minimum track length D is equal to the distance-to-surface crossing DLS, the particle is

transported distance D to surface JSU and SURFAC is called to cross the surface and do any

surface tallies (by calling TALLY) and to process the particle across the surface into the next cell

by calling NEWCEL. It is in SURFAC that reﬂecting surfaces, periodic boundaries, geometry

splitting, Russian roulette from importance sampling, and loss to escape are treated. For

splitting, one bank entry of NPA particle tracks is made in BANKIT for an (NPA+1)-for-1 split.

The bank is the IBNK array, and entries or retrievals are made with the GPBLCM and JPBLCM

arrays (the bank operates strictly on a last-in, ﬁrst-out basis). The history is continued by going

back to HSTORY and calling TRACK.

If the distance to collision PMF is less than the distance to surface DLS, or if a multigroup

charged particle reaches the distance to energy boundary DEB, the particle undergoes a

collision. Everything about the collision is determined in COLIDN for neutrons and COLIDP

for photons. COLIDN determines which nuclide is involved in the collision, samples the target

velocity of the collision nuclide by calling TGTVEL for the free gas thermal treatment,

generates and banks any photons (ACEGAM), handles analog capture or capture by weight

reduction, plays the weight cutoff game, handles thermal collisions (SABCOL) and

elastic or inelastic scattering (ACECOL). For criticality problems, COLIDK is called to store

ﬁssion sites for subsequent generations. Any additional tracks generated in the collision are put

in the bank. ACECAS and ACECOS determine the energies and directions of particles exiting

the collision. Multigroup and multigroup/adjoint collisions are treated separately in MGCOLN

and MGACOL that are called from COLIDN. The collision process and thermal treatments are

described in more detail later in this chapter (see page 2–28).

COLIDP for photons is similar to COLIDN, and it covers the simple or the detailed physics

treatments. The simple physics treatment is better for free electrons; the detailed treatment is the

default and includes form factors for electron binding effects, coherent (Thomson) scatter, and

ﬂuorescence from photoelectric capture (see page 2–55). COLIDP samples for the collision

nuclide, treats photoelectric absorption, or capture (with ﬂuorescence in the detailed physics

treatment), incoherent (Compton) scatter (with form factors in the detailed physics treatment to

account for electron binding), coherent (Thomson) scatter for the detailed physics treatment only

(again with form factors), and pair production. Electrons are generated (EMAKER) for

Sαβ,()

April 10, 2000 2-9

CHAPTER 2

GEOMETRY

incoherent scatter, pair production, and photoelectric absorption. These electrons may be

assumed to deposit all their energy instantly if IDES=1 on the PHYS:P card, or they may

produce electrons with the thick−target bremsstrahlung approximation (default for MODE P

problems, IDES=0 on the PHYS:P card), or they may undergo full electron transport (default for

MODE P E problems, IDES=0 on the PHYS:P card.) Multigroup or multigroup/adjoint photons

are treated separately in MGCOLP or MGACOL.

After the surface crossing or collision is processed, control returns to HSTORY and transport

continues by calling TRACK, where the distance to cell boundary is calculated. Or if the particle

involved in the collision was killed by capture or variance reduction, the bank is checked for any

remaining progeny, and if none exists, the history is terminated. Appropriate summary

information is incremented, the tallies of this particular history are added to the total tally data

by TALSHF, and a return is made to TRNSPT.

In TRNSPT, checks are made to see if output is required or if the job should be terminated

because enough histories have been run or too little time remains to continue. For continuation,

HSTORY is called again. Otherwise a return is made to MCRUN. MCRUN calls OUTPUT,

which calls SUMARY to print the summary information. Then SUMARY calls TALLYP to print

the tally data. Appendix E deﬁnes all of the MCNP variables that are in COMMON as well as

detailed descriptions of some important arrays.

II. GEOMETRY

The basic MCNP geometry concepts, discussed in Chapter 1, include the sense of a cell, the

intersection and union operators, and surface speciﬁcation. Covered in this section are the

complement operator; the repeated structure capability; an explanation of two surfaces, the cone

and the torus; and a description of ambiguity, reﬂecting, white, and periodic boundary surfaces.

A. Complement Operator

This operator provides no new capability over the intersection and union operators; it is just a

shorthand cell-specifying method that implicitly uses the intersection and union operators.

The symbol # is the complement operator and can be thought of as standing for not in. There

are two basic uses of the operator:

#nmeans that the description of the current cell is the complement of the description of cell n.

#(...) means complement the portion of the cell description in the parentheses (usually just a

list of surfaces describing another cell).

In the ﬁrst of the two above forms, MCNP performs ﬁve operations: (1) the symbol # is removed,

(2) parentheses are placed around n, (3) any intersections in nbecome unions, (4) any unions in

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GEOMETRY

nare replaced by back-to-back parentheses, “)(“, which is an intersection, and (5) the senses of

the surfaces deﬁning nare reversed.

A simple example is a cube. We deﬁne a two−cell geometry with six surfaces, where cell 1 is the

cube and cell 2 is the outside world:

1 0 −1 2 −3 4 −5 6

2 0 1:−2: 3:−4: 5:−6

Note that cell 2 is everything in the universe that is not in cell 1, or

2 0 #1

The form #(n) is not allowed; it is functionally available as the equivalent of −n.

CAUTION: Using the complement operator can destroy some of the necessary conditions for

some cell volume and surface area calculations by MCNP. See page 4–15 for an example.

The complement operator can be easily abused if it is used indiscriminately. A simple example

can best illustrate the problems. Fig. 2-1 consists of two concentric spheres inside a box. Cell 4

can be described using the complement operator as

4 0 #3 #2 #1

Although cells 1 and 2 do not touch cell 4, to omit them would be incorrect. If they were omitted,

the description of cell 4 would be everything in the universe that is not in cell 3. Since cells 1

and 2 are not part of cell 3, they would be included in cell 4. Even though surfaces 1 and 2 do

not physically bound cell 4, using the complement operator as in this example causes MCNP to

think that all surfaces involved with the complement do bound the cell. Even though this

speciﬁcation is correct and required by MCNP, the disadvantage is that when a particle enters

cell 4 or has a collision in cell 4, MCNP must calculate the intersection of the particle's trajectory

with all real bounding surfaces of cell 4 plus any extraneous ones brought in by the complement

operator. This intersection calculation is very expensive and can add signiﬁcantly to the required

computer time.

Figure 2-1.

April 10, 2000 2-11

CHAPTER 2

GEOMETRY

A better description of cell 4 would be to complement the description of cell 3 (omitting surface

2) by reversing the senses and interchanging union and intersection operators as illustrated in the

cell cards that describe the simple cube in the preceding paragraphs.

B. Repeated Structure Geometry

The repeated structure geometry feature is explained in detail starting on page 3–25. The

capabilities are only introduced here. Examples are shown in Chapter 4. The cards associated

with the repeated structure feature are U (universe), FILL, TRCL, and LAT (lattice) and cell

cards with LIKE m BUT.

The repeated structure feature makes it possible to describe only once the cells and surfaces of

any structure that appears more than once in a geometry. This unit then can be replicated at other

xyz locations by using the “LIKE m BUT” construct on a cell card. The user speciﬁes that a cell

is ﬁlled with something called a universe. The U card identiﬁes the universe, if any, to which a

cell belongs. The FILL card speciﬁes with which universe a cell is to be ﬁlled. A universe is

either a lattice or an arbitrary collection of cells. The two types of lattice shapes, hexagonal

prisms and hexahedra, need not be rectangular nor regular, but they must ﬁll space exactly.

Several concepts and cards combine in order to use this capability.

C. Surfaces

1. Explanation of Cone and Torus

Two surfaces, the cone and torus, require more explanation. The quadratic equation for a cone

describes a cone of two sheets (just like a hyperboloid of two sheets)−one sheet is a cone of

positive slope, and the other has a negative slope. A cell whose description contains a two−

sheeted cone may require an ambiguity surface to distinguish between the two sheets. MCNP

provides the option to select either of the two sheets; this option frequently simpliﬁes geometry

setups and eliminates any ambiguity. The +1 or the −1 entry on the cone surface card causes the

one sheet cone treatment to be used. If the sign of the entry is positive, the speciﬁed sheet is the

one that extends to inﬁnity in the positive direction of the coordinate axis to which the cone axis

is parallel. The converse is true for a negative entry. This feature is available only for cones

whose axes are parallel to the coordinate axes of the problem.

The treatment of fourth degree surfaces in Monte Carlo calculations has always been difﬁcult

because of the resulting fourth order polynomial (“quartic”) equations. These equations must be

solved to ﬁnd the intersection of a particle’s line of ﬂight with a toroidal surface. In MCNP these

equations must also be solved to ﬁnd the intersection of surfaces in order to compute the volumes

and surface areas of geometric regions of a given problem. In either case, the quartic equation,

2-12 April 10, 2000

CHAPTER 2

GEOMETRY

is difﬁcult to solve on a computer because of roundoff errors. For many years the MCNP toroidal

treatment required 30 decimal digits (CDC double-precision) accuracy to solve quartic

equations. Even then there were roundoff errors that had to be corrected by Newton-Raphson

iterations. Schemes using a single-precision quartic formula solver followed by a Newton-

Raphson iteration were inadequate because if the initial guess of roots supplied to the Newton-

Raphson iteration is too inaccurate, the iteration will often diverge when the roots are close

together.

The single-precision quartic algorithm in MCNP basically follows the quartic solution of

Cashwell and Everett.22 When roots of the quartic equation are well separated, a modiﬁed

Newton-Raphson iteration quickly achieves convergence. But the key to this method is that if

the roots are double roots or very close together, they are simply thrown out because a double

root corresponds to a particle’s trajectory being tangent to a toroidal surface, and it is a very good

approximation to assume that the particle then has no contact with the toroidal surface. In

extraordinarily rare cases where this is not a good assumption, the particle would become “lost.”

Additional reﬁnements to the quartic solver include a carefully selected ﬁnite size of zero, the

use of a cubic rather than a quartic equation solver whenever a particle is transported from the

surface of a torus, and a gross quartic coefﬁcient check to ascertain the existence of any real

positive roots. As a result, the single-precision quartic solver is substantially faster than double-

precision schemes, portable, and also somewhat more accurate.

In MCNP, elliptical tori symmetric about any axis parallel to a coordinate axis may be speciﬁed.

The volume and surface area of various tallying segments of a torus usually will be calculated

automatically.

2. Ambiguity Surfaces

The description of the geometry of a cell must eliminate any ambiguities as to which region of

space is included in the cell. That is, a particle entering a cell should be able to determine

uniquely which cell it is in from the senses of the bounding surfaces. This is not possible in a a

geometry such as shown in Fig. 2-2 unless an ambiguity surface is speciﬁed. Suppose the ﬁgure

is rotationally symmetric about the y−axis.

A particle entering cell 2 from the inner spherical region might think it was entering cell 1

because a test of the senses of its coordinates would satisfy the description of cell 1 as well as

that of cell 2. In such cases, an ambiguity surface is introduced such as a, the plane y = 0. An

ambiguity surface need not be a bounding surface of a cell, but it may be and frequently is. It

can also be the bounding surface of some cell other than the one in question. However, the

surface must be listed among those in the problem and must not be a reﬂecting surface (see page

2–14). The description of cells 1 and 2 in Fig. 2-2 is augmented by listing for each its sense

x4Bx3Cx2Dx E++++ 0=

April 10, 2000 2-13

CHAPTER 2

GEOMETRY

Figure 2-2.

relative to surface aas well as that of each of its other bounding surfaces. A particle in cell 1

cannot have the same sense relative to surface a as does a particle in cell 2. More than one

ambiguity surface may be required to deﬁne a particular cell.

A second example may help to clarify the signiﬁcance of ambiguity surfaces. We would like to

describe the geometry of Fig. 2-3a. Without the use of an ambiguity surface, the result will be

Fig. 2-3b. Surfaces 1 and 3 are spheres about the origin, and surface 2 is a cylinder around the

y−axis. Cell 1 is both the center and outside world of the geometry connected by the region

interior to surface 2.

Figure 2-3.

At ﬁrst glance it may appear that cell 1 can easily be speciﬁed by −1 : −2 : 3 whereas cell 2 is

simply #1. This results in Figure 2.3b, in which cell 1 is everything in the universe interior to

surface 1 plus everything in the universe interior to surface 2 (remember the cylinder goes to plus

and minus inﬁnity) plus everything in the universe exterior to surface 3.

An ambiguity surface (a plane at y=0) will solve the problem. Everything in the universe to the

right of the ambiguity surface (call it surface 4) intersected with everything in the universe

2 1

(a)

2 1

(b)

2-14 April 10, 2000

CHAPTER 2

GEOMETRY

interior to the cylinder is a cylindrical region that goes to plus inﬁnity but terminates at y=0.

Therefore, −1 : (4 −2) : 3 deﬁnes cell 1 as desired in Figure 2.3a. The parentheses in this last

expression are not required because intersections are done before unions. Another expression for

cell 2 rather than #1 is 1 −3 #(4 −2).

For the user, ambiguity surfaces are speciﬁed the same way as any other surface–simply list the

signed surface number as an entry on the cell card. For MCNP, if a particular ambiguity surface

appears on cell cards with only one sense, it is treated as a true ambiguity surface. Otherwise, it

still functions as an ambiguity surface but the TRACK subroutine will try to ﬁnd intersections

with it, thereby using a little more computer time.

3. Reflecting Surfaces

A surface can be designated a reﬂecting surface by preceding its number on the surface card with

an asterisk. Any particle hitting a reﬂecting surface is specularly (mirror) reﬂected. Reﬂecting

planes are valuable because they can simplify a geometry setup (and also tracking) in a problem.

They can, however, make it difﬁcult (or even impossible) to get the correct answer. The user is

cautioned to check the source weight and tallies to ensure that the desired result is achieved. Any

tally in a problem with reﬂecting planes should have the same expected result as the tally in the

same problem without reﬂecting planes. Detectors or DXTRAN used with reﬂecting surfaces

give WRONG answers (see page 2–92).

The following example illustrates the above points and hopefully makes you very cautious in the

use of reﬂecting surfaces; they should never be used in any situation without a lot of thought.

Consider a cube of carbon 10 cm on a side sitting on top of a 5-MeV neutron source distributed

uniformly in volume. The source cell is a 1-cm-thick void completely covering the bottom of the

carbon cube and no more. The average neutron ﬂux across any one of the sides (but not top or

bottom) is calculated to be 0.150 (±0.5%) per cm2per starting neutron from an MCNP F2 tally,

and the ﬂux at a point at the center of the same side is 1.55e-03 n/cm2(±1%) from an MCNP F5

tally.

The cube can be modeled by half a cube and a reﬂecting surface. All dimensions remain the same

except the distance from the tally surface to the opposite surface (which becomes the reﬂecting

surface) is 5 cm. The source cell is cut in half also. Without any source normalization, the ﬂux

across the surface is now 0.302 ( %), which is twice the ﬂux in the nonreﬂecting geometry.

The detector ﬂux is 2.58E −03 ( %), which is less than twice the point detector ﬂux in the

nonreﬂecting problem.

The problem is that for the surface tally to be correct, the starting weight of the source particles

has to be normalized; it should be half the weight of the nonreﬂected source particles. The

detector results will always be wrong (and lower) for the reason discussed on page 2–92.

0.5±1±

April 10, 2000 2-15

CHAPTER 2

GEOMETRY

In this particular example, the normalization factor for the starting weight of source particles

should be 0.5 because the source volume is half of the original volume. Without the

normalization, the full weight of source particles is started in only half the volume. These

normalization factors are problem dependent and should be derived very carefully.

Another way to view this problem is that the tally surface has doubled because of the reﬂecting

surface; two scores are being made across the tally surface when one is made across each of two

opposite surfaces in the nonreﬂecting problem. The detector has doubled too,except that the

contributions to it from beyond the reﬂecting surface are not being made, see page 2–92.

4. White Boundaries

A surface can be designated a white boundary surface by preceding its number on the surface

card with a plus. A particle hitting a white boundary is reﬂected with a cosine distribution,

p(µ)=µ, relative to the surface normal; that is, µ= , where ξ is a random number. White

boundary surfaces are useful for comparing MCNP results with other codes that have white

boundary conditions. They also can be used to approximate a boundary with an inﬁnite scatterer.

They make absolutely no sense in problems with next-event estimators such as detectors or

DXTRAN (see page 2–92) and should always be used with caution.

5. Periodic Boundaries

Periodic boundary conditions can be applied to pairs of planes to simulate an inﬁnite lattice.

Although the same effect can be achieved with an inﬁnite lattice, the periodic boundary is easier

to use, simpliﬁes comparison with other codes having periodic boundaries, and can save

considerable computation time. There is approximately a 55% run-time penalty associated with

repeated structures and lattices that can be avoided with periodic boundaries. However,

collisions and other aspects of the Monte Carlo random walk usually dominate running time, so

the savings realized by using periodic boundaries are usually much smaller. A simple periodic

boundary problem is illustrated in Figure 2.3c.

Figure 2-3(c).

2-16 April 10, 2000

CHAPTER 2

CROSS SECTIONS

It consists of a square reactor lattice inﬁnite in the z direction and 10 cm on a side in the x and

y directions with an off-center 1 cm−radius cylindrical fuel pin. The MCNP surface cards are:

1−2px −5

2−1px 5

3−4py −5

4−3py 5

5 c/z −241

The negative entries before the surface mnemonics specify periodic boundaries. Card one says

that surface 1 is periodic with surface 2 and is a px plane. Card two says that surface 2 is periodic

with surface 1 and is a px plane. Card three says that surface 3 is periodic with surface 4 and is

a py plane. Card four says that surface 4 is periodic with surface 3 and is a py plane. Card ﬁve

says that surface 5 is an inﬁnite cylinder parallel to the z−axis. A particle leaving the lattice out

the left side (surface 1) re-enters on the right side (surface 2). If the surfaces were reﬂecting, the

re-entering particle would miss the cylinder, shown by the dotted line. In a fully speciﬁed lattice

and in the periodic geometry, the re-entering particle will hit the cylinder as it should.

Much more complicated examples are possible, particularly hexagonal prism lattices. In all

cases, MCNP checks that the periodic surface pair matches properly and performs all the

necessary surface rotations and translations to put the particle in the proper place on the

corresponding periodic plane.

The following limitations apply:

• Periodic boundaries cannot be used with next event estimators such as detectors or

DXTRAN (see page 2–92);

• All periodic surfaces must be planes;

• Periodic planes cannot also have a surface transformation;

• The periodic cells may be inﬁnite or bounded by planes on the top or bottom that must be

reﬂecting or white boundaries but not periodic;

• Periodic planes can only bound other periodic planes or top and bottom planes;

• A single zero-importance cell must be on one side of each periodic plane;

• All periodic planes must have a common rotational vector normal to the geometry top and

bottom.

III. CROSS SECTIONS

The MCNP code package is incomplete without the associated nuclear data tables. The kinds of

tables available and their general features are outlined in this section. The manner in which

information contained on nuclear data tables is used in MCNP is described in Sec. IV of this

chapter.

April 10, 2000 2-17

CHAPTER 2

CROSS SECTIONS

There are two broad objectives in preparing nuclear data tables for MCNP. First, it is our

responsibility to ensure that the data available to MCNP reproduce the original evaluated data as

much as is practical. Second, new data should be brought into the MCNP package in a timely

fashion, thereby giving users access to the most recent evaluations.

Eight classes of nuclear data tables exist for MCNP. They are: (1) continuous-energy neutron

interaction data, (2) discrete reaction neutron interaction data, (3) photon interaction data, (4)

neutron dosimetry cross sections, (5) neutron S(α,β) thermal data (6) multigroup neutron,

coupled neutron/photon, and charged particles masquerading as neutrons, (7) multigroup

photon, and (8) electron interaction data. It is understood that photon and electron data are

atomic rather than nuclear. In Mode N problems, one continuous-energy or discrete-reaction

neutron interaction table is required for each isotope or element in the problem. Likewise, one

photon interaction table is required for each element in a Mode P problem, and one electron

interaction table is required for each element in a Mode E problem. Dosimetry and thermal data

are optional. Cross sections from dosimetry tables can be used as response functions with the

FM card to determine reaction rates. Thermal S(α,β) tables are appropriate if the neutrons are

transported at sufﬁciently low energies where molecular binding effects are important.

MCNP can read from data tables in two formats. Data tables are transmitted between computer

installations in 80-column card-image BCD format (Type-1 format). An auxiliary processing

code, MAKXSF, converts the BCD ﬁles to standard unformatted binary ﬁles (Type-2 format),

allowing more economical access during execution of MCNP. The data contained on a table for

a speciﬁc ZAID (10-character name for a nuclear data table) are independent of the format of

the table.

The format of nuclear data tables is given in considerable detail in Appendix F. This appendix

may be useful for users making extensive modiﬁcations to MCNP involving cross sections or for

users debugging MCNP at a fairly high level.

The available nuclear data tables are listed in Appendix G. Each nuclear data table is identiﬁed

by a ZAID. The general form of a ZAID is ZZZAAA.nnX, where ZZZ is the atomic number,

AAA is the atomic weight, nn is the evaluation identiﬁer, and X indicates the class of data. For

elemental evaluations AAA=000. Nuclear data tables are selected by the user with the Mn and

MTn cards.

In the remainder of this section we describe several characteristics of each class of data such as

evaluated sources, processing tools, and any differences between data on the original evaluation

and on the MCNP data tables. The means of accessing each class of data through MCNP input

will be detailed and some hints will be provided on how to select the appropriate data tables.

2-18 April 10, 2000

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CROSS SECTIONS

A. Neutron Interaction Data: Continuous-Energy and Discrete-Reaction

In neutron problems, one neutron interaction table is required for each isotope or element in the

problem. The form of the ZAIDs is ZZZAAA.nnC for a continuous-energy table and

ZZZAAA.nnD for a discrete reaction table. The neutron interaction tables available to MCNP

are listed in Table G.2 of Appendix G. (It should be noted that although all nuclear data tables

in Appendix G are available to users at Los Alamos, users at other installations will generally

have only a subset of the tables available.)

For most materials there are many cross-section sets available (represented by different values

of nn in the ZAIDs) because of multiple sources of evaluated data and different parameters used

in processing the data. An evaluated nuclear data set is produced by analyzing experimentally

measured cross sections and combining those data with the predictions of nuclear model

calculations in an attempt to extract the most accurate cross-section information. Preparing

evaluated cross-section sets has become a discipline in itself and has developed since the early

1960s. People in most of the national laboratories and several of the commercial reactor design

ﬁrms are involved in such work. American evaluators joined forces in the mid-1960s to create

the national ENDF system.23 The ENDF contributors collaborate through the Cross Section

Evaluation Working Group (CSEWG).

In recent years the primary evaluated source of neutron interaction data for MCNP has been the

ENDF/B system. Recently evaluated neutron interaction data tables are also extracted from two

other sources: Lawrence Livermore National Laboratory's Evaluated Nuclear Data Library

(ENDL),24 and supplemental evaluations performed in the Nuclear Theory and Applications

Group at Los Alamos.25,26,27 Older evaluations come from previous versions of ENDF/B and

ENDL, the Los Alamos Master Data File,28 and the Atomic Weapons Research Establishment

in Great Britain.

MCNP does not access evaluated data directly; these data must ﬁrst be processed into ACE

format. The very complex processing codes used for this purpose include NJOY29 for evaluated

data in ENDF/B format and MCPOINT30 for ENDL data.

Data on the MCNP neutron interaction tables include cross sections and much more. Cross

sections for all reactions given in the evaluated data are speciﬁed. For a particular table, the cross

sections for each reaction are given on one energy grid that is sufﬁciently dense that linear-linear

interpolation between points reproduces the evaluated cross sections within a speciﬁed tolerance

that is generally 1% or less. Depending primarily on the number of resolved resonances for each

isotope, the resulting energy grid may contain as few as ∼250 points (for example, H-1) or as

many as ∼22,500 points (for example, the ENDF/B-V version of AU-197). Other information,

including the total absorption cross section, the total photon production cross section, and the

average heating number (for energy deposition calculations), is also tabulated on the same

energy grid.

April 10, 2000 2-19

CHAPTER 2

CROSS SECTIONS

Angular distributions of scattered neutrons are included in the neutron interaction tables for all

reactions emitting neutrons. The distributions are given in the center-of-mass system for elastic

scattering, discrete-level inelastic scattering, and for some ENDF/B-VI scattering laws, and are

given in the laboratory system for all other inelastic reactions. Angular distributions are given

on a reaction-dependent grid of incident neutron energies. These tables are sampled to conserve

energy for many collisions but will not necessarily conserve energy for a single collision; that is,

energy is conserved on average.

The sampled angle of scattering uniquely determines the secondary energy for elastic scattering

and discrete-level inelastic scattering. For other inelastic reactions, energy distributions of the

scattered neutrons are provided in the neutron interaction tables. As with angular distributions,

the energy distributions are given on a reaction-dependent grid of incident neutron energies.

When evaluations contain data about secondary photon production, that information appears in

the MCNP neutron interaction tables. Many processed data sets contain photon production cross

sections, photon angular distributions, and photon energy distributions for each neutron reaction

that produces secondary photons. The information is given in a manner similar to that described

in the last few paragraphs for neutron cross sections and secondary neutron distributions.

Other miscellaneous information on the neutron interaction tables includes the atomic weight

ratio of the target nucleus, the Q-values of each reaction, and nubar, , data (the average number

of neutrons per ﬁssion) for ﬁssionable isotopes. In many cases both prompt and total are given.

Prompt is the default for all but KCODE criticality problems and total is the default for

KCODE criticality problems. The TOTNU input card can be used to change the default.

Approximations must be made when processing an evaluated data set into ACE format. As

mentioned above, cross sections are reproduced only within a certain tolerance, generally <1%;

to decrease it further would result in excessively large data tables. For many nuclides, a

“thinned” neutron interaction table is available with a coarse tolerance, > 1%, that greatly

reduces the library size. Smaller library sizes also can be obtained by using discrete reaction

tables or higher temperature data. Evaluated angular distributions for secondary neutrons and

photons are approximated on MCNP data tables by 32 equally probable cosine bins. This

approximation is clearly necessary when contrasted to the alternative that might involve

sampling from a 20th-order Legendre polynomial distribution. Secondary neutron energy

distributions given in tabular form by evaluators are sometimes approximated on MCNP data

tables by 32 equally probable energy bins. Older cross-section tables include a 30x20 matrix

approximation of the secondary photon energy spectra (described on page 2–34). On the whole,

the approximations are small, and MCNP neutron interaction data tables are extremely faithful

representations of evaluated data.

Discrete-reaction tables are identical to continuous-energy tables except that in the discrete

reaction tables all cross sections have been averaged into 262 groups. The averaging is done with

υ υ

2-20 April 10, 2000

CHAPTER 2

CROSS SECTIONS

a ﬂat weighting function. This is not a multigroup representation; the cross sections are simply

given as histograms rather than as continuous curves. The remaining data (angular distributions,

energy distributions, , etc.) are identical in discrete-reaction and continuous-energy tables.

Discrete-reaction tables are provided primarily as a method of shrinking the required data

storage to enhance the ability to run MCNP on small machines or in a time-sharing environment.

The tables may also be useful for preliminary scoping studies or for isotopes that exist only in

trace quantities in a problem. They are not, however, recommended as a substitute for the

continuous-energy tables when performing ﬁnal calculations, particularly for problems

involving transport through the resonance region.

The matter of how to select the appropriate neutron interaction tables for your calculation is now

discussed. Multiple tables for the same isotope are differentiated by the “nn” portion of the

ZAID. The easiest choice for the user, although by no means the recommended one, is not to

enter the nn at all. MCNP will select the ﬁrst match found in the directory ﬁle XSDIR. The

default nnX can be changed for all isotopes of a material by the NLIB keyword entry on the Mm

card. The default will be overridden by fully specifying the ZAID. Default continuous-energy

neutron interaction tables are accessed by entering ZZZAAA for the ZAID\null. Including a

DRXS card in the input ﬁle will force MCNP to choose the default discrete reaction tables.

Careful users will want to think about what neutron interaction tables to choose. There is,

unfortunately, no strict formula for choosing the tables. The following guidelines and

observations are the best that can be offered:

1. Users should be aware of the differences between the “.50C” series of data tables and

the “.51C” series. Both are derived from ENDF/B-V. The “.50C” series is the most

faithful reproduction of the evaluated data. The “.51C” series, also called the

“thinned” series, has been processed with a less rigid tolerance than the “.50C” series.

As with discrete reaction data tables, although by no means to the same extent, users

should be careful when using the “thinned” data for transport through the resonance

region.

2. Consider differences in evaluators' philosophies. The Physical Data Group at

Livermore is justly proud of its extensive cross-section efforts; their evaluations

manifest a philosophy of reproducing the data with the fewest number of points.

Livermore evaluations are available mainly in the “.40C” series. We at Los Alamos

are particularly proud of the evaluation work being carried out in the Nuclear Theory

and Applications Group T-2; generally, these evaluations are the most complex

because they are the most thorough. Recent evaluations from Los Alamos are

available in the “.55C” series.

3. Be aware of the neutron energy spectrum in your problem. For high-energy problems,

the “thinned” and discrete reaction data are probably not bad approximations.

April 10, 2000 2-21

CHAPTER 2

CROSS SECTIONS

Conversely, it is essential to use the most detailed continuous-energy set available for

problems influenced strongly by transport through the resonance region.

4. Check the temperature at which various data tables have been processed. Do not use

a set that is Doppler broadened to 12,000,000 K for a room temperature calculation.

5. Consider checking the sensitivity of the results to various sets of nuclear data. Try, for

example, a calculation with ENDF/B-V cross sections, and then another with ENDL

cross sections. If the results of a problem are extremely sensitive to the choice of

nuclear data, it is advisable to find out why.

6. For a coupled neutron/photon problem, be careful that the tables you choose have

photon production data available. If possible, use the more-recent sets that have been

processed into expanded photon production format.

7. In general, use the best data you can afford. It is understood that the latest evaluations

tend to be more complex and therefore require more memory and longer execution

times. If you are limited by available memory, try to use smaller data tables such as

thinned or discrete-reaction for the minor isotopes in the calculation. Discrete reaction

data tables might be used for a parameter study, followed by a calculation with the full

continuous-energy data tables for confirmation.

To select the neutron interaction data tables, the nn portion of the ZAIDs must be entered on the

Mn card(s). For a continuous-energy set, ZZZAAA.nn is equivalent to ZZZAAA.nnC. To use a

discrete-reaction table (unless there is a DRXS card in the input) the full ZAID, ZZZAAA.nnD,

must be entered.

If only the integer portion of the ZAID is entered (ZZZAAA), MCNP will choose the cross−

section table that it will use. Based on other cards (i.e., MODE, MGOPT, DRXS), MCNP knows

which class of data is required. The code then “reads” the cross-section directory ﬁle (XSDIR)

and selects the ﬁrst table it ﬁnds that meets the ZZZAAA and class criteria. Thus, default cross

sections are based entirely on the ordering of the entries in the XSDIR ﬁle you are using at your

installation.

In conclusion, the additional time necessary to choose appropriate neutron interaction data tables

rather than simply to accept the defaults often will be rewarded by increased understanding of

your calculation.

B. Photon Interaction Data

Photon interaction cross sections are required for all photon problems. The form of the ZAID is

ZZZ000.nnP. There are two photon interaction data libraries: nn = 01 and nn = 02.

For the ZAID=ZZZ000.01P library, the photon interaction tables for Z = 84, 85, 87, 88, 89, 91,

and 93 are based on the compilation of Storm and Israel31 from 1 keV to 15 MeV. For all other

2-22 April 10, 2000

CHAPTER 2

CROSS SECTIONS

elements from Z = 1 through Z = 94 the photon interaction tables are based on evaluated data

from ENDF32 from 1 keV to 100 MeV. Fluorescence data are taken from work by Everett and

Cashwell.33 Energy grids are tailored speciﬁcally for each element and contain approximately

40 to 60 points.

The ZAID = ZZZ000.02P library is a superset of the ZAID = ZZZ000.01P library with pair

production thresholds added for the Storm-Israel data. Data above 15 MeV for the Storm-Israel

data and above 100 MeV for the ENDF data come from adaptation of the Livermore Evaluated

Photon Data Library (EPDL)34 and go up to 100 GeV. However, it usually is impractical to run

above 1 GeV with MCNP because electron data only go to 1 GeV. The energy grid for the ZAID

= ZZZ000.02P library contains approximately 100 points.

For each nuclide the photon interaction libraries contain an energy grid (logarithms of energies),

including the photoelectric edges and the pair production threshold. These energies are tailored

speciﬁcally for each element. The logarithmic energies are followed by tables of incoherent and

coherent form factors that are tabulated as a function of momentum transfer. The next tables are

logarithms of the incoherent scattering, coherent scattering, photoelectric, and pair production

cross sections, followed by the photon heating numbers. The total cross section is not stored, but

rather summed from the other cross sections during transport.

The determination of directions and energies of scattered photons requires information different

from the sets of angular and energy distributions found on neutron interaction tables. Angular

distributions of secondary photons are isotropic for photoelectric effect, ﬂuorescence, and pair

production, and come from sampling the well-known Thomson and Klein-Nishina formulas for

coherent and incoherent scattering. The energy of an incoherently scattered photon is calculated

from the sampled scattering angle. Values of the integrated coherent form factor are tabulated on

the photon interaction tables for use with next event estimators such as point detectors.

Very few approximations are made in the various processing codes used to transfer photon data

from ENDF into the format of MCNP photon interaction tables. Cross sections are reproduced

exactly as given. Form factors and scattering functions are reproduced as given; however, the

momentum transfer grid on which they are tabulated may be different from that of the original

evaluation. Heating numbers are calculated values, not given in evaluated sets, but inferred from

them. Fluorescence data are not provided in ENDF; therefore the data for MCNP are extracted

from a variety of sources as described in Ref. 31.

To select photon interaction data, speciﬁc ZAID identiﬁers can be used, such as

ZAID = ZZZ000.02P, or selections from a library can be used by specifying PLIB=nnP on the

M card. The PLIB = speciﬁcation on the M card is the preferred method because the ZAID

entries may already be used to specify neutron libraries and, unlike neutrons, it usually is

desirable to pick all photon data from the same library. A speciﬁcation on the Mn card for a

neutron interaction table with ZAID = ZZZAAA.nnC or ZAID = ZZZAAA.nnD immediately

April 10, 2000 2-23

CHAPTER 2

CROSS SECTIONS

causes a photon interaction table with ZAID = ZZZ000.nnP to be accessed as well, where nn is

the ﬁrst photon data encountered for ZZZ000 on the XSDIR cross section directory ﬁle or nn

comes from PLIB = nn. The data table required for ZAID = ZZZAAA.nnP is identical to that

required for ZAID = ZZZ000.nnP; however, the atomic weight used in the calculation will likely

be different.

C. Electron Interaction Data

Electron interaction data tables are required both for problems in which electrons are actually

transported, and for photon problems in which the thick-target bremsstrahlung model is used.

Electron data tables are identiﬁed by ZAIDs of the form ZZZ000.nnE, and are selected by

default when the problem mode requires them. There are two electron interaction data libraries:

nn=03 and nn = 01.

The electron library contains data on an element-by-element basis for atomic numbers Z=1–94.

As is the case with photons, there is no distinction between isotopes for a given element. The

library data contain energies for tabulation, radiative stopping power parameters,

bremsstrahlung production cross sections, bremsstrahlung energy distributions, K-edge

energies, Auger electron production energies, parameters for the evaluation of the Goudsmit-

Saunderson theory for angular deﬂections based on the Riley cross section calculation, and Mott

correction factors to the Rutherford cross sections also used in the Goudsmit-Saunderson theory.

The el03 database also includes the atomic data of Carlson used in the density effect calculation.

Internally, calculated data are electron stopping powers and ranges, K x-ray production

probabilities, knock-on probabilities, bremsstrahlung angular distributions, and the Landau-

Blunck-Leisegang theory of energy-loss ﬂuctuations. The el03 evaluation is derived from the

ITS3.0 code system.35 Discussions of the theoretical basis for these data and references to the

relevant literature are presented in Section IV-E of this chapter.

The hierarchy rules for electron cross sections require that each material must use the same

electron library. If a speciﬁc ZAID is selected, such as ZZZ000.01E, that choice will override

any defaults. Alternatively, a default electron library for a given material can be chosen by

specifying ELIB = nnE on the M card. However, one can not specify different libraries, nn=01

and nn=03, by any means; overriding this with a fatal option will result in unreliable results. In

the absence of either of these speciﬁcations, MCNP will use the ﬁrst electron data table listed in

the XSDIR cross section directory ﬁle for the relevant element.

D. Neutron Dosimetry Cross Sections

Dosimetry cross-section tables cannot be used for transport through material. These incomplete

cross-section sets provide energy-dependent neutron cross sections to MCNP for use as response

2-24 April 10, 2000

CHAPTER 2

CROSS SECTIONS

functions with the FM tally feature. ZAIDs of dosimetry tables are of the form ZZZAAA.nnY.

Remember, dosimetry cross-section tables have no effect on the particle transport of a problem.

The available dosimetry cross sections are from three sources: ENDF/B−V Dosimetry Tape 531,

ENDF/B−V Activation Tape 532, and ACTL36–an evaluated neutron activation cross-section

library from the Lawrence Livermore National Laboratory. Various codes have been used to

process evaluated dosimetry data into the format of MCNP dosimetry tables.

Data on dosimetry tables are simply energy-cross-section pairs for one or more reactions. The

energy grids for all reactions are independent of each other. Interpolation between adjacent

energy points can be speciﬁed as histogram, linear-linear, linear-log, log-linear, or log-log. With

the exception of the tolerance involved in any reconstruction of pointwise cross sections from

resonance parameters, evaluated dosimetry cross sections can be reproduced on the MCNP data

tables with no approximation.

ZAIDs for dosimetry tables must be entered on material cards that are referenced by FM cards,

not on Mm cards referenced by cell cards. The complete ZAID, ZZZAAA.nnY, must be given;

there are no defaults for dosimetry tables.

E. Neutron Thermal S(α,β) Tables

Thermal S(α,β) tables are not required, but they are absolutely essential to get correct answers

in problems involving neutron thermalization. Thermal tables have ZAIDs of the form

XXXXXX.nnT, where XXXXXX is a mnemonic character string. The data on these tables

encompass those required for a complete representation of thermal neutron scattering by

molecules and crystalline solids. The source of S(α,β) data is a special set of ENDF tapes.37 The

THERMR and ACER modules of the NJOY29 system have been used to process the evaluated

thermal data into a format appropriate for MCNP.

Data are for neutron energies generally less than 4 eV. Cross sections are tabulated on table-

dependent energy grids; inelastic scattering cross sections are always given and elastic scattering

cross sections are sometimes given. Correlated energy-angle distributions are provided for

inelastically scattered neutrons. A set of equally probable ﬁnal energies is tabulated for each of

several initial energies. Further, a set of equally probable cosines or cosine bins is tabulated for

each combination of initial and ﬁnal energies. Elastic scattering data can be derived from either

an incoherent or a coherent approximation. In the incoherent case, equally probable cosines or

cosine bins are tabulated for each of several incident neutron energies. In the coherent case,

scattering cosines are determined from a set of Bragg energies derived from the lattice

parameters. During processing, approximations to the evaluated data are made when

constructing equally probable energy and cosine distributions.

April 10, 2000 2-25

CHAPTER 2

PHYSICS

ZAIDs for the thermal tables are entered on an MTn card that is associated with an existing Mn

card. The thermal table generally will provide data for one component of a material–for

example, hydrogen in light water. Thermal ZAIDs may be entered on the MTn card(s) as

XXXXXX, XXXXXX.nn, or XXXXXX.nnT.

F. Multigroup Tables

Multigroup cross section libraries are the only libraries allowed in multigroup/adjoint problems.

Neutron multigroup problems cannot be supplemented with S(α,β) thermal libraries; the thermal

effects must be included in the multigroup neutron library. Photon problems cannot be

supplemented with electron libraries; the electrons must be part of the multigroup photon library.

The form of the ZAID is ZZZAAA.nnM or ZZZAAA.nnG for photons only.

Although continuous-energy data are more accurate than multigroup data, the multigroup option

is useful for a number of important applications: (1) comparison of deterministic (Sn) transport

codes to Monte Carlo; (2) use of adjoint calculations in problems where the adjoint method is

more efﬁcient; (3) generation of adjoint importance functions; (4) cross section sensitivity

studies; (5) solution of problems for which continuous-cross sections are unavailable; and (6)

charged particle transport using the Boltzmann-Fokker-Planck algorithm in which charged

particles masquerade as neutrons.

Multigroup cross sections are very problem dependent. Some multigroup libraries are available

from the Transport Methods Group at Los Alamos but must be used with caution. Users are

encouraged to generate or get their own multigroup libraries and then use the supplementary

code CRSRD38 to convert them to MCNP format. Reference 38 describes the conversion

procedure. This report also describes how to use both the multigroup and adjoint methods in

MCNP and presents several benchmark calculations demonstrating the validity and

effectiveness of the multigroup/adjoint method.

To generate cross-section tables for electron/photon transport problems that will use the

multigroup Boltzmann-Fokker-Planck algorithm,39 the CEPXS40 code developed by Sandia

National Laboratory and available from RSICC can be used. The CEPXS manuals describe the

algorithms and physics database upon which the code is based; the physics package is essentially

the same as ITS version 2.1. The keyword “MONTE-CARLO” is needed in the CEPXS input

ﬁle to generate a cross-section library suitable for input into CRSRD; this undocumented feature

of the CEPXS code should be approached with caution.

IV. PHYSICS

The physics of neutron, photon, and electron interactions is the very essence of MCNP. This

section may be considered a software requirements document in that it describes the equations

2-26 April 10, 2000

CHAPTER 2

PHYSICS

MCNP is intended to solve. All the sampling schemes essential to the random walk are presented

or referenced. But ﬁrst, particle weight and particle tracks, two concepts that are important for

setting up the input and for understanding the output, are discussed in the following sections.

A. Particle Weight

If MCNP were used only to simulate exactly physical transport, then each MCNP particle would

represent one physical particle and would have unit weight. However, for computational

efﬁciency, MCNP allows many techniques that do not exactly simulate physical transport. For

instance, each MCNP particle might represent a number w of particles emitted from a source.

This number w is the initial weight of the MCNP particle. The w physical particles all would

have different random walks, but the one MCNP particle representing these w physical particles

will only have one random walk. Clearly this is not an exact simulation; however, the true

number of physical particles is preserved in MCNP in the sense of statistical averages and

therefore in the limit of large particle numbers (of course including particle production or loss if

they occur). Each MCNP particle result is multiplied by the weight so that the full results of the

w physical particles represented by each MCNP particle are exhibited in the ﬁnal results (tallies).

This procedure allows users to normalize their calculations to whatever source strength they

desire. The default normalization is to weight one per MCNP particle. A second normalization

to the number of Monte Carlo histories is made in the results so that the expected means will be

independent of the number of source particles actually initiated in the MCNP calculation.

The utility of particle weight, however, goes far beyond simply normalizing the source. Every

Monte Carlo biasing technique alters the probabilities of random walks executed by the

particles. The purpose of such biasing techniques is to increase the number of particles that

sample some part of the problem of special interest (1) without increasing (sometimes actually

decreasing) the sampling of less interesting parts of the problem, and (2) without erroneously

affecting the expected mean physical result (tally). This procedure, properly applied, increases

precision in the desired result compared to an unbiased calculation taking the same computing

time. For example, if an event is made times as likely to occur (as it would occur without

biasing), the tally ought to be multiplied by so that the expected average tally is unaffected.

This tally multiplication can be accomplished by multiplying the particle weight by

because the tally contribution by a particle is always multiplied by the particle weight in MCNP.

Note that weights need not be integers.

In short, particle weight is a number carried along with each MCNP particle, representing that

particle's relative contribution to the ﬁnal tallies. Its magnitude is determined to ensure that

whenever MCNP deviates from an exact simulation of the physics, the expected physical result

nonetheless is preserved in the sense of statistical averages, and therefore in the limit of large

MCNP particle numbers. Its utility is in the manipulation of the number of particles, sampling

just a part of the problem to improve the precision of selected results obviating a full unbiased

calculation−with its added cost in computing time−to achieve the same results and precision.

212 12

April 10, 2000 2-27

CHAPTER 2

PHYSICS

B. Particle Tracks

When a particle starts out from a source, a particle track is created. If that track is split 2 for 1 at

a splitting surface, a second track is created and there are now two tracks from the original source

particle, each with half the single track weight. If one of the tracks has an (n,2n) reaction, one

more track is started for a total of three. A track refers to each component of a source particle

during its history. Track length tallies use the length of a track in a given cell to determine a

quantity of interest, such as ﬂuence, ﬂux, or energy deposition. Tracks crossing surfaces are used

to calculate ﬂuence, ﬂux, or pulse-height energy deposition (surface estimators). Tracks

undergoing collisions are used to calculate multiplication and criticality (collision estimators).

Within a given cell of ﬁxed composition, the method of sampling a collision along the track is

determined using the following theory. The probability of a ﬁrst collision for a particle between

l and l + dl along its line of ﬂight is given by

where is the macroscopic total cross section of the medium and is interpreted as the

probability per unit length of a collision. Setting ξ the random number on [0,1), to be

it follows that

But, because is distributed in the same manner as ξand hence may be replaced by ξ,we

obtain the well-known expression for the distance to collision,

C. Neutron Interactions

When a particle (representing any number of neutrons, depending upon the particle weight)

collides with a nucleus, the following sequence occurs:

1. the collision nuclide is identified;

pl()dl e Σt

–Σt

=dl

Σt

ξeΣts–

∫

=Σtds 1eΣ–tl

–=

Σt

-----ln 1ξ–()–=

1ξ–

Σt

-----ξ()ln=

2-28 April 10, 2000

CHAPTER 2

PHYSICS

2. either the S(a,b) treatment is used or the velocity of the target nucleus is sampled for

low−energy neutrons;

3. photons are optionally generated for later transport;

4. neutron capture (that is, neutron disappearance by any process) is modeled;

5. unless the S(a,b) treatment is used, either elastic scattering or an inelastic reaction is

selected, and the new energy and direction of the outgoing track(s) are determined;

6. if the energy of the neutron is low enough and an appropriate S(a,b) table is present,

the collision is modeled by the S(a,b) treatment instead of by step 5.

1. Section of Collision Nuclide

If there are ndifferent nuclides forming the material in which the collision occurred, and if ξis

a random number on the unit interval [0,1), then the kth nuclide is chosen as the collision nuclide

where is the macroscopic total cross section of nuclide . If the energy of the neutron is low

enough (below about 4 eV) and the appropriate table is present, the total cross section is

the sum of the capture cross section from the regular cross-section table and the elastic and

inelastic scattering cross sections from the table. Otherwise, the total cross section is taken

from the regular cross-section table and is adjusted for thermal effects as described below.

2. Free Gas Thermal Treatment

A collision between a neutron and an atom is affected by the thermal motion of the atom, and in

most cases, the collision is also affected by the presence of other atoms nearby. The thermal

motion cannot be ignored in many applications of MCNP without serious error. The effects of

nearby atoms are also important in some applications. MCNP uses a thermal treatment based

on the free gas approximation to account for the thermal motion. It also has an explicit S(a,b)

capability that takes into account the effects of chemical binding and crystal structure for

incident neutron energies below about 4 eV, but is available for only a limited number of

substances and temperatures. The S(a,b) capability is described later on page 2–53.

The free gas thermal treatment in MCNP assumes that the medium is a free gas and also that, in

the range of atomic weight and neutron energy where thermal effects are signiﬁcant, the elastic

scattering cross section at zero temperature is nearly independent of the energy of the neutron,

and that the reaction cross sections are nearly independent of temperature. These assumptions

allow MCNP to have a thermal treatment of neutron collisions that runs almost as fast as a

completely nonthermal treatment and that is adequate for most practical problems.

Σti

i1=

k1–

∑ξΣ

i1=

∑

<Σ

i1=

∑

≤

Σti i

Sαβ,()

April 10, 2000 2-29

CHAPTER 2

PHYSICS

With the above assumptions, the free gas thermal treatment consists of adjusting the elastic cross

section and taking into account the velocity of the target nucleus when the kinematics of a

collision are being calculated. Note that Doppler broadening of the inelastic cross sections is

assumed to have already been done in the processing of the cross section libraries. The free gas

thermal treatment effectively applies to elastic scattering only.

a. Adjusting the Elastic Cross Section: The ﬁrst aspect of the free gas thermal treatment

is to adjust the zero-temperature elastic cross section by raising it by the factor

where , A = atomic weight, E= neutron energy, and T= temperature. For speed,

F is approximated by F=1+ 0.5/a2 when and by linear interpolation in a table of 51

values of aF when a < 2. Both approximations have relative errors less than 0.0001. The total

cross section also is increased by the amount of the increase in the elastic cross section.

The adjustment to the elastic and total cross sections is done partly in the setup of a problem and

partly during the actual transport calculation. No adjustment is made if the elastic cross section

in the data library was already processed to the temperature that is needed in the problem. If all

of the cells that contain a particular nuclide have the same temperature, constant in time, that is

different from the temperature of the library, the elastic and total cross sections for that nuclide

are adjusted to that temperature during the setup so that the transport will run a little faster.

Otherwise, these cross sections are reduced, if necessary, to zero temperature during the setup

and the thermal adjustment is made when the cross sections are used. For speed, the thermal

adjustment is omitted if the neutron energy is greater than 500 kT/A. At that energy the

adjustment of the elastic cross section would be less than 0.1%.

b. Sampling the Velocity of the Target Nucleus: The second aspect of the free gas thermal

treatment consists of taking into account the velocity of the target nucleus when the kinematics

of a collision are being calculated. The target velocity is sampled and subtracted from the

velocity of the neutron to get the relative velocity. The collision is sampled in the target-at-rest

frame and the outgoing velocities are transformed to the laboratory frame by adding the target

velocity.

There are different schools of thought as to whether the relative energy between the neutron and

target, Er, or the laboratory frame incident neutron energy (target-at-rest), Eo, should be used for

all the kinematics of the collision. Eois used in MCNP to obtain the distance-to-collision, select

the collision nuclide, determine energy cutoffs, generate photons, generate ﬁssion sites for the

next generation of a KCODE criticality problem, for S(α,β) scattering, and for capture. Er is

used for everything else in the collision process, namely elastic and inelastic scattering,

including ﬁssion and (n,xn) reactions. It is shown in Eqn. 2.1 that Er is based upon vrel that is

based upon the elastic scattering cross section, and, therefore, Er is truly valid only for elastic

F1 0.5 a2

⁄+()erf a() a2

–()aπ()⁄exp+=

aAEkT⁄=a2≥

2-30 April 10, 2000

CHAPTER 2

PHYSICS

scatter. However, the only signiﬁcant thermal reactions for stable isotopes are absorption, elastic

scattering, and ﬁssion. 181Ta has a 6 keV threshold inelastic reaction; all other stable isotopes

have higher inelastic thresholds. Metastable nuclides like 242mAm have inelastic reactions all the

way down to zero, but these inelastic reaction cross sections are neither constant nor 1/v cross

sections and these nuclides are generally too massive to be affected by the thermal treatment

anyway. Furthermore, ﬁssion is very insensitive to incident neutron energy at low energies. The

ﬁssion secondary energy and angle distributions are nearly ﬂat or constant for incident energies

below about 500 keV. Therefore, it makes no signiﬁcant difference if Eris used only for elastic

scatter or for other inelastic collisions as well. At thermal energies, whether Eror Eois used only

makes a difference for elastic scattering.

If the energy of the neutron is greater than 400 kT and the target is not 1H, the velocity of the

target is set to zero. Otherwise, the target velocity is sampled as follows. The free-gas kernel is

a thermal interaction model that results in a good approximation to the thermal ﬂux spectrum in

a variety of applications and can be sampled without tables. The effective scattering cross section

in the laboratory system for a neutron of kinetic energy E is

(E) = (2.1)

Here, vrel is the relative velocity between a neutron moving with a scalar velocity vnand a target nucleus moving

with a scalar velocity V, and µt is the cosine of the angle between the neutron and the target direction-of-ﬂight

vectors. The equation for vrel is

The scattering cross section at the relative velocity is denoted by σs(vrel), and p(V) is the

probability density function for the Maxwellian distribution of target velocities,

with β deﬁned as

where A is the mass of a target nucleus in units of the neutron mass, Mn is the neutron mass in

MeV-sh2/cm2, and kT is the equilibrium temperature of the target nuclei in MeV.

σs

eff 1

-----σs

∫∫ vrel

()vrelpV()dvdµt

--------

vrel vn

2V22vnVµt–+()

---

pV() 4

π12⁄

-----------β3V2eβ–2V2

βAMn

2kT

------------





---

April 10, 2000 2-31

CHAPTER 2

PHYSICS

The most probable scalar velocity V of the target nuclei is 1/β, which corresponds to a kinetic

energy of kT for the target nuclei. This is not the average kinetic energy of the nuclei, which is

3kT/2. The quantity that MCNP expects on the TMPn input card is kT and not just T (see page

3–121). Note that kT is not a function of the particle mass and is therefore the kinetic energy at

the most probable velocity for particles of any mass.

Equation (2.1) implies that the probability distribution for a target velocity n and cosine is

It is assumed that the variation of with target velocity can be ignored. The justiﬁcation

for this approximation is that (1) for light nuclei, is slowly varying with velocity, and

(2) for heavy nuclei, where can vary rapidly, the moderating effect of scattering is small

so that the consequences of the approximation will be negligible. As a result of the

approximation, the probability distribution actually used is

Note that the above expression can be written as

As a consequence, the following algorithm is used to sample the target velocity.

1. With probability , the target velocity Vis sampled from the

distribution . The transformation reduces this

distribution to the sampling distribution for . MCNP actually codes

2. With probability 1 − α, the target velocity is sampled from the distribution

. Substituting V = y/β reduces the distribution to the

sampling distribution for y: .

3. The cosine of the angle between the neutron velocity and the target velocity is sampled

uniformly on the interval + 1.

µt

PVµt

,()

σsvrel

()vrelPV()

2σs

eff E()vn

-----------------------------------------=

σsv() σsvrel

()

σsvrel

()

PVµt

,()ν

2V22Vνnµt

–V2eβ2V

–

∝

Vµt

,()

νn

2V22Vνnµt

–+

νnV+

----------------------------------------------- V3eβ2V2

–νnV2eβ2V

–

+(∝

α11 πβvn2⁄()+()⁄=

P1V() 2β4V3eβ2V2

–

=Vyβ⁄=

Py() ye y–

1α–

P2V() 4β3π⁄()V2eβ2V2

–

Py() 4π⁄()y2ey2

–

1µt

≤≤–

2-32 April 10, 2000

CHAPTER 2

PHYSICS

4. The rejection function R(V, µt) is computed using

With probability R(V,µt), the sampling is accepted; otherwise, the sampling is rejected and the

procedure is repeated. The minimum efﬁciency of this rejection algorithm averaged over µtis

68% and approaches 100% as either the incident neutron energy approaches zero or becomes

much larger than kT.

3. Optional Generation of Photons

Photons are generated if the problem is a combined neutron/photon run and if the collision

nuclide has a nonzero photon production cross section. The number of photons produced is a

function of neutron weight, neutron source weight, photon weight limits (entries on the PWT

card), photon production cross section, neutron total cross section, cell importance, and the

importance of the neutron source cell. No more than 10 photons may be born from any neutron

collision. In a KCODE calculation, secondary photon production from neutrons is turned off

during the inactive cycles.

Because of the many low-weight photons typically created by neutron collisions, Russian

roulette is played for particles with weight below the bounds speciﬁed on the PWT card,

resulting in fewer particles, each having a larger weight. The created photon weight before

Russian roulette is

where Wp= photon weight

Wn= neutron weight

= photon production cross section

σT= total neutron cross section.

Both and σTare evaluated at the incoming neutron energy without the effects of the thermal

free gas treatment because nonelastic cross sections are assumed independent of temperature.

The Russian roulette game is played according to neutron cell importances for the collision and

source cell. For a photon produced in cell iwhere the minimum weight set on the PWT card is

Vµt

,()

2V22Vvnµt

–+

vnV+

----------------------------------------------=1≤

Wnσγ

σT

--------------=

σγ

April 10, 2000 2-33

CHAPTER 2

PHYSICS

, let Iibe the neutron importance in cell iand let Isbe the neutron importance in the source

cell. If , one or more photons will be produced. The number of photons

created is Np,where Np = (Wp∗ Ii)/(5 * ∗ Is)+1. . Each photon is stored in the

bank with weight Wp/Np. If ∗Is/Ii, Russian roulette is played and the photon

survives with probability Wp∗and is given the weight, ∗Is/Ii.

If weight windows are not used and if the weight of the starting neutrons is not unity, setting all

the on the PWT card to negative values will make the photon minimum weight relative to

the neutron source weight. This will make the number of photons being created roughly

proportional to the biased collision rate of neutrons. It is recommended for most applications that

negative numbers be used and be chosen to produce from one to four photons per source neutron.

The default values for on the PWT card are −1, which should be adequate for most

problems using cell importances.

If energy−independent weight windows are used, the entries on the PWT card should be the

same as on the WWN1:P card. If energy−dependent photon weight windows are used, the entries

on the PWT card should be the minimum WWNn:P entry for each cell, where n refers to the

photon weight window energy group. This will cause most photons to be born within the weight

window bounds.

Any photons generated at neutron collision sites are temporarily stored in the bank. There are

two methods for determining the exiting energies and directions, depending on the form in which

the processed photon production data are stored in a library. The ﬁrst method has the evaluated

photon production data processed into an “expanded format.”41 In this format, energy−

dependent cross sections, energy distributions, and angular distributions are explicitly provided

for every photon−producing neutron interaction. In the second method, used with data

processed from older evaluations, the evaluated photon production data have been collapsed so

that the only information about secondary photons is in a matrix of 20 equally probable photon

energies for each of 30 incident neutron energy groups. The sampling techniques used in each

method are now described.

a. Expanded Photon Production Method: In the expanded photon production method,

the reaction n responsible for producing the photon is sampled from

where ξ is a random number on the interval (0,1), N is the number of photon production

reactions, and σiis the photon production cross section for reaction i at the incident neutron

energy. Note that there is no correlation between the sampling of the type of photon production

reaction and the sampling of the type of neutron reaction described on page 2–36.

min

WpWi

min IsIi

⁄> ∗

min Np10≤

WpWi

min

IiWi

min Is

()⁄∗Wi

min

σiξσ

iσi

i1=

∑

≤

i1=

∑

i1=

n1–

∑

2-34 April 10, 2000

CHAPTER 2

PHYSICS

Just as every neutron reaction (for example, (n,2n)) has associated energy-dependent angular and

energy distributions for the secondary neutrons, every photon production reaction (for example,

(n,pγ)) has associated energy-dependent angular and energy distributions for the secondary

photons. The photon distributions are sampled in much the same manner as their counterpart

neutron distributions.

All nonisotropic secondary photon angular distributions are represented by 32 equiprobable

cosine bins. The distributions are given at a number of incident neutron energies. All photon-

scattering cosines are sampled in the laboratory system. The sampling procedure is identical to

that described for secondary neutrons on page 2–37.

Secondary photon energy distributions are also a function of incident neutron energy. There are

two representations of secondary photon energy distributions allowed in ENDF/B format:

tabulated spectra and discrete (line) photons. Correspondingly, there are three laws used in

MCNP for the determination of secondary photon energies. Law 4 is an exact representation of

tabulated photon spectra. Law 2 is used for discrete photons. Law 44 is for discrete photon lines

with a continuous background. These laws are described beginning on page 2–41.

The expanded photon production method has clear advantages over the original 30 x 20 matrix

method described below. In coupled neutron/photon problems, users should attempt to specify

data sets that contain photon production data in expanded format. Such data sets are identiﬁed

by “YES P(E)” entries in the GPD column in Table G.2 in Appendix G.

b. 30 x 20 Photon Production Method: For lack of better terminology, we will refer to the

photon production data contained on older libraries as “30 x 20 photon production” data. In

contrast to expanded photon production data, there is no information about individual photon

production reactions in the 30 x 20 data.

The only secondary photon data are a 30 x 20 matrix of photon energies; that is, for each of 30

incident neutron energy groups there are 20 equally probable exiting photon energies. There is

no information regarding secondary photon angular distributions; therefore, all photons are

taken to be produced isotropically in the laboratory system.

There are several problems associated with 30 x 20 photon production data. The 30 x 20 matrix

is an inadequate representation of the actual spectrum of photons produced. In particular,

discrete photon lines are not well represented, and the high-energy tail of a photon continuum

energy distribution is not well sampled. Also, the multigroup representation is not consistent

with the continuous-energy nature of MCNP. Finally, not all photons should be produced

isotropically. None of these problems exists for data processed into the expanded photon

production format.

April 10, 2000 2-35

CHAPTER 2

PHYSICS

4. Capture

Capture is treated in one of two ways: analog or implicit. Either way, the incident incoming

neutron energy does not include the relative velocity of the target nucleus from the free gas

thermal treatment because nonelastic reaction cross sections are assumed to be nearly

independent of temperature. That is, only the scattering cross section is affected by the free gas

thermal treatment. In MCNP, “absorption” and “capture” are used interchangeably, both

meaning (n,0n), and σc and σa are used interchangeably also.

a. Analog Capture: In analog capture, the particle is killed with probability σa/σT, where

σaand σT are the absorption and total cross sections of the collision nuclide at the incoming

neutron energy. The absorption cross section is specially deﬁned for MCNP as the sum of all

(n,x) cross sections, where xis anything except neutrons. Thus σais the sum of σn,g,σn,a,σn,d,

etc. For all particles killed by analog capture, the entire particle energy and weight are

deposited in the collision cell.

b. Implicit Capture: For implicit capture, the neutron weight Wn is reduced to Wn as

follows:

If the new weight Wnis below the problem weight cutoff (specified on the CUT card), Russian

roulette is played, resulting overall in fewer particles with larger weight.

For implicit capture, a fraction σa/σT of the incident particle weight and energy is deposited in

the collision cell corresponding to that portion of the particle that was captured. Implicit capture

is the default method of neutron capture in MCNP.

c. Implicit Capture Along a Flight Path: Implicit capture also can be done continuously

along the ﬂight path of a particle trajectory as is the common practice in astrophysics. In this

case, the distance to scatter, rather than the distance to collision, is sampled. The distance to

scatter is

The particle weight at the scattering point is reduced by the capture loss,

…

Wn1σa

σT

------–





*Wn

‘

Σs

-----–1ξ–(

)

ln=

W′We Σal–

2-36 April 10, 2000

CHAPTER 2

PHYSICS

where W’ = reduced weight after capture loss,

W = weight before capture along ﬂight path,

σa= absorption cross section,

σs= scattering cross section,

σt=σs + σa = total cross section,

l= distance to scatter, and

ξ= random number.

Implicit capture along a ﬂight path is a special form of the exponential transformation coupled

with implicit capture at collisions. (See the description of the exponential transform on

page 2–141.) The path length is stretched in the direction of the particle, µ= 1, and the stretching

parameter is p = Σa/Σt. Using these values the exponential transform and implicit capture at

collisions yield the identical equations as does implicit capture along a ﬂight path.

Implicit capture along a ﬂight path is invoked in MCNP as a special option of the exponential

transform variance reduction method. It is most useful in highly absorbing media, that is, Σa/Σt

approaches 1. When almost every collision results in capture, it is very inefﬁcient to sample

distance to collision. However, implicit capture along a ﬂight path is discouraged. In highly

absorbing media, there is usually a superior set of exponential transform parameters. In

relatively nonabsorbing media, it is better to sample the distance to collision than the distance to

scatter.

5. Elastic and Inelastic Scattering

If the conditions for the S(α,β)treatment are not met, the particle undergoes either an elastic or

inelastic collision. The selection of an elastic collision is made with probability

where

σel is the elastic scattering cross section.

σin is the inelastic cross section; includes any neutron-out process−(n,n'), (n,f), (n,np), etc.

σais the absorption cross section; , that is, all neutron disappearing

reactions.

σTis the total cross section, σT=σel + σin + σa.

Both σel and σTare adjusted for the free gas thermal treatment at thermal energies.

The selection of an inelastic collision is made with the remaining probability

σel

σin σel

---------------------σel

σTσa

–

------------------=

Σσ nx,() n≠,

‘

April 10, 2000 2-37

CHAPTER 2

PHYSICS

If the collision is determined to be inelastic, the type of inelastic reaction, n, is sampled from

where ξis a random number on the interval [0,1), Nis the number of inelastic reactions, and the

σi's are the inelastic reaction cross sections at the incident neutron energy.

For both elastic and inelastic scattering, the direction of exiting particles usually is determined

by sampling angular distribution tables from the cross-section ﬁles. This process is described

shortly. For elastic collisions and discrete inelastic scattering from levels, the exiting particle

energy is determined from two body kinematics based upon the center-of-mass cosine of the

scattering angle. For other inelastic processes, the energy of exiting particles is determined from

secondary energy distribution laws from the cross-section ﬁles, which vary according to the

particular inelastic collision modeled.

a. Sampling of Angular Distributions: The direction of emitted particles is sampled in

the same way for most elastic and inelastic collisions. The cosine of the angle between incident

and exiting particle directions, µ, is sampled from angular distribution tables in the collision

nuclide's cross-section library. The angular distribution tables consist of 32 equiprobable cosine

bins and are given at a number of different incident neutron energies. The cosines are either in

the center-of-mass or target-at-rest system, depending on the type of reaction. If Eis the incident

neutron energy, if Enis the energy of table n, and if En+1 is the energy of table n+1, then a value

of µ is sampled from table n+ 1 with probability (E−En)/(En + 1 −En) and from table nwith

probability (En + 1 −E)/(En+1 −En). A random number ξ on the interval [0,1) is then used to

select the ith cosine bin such that i= 32 ξ+ 1. The value of µ is then computed as

µ=µi+ (32ξ−i)(µi+1 −µ

i).

If, for some incident neutron energy, the emitted angular distribution is isotropic, µis chosen

from µ = ξ', where ξ' is a random number on the interval [−1,1). (Strictly, in MCNP random

numbers are always furnished on the interval [0,1). Thus, to compute ξ'on[−1,1) we calculate

ξ'= 2 ξ − 1, where ξ is a random number on [0,1).)

For elastic scattering, inelastic level scattering, and some ENDF/B−VI inelastic reactions, the

scattering cosine is chosen in the center-of-mass system. Conversion must then be made to µlab,

the cosine in the target-at-rest system. For other inelastic reactions, the scattering cosine is

sampled directly in the target-at-rest system.

σin

σTσa

–

------------------

σi

i1=

n1–

∑ξσ

i1=

∑σi

i1=

∑

≤<

2-38 April 10, 2000

CHAPTER 2

PHYSICS

The incident particle direction cosines, (uo,vo,wo), are rotated to new outgoing target-at-rest

system cosines, (u, v, w), through a polar angle whose cosine is µlab, and through an azimuthal

angle sampled uniformly. For random numbers ξ1 and ξ2 on the interval [−1,1) with rejection

criterion , the rotation scheme is (Ref. 2, pg. 54):

If , then

If the scattering distribution is isotropic in the target-at-rest system,it is possible to use an even

simpler formulation that takes advantage of the exiting direction cosines, (u,v,w), being

independent of the incident direction cosines, (uo,vo,wo). In this case,

ξ1

2ξ2

21≤

oµlab

1µlab

–ξ1uowoξ2o–()

ξ1

2ξ2

+()1wo

–()

-------------------------------------------------------------+=

oµlab

1ulab

–ξ1vowoξ2uo

+()

ξ1

2ξ2

+()1wo

–()

---------------------------------------------------------------+=

oµlab

ξ11µlab

–()1wo

–()

ξ1

2ξ2

+()

----------------------------------------------------–=

1wo

20∼–

oµlab

1µlab

–ξ1uovoξ2wo

+()

ξ1

2ξ2

+()1υo

–()

---------------------------------------------------------------+=

oµlab

ξ11µlab

–()1vo

–()

ξ1

2ξ2

+()

-----------------------------------------------------–=

oµlab

=1µlab

–ξ1wovoξ2uo

–()

ξ1

2ξ2

+()1vo

–()

---------------------------------------------------------------+

u2ξ1

22ξ2

21–+=

vξ11u2

–

ξ1

2ξ2

-----------------=

April 10, 2000 2-39

CHAPTER 2

PHYSICS

where ξ1 and ξ2 are rejected if .

b. Elastic Scattering: The particle direction is sampled from the appropriate angular

distributiontables, and the exiting energy, Eout, is dictated by two-body kinematics:

where Ein =incident neutron energy, µcm =center-of-mass cosine of the angle between incident

and exiting particle directions,

and A= mass of collision nuclide in units of the mass of a neutron (atomic weight ratio).

c. Inelastic Scattering: The treatment of inelastic scattering depends upon the particular

inelastic reaction chosen. Inelastic reactions are deﬁned as (n,y) reactions such as (n,n'),(n,2n),

(n,f), (n,n'α) in which y includes at least one neutron.

For many inelastic reactions, such as (n,2n), more than one neutron can be emitted for each

incident neutron. The weight of each exiting particle is always the same as the weight of the

incident particle minus any implicit capture. The energy of exiting particles is governed by

various scattering laws that are sampled independently from the cross-section ﬁles for each

exiting particle. Which law is used is prescribed by the particular cross-section evaluation used.

In fact, more than one law can be speciﬁed, and the particular one used at a particular time is

decided with a random number. In an (n,2n) reaction, for example, the ﬁrst particle emitted may

have an energy sampled from one or more laws, but the second particle emitted may have an

energy sampled from one or more different laws, depending upon speciﬁcations in the nuclear

data library. Because emerging energy and scattering angle is sampled independently for each

particle, there is no correlation between the emerging particles. Hence energy is not conserved

in an individual reaction because, for example, a 14-MeV particle could conceivably produce

two 12-MeV particles in a single reaction. But the net effect of many particle histories is

unbiased because on the average the correct amount of energy is emitted. Results are biased only

wξ21u2

–

ξ1

2ξ2

-----------------=

ξ1

2ξ2

21>+

Eout 1

---Ein 1α–()µ

cm 1α++[]=

Ein

1A22Aµcm

1A+()

---------------------------------------

αA1–

A1+

-------------





2-40 April 10, 2000

CHAPTER 2

PHYSICS

when quantities that depend upon the correlation between the emerging particles are being

estimated.

Users should note that MCNP follows a very particular convention. The exiting particle energy

and direction are always given in the target-at-rest (laboratory) coordinate system. For the

kinematical calculations in MCNP to be performed correctly, the angular distributions for

elastic, discrete inelastic level scattering, and some ENDF/B−VI inelastic reactions must be

given in the center-of-mass system, and the angular distributions for all other reactions {\it must}

be given in the target-at-rest system. MCNP does not stop if this convention is not adhered to,

but the results will be erroneous. In the checking of the cross-section libraries prepared for

MCNP at Los Alamos, however, careful attention has been paid to ensure that these conventions

are followed.

The exiting particle energy and direction in the target–at–rest (laboratory) coordinate system are

related to the center−of−mass energy and direction as follows:1

; and

where

= exiting particle energy (laboratory),

= exiting particle energy (center-of-mass),

E= incident particle energy (laboratory),

µcm = cosine of center−of−mass scattering angle,

µlab = cosine of laboratory scattering angle,

A= atomic weight ratio (mass of nucleus divided by mass of incident particle.)

For point detectors it is necessary to convert

where

′E′cm

E2µcm A1+()EE′cm

A1+()

-------------------------------------------------------------

µlab µcm

E′cm

E′

-----------1

A1+

-------------E

E′

-----+=

E′

E′cm

pµlab

()pµcm

()

dµcm

dµlab

-------------

April 10, 2000 2-41

CHAPTER 2

PHYSICS

and1

d. Nonﬁssion Inelastic Scattering and Emission Laws: Nonﬁssion inelastic reactions are

handled differently from ﬁssion inelastic reactions. For each nonﬁssion reaction Npparticles are

emitted, where Np is an integer quantity speciﬁed for each reaction in the cross-section data

library of the collision nuclide. The direction of each emitted particle is independently sampled

from the appropriate angular distribution table, as was described earlier. The energy of each

emitted particle is independently sampled from one of the following scattering or emission laws.

Energy and angle are correlated only for ENDF/B--VI laws 44 and 67. For completeness and

convenience we list all the laws together, regardless of whether the law is appropriate for

nonﬁssion inelastic scattering (for example, Law~3), ﬁssion spectra (for example, Law 11), both

(for example, Law 9), or neutron-induced photon production (for example, Law 2). The

conversion from center−of−mass to target−at−rest (laboratory) coordinate systems is as above.

Law 1 (ENDF law 1): Equiprobable energy bins.

The index iand the interpolation fraction rare found on the incident energy grid for the

incident energy Ein such that

and

A random number on the unit interval ξ1is used to select an equiprobable energy bin k

from the K equiprobable outgoing energies Eik

Then scaled interpolation is used with random numbers ξ2 and ξ3 on the unit interval.

Let

and

µcm µlab E′

Ecm

′

--------- 1

A1+

-------------E

Ecm

′

---------–=

dµcm

dµlab

-------------E′E′cm

⁄

E′

E′cm

-----------µlab

A1+

-------------E

E′cm

-----------–

---------------------------------------------------=

E′

E′cm

-----------

1µlab

A1+

-------------E

E′

-----–

--------------------------------=

EiEin Ei1+

Ein EirE

i1+ Ei

–()+=

kξiK1+=

E1Ei1,rE

i11,+Ei1,

–()+=

2-42 April 10, 2000

CHAPTER 2

PHYSICS

; and

if and

; then

Law 2 Discrete photon energy.

The value provided in the library is Eg. The secondary photon energy

Eout is either

Eout =Eg for non-primary photons or

Eout = Eg + [A/(A+1)]Ein for primary photons,

where A is the atomic weight to neutron weight ratio of the target and Ein

is the incident neutron energy.

Law 3 (ENDF law 3): Inelastic scattering (n,n') from nuclear levels.

The value provided in the library is Q.

Law 4 Tabular distribution (ENDF law 4).

For each incident neutron energy Ei there is a pointer to a table of secondary energies

Ei,k, probability density functions pi,k, and cumulative density functions ci,k. The index

i and the interpolation fraction r are found on the incident energy grid for the incident

energy Ein such that

and

A random number on the unit interval ξ1 is used to sample a secondary energy bin k

from the cumulative density function

EKEiK,rE

i1K,+EiK,

–()+=

liif ξ3ror>=

li1+= ξ3r<

E′Elk,ξ2Elk 1+,Elk,

–()+=

Eout E1

E′El1,

–()EKE1

–()

ElK,El1,

–

--------------------------------------------------+=

Eout A

A1+

-------------





2Ein QA 1+()

----------------------–=

EiEin Ei1+

Ein EirE

i1+ Ei

–()+=

cik,rc

i1k,+cik,

–()ξ

1cik 1+,rc

i1k1+,+cik 1+,

–()+<<+

April 10, 2000 2-43

CHAPTER 2

PHYSICS

If these are discrete line spectra, then the sampled energy E' is interpolated between

incident energy grids as

It is possible to have all discrete lines, all continuous spectra, or a mixture of discrete

lines superimposed on a continuous background. For continuous distributions, the

secondary energy bin k is sampled from

where l=iifξ2>rand l=i+1if ξ2<r, and ξ2is a random number on the unit interval.

For histogram interpolation the sampled energy is

For linear-linear interpolation the sampled energy is

For neutron–induced photons, Eout = E' and the angle is selected as described on

page 2–37. That is, the photon secondary energy is sampled from either of the two

bracketing incident energy bins, l = i or l = i + 1.

The neutron secondary energy must be interpolated between the incident energy bins i

and i + 1 to properly preserve thresholds. Let

and

; then

E′Eik,rE

i1k,+Eik,

–()+=

clk,ξ1clk 1+,

E′Elk,

ξ1clk,

–()

plk,

------------------------+=

E′Elk,

Plk,

22plk 1+,p–lk,

Elk 1+,Elk,

–

--------------------------------ξ1clk,

–()+plk,

–

plk 1+,plk,

–

Elk 1+,Elk,

–

--------------------------------

-----------------------------------------------------------------------------------------------------











E1Ei1,rE

i11,+Ei1,

–()+=

EKEiK,rE

i1K,+EiK,

–()+=

Eout E1

E′El1,

–()EKE1

–()

ElK,El1,

–()

--------------------------------------------------+=

2-44 April 10, 2000

CHAPTER 2

PHYSICS

The outgoing neutron energy is then adjusted to the laboratory system, if it is in the

center-of-mass system, and the outgoing angle is selected as described on page 2–37.

Law 5 (ENDF law 5): General evaporation spectrum.

The function g(x) is tabulated versus χ and the energy is tabulated versus incident

energy Ein. The law is then

This density function is sampled by

Eout = χ(ξ) T(Ein),

where T(Ein) is a tabulated function of the incident energy and

c(ξ) is a table of equiprobable χ values.

Law 7 (ENDF law 7): Simple Maxwell Fission Spectrum.

The nuclear temperature T(Ein)is a tabulated function of the incident energy. The

normalization constant C is given by

U is a constant provided in the library and limits Eout to . In MCNP

this density function is sampled by the rejection scheme

where ξ1,ξ2,ξ3, and ξ4are random numbers on the unit interval. ξ1and ξ2are rejected

Law 9 (ENDF law 9): Evaporation spectrum.

in Eout

→()gEout

()

----------------





in Eout

→()C=EouteE–out TE

()⁄

C1– T32⁄π

-------





erf Ein U–()

-----------------------





Ein U–()

-----------------------eEin U–()T⁄–

–=

0Eout Ein U–≤≤

Eout TE

()

ξ1

2ξ3

ξ1

2ξ2

-----------------ξ4

ln+–=

ξ1

2ξ2

21>+

in Eout

→()C=EouteEout TE

()⁄–

April 10, 2000 2-45

CHAPTER 2

PHYSICS

where the nuclear temperature T(Ein)is a tabulated function of the incident energy. The

energy U is provided in the library and is assigned so that Eout is limited by

. The normalization constant Cis given by

In MCNP this density function is sampled by

where ξ1and ξ2 are random numbers on the unit interval, and ξ1 and ξ2 are rejected if

Eout > Ein −U.

Law 11 (ENDF law 11): Energy Dependent Watt Spectrum.

The constants a and b are tabulated functions of incident energy and U is a constant

from the library. The normalization constant C is given by

where the constant U limits the range of outgoing energy so that .

This density function is sampled as follows. Let

. Then Eout = −ag ln ξ1.

Eout is rejected if

where ξ1 and ξ2 are random numbers on the unit interval.

0Eout Ein U–≤≤

C1– T21eEin U–()T⁄–1Ein U–()T⁄+()–[]=

Eout TE

()ξ

1ξ2

()ln–=

in Eout

→()Ce Eout aE

()⁄–bE

()Eout

sinh=

c1– 1

---πa3b

------------ab

------





erf Ein U–()

-----------------------ab

------–





erf Ein U–()

-----------------------ab

------+





aEin U–()

-----------------------– bE

in U–()sinhexp–

exp=

0Eout Ein U–≤≤

g1ab

------+





21–1

------+





1g–()1ξ1

ln–()ξ

ln–[]

2bEout

2-46 April 10, 2000

CHAPTER 2

PHYSICS

Law 22 (UK law 2): Tabular linear functions of incident energy out.

Tables of Pij,Cij, and Tij are given at a number of incident energies Ei. If

then the ith Pij, Cij, and Tij tables are used.

where kis chosen according to

where ξ is a random number on the unit interval [0,1).

Law 24 (UK law 6): Equiprobable energy multipliers. The law is

The library provides a table of K equiprobable energy multipliers Ti,k for a grid of

incident neutron energies Ei. For incident energy Ein such that

the random numbers ξ1 and ξ2 on the unit interval are used to ﬁnd T:

and then

Law 44 Tabular Distribution (ENDF/B-VI ﬁle 6 law=1 lang=2, Kalbach-87 correlated energy-

angle scattering). Law 44 is a generalization of law 4. For each incident neutron energy

Ei there is a pointer to a table of secondary energies Ei,k, probability density functions

pi,k, cumulative density functions ci,k, precompound fractions Ri,k, and angular

distribution slope values Ai,k. The index i and the interpolation fraction r are found on

the incident energy grid for the incident energy Ein such that

Ei < Ein < Ei+1 and

Ein = Ei + r(Ei + 1 − Ei ) .

A random number on the unit interval ξ1 is used to sample a secondary energy bin k

from the cumulative density function

EiEin Ei1+

<≤

Eout Cik Ein Tik

–()=

Pij ξPij

j1=

k1+

∑

≤<

j1=

∑

Eout EinTE

()=

EiEin Ei1+

kξ1K1+=

ik,ξ2Tik 1+,Tik,

–()+=

Eout EinT=

April 10, 2000 2-47

CHAPTER 2

PHYSICS

ci,k + r (ci+1,k − ci,k)<ξ

1< ci,k+1 + r (ci+1,k+1 − ci,k+1).

If these are discrete line spectra, then the sampled energy E' is interpolated between

incident energy grids as

It is possible to have all discrete lines, all continuous spectra, or a mixture of discrete

lines superimposed on a continuous background. For continuous distributions, the

secondary energy bin k is sampled from

where l=iifξ2>rand l=i+1if ξ2<r, and ξ2is a random number on the unit interval.

For histogram interpolation the sampled energy is

For linear-linear interpolation the sampled energy is

Unlike Law 4, the sampled energy is interpolated between the incident energy bins iand

i + 1 for both neutron-induced photons and neutrons. Let

and

; then

For neutron-induced photons, the outgoing angle is selected as described on

page 2–37. For neutrons, Eout is always in the center-of-mass system and must be

adjusted to the laboratory system. The outgoing neutron center-of-mass scattering angle

µ is sampled from the Kalbach-87 density function

E′Eik,rE

i1k,+Eik,

–()+=

clk,ξ1clk 1+,

E′Elk,

ξ1clk,

–()

plk,

------------------------+=

′Elk,

plk,

22plk 1+,plk,

–

Elk 1+,Elk,

–

--------------------------------ξ1clk,

–()+plk,

–

plk 1+,plk,

–

Elk 1+,Elk,

–

--------------------------------

-----------------------------------------------------------------------------------------------------













E1Ei1,rE

i11,+Ei1,

–()+=

EKEiK,rE

i1K,+EiK,

–()+=

Eout E1

E′El1,

–()EKE1

–()

ElK,El1,

–()

--------------------------------------------------+=

2-48 April 10, 2000

CHAPTER 2

PHYSICS

using the random numbers ξ3 and ξ4 on the unit interval as follows. If ξ3> R, then let

or if ξ3< R, then

R and A are interpolated on both the incident and outgoing energy grids. For discrete

spectra,

For continuous spectra with histogram interpolation,

For continuous spectra with linear-linear interpolation,

The Kalbach-87 formalism (Law 44) is also characterized by an energy-dependent

multiplicity in which the number of neutrons emerging from a collision varies. If the

number is less than one, Russian roulette is played and the collision can result in a

capture. If the number is greater than one, the usual MCNP approach is taken whereby

the additional particles are banked and only the ﬁrst one contributes to detectors and

DXTRAN.

Law 66 N-body phase space distribution (ENDF/B-VI ﬁle 6 law 6).

The phase space distribution for particle i in the center-of-mass coordinate system is:

pµEin Eout

,,()

---= A

A()sinh

------------------- Aµ()RAµ()sinh+cosh[]

T2ξ41–()A()andsinh=

µTT

21++()A⁄ln=

µξ4eA1ξ4

–()eA–

+A⁄ln=

Aik,rA

i1k,+Aik,

–()+=

ik,rR

i1k,+RIk,

–()+=

lk,

Rlk,⋅

lk,Alk 1+,Alk,

–()E′Elk,

–()Elk 1+,Elk,

–()⁄

Rlk,Rlk 1+,Rlk,

–()E′Elk,

–()Elk 1+,Elk,

–()⁄⋅+

April 10, 2000 2-49

CHAPTER 2

PHYSICS

where all energies and angles are also in the center-of-mass system and is the

maximum possible energy for particle i,µand T.Tis used for calculating Eout. The Cn

normalization constants for n = 3, 4, 5 are:

and

Eimax is a fraction of the energy available, Ea,

where M is the total mass of the nparticles being treated, miis the mass of particle i, and

where mT is the target mass and mp is the projectile mass. For neutrons,

and for a total mass ratio Ap = M/mi,

Thus,

PiµEin T,,()CnTE

max T–()

3n24–⁄,=

max

C34

πEi

max

()

-----------------------,=

C4105

32 Ei

max

()

72⁄

-------------------------------,=

C5256

14πEi

max

()

-----------------------------⋅=

max Mm

–

-----------------Ea,=

mpmT

-------------------- Ein Q,+=

mpmT

-------------------- A

A1+

-------------=

–

-----------------Ap1–

---------------=

max Ap1–

--------------- A

A1+

-------------Ein Q+





⋅=

2-50 April 10, 2000

CHAPTER 2

PHYSICS

The total mass Apand the number of particles in the reaction nare provided in the data

library. The outgoing energy is sampled as follows.

Let ξi, i = 1,9 be random numbers on the unit interval. Then from rejection technique

R28 from the Monte Carlo Sampler,3 accept ξ1 and ξ2 if

and accept ξ3and ξ4 if

Then let

and

and let

and

then

The cosine of the scattering angle is always sampled isotropically in the center-of-mass

system using another random number ξ2 on the unit interval:

ξ1

2ξ2

+1≤

ξ3

2ξ4

21⋅≤+

pξ5if n3,==

pξ5ξ6if n4,==

pξ5ξ6ξ7ξ8if n5,==

xξ1ξ1

2ξ2

+()ln–

ξ1

2ξ2

+()

------------------------------------ ξ9,ln–=

yξ3ξ3

2ξ4

+()ln–

ξ3

2ξ4

+()

------------------------------------ pln ,–=

xy+

------------;=

Eout TEi

max ⋅=

µ2ξ21⋅–=

April 10, 2000 2-51

CHAPTER 2

PHYSICS

Law 67 Correlated energy-angle scattering (ENDF/B-VI ﬁle 6 law 7).

For each incident neutron energy, ﬁrst the exiting particle direction µis sampled as

described on page 2–37. In other Law data, ﬁrst the exiting particle energy is sampled

and then the angle is sampled. The index iand the interpolation fraction rare found on

the incident energy grid for the incident energy Ein, such that

For each incident energy Eithere is a table of exiting particle direction cosines µi,j and

locators Li,j. This table is searched to ﬁnd which ones bracket µ, namely,

Then the secondary energy tables at Li,j and Li,j+1 are sampled for the outgoing particle energy.

The secondary energy tables consist of a secondary energy grid Ei,j,k, probability density

functions pi,j,k, and cumulative density functions ci,j,k. A random number ξ1on the unit interval

is used to pick between incident energy indices: if ξ1< r then l = i + 1; otherwise, l = i. Two

more random numbers ξ2and ξ3on the unit interval are used to determine interpolation energies.

If , then

Otherwise,

If ξ3 < (µ−µ

i+1,j)/(µi+1,j+1 −µ

i+1,j), then

Otherwise,

A random number ξ4 on the unit interval is used to sample a secondary energy bin k

from the cumulative density function

For histogram interpolation the sampled energy is

EiEin Ei1+ and<<

Ein EirE

i1+ Ei

–()⋅+=

µij,µµ

ij,1+ ⋅<<

ξ2µµ

1j,

–()µ

1j1+,µij,

–()⁄<

Eik,Eij 1k,+,and mj1,+==ifli⋅=

Eik,Eijk,, and mj,==ifli⋅=

Ei1k,+Ei1j1k,+,+and mj1,+==ifli1⋅+=

Ei1k,+Ei1jk,,+and mj if l,i1⋅+===

clmk,, ξ4clmk 1+,,

2-52 April 10, 2000

CHAPTER 2

PHYSICS

For linear-linear interpolation the sampled energy is

The ﬁnal outgoing energy Eout uses scaled interpolation. Let

e. Fission Inelastic Scattering: For any ﬁssion reaction a number of neutrons, Np, are

emitted according to the value of . The average number of neutrons per ﬁssion, ,

is either a tabulated function of energy or a polynomial function of energy. If I is the largest

integer less than , then

, where ξis a random number.

The type of emitted neutron, either delayed or prompt, is then determined from the ratio of

delayed to total as

, produce a delayed neutron, or

, produce a prompt neutron.

Each delayed ﬁssion neutron energy and time of emission is determined by sampling from the

abundance of each decay group and then the appropriate decay constant for time and tabular

emission distribution as speciﬁed in the evaluation is used.

E′Elmk,,

ξ4clmk,,

–()

plmk,,

------------------------------⋅+=

E′Elmk,,

Plmk,,

22plmk 1+,, plmk,,

–

Elmk 1+,, Elmk,,

–

-------------------------------------------ξ4clmk,,

–()+plmk,,

–

plmk 1+,, plmk,,

–

Elmk 1+,, Elmk,,

–

-------------------------------------------

--------------------------------------------------------------------------------------------------------------------------------











E1Ei1,rE

i11,+Ei1,

–()+=

and EKEiK,rE

i1K,+EiK,

–()⋅+=

Then Eout E1

E′El1,

–()EKE1

–()

ElK,El1,

–()

--------------------------------------------------+=

νEin

() νEin

()

νEin

()

NpI–1ifξνEin

()1–≤+

NpIif ξνEin

()I–>=

νDEin

() νtot Ein

()

if ξν

DEin

()ν

tot

⁄Ein

()≤

if ξν

DEin

()ν

tot

⁄Ein

()>

April 10, 2000 2-53

CHAPTER 2

PHYSICS

The energy of each prompt ﬁssion neutron is determined from the emission law as speciﬁed in

the evaluation. The three laws used for prompt ﬁssion neutron spectra are 7, 9, and 11. These

laws are discussed in the preceding section, starting on page 2–44.

The direction of each emitted neutron is sampled independently from the appropriate angular

distribution table by the procedure described on page 2–37.

The energy of each ﬁssion neutron is determined from the appropriate (that is, as speciﬁed in the

evaluation) emission law. The three laws used for ﬁssion neutron spectra are 7, 9, and 11. These

laws are discussed in the preceding section, starting on page 2–44. MCNP then models the

transport of the ﬁrst neutron out after storing all other neutrons in the bank.

6. The S(α,β) treatment

The S(α,β) thermal scattering treatment is a complete representation of thermal neutron

scattering by molecules and crystalline solids. Two processes are allowed: (1) inelastic scattering

with cross section σin and a coupled energy-angle representation derived from an

ENDF/B S(α,β) scattering law, and (2) elastic scattering with no change in the outgoing neutron

energy for solids with cross section σel and an angular treatment derived from lattice parameters.

The elastic scattering treatment is chosen with probability σel/(σel +σin). This thermal scattering

treatment also allows the representation of scattering by multiatomic molecules (for example,

BeO).

For the inelastic treatment, the distribution of secondary energies is represented by a set of

equally probable ﬁnal energies (typically 16 or 32) for each member of a grid of initial energies

from an upper limit of typically 4 eV down to 10−5eV, along with a set of angular data for each

initial and ﬁnal energy. The selection of a ﬁnal energy E' given an initial energy E can be

characterized by sampling from the distribution

where Ei and Ei+1 are adjacent elements on the initial energy grid,

N is the number of equally probable ﬁnal energies, and Eij is the jth discrete ﬁnal energy for

incident energy Ei.

pE′EiEE

i1+

<<()

----δE′ρEij,1ρ–()Ei1j,+

––[],

i1=

∑

ρEi1+ E–

Ei1+ Ei

–

----------------------- ,=

2-54 April 10, 2000

CHAPTER 2

PHYSICS

There are two allowed schemes for the selection of a scattering cosine following selection of a

ﬁnal energy and ﬁnal energy index j. In each case, the (i,j)th set of angular data is associated with

the energy transition .

(1.) The data consist of sets of equally probable discrete cosines µi,j,k for k = 1,...,νwith ν

typically 4 or 8. An index k is selected with probability 1/ν, and µ is obtained by the relation

(2.) The data consist of bin boundaries of equally probable cosine bins. In this case,

random linear interpolation is used to select one set or the other, with ρbeing the probability of

selecting the set corresponding to incident energy Ei. The subsequent procedure consists of

sampling for one of the equally probable bins and then choosing µuniformly in the bin.

For elastic scattering, the above two angular representations are allowed for data derived by an

incoherent approximation. In this case, one set of angular data appears for each incident energy

and is used with the interpolation procedures on incident energy described above.

For elastic scattering, when the data have been derived in the coherent approximation, a

completely different representation occurs. In this case, the data actually stored are the set of

parameters Dk, where

and EBk are Bragg energies derived from the lattice parameters. For incident energy Esuch that

represents a discrete cumulative probability distribution that is sampled to obtain index i,

representing scattering from the ith Bragg edge. The scattering cosine is then obtained from the

relationship

Using next event estimators such as point detectors with S(α,β) scattering cannot be done

exactly because of the discrete scattering angles. MCNP uses an approximate scheme42,43 that

in the next event estimation calculation replaces discrete lines with histograms of width

< .1 .

See also page 2–95.

iE′→ Eij,

µρµ

ijk,, 1ρ–()µ

i1jk,,+⋅+=

σeI DkEfor⁄Ebk EE

bk 1+

<≤=

σeI 0()Efor⁄EE

EBk EE

Bk 1+

≤≤

PiDiDk

⁄for i1…k,,==

µ12EBi E⁄⋅–=

δµ

April 10, 2000 2-55

CHAPTER 2

PHYSICS

7. Unresolved Resonance Range Probability Tables

Above the resonance range ( 2 - 25 keV for 235U in ENDF/B-VI, 10 - 300 keV for 238U in

ENDF/B-VI), continuous-energy neutron cross sections appear to be smooth as a function of

energy. This is not because the resonances end, but rather because the resonances are so close

together that they are unresolved. The cross section can, however, be represented by

probabilities. The unresolved resonance range probability table method provides a table of

probabilities for the cross sections in the unresolved resonance energy range. Properly sampling

unresolved resonances is important to properly model resonance self-shielding effects,

particularly for fast-spectra nuclear systems such as unmoderated critical assemblies.

Sampling cross sections from probability tables is straightforward. At each of a number of

incident energies there is a table of cumulative probabilities (typically 20) and the value of the

near-total, elastic, ﬁssion, and radiative capture cross sections and heat deposition numbers

corresponding to those probabilities. These data supplement the usual continuous data; if

probability tables are turned off (PHYS:N card), then the usual smooth cross section is used. But

if the probability tables are turned on (default), if they exist for the nuclide of a collision, and if

the energy of the collision is in the unresolved resonance energy range of the probability tables,

then the cross sections are sampled from the tables. The near-total is the total of the elastic,

ﬁssion, and radiative capture cross sections; it is not the total cross section, which may include

other absorption or inelastic scatter in addition to the near-total. The radiative capture cross

section is not the same as the usual MCNP capture cross section, which is more properly called

“destruction” or absorption and includes not only radiative capture but all other reactions not

emitting a neutron. Sometimes the probability tables are provided as factors (multipliers of the

average or underlying smooth cross section) which adds computational complexity but now

includes any structure in the underlying smooth cross section.

It is essential to maintain correlations in the random walk when using probability tables to

properly model resonance self-shielding. Suppose we sample the 17th level (probability) from

the table for a given collision. This position in the probability table must be maintained for the

neutron trajectory until the next collision, regardless of particle splitting for variance reduction

or surface crossings into various other materials whose nuclides may or may not have probability

table data. Correlation must also be retained in the unresolved energy range when two or more

cross-section sets for an isotope that utilize probability tables are at different temperatures.

D. Photon Interactions

Sampling of a collision nuclide, analog capture, implicit capture, and many other aspects of

photon interactions such as variance reduction, are the same as for neutrons. The collision

physics are completely different.

MCNP has two photon interaction models: simple and detailed.

2-56 April 10, 2000

CHAPTER 2

PHYSICS

The simple physics treatment ignores coherent (Thomson) scattering and ﬂuorescent photons

from photoelectric absorption. It is intended for high-energy photon problems or problems

where electrons are free and is also important for next event estimators such as point detectors,

where scattering can be nearly straight ahead with coherent scatter. The simple physics treatment

uses implicit capture unless overridden with the CUT:P card, in which case it uses analog

capture.

The detailed physics treatment includes coherent (Thomson) scattering and accounts for

ﬂuorescent photons after photoelectric absorption. Form factors are used to account for electron

binding effects. Analog capture is always used. The detailed physics treatment is used below

energy EMCPF on the PHYS:P card, and because the default value of EMCPF is 100 MeV, that

means it is almost always used by default. It is the best treatment for most applications,

particularly for high Z nuclides or deep penetration problems.

The generation of electrons from photons is handled three ways. These three ways are the same

for both the simple and detailed photon physics treatments. (1) If electron transport is turned on

(Mode P E), then all photon collisions except coherent scatter can create electrons that are

banked for later transport. (2) If electron transport is turned off (no E on the Mode card), then a

thick-target bremsstrahlung model (TTB) is used. This model generates electrons, but assumes

that they travel in the direction of the incident photon and that they are immediately annihilated.

Any bremsstrahlung photons produced by the nontransported electrons are then banked for later

transport. Thus electron-induced photons are not neglected, but the expensive electron transport

step is omitted. (3) If IDES = 1 on the PHYS:P card, then all electron production is turned off,

no electron-induced photons are created, and all electron energy is assumed to be locally

deposited.

The TTB approximation cannot be used in Mode P E problems, but it is the default for Mode P

problems.

1. Simple Physics Treatment

The simple physics treatment is intended primarily for higher energy photons. It is inadequate

for high Z nuclides or deep penetration problems. The physical processes treated are

photoelectric effect, pair production, and Compton scattering on free electrons. The

photoelectric effect is regarded as an absorption (without ﬂuorescence), scattering (Compton) is

regarded to be on free electrons (without use of form factors), and the highly forward coherent

Thomson scattering is ignored. Thus the total cross section σt is regarded as the sum of three

components:

σtσpe σpp σs⋅++=

April 10, 2000 2-57

CHAPTER 2

PHYSICS

a. Photoelectric effect: This is treated as a pure absorption by implicit capture with a

corresponding reduction in the photon weight WGT, and hence does not result in the loss of a

particle history except for Russian roulette played on the weight cutoff. The noncaptured weight

WGT(1 −σ

pe/σt)is then forced to undergo either pair production or Compton scattering. The

captured weight either is assumed to be locally deposited or becomes a photoelectron for

electron transport or for the TTB approximation.

b. Pair production: In a collision resulting in pair production [probability σpp/(σt−σ

pe)],

either an electron-positron pair are created for further transport (or the TTB treatment) and the

photon disappears, or it is assumed that the kinetic energy WGT(E−1.022) MeV of the electron-

positron pair produced is deposited as thermal energy at the time and point of collision, with

isotropic production of one photon of energy 0.511 MeV headed in one direction and another

photon of energy 0.511 MeV headed in the opposite direction. The rare single 1.022−MeV

annihilation photon is ignored. The simple physics treatment for pair production is the same as

the detailed physics treatment that is described in detail below.

c. Compton scattering: The alternative to pair production is Compton scattering on a free

electron, with probabilityσs/(σt−σ

pe). In the event of such a collision, the objective is to

determine the energy E' of the scattered photon, and for the angle of deﬂection

from the line of ﬂight. This yields at once the energy deposited at the point of

collision and the new direction of the scattered photon. The energy deposited at the point of

collision can then be used to make a Compton recoil electron for further transport or for the TTB

approximation.

The differential cross section for the process is given by the Klein-Nishina formula1

(2.2)

where rois the classical electron radius , αand are the incident and ﬁnal

photon energies in units of 0.511 MeV , where mis the mass of the electron and

c is the speed of light], and .

The Compton scattering process is sampled exactly by Kahn's method44 below 1.5 MeV and by

Koblinger's method45 above 1.5 MeV as analyzed and recommended by Blomquist and

Gelbard.46

For next event estimators such as detectors and DXTRAN, the probability density for scattering

toward the detector point must be calculated:

µθcos=θ

WGT E E′–()

Kαµ,()dµπro

2α′

-----





2α′

----- α

α′

----- µ21–++ dµ,=

2.817938 10 13– cm×α′

α[ Emc

()⁄=

α′ α 1α1µ–()+[]⁄=

2-58 April 10, 2000

CHAPTER 2

PHYSICS

where is the total Klein-Nishina cross section obtained by integrating K(α,µ) over all

angles for energy α. This is a difﬁcult integration, so the empirical formula of Hastings2is used:

where η = 1 + .222037a, c1 = 1.651035, c2 = 9.340220, c3 = -8.325004, d1 = 12.501332,

d2 = -14.200407, and d3 = 1.699075. Thus,

Above 100 MeV, where the Hastings ﬁt is no longer valid, the approximation

is made so that

2. Detailed Physics Treatment

The detailed physics treatment includes coherent (Thomson) scattering and accounts for

ﬂuorescent photons after photoelectric absorption. Form factors are used with coherent and

incoherent scattering to account for electron binding effects. Analog capture is always used, as

described below under photoelectric effect. The detailed physics treatment is used below energy

EMCPF on the PHYS:P card, and because the default value of EMCPF is 100 MeV, that means

it is almost always used by default. It is the best treatment for most applications, particularly for

high Z nuclides or deep penetration problems.

The detailed physics treatment for next event estimators such as point detectors is inadvisable,

as explained on page 2–62, unless the NOCOH=1 option is used on the PHYS:P card to turn off

coherent scattering.

a. Incoherent (Compton) scattering: To model Compton scattering it is necessary to

determine the angle θof scattering from the incident line of ﬂight (and thus the new direction),

pµ() 1

σ1

KZα,()

---------------------- Kαµ,(),=

σt

KZα,()

σ1

KZα,()πro

2c1η2c2ηc3

η3d1η2d2ηd3

+++

----------------------------------------------------

pµ() η3d1η2d2ηd3

+++

c1η2c2ηc3

---------------------------------------------------- α′

-----





2α

α′

----- α′

----- µ21–++





⋅=

σ1

KZα,()σ

1Zα,()Z⁄=

pµ() Zπr0

σ1Zα,()

--------------------- α′

-----





2α

α′

----- α′

----- µ21–++





April 10, 2000 2-59

CHAPTER 2

PHYSICS

the new energy Eof the photon, and the recoil kinetic energy of the electron, E−E. The recoil

kinetic energy can be deposited locally, can be transported in Mode P E problems, or (default)

can be treated with the TTB approximation.

Incoherent scattering is assumed to have the differential cross section

, where I(Z,v) is an appropriate scattering factor

modifying the Klein-Nishina cross section in Eq. (2.2).

Qualitatively, the effect of I(Z,v) is to decrease the Klein-Nishina cross section (per electron)

more extremely in the forward direction, for low E and for high Z independently. For any Z,

I(Z,v) increases from to . The parameter v is the inverse length

. The

maximum value of νis at µ=−1. The essential features of I(Z,v) are

indicated in Fig. 2-4.

For hydrogen, an exact expression for the form factor is used:47

where f is the inverse ﬁne structure constant, f = 137.0393, and .

The Klein-Nishina formula is sampled exactly by Kahn's method44 below 1.5 MeV and by

Koblinger's method45 above 1.5 MeV as analyzed and recommended by Blomquist and

Gelbard.46 The outgoing energy and angle µ are rejected according to the form factors.

For next event estimators such as detectors and DXTRAN, the probability density for scattering

toward the detector point must be calculated:

′′

σIZαµ,,()dµIZv()Kαµ,()dµ=

IZ0,()0= IZ∞,()Z=

vθ2⁄()λ⁄sin κα 1µ– where κ10 8– moch2()⁄29.1445cm 1–

== = =

ax kα2 41.2166

Figure 2-4.

I1v,()11

---f2v2





--------------------------------–=

f2⁄96.9014=

2-60 April 10, 2000

CHAPTER 2

PHYSICS

where and σ1(Z, α) and are looked up in the data library.

b. Coherent (Thomson) scattering: Thomson scattering involves no energy loss, and thus

is the only photon process that cannot produce electrons for further transport and that cannot use

the TTB approximation. Only the scattering angle θis computed, and then the transport of the

photon continues.

The differential cross section is σ2(Z, α,µ)dµ=C

2(Z, v)T(µ)dµ, where C(Z, v) is a form factor

modifying the energy-independent Thomson cross section .

The general effect of C2(Z, v)/Z2 is to decrease the Thomson cross section more extremely for

backward scattering, for high E, and low Z. This effect is opposite in these respects to the effect

of I(Z,v)/Z on K(α,µ)in incoherent (Compton) scattering. For a given Z, C(Z,v) decreases from

to . For example, C(Z, v) is a rapidly decreasing function of µas µ

varies from +1to−1, and therefore the coherent cross section is peaked in the forward direction.

At high energies of the incoming photon, coherent scattering is strongly forward and can be

ignored. The parameter v is the inverse length where

. The maximum value of v is

at µ = −1. The square of the maximum value is

. The qualitative features of C(Z,v) are shown in Fig. 2-5.

For next event estimators, one must evaluate the probability density function

for given µ. Here σ2(Z,α) is the integrated coherent

cross section. The value of at must be interpolated in the original

C2(Z,vi) tables separately stored on the cross-section library for this purpose.

pµ() 1

σ1Zα,()

--------------------- IZv,()Kαµ,() πro

σ1Zα,()

--------------------- IZv,()

α′

-----





2α

α′

----- α′

----- µ21–++





πro

22494351= IZv,()

Tµ() πr0

21µ2

+()dµ=

CZ0,()Z=CZ∞,()0=

υθ2⁄()λ⁄sin κα 1µ–==

κ10 8– moch2()⁄29.1445cm 1–

υmax κα 2 41.2166α==

υmax

21698.8038α2

Figure 2-5.

pµ() πr0

21µ2

+()C2Zv,()σ

2Zα,()⁄=C2Zv,()vκα 1µ–=

April 10, 2000 2-61

CHAPTER 2

PHYSICS

Note that at high energies, coherent scattering is virtually straight ahead with no energy loss;

thus, it appears from a transport viewpoint that no scattering took place. For a point detector to

sample this scattering, the point must lie on the original track , which is seldom the case.

Thus, photon point detector variances generally will be much greater with detailed photon

physics than with simple physics unless coherent scattering is turned off with NOCOH = 1 on

the PHYS:P card, as explained on page 2–62.

c. Photoelectric effect: The photoelectric effect consists of the absorption of the incident

photon of energy E, with the consequent emission of several ﬂuorescent photons and the ejection

(or excitation) of an orbital electron of binding energy e<E, giving the electron a kinetic energy

of E−e. Zero, one, or two ﬂuorescent photons are emitted. These three cases are now described.

(1) Zero photons greater than 1 keV are emitted. In this event, the cascade of

electrons that ﬁlls up the orbital vacancy left by the photoelectric ejection produces electrons and

low-energy photons (Auger effect). These particles can be followed in Mode P E problems, or

be treated with the TTB approximation, or be assumed to deposit energy locally. Because no

photons are emitted by ﬂuorescence (some may be produced by electron transport or the TTB

model), the photon track is terminated. This photoelectric “capture” of the photon is scored like

analog capture in the summary table of the output ﬁle. Implicit capture is not possible.

(2) One ﬂuorescent photon of energy greater than 1 keV is emitted. The photon

energy is the difference in incident photon energy E, less the ejected electron kinetic energy

E−e, less a residual excitation energy that is ultimately dissipated by further Auger processes.

This dissipation leads to additional electrons or photons of still lower energy. The ejected

electron and any Auger electrons can be transported or treated with the TTB approximation. In

general,

These primary transactions are taken to have the full ﬂuorescent yield from all possible upper

levels , but are apportioned among the x−ray lines Kα1,,

(mean ); and , (mean ).

(3) Two ﬂuorescence photons can occur if the residual excitation of process (2)

exceeds 1 keV. An electron of binding energy can ﬁll the orbit of binding energy , emitting

a second ﬂuorescent photon of energy . As before, the residual excitation is

dissipated by further Auger events and electron production that can be modeled with electron

transport in Mode P E calculations, approximated with the TTB model, or assumed to deposit

all energy locally. These secondary transitions come from all upper shells and go to Lshells.

Thus the primary transitions must be Kα1 or Kα2 to leave an L shell vacancy.

Each ﬂuorescent photon born as discussed above is assumed to be emitted isotropically and is

transported, provided that , . The binding energies e, , and are very nearly

µ1≅()

E′e′

E′EEe–()–e′–ee′–==

e′L3K→()Kα2L2K→()Kβ′1

;,;

→kβ2

′NK→

e′

e′′ e′

E′′ e′e′′–= e′′

E′

E′′ 1keV>e′e′′

2-62 April 10, 2000

CHAPTER 2

PHYSICS

the x−ray absorption edges because the x−ray absorption cross section takes an abrupt jump as

it becomes energetically possible to eject (or excite) the electron of energy ﬁrst , then ,

then e, etc. The jump can be as much as a factor of 20 (for example, K-carbon).

A photoelectric event is terminal for elements Z<12 because the possible ﬂuorescence energy

is below 1 keV. The event is only a single ﬂuorescence of energy above 1 keV for ,

but double ﬂuorescence (each above 1 keV) is possible for . For , primary lines

Kα1,Kα2, and are possible and, in addition, for , the line is possible.

In all photoelectric cases where the photon track is terminated because either no ﬂuorescent

photons are emitted or the ones emitted are below the energy cutoff, the termination is

considered to be caused by analog capture in the output ﬁle summary table (and not energy

cutoff).

d. Pair Production: This process is considered only in the ﬁeld of a nucleus. The

threshold is MeV, where Mis the nuclear mass and m is the mass

of the electron. There are three cases:

(1) In the case of electron transport (Mode P E), the electron and positron are created

and banked and the photon track terminates.

(2) For Mode P problems with the TTB approximation, both an electron and positron

are produced but not transported. Both particles can make TTB approximation photons. If the

positron is below the electron energy cutoff, then it is not created and a photon pair is created as

in case (3).

(3) For Mode P problems when positrons are not created by the TTB approximation,

the incident photon of energy E vanishes. The kinetic energy of the created positron/electron

pair, assumed to be E− 2mc2, is deposited locally at the collision point. The positron is

considered to be annihilated with an electron at the point of collision, resulting in a pair of

photons, each with the incoming photon weight, and each with an energy of mc2=0.511 MeV.

The ﬁrst photon is emitted isotropically, and the second is emitted in the opposite direction. The

very rare single-annihilation photon of 1.022 MeV is omitted.

e. Caution for detectors and coherent scattering: The use of the detailed photon physics

treatment is not recommended for photon next event estimators (such as point detectors and ring

detectors) nor for DXTRAN, unless coherent scatter is turned off with the NOCOH = 1 option

on the PHYS:P card. Alternatively, the simple physics treatment (EMCPF < .001 on the

PHYS:P card) can be used. Turning off coherent scattering can improve the ﬁgure of merit (see

page 2–108) by more than a factor of 10 for tallies with small relative errors because coherent

scattering is highly peaked in the forward direction. Consequently, coherent scattering becomes

undersampled because the photon must be traveling directly at the detector point and undergo a

Ee′′≅e′

31 Z12≥>

Z31≥Z31≥

Kβ′

1Z37≥Kβ2

‘

2mc

1mM⁄()+[]1.022≅

April 10, 2000 2-63

CHAPTER 2

PHYSICS

coherent scattering event. When the photon is traveling nearly in the direction of the point

detector or the chosen point on a ring detector or DXTRAN sphere, the PSC term, p(µ), of the

point detector (see page 2–85) becomes very large, causing a huge score for the event and

severely affecting the tally. Remember that p(µ) is not a probability (that can be no larger than

unity); it is a probability density function (the derivative of the probability) and can approach

inﬁnity for highly forward-peaked scattering. Thus the undersampled coherent scattering event

is characterized by many low scores to the detector when the photon trajectory is away from the

detector (p(µ)= small) and a very few very large scores (p(µ)= huge) when the trajectory is

nearly aimed at the detector. Such undersampled events cause a sudden increase in both the tally

and the variance, a sudden drop in the ﬁgure of merit, and a failure to pass the statistical checks

for the tally as described on page 2–121.

E. Electron Interactions

The transport of electrons and other charged particles is fundamentally different from that of

neutrons and photons. The interaction of neutral particles is characterized by relatively

infrequent isolated collisions, with simple free ﬂight between collisions. By contrast, the

transport of electrons is dominated by the long-range Coulomb force, resulting in large numbers

of small interactions. As an example, a neutron in aluminum slowing down from 0.5 MeV to

0.0625 MeV will have about 30 collisions, while a photon in the same circumstances will

experience fewer than ten. An electron accomplishing the same energy loss will undergo about

105 individual interactions. This great increase in computational complexity makes a single-

collision Monte Carlo approach to electron transport unfeasible for most situations of practical

interest.

Considerable theoretical work has been done to develop a variety of analytic and semi-analytic

multiple-scattering theories for the transport of charged particles. These theories attempt to use

the fundamental cross sections and the statistical nature of the transport process to predict

probability distributions for signiﬁcant quantities, such as energy loss and angular deﬂection.

The most important of these theories for the algorithms in MCNP are the Goudsmit-

Saunderson48 theory for angular deﬂections, the Landau49 theory of energy-loss ﬂuctuations,

and the Blunck-Leisegang50 enhancements of the Landau theory. These theories rely on a variety

of approximations that restrict their applicability, so that they cannot solve the entire transport

problem. In particular, it is assumed that the energy loss is small compared to the kinetic energy

of the electron.

In order to follow an electron through a signiﬁcant energy loss, it is necessary to break the

electron's path into many steps. These steps are chosen to be long enough to encompass many

collisions (so that multiple-scattering theories are valid) but short enough that the mean energy

loss in any one step is small (so that the approximations necessary for the multiple-scattering

theories are satisﬁed). The energy loss and angular deﬂection of the electron during each of the

steps can then be sampled from probability distributions based on the appropriate multiple-

2-64 April 10, 2000

CHAPTER 2

PHYSICS

scattering theories. This subsumption of the effects of many individual collisions into single

steps that are sampled probabilistically constitutes the “condensed history” Monte Carlo

method.

The most inﬂuential reference for the condensed history method is the 1963 paper by Martin J.

Berger.51 Based on the techniques described in that work, Berger and Stephen M. Seltzer

developed the ETRAN series of electron/photon transport codes.52 These codes have been

maintained and enhanced for many years at the National Bureau of Standards (now the National

Institute of Standards and Technology). The ETRAN codes are also the basis for the Integrated

TIGER Series,53 a system of general-purpose, application-oriented electron/photon transport

codes developed and maintained by John A. Halbleib and his collaborators at Sandia National

Laboratories in Albuquerque, New Mexico. The electron physics in MCNP is essentially that of

the Integrated TIGER Series.

1. Electron Steps and Substeps

The condensed random walk for electrons can be considered in terms of a sequence of sets of

values

(0,E0,t0,u0,r0), (s1,E1,t1,u1,r1), (s2,E2,t2,u2,r2), ...

where sn,En,tn,un, and rn are the total path length, energy, time, direction, and position of the

electron at the end of n steps. On the average, the energy and path length are related by

, (2.3)

where −dE/ds is the total stopping power in energy per unit length. This quantity depends on

energy and on the material in which the electron is moving. ETRAN-based codes customarily

choose the sequence of path lengths {sn} such that

, (2.4)

for a constant k. The most commonly used value is k=2−1/8, which results in an average energy

loss per step of 8.3%.

Electron steps with (energy-dependent) path lengths s=s

n−sn-1 determined by Eqs. 2.3-2.4 are

called major steps or energy steps. The condensed random walk for electrons is structured in

terms of these energy steps. For example, all precalculated and tabulated data for electrons are

stored on an energy grid whose consecutive energy values obey the ratio in Eq. 2.4. In addition,

the Landau and Blunck-Leisegang theories for energy straggling are applied once per energy

En1– En

–sn1–

∫

–= dE

-------ds

En1–

-------------k=

April 10, 2000 2-65

CHAPTER 2

PHYSICS

step. For a single step, the angular scattering could also be calculated with satisfactory accuracy,

since the Goudsmit-Saunderson theory is valid for arbitrary angular deﬂections. However, the

representation of the electron's trajectory as the result of many small steps will be more accurate

if the angular deﬂections are also required to be small. Therefore, the ETRAN codes and MCNP

further break the electron steps into smaller substeps. A major step of path length s is divided

into m substeps, each of path length s/m. Angular deﬂections and the production of secondary

particles are sampled at the level of these substeps. The integer m depends only on material

(average atomic number Z). Appropriate values for m have been determined empirically, and

range from m= 2 for Z< 6 to m= 15 for Z> 91.

In some circumstances, it may be desirable to increase the value of mfor a given material. In

particular, a very small material region may not accommodate enough substeps for an accurate

simulation of the electron's trajectory. In such cases, the user can increase the value of m with

the ESTEP option on the material card. The user can gain some insight into the selection of m

by consulting Print Table 85 in the MCNP output. Among other information, this table presents

a quantity called DRANGE as a function of energy. DRANGE is the size of an energy step in

g/cm2. Therefore, DRANGE/mis the size of a substep in the same units, and if ρis the material

density in g/cm3, then DRANGE/(mρ) is the length of a substep in cm. This quantity can be

compared with the smallest dimension of a material region. A reasonable rule of thumb is that

an electron should make at least ten substeps in any material of importance to the transport

problem.

2. Condensed Random Walk

In the initiation phase of a transport calculation involving electrons, all relevant data are either

precalculated or read from the electron data ﬁle and processed. These data include the electron

energy grid, stopping powers, electron ranges, energy step ranges, substep lengths, and

probability distributions for angular deﬂections and the production of secondary particles.

Although the energy grid and electron steps are selected according to Eqs. 2.3-2.4, energy

straggling, the analog production of bremsstrahlung, and the intervention of geometric

boundaries and the problem time cutoff will cause the electron’s energy to depart from a simple

sequence sn satisfying Eq. 2.4. Therefore, the necessary parameters for sampling the random

walk will be interpolated from the points on the energy grid.

At the beginning of each major step, the collisional energy loss rate is sampled. In the absence

of energy straggling, this will be a simple average value based on the nonradiative stopping

power described in the next section. In general, however, ﬂuctuations in the energy loss rate will

occur. The number of substeps m per energy step will have been preset, either from the

empirically-determined default values, or by the user, based on geometric considerations. At

most msubsteps will be taken in the current major step, i. e., with the current value for the energy

loss rate. The number of substeps may be reduced if the electron's energy falls below the

boundary of the current major step, or if the electron reaches a geometric boundary. In these

2-66 April 10, 2000

CHAPTER 2

PHYSICS

circumstances, or upon the completion of msubsteps, a new major step is begun, and the energy

loss rate is resampled.

Except for the energy loss and straggling calculation, the detailed simulation of the electron

history takes place in the sampling of the substeps. The Goudsmit-Saunderson48 theory is used

to sample from the distribution of angular deﬂections, so that the direction of the electron can

change at the end of each substep. Based on the current energy loss rate and the substep length,

the projected energy for the electron at the end of the substep is calculated. Finally, appropriate

probability distributions are sampled for the production of secondary particles. These include

electron-induced ﬂuorescent X−rays, “knock-on” electrons (from electron-impact ionization),

and bremsstrahlung photons.

Note that the length of the substep ultimately derives from the total stopping power used in

Eq. 2.3, but the projected energy loss for the substep is based on the nonradiative stopping

power. The reason for this difference is that the sampling of bremsstrahlung photons is treated

as an essentially analog process. When a bremsstrahlung photon is generated during a substep,

the photon energy is subtracted from the projected electron energy at the end of the substep.

Thus the radiative energy loss is explicitly taken into account, in contrast to the collisional

(nonradiative) energy loss, which is treated probabilistically and is not correlated with the

energetics of the substep. Two biasing techniques are available to modify the sampling of

bremsstrahlung photons for subsequent transport. However, these biasing methods do not alter

the linkage between the analog bremsstrahlung energy and the energetics of the substep.

MCNP uses identical physics for the transport of electrons and positrons, but distinguishes

between them for tallying purposes, and for terminal processing. Electron and positron tracks

are subject to the usual collection of terminal conditions, including escape (entering a region of

zero importance), loss to time cutoff, loss to a variety of variance-reduction processes, and loss

to energy cutoff. The case of energy cutoff requires special processing for positrons, which will

annihilate at rest to produce two photons, each with energy m c2= 0.511008 MeV.

3. Stopping Power

3a. Collisional Stopping Power

Berger51 gives the restricted electron collisional stopping power, i. e., the energy loss per unit

path length to collisions resulting in fractional energy transfers εless than an arbitrary maximum

value εm, in the form

, (2.5)

-------





εm

–

NZC E2τ2+()

2I2

----------------------- f–τε

,()δ–+ln









April 10, 2000 2-67

CHAPTER 2

PHYSICS

where

(2.6)

Here ε and εm represent energy transfers as fractions of the electron kinetic energy E;I is the

mean ionization potential in the same units as E;βis v/c;τis the electron kinetic energy in units

of the electron rest mass; δ is the density effect correction (related to the polarization of the

medium); Zis the average atomic number of the medium; Nis the atom density of the medium

in cm−3; and the coefﬁcient C is given by

, (2.7)

where m,e, and vare the rest mass, charge, and speed of the electron, respectively. The density

effect correction δis calculated using the prescriptions of Sternheimer, Berger and Seltzer54 in

the el03 evaluation and using the method of Sternheimer and Peierls55 for the el1 evaluation.

The ETRAN codes and MCNP do not make use of restricted stopping powers, but rather treat

all collisional events in an uncorrelated, probabilistic way. Thus, only the total energy loss to

collisions is needed, and Eqs. 2.5-2-6 can be evaluated for the special value εm=1/2. The rea-

son for the 1/2 is the indistinguishability of the two outgoing electrons. The electron with the

larger energy is, by deﬁnition, the primary. Therefore, only the range ε< 1/2 is of interest. With

εm=1/2, Eq. 2.6 becomes

. (2.8)

On the right side of Eq. 2.5, we can express both Eand Iin units of the electron rest mass. Then

E can be replaced by τon the right side of the equation. We also introduce supplementary

constants

(2.9)

f–τε

,() 1– β2

–τ

τ1+

------------





2ε2

-------- 2τ1+

τ1+()

-------------------1εm

–()ln++=

4εm1εm

–()[]

1εm

–

---------------+ln

C2πe4

mv2

------------=

f–τε

,() β

–12ln–()

---2ln+





τ1+

------------





++=

C22I2

(),ln=

C31 2,ln–=

C41

---2,ln+=

2-68 April 10, 2000

CHAPTER 2

PHYSICS

so that Eq. 2.5 becomes

(2.10)

This is the collisional energy loss rate in MeV/cm in a particular medium. In MCNP, we are

actually interested in the energy loss rate in units of MeV barns (so that different cells containing

the same material need not have the same density). Therefore, we divide Eq. 2.10 by N and

multiply by the conversion factor 1024 barns/cm2. We also use the deﬁnition of the ﬁne structure

constant

where his Planck's constant, to eliminate the electronic charge efrom Eq. 2.10. The result is as

follows:

(2.11)

This is the form actually used in MCNP to preset the collisional stopping powers at the energy

boundaries of the major energy steps.

The mean ionization potential and density effect correction depend upon the state of the

material, either gas or solid. In the ﬁt of Sternheimer and Peierls55 the physical state of the

material also modiﬁes the density effect calculation. In the Sternheimer, Berger and Seltzer54

treatment, the calculation of the density effect uses the conduction state of the material to

determine the contribution of the outermost conduction electron to the ionization potential. The

occupation numbers and atomic binding energies used in the calculation are from Carlson.56

3b. Radiative Stopping Power

The radiative stopping power is

where is the scaled electron-nucleus radiative energy-loss cross section based upon

evaluations by Berger and Seltzer for either el1 and el03 (details of the numerical values of the

-------





–NZ2πe4

mv2

------------τ2τ2+()[]C2– C3β2

–C4τ

τ1+

------------





2δ–++ln







α2πe2

------------=

-------





–1024α2h2c2

2πmc2

---------------------------- Zτ2τ2+()[]C2– C3β2

–C4τ

τ1+

------------





2δ–++ln







β2

-----

-------

rad

–10

24ZZ η+()αre

()Tmc

+()Φ

rad

n()

Φrad

n()

April 10, 2000 2-69

CHAPTER 2

PHYSICS

el03 evaluation can be found in Ref. 57, Ref. 58, and Ref. 59; is a parameter to account for

the effect of electron-electron bremsstrahlung (it is unity in the el1 evaluation and, in the el03

evaluation, it is based upon the work of S. Seltzer and M. Berger57,58,59 and can be different from

unity); α is the ﬁne structure constant, mc2 is the mass energy of an electron, and re is the

classical electron radius. The dimensions of the radiative stopping power are the same as the

collisional stopping power.

4. Energy Straggling

Because an energy step represents the cumulative effect of many individual random collisions,

ﬂuctuations in the energy loss rate will occur. Thus the energy loss will not be a simple average

; rather there will be a probability distribution f(s,∆)d∆from which the energy loss ∆for the

step of length s can be sampled. Landau49 studied this situation under the simplifying

assumptions that the mean energy loss for a step is small compared with the electron’s energy,

that the energy parameter ξdeﬁned below is large compared with the mean excitation energy of

the medium, that the energy loss can be adequately computed from the Rutherford60 cross

section, and that the formal upper limit of energy loss can be extended to inﬁnity. With these

simpliﬁcations, Landau found that the energy loss distribution can be expressed as

in terms of , a universal function of a single scaled variable

Here mand vare the mass and speed of the electron, δis the density effect correction, βis v/c,I

is the mean excitation energy of the medium, and γis Euler’s constant . The

parameter ξ is deﬁned by

where eis the charge of the electron and NZis the number density of atomic electrons, and the

universal function is

where x is a positive real number specifying the line of integration.

∆

fs∆,()d∆φλ()dλ=

φλ()

λ∆

--- 2ξmv2

1β2

–()I2

------------------------ δβ

21– γ⋅++ +ln–=

γ0.5772157…=()

ξ2πe4NZ

mv2

-------------------- s,=

φλ() 1

2πi

-------- eµµλµ+ln µ,d

xi∞–

xi∞+

∫

2-70 April 10, 2000

CHAPTER 2

PHYSICS

For purposes of sampling, is negligible for , so that this range is ignored. B rsch -

Supan61 originally tabulated in the range , and derived for the range

the asymptotic form

in terms of the auxiliary variable w, where

Recent extensions62 of B rsch-Supan's tabulation have provided a representation of the function

in the range in the form of ﬁve thousand equally probable bins in λ. In MCNP, the

boundaries of these bins are saved in the array eqlm(mlam), where mlam =5001. Sampling from

this tabular distribution accounts for approximately 98.96% of the cumulative probability for

. For the remaining large-λ tail of the distribution, MCNP uses the approximate form

, which is easier to sample than (w2 + π2)−1, but is still quite accurate for λ> 100.

Blunck and Leisegang50 have extended Landau’s result to include the second moment of the

expansion of the cross section. Their result can be expressed as a convolution of Landau's

distribution with a Gaussian distribution:

Blunck and Westphal63 provided a simple form for the variance of the Gaussian:

Subsequently, Chechin and Ermilova64 investigated the Landau/Blunck-Leisegang theory, and

derived an estimate for the relative error

caused by the neglect of higher-order moments. Based on this work, Seltzer65 describes and

recommends a correction to the Blunck-Westphal variance:

φλ() λ 4–<o

˙˙

φλ() 4λ100≤≤–λ100>

φλ() 1

w2π2

------------------ ,≈

λwwγ3

---–+ln+=

˙˙

4λ100≤≤–

φλ()

φλ() w2–

≈

f∗s∆,() 1

2πσ

-------------- fs∆′,() ∆∆′–()

2σ2

----------------------

exp ∆′d

∞–

∞+

∫

σBW

210eV Z43⁄∆⋅=

εCE 10ξ

---------1ξ

10I

--------+





---–

≈

April 10, 2000 2-71

CHAPTER 2

PHYSICS

This value for the variance of the Gaussian is used in MCNP.

Examination of the asymptotic form for shows that unrestricted sampling of λ will not

result in a ﬁnite mean energy loss. Therefore, a material− and energy−dependent cutoff λcis

imposed on the sampling of λ. In the initiation phase of an MCNP calculation, the code makes

use of two preset arrays, flam(mlanc) and avlm(mlanc), with mlanc =1591. The array flam contains

candidate values for λcin the range ; the array avlm contains the corresponding

expected mean values for the sampling of λ. For each material and electron energy, the code uses

the known mean collisional energy loss , interpolating in this tabular function to select a

suitable value for λc, which is then stored in the dynamically-allocated array flc. During the

transport phase of the calculation, the value of flc applicable to the current material and electron

energy is used as an upper limit, and any sampled value of λgreater than the limit is rejected. In

this way, the correct mean energy loss is preserved.

5. Angular Deflections

The ETRAN codes and MCNP rely on the Goudsmit-Saunderson48 theory for the probability

distribution of angular deﬂections. The angular deﬂection of the electron is sampled once per

substep according to the distribution

where sis the length of the substep, is the angular deﬂection from the direction at the

beginning of the substep, Pl(µ) is the lth Legendre polynomial, and Gl is

in terms of the microscopic cross section , and the atom density N of the medium.

For electrons with energies below 0.256 MeV, the microscopic cross section is taken from

numerical tabulations developed from the work of Riley.66 For higher-energy electrons, the

microscopic cross section is approximated as a combination of the Mott67 and Rutherford60

cross sections, with a screening correction. Seltzer52 presents this “factored cross section” in the

form

σσBW

13εCE

---------------------=

φλ()

4λc50000≤≤–

∆

Fsµ,() l1

---+



 sGl

–()Plµ()exp

l0=

∞

∑

µθcos=

Gl2πN1–

∫dσ

dΩ

------- 1Plµ()–[]dµ=

dσdΩ⁄

2-72 April 10, 2000

CHAPTER 2

PHYSICS

where e,p, and v are the charge, momentum, and speed of the electron, respectively. The

screening correction was originally given by Molière68 as

where αis the ﬁne structure constant, mis the rest mass of the electron, and β=v/c. MCNP now

follows the recommendation of Seltzer,52 and the implementation in the Integrated TIGER

Series, by using the slightly modiﬁed form

where τis the electron energy in units of electron rest mass. The multiplicative factor in the ﬁnal

term is an empirical correction which improves the agreement at low energies between the

factored cross section and the more accurate partial-wave cross sections of Riley.

6. Bremsstrahlung

In the el1 evaluation, for the sampling of bremsstrahlung photons, MCNP relies primarily on the

Bethe-Heitler69 Born-approximation results that have been used until rather recently57 in

ETRAN. A comprehensive review of bremsstrahlung formulas and approximations relevant to

the present level of the theory in MCNP can be found in the paper of Koch and Motz.70 Particular

prescriptions appropriate to Monte Carlo calculations have been developed by Berger and

Seltzer.71 For the ETRAN-based codes, this body of data has been converted to tables including

bremsstrahlung production probabilities, photon energy distributions, and photon angular

distributions.

In the el03 evaluation, the production cross section for bremsstrahlung photons and energy

spectra are from the evaluation by Seltzer and Berger.57,58,59 We summarize the salient features

of the evaluation below; more details can be found in the evaluators’ documentation. The

evaluation uses detailed calculations of the electron-nucleus bremsstrahlung cross section for

electrons with energies below 2 MeV and above 50 MeV. The evaluation below 2 MeV uses the

results of Pratt, Tseng, and collaborators, based on numerical phase-shift calculations.72,73,74 For

50 MeV and above, the analytical theory of Davies, Bethe, Maximom, and Olsen75 is used and

is supplemented by the Elwert Coulomb76 correction factor and the theory of the high-frequency

limit or tip region given by Jabbur and Pratt.77 Screening effects are accounted for by the use of

Hartree-Fock atomic form factors.78 The values between these ﬁrmly grounded theoretical limits

dσ

dΩ

------- Z2e2

p2v21µ–2η+()

-------------------------------------------- dσdΩ⁄()

Mott

dσdΩ⁄()

Rutherford

----------------------------------------------

η1

---αmc

0.885p

-----------------





2Z23⁄1.13 3.76 αZβ⁄()

+[]=

η1

---αmc

0.885p

-----------------





2Z23⁄1.13 3.76 αZβ⁄()

τ1+

------------+=

April 10, 2000 2-73

CHAPTER 2

PHYSICS

are found by a cubic-spline interpolation as described in Ref. 57 and Ref. 58. Seltzer reports

good agreement between interpolated values and those calculated by Tseng and Pratt79 for 5 and

10 MeV electrons in aluminum and uranium. Electron-electron bremsstrahlung is also included

in the cross section evaluation based on the theory of Haug80 with screening corrections derived

from Hartree-Fock incoherent scattering factors.78 The energy spectra for the bremsstrahlung

photons are provided in the evaluation. No major changes were made to the tabular angular

distributions, which are internally calculated when using the el1 evaluation, except to make ﬁner

energy bins over which the distribution is calculated.

MCNP addresses the sampling of bremsstrahlung photons at each electron substep. The tables

of production probabilities are used to determine whether a bremsstrahlung photon will be

created. In the el03 evaluation, the bremsstrahlung production is sampled according to a Poisson

distribution along the step so that none, one or more photons could be produced; the el1

evaluation allows for either none or one bremsstrahlung photon in a substep. If a photon is

produced, the new photon energy is sampled from the energy distribution tables. By default, the

angular deﬂection of the photon from the direction of the electron is also sampled from the

tabular data. The direction of the electron is unaffected by the generation of the photon, because

the angular deﬂection of the electron is controlled by the multiple scattering theory. However,

the energy of the electron at the end of the substep is reduced by the energy of the sampled

photon, because the treatment of electron energy loss, with or without straggling, is based only

on nonradiative processes.

There is an alternative to the use of tabular data for the angular distribution of bremsstrahlung

photons. If the fourth entry on the PHYS:E card is 1, then the simple, material-independent

probability distribution

, (2.12)

where and β=v/c, will be used to sample for the angle of the photon relative to the

direction of the electron according to the formula

where ξis a random number. This sampling method is of interest only in the context of detectors

and DXTRAN spheres. A set of source contribution probabilities p(µ) consistent with the tabular

data is not available. Therefore, detector and DXTRAN source contributions are made using

Eq. 2.12. Specifying that the generation of bremsstrahlung photons rely on Eq. 2.12 allows the

user to force the actual transport to be consistent with the source contributions to detectors and

DXTRAN.

pµ()dµ1β2

–

21 βµ–()

-------------------------- dµ=

µθcos=

µ2ξ1– β–

2ξβ 1– β–

----------------------------=

2-74 April 10, 2000

CHAPTER 2

PHYSICS

7. K-shell electron impact ionization and Auger transitions

The el03 evaluation does not change the K-shell impact ionization calculation (based upon

ITS1.0) except for how the emission of relaxation photons is treated; the el03 evaluation model

has been modiﬁed to be consistent with the photo-ionization relaxation model. In the el1

evaluation, a K-shell impact ionization event generated a photon with the average K-shell

energy. The el03 evaluation generates photons with energies given by Everett and Cashwell.33

Both el03 and el1 treatments only take into account the highest Z component of a material. Thus

inclusion of trace high Z impurities could mask K-shell impact ionization from other dominant

components.

Auger transitions are handled the same in the el03 and el1 evaluations. If an atom has undergone

an ionizing transition and can undergo a relaxation, if it does not emit a photon it will emit an

Auger electron. The difference between el1 and el03 is the energy with which an Auger electron

is emitted, given by or for el1 or e03, respectively. The el1

value is that of the highest energy Auger electron while the el03 value is the energy of the most

probable Auger electron. It should be noted that both models are somewhat crude.

8. Knock-On Electrons

The Møller cross section81 for scattering of an electron by an electron is

, (2.13)

where ∈,τ,E, and C have the same meanings as in Eqs. 2.5-2.7. When calculating stopping

powers, one is interested in all possible energy transfers. However, for the sampling of

transportable secondary particles, one wants the probability of energy transfers greater than

some εc representing an energy cutoff, below which secondary particles will not be followed.

This probability can be written

The reason for the upper limit of 1/2 is the same as in the discussion of Eq. 2.8. Explicit

integration of Eq. 2.13 leads to

EAEK

AEK2EL

–=

εd

dσC

----1

ε2

-----1

1ε–()

------------------ τ

τ1+

------------





22τ1+

τ1+()

-------------------1

ε1ε–()

-------------------

–++







σε

() εd

dσεd

εc

12⁄

∫

σε

() C

----1

εc

---- 1

1εc

–

-------------–τ

τ1+

------------





---εc

–





2τ1+

τ1+()

-------------------1εc

–

εc

-------------

ln–+







April 10, 2000 2-75

CHAPTER 2

PHYSICS

Then the normalized probability distribution for the generation of secondary electrons with

ε>ε

cis given by

. (2.14)

At each electron substep, MCNP uses σ(εc) to determine randomly whether knock-on electrons

will be generated. If so, the distribution of Eq. 2.14 is used to sample the energy of each

secondary electron. Once an energy has been sampled, the angle between the primary direction

and the direction of the newly generated secondary particle is determined by momentum

conservation. This angular deﬂection is used for the subsequent transport of the secondary

electron. However, neither the energy nor the direction of the primary electron is altered by the

sampling of the secondary particle. On the average, both the energy loss and the angular

deﬂection of the primary electron have been taken into account by the multiple scattering

theories.

9. Multigroup Boltzmann−Fokker−Planck Electron Transport

The electron physics described above can be implemented into a multigroup form using a hybrid

multigroup/continuous-energy method for solving the Boltzmann−Fokker−Planck equation as

described by Morel.39 The multigroup formalism for performing charged charged particle

transport was pioneered by Morel and Lorence40 for use in deterministic transport codes. With

a ﬁrst order treatment for the continuous slowing down approximation (CSDA) operator, this

formalism is equally applicable to a standard Monte Carlo multigroup transport code as

discussed by Sloan.82 Unfortunately, a ﬁrst order treatment is not adequate for many

applications. Morel, et.al. have addressed this difﬁculty by developing a hybrid multigroup/

continuousenergy algorithm for charged particles that retains the standard multigroup treatment

for large-angle scattering, but treats exactly the CSDA operator. As with standard multigroup

algorithms, adjoint calculations are performed readily with the hybrid scheme.

The process for performing an MCNP/MGBFP calculation for electron/photon transport

problems involves executing three codes. First the CEPXS40 code is used to generate coupled

electron−photon multigroup cross sections. Next the CRSRD code casts these cross sections into

a form suitable for use in MCNP by adjusting the discrete ordinate moments into a Radau

quadrature form that can be used by a Monte Carlo code. CRSRD also generates a set of

multigroup response functions for dose or charge deposition that can be used for response

estimates for a forward calculation or for sources in an adjoint calculation. Finally, MCNP is

executed using these adjusted multigroup cross sections. Some applications of this capability for

electron/photon transport have been presented in Ref. 83.

gεε

,()dε1

σε

()

------------- εd

dσεd=

2-76 April 10, 2000

CHAPTER 2

TALLIES

V. TALLIES

MCNP provides seven standard neutron tallies, six standard photon tallies, and four standard

electron tallies. These basic tallies can be modiﬁed by the user in many ways. All tallies are

normalized to be per starting particle except in KCODE criticality problems.

The above seven tally categories represent the basic MCNP tally types. To have many tallies of

a given type, add multiples of 10 to the tally number. For example, F1, F11, F21...F981, F991

are all type F1 tallies. Particle type is speciﬁed by appending a colon and the particle designator.

For example, F11:N and F96:N are neutron tallies and F2:P and F25:P are photon tallies. F6

tallies can be for both neutrons and photons − F16:N,P. F8 tallies are for both photons and

electrons: F8:P, F8:E, and F8:P,E are all identical. F8:N is also allowed, though not advised,

because MCNP neutron transport does not currently sample joint collision exit densities in an

analog way.

Thought should be given to selecting a tally and to comparing one tally with another. For

example, if the ﬂux is varying as 1/R2in a cell, an average ﬂux in the cell determined by the F4

tally will be higher than the ﬂux at a point in the center of the cell determined by a detector. This

same consideration applies to the average ﬂux provided by DXTRAN spheres (see page 2–150).

Standard summary information that gives the user a better insight into the physics of the problem

and the adequacy of the Monte Carlo simulation includes a complete accounting of the creation

and loss of all tracks and their energy; the number of tracks entering and reentering a cell plus

the track population in the cell; the number of collisions in a cell; the average weight, mean free

path, and energy of tracks in a cell; the activity of each nuclide in a cell; and a complete weight

balance for each cell.

Tally Mnemonic Description

F1:N or F1:P or F1:E Surface current

F2:N or F2:P or F2:E Surface ﬂux

F4:N or F4:P or F4:E Track length estimate of cell ﬂux

F5a:N or F5a:P Flux at a point or ring detector

F6:N or F6:P or F6:N,P Track length estimate of energy deposition

F7:N Track length estimate of ﬁssion energy deposition

F8:N or F8:P or F8:E Pulse height tally

or F8:P,E

April 10, 2000 2-77

CHAPTER 2

TALLIES

The quantities actually scored in MCNP before the ﬁnal normalization per starting particle are

presented in Table 2.1. Note that adding an asterisk (∗Fn) changes the units and multiplies the

tally as indicated in the last column of Table 2.1. For an F8 pulse height tally the asterisk changes

the tally from deposition of pulses to an energy deposition tally. Table 2.1 also deﬁnes much of

the notation used in the remainder of this section.

Extensive statistical analysis of tally convergence also is applied to one bin of each tally. Ten

statistical checks are made, including the variance of the variance and the Pareto slope of the

tally density function. These are described in detail starting on page 2–99.

W=particle weight

Ws=source weight

E= particle energy (MeV)

|µ| = absolute value of cosine of angle between surface normal and particle trajectory.

If |µ| < .1, set |µ| = .05.

A= surface area (cm2)

Tl= track length (cm) = transit time ∗ velocity

V= volume (cm3)

p(µ) = probability density function: µ= cosine of angle between particle trajectory and

detector

λ= total mean free path to detector

R= distance to detector (cm)

σT(E) = microscopic total cross section (barns)

H(E) = heating number (MeV/collision)

ρa= atom density (atoms/barn-cm)

m= cell mass (gm)

TABLE 2.1:

Tally Quantities Scored

Tally Fn

Quantity Fn

Units ∗Fn

Multiplier ∗Fn

Units

F1 W E MeV

F2 W/(|µ| ∗ A) 1/cm2E MeV/cm2

F4 W∗ Tl/V1/cm2E MeV/cm2

F5 W∗ p(µ)∗ exp(−λ)/(2π R2) 1/cm2E MeV/cm2

F6 W ∗ Tl∗ σT(Ε) ∗ Η(Ε) ∗ ρa/m MeV/gm 1.60219E−22 jerks/gm

F7 W∗ Tl∗σ

f(E)*Q ∗ρ

a/mMeV/gm 1.60219E−22 jerks/gm

F8 Ws put in bin E∗ W/Wspulses E MeV

2-78 April 10, 2000

CHAPTER 2

TALLIES

σf(E) = microscopic ﬁssion cross section (barns)

Q= ﬁssion heating Q-value (MeV)

The following MCNP deﬁnitions of current and ﬂux come from reactor theory but are related to

similar quantities in radiative transfer theory. The MCNP particle angular ﬂux multiplied by the

particle energy is the same as the intensity in radiative transfer theory. The MCNP particle total

ﬂux at energy Emultiplied by the particle energy equals the integrated energy density times the

speed of light in radiative transfer theory. The MCNP particle current multiplied by the particle

energy is analogous to the radiative ﬂux crossing an area in radiative transfer theory. The MCNP

particle current uses |µ| in the deﬁnition, whereas the radiative transfer ﬂux uses µ in its

deﬁnition. MCNP current is neither net nor positive nor negative current; it is the number of

particles crossing a surface in a particular direction. The MCNP particle ﬂuence multiplied by

the particle energy is the same as the ﬂuence in radiative transfer theory.

A. Surface Current Tally

The F1 surface current tally estimates the following quantity:

This tally is the number of particles (quantity of energy for ∗F1) crossing a surface. The scalar

current is related to the ﬂux as . The range of integration over

area, energy, time, and angle (A,E,t,µ) can be controlled by FS, E, T, and C cards, respectively.

The FT card can be used to change the vector relative to which µis calculated (FRV option) or

to segregate electron current tallies by charge (ELC option).

B. Flux Tallies

The F2, F4 and F5 ﬂux tallies are estimates of

∗

F1Jr Etµ,,,()

∫

=dE dt dµdA

∗F1ΕJr Etµ,,,()EdtdµAdd

∫

=∗

Jr Etµ,,,()µΦrEt,,()A=

F2ΦrEt,,()td Ad

------

∫

F2E∗ΦrEt,,()EdtdAd

------

∫

April 10, 2000 2-79

CHAPTER 2

TALLIES

∗

The range of integration over energy and time is controlled by E and T cards. The F2 surface ﬂux

and F4 cell ﬂux tallies are discussed below. The F5 detector ﬂux tally, a major topic, is discussed

on page 2–85.

The units of the ﬂux tally are the units of the source. If the source has units of particles per unit

time, the tally is also particles per unit time. When the source has units of particles, this tally

represents a ﬂuence tally. A steady-state ﬂux solution can be obtained by having a source with

units of particles per unit time and integrating over all time (that is, omitting the Tn card). The

ﬂux can be obtained from the ﬂuence tally for a time-dependent source by dividing the tally by

the time bin width. These tallies can all be made per unit energy by dividing by the energy bin

width.

1. Track Length Estimate of Cell Flux (F4)

The deﬁnition of particle ﬂux is , where v= particle velocity and

N=particle density = particle weight/unit volume. Roughly speaking, the time integrated ﬂux is

More precisely, let ds = vdt. Then the time-integrated ﬂux is

Because is a track length density, MCNP estimates this integral by summing

WTl/V for all particle tracks in the cell, time range, and energy range. Because of the track length

term Tlin the numerator, this tally is known as a track length estimate of the ﬂux. It is generally

quite reliable because there are frequently many tracks in a cell (compared to the number of

collisions), leading to many contributions to this tally.

F4ΦrEt,,()EdtdVd

-------

∫

F4E∗ΦrEt,,()EdtdVd

-------

∫

F5ΦrEt,,()Edtd

∫

5EΦrEt,,()Edtd

∫

=∗

ΦrEt,,()vN r E t,,()=

ΦrEt,,()EdtdVd

-------

∫

∫Wv t V⁄WTlV⁄==

ΦrEt,,()tdEdVd

-------





∫

∫NrEt,,()sdEdVd

-------

∫

NrEt,,()ds

2-80 April 10, 2000

CHAPTER 2

TALLIES

2. Surface Flux (F2)

The surface ﬂux is a surface estimator but can be thought of as the limiting case of the cell ﬂux

or track length estimator when the cell becomes inﬁnitely thin as illustrated in Fig. 2-6.

Figure 2-6.

As the cell thickness δapproaches zero, the volume approaches Aδand the track length

approaches δ/|µ|, where , the angle between the surface normal and the particle

trajectory. This deﬁnition of ﬂux also follows directly from the relation between ﬂux and current,

. MCNP sets |µ| = .05 when |µ| < .1. The F2 tally is essential for

stochastic calculation of surface areas when the normal analytic procedure fails.

C. Track Length Cell Energy Deposition Tallies

The F6 and F7 cell heating and energy deposition tallies are the following track length estimates

where

ρa=atom density (atoms/barn-cm)

ρg= gram density (grams/cm3)

H(E) =heating response (summed over nuclides in a material)

The units of the heating tally are MeV/gm. An asterisk (∗F6,7)changes the units to jerks/gm (1

MeV =1.60219E-22 jerks). The asterisk causes the F6,7 tally to be modiﬁed by a constant rather

Θ



δ

F2WTl

δ0→

lim V⁄=

Wδθcos⁄()Aδ()⁄=WAµ()⁄=

µθcos=

JrEtµ,,,()µΦrEt,,()A=

67,ρaρg

⁄HE()ΦrEt,,()EdtdVd

-------

∫

April 10, 2000 2-81

CHAPTER 2

TALLIES

than by energy as in other tallies. Note that the heating tallies are merely ﬂux tallies (F4)

multiplied by an energy-dependent multiplier (FM card).

Energy deposition for photons and electrons can be computed with the ∗F8 tally, which is a

surface estimator rather than a track length estimator. See page 2–83 .

The F7 tally includes the gamma heating because the photons are deposited locally. The F6 tally

deposits the photons elsewhere, so it does not include gamma heating. Thus for ﬁssionable

materials, the F7 result often will be greater than the F6 result even though F7 includes only

ﬁssion and F6 includes all reactions. The true heating is found by summing the neutron and

photon F6 tallies in a coupled neutron/photon calculation. In a neutron-only problem, F6 will

give the right heating of light materials only if all photons escape the geometry. F7 will give

about the right heating of ﬁssionable materials only if no photons come from elsewhere, all

ﬁssion photons are immediately captured, and nonﬁssion reactions can be ignored. The F7 tally

cannot be used for photons. Examples of combining the neutron and photon F6 tallies are F6:N,P

and F516:P,N

The heating response H(E) has different meanings, depending upon context as follows:

1. F6 Neutrons

H(E) =σ

T(E) Havg(E), where the heating number is

and

σT= total neutron cross section,

E= incident neutron energy,

pi(E) = probability of reaction i,

= average exiting neutron energy for reaction i,

Qi= Q-value of reaction i,

= average energy of exiting gammas for reaction i.

2. F6 Photons

H(E) =σ

T(E)Havg(E), where the heating number is

∗

avg E() Ep

iE()EoutiE() QiEγiE()+–[]

∑

–=

Eouti

Eγi

Havg E() piE()

i1=

∑

=EE

out

–()

2-82 April 10, 2000

CHAPTER 2

TALLIES

i= 1 incoherent (Compton) scattering with form factors

i= 2 pair production

i = 3 photoelectric.

All energy transferred to electrons is assumed to be deposited locally.

3. F7 Neutrons

where

σf(E) = total ﬁssion cross section and

Q= ﬁssion Q-value (MeV).

The Q-values as tabulated represent the total prompt energy release per ﬁssion and are printed

in optional PRINT TABLE 98. The total ﬁssion cross section is (n,f) + (n,nf) + .

4. F7 Photons

H(E) is undeﬁned because photoﬁssion is not included in MCNP.

5. Equivalence of F4, F6, and F7 Tallies

The F6 and F7 heating tallies are special cases of the F4 track length estimate of cell ﬂux with

energy-dependent multipliers. The following F4 and FM4 combinations give exactly the same

results as the F6 and F7 tallies. In this example, material 9 in cell 1 is 235U with an atom density

(ρa) of .02 atoms/barn-cm and a gram density (ρg) of 7.80612 g/cm3 for an atom/gram ratio of

.0025621.

F4:N 1

FM4 .0025621 9 1 4 gives the same result as F6:N 1

F14:N 1

FM14 .0025621 9 −6 8 gives the same result as F17:N 1

F24:P 1

FM24 .0025621 9 5 6 gives the same result as F26:P 1

For the photon results to be identical, both electron transport and the thick target bremsstrahlung

approximation must be turned off by PHYS:P j 1. In the F6 tally, if a photon produces an electron

that produces a photon, the second photon is not counted again. It is already tallied in the ﬁrst

photon heating. In the F4 tally, the second photon track is counted, so the F4 tally will slightly

overpredict the tally.

Eout 1.022016 2moc2

HE() σ

fE()Q=

…

April 10, 2000 2-83

CHAPTER 2

TALLIES

The photon heating tally also can be checked against the ∗F8 energy deposition tally (divided by

cell mass to give answers in MeV per gram). Results will not be identical because the tallies are

totally independent and use different estimators.

The FM card can be used to make the surface ﬂux tally (F2) and point and ring detector tallies

(F5) calculate heating as well.

D. Pulse Height Tallies

The pulse height tally provides the energy distribution of pulses created in a cell that models a

physical detector. It also can provide the energy deposition in a cell. Although the entries on the

F8 card are cells, this is not a track length cell tally. F8 tallies are made at source points and at

surface crossings.

The pulse height tally is analogous to a physical detector. The F8 energy bins correspond to the

total energy deposited in a detector in the speciﬁed channels by each physical particle. All the

other MCNP tallies record the energy of a scoring track in the energy bin.

In an experimental conﬁguration, suppose a source emits 100 photons at 10 MeV, and ten of

these get to the detector cell. Further, suppose that the ﬁrst photon (and any of its progeny created

in the cell) deposits 1 keV in the detector before escaping, the second deposits 2 keV, and so on

up to the tenth photon which deposits 10 keV. Then the pulse height measurement at the detector

would be one pulse in the 1 keV energy bin, 1 pulse in the 2 keV energy bin, and so on up to 1

pulse in the 10 keV bin.

In the analogous MCNP pulse height tally, the source cell is credited with the energy times the

weight of the source particle. When a particle crosses a surface, the energy times the weight of

the particle is subtracted from the account of the cell that it is leaving and is added to the account

of the cell that it is entering. The energy is the kinetic energy of the particle plus

2moc2=1.022016 if the particle is a positron. At the end of the history, the account in each tally

cell is divided by the source weight. The resulting energy determines which energy bin the score

is put in. The value of the score is the source weight for an F8 tally and the source weight times

the energy in the account for a ∗F8 tally. The value of the score is zero if no track entered the

cell during the history.

The pulse height tally is an inherently analog process. Therefore, it does not work well with

neutrons, which are inherently non analog, and it does not work at all with most variance

reduction schemes. The pulse height tally depends on sampling the joint density of all particles

exiting a collision event. MCNP does not currently sample this joint density for neutron

collisions. Thus neutron F8 tallies must be done with extreme caution when more than one

neutron can exit a collision. Suppose in the above example, the photon that deposited 10 keV in

the detector cell underwent a 2−for−1 split. Then if only one of the split halves entered the cell,

2-84 April 10, 2000

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the tally would be incorrectly put in the 5 keV bin rather than the 10 keV bin. Or if the particle

survived a Russian roulette event, its weight would be double and the score would be put into

the 20 keV bin. Similar scenarios can be given for other variance reduction methods. The MCNP

pulse height tally will not work with any variance reduction other than source biasing. It doesn't

work well with neutrons even without variance reduction because the MCNP neutron physics is

nonanalog (in the joint density sampling), particularly in the way that multiple neutrons exiting

a collision are totally uncorrelated and don't even conserve energy except in an average sense

over many neutron histories.

Another aspect of the pulse height tally that is different from other MCNP tallies is that F8:P,

F8:E and F8:P,E are all equivalent. All the energy from both photons and electrons, if present,

will be deposited in the cell, no matter which tally is speciﬁed.

When the pulse height tally is used with energy bins, care must be taken because of negative

scores from nonanalog processes and zero scores caused by particles passing through the pulse

height cell without depositing energy. In some codes, like the Integrated Tiger Series, these

events cause large contributions to the lowest energy bin pulse height score. In other codes no

contribution is made. MCNP compromises by counting these events in a zero bin and an epsilon

bin so that these scores can be segregated out. It is recommended that your energy binning for

an F8 tally be something like

E8 0 1 E -5 E1 E2 E3 E4 E5

Knock-on electrons in MCNP are nonanalog in that the energy loss is included in the multiple

scattering energy loss rate rather than subtracted out at each knock−on event. Thus knock−ons

can cause negative energy pulse height scores. These scores will be caught in the 0 energy bin.

If they are a large fraction of the total F8 tally, then the tally is invalid because of nonanalog

events. Another situation is differentiating zero contributions from particles not entering the cell

and particles entering the cell but not depositing any energy. These are differentiated in MCNP

by causing an arbitrary 1.E-12 energy loss for particles just passing through the cell. These will

appear in the 0-epsilon bin.

When the ∗F8 energy deposition tally is used and no energy bins are speciﬁed, variance

reduction of all kinds is allowed. The analog requirement to put a score in the proper energy bin

is removed in this special case of ∗F8 with no energy binning. If the tally had energy bins, the

total energy deposition is correct even though the tallies in the energy bins are wrong. When

Russian roulette is played at a surface bounding a pulse height tally, the variance can become

large because the roulette is played after the energy-times-weight entering the cell is recorded.

Particles terminated by roulette deposit all their energy in the cell. Particles surviving the roulette

have increased weight that can now record more energy-times-weight leaving the cell than

entered. On average, the total energy deposition is correct, but the negative and positive scores

cause an unbounded variance. Therefore, do not play roulette at pulse height cell boundaries.

…

April 10, 2000 2-85

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E. Flux at a Detector

Flux can be estimated at a point with either point or ring detector next-event estimators.

Detectors can yield anomalous statistics and must be used with caution. Detectors also have

special variance reduction features, such as a highly advantageous DD card Russian roulette

game. Whenever a user-supplied source is speciﬁed, a user-supplied source angle probability

density function must be provided also.

1. Point Detector

A point detector is a deterministic estimate (from the current event point) of the ﬂux at a point

in space. Contributions to the point detector tally are made at source and collision events

throughout the random walk. Suppose is the probability of the particle’s scattering

or being born into the solid angle about the direction , where is the azimuthal angle

and µis the cosine of the angle between the incident particle direction and the direction from the

collision point to the detector. If R is the distance to the detector from the collision or source

point, then

yields the probability of scattering into about and arriving at the detector point with

no further collisions. The attenuation of a beam of monoenergetic particles passing through a

material medium is given by where sis measured along the direction from the

collision or source point to the detector and Σt(s) is the macroscopic total cross section at s. If

dA is an element of area normal to the scattered line of ﬂight to the detector, and

therefore

is the expression giving the probability of scattering toward the detector and passing through the

element of area dA normal to the line of ﬂight to the detector. Because the ﬂux is by deﬁnition

the number of particles passing through a unit area normal to the scattered direction, the general

expression for the contribution to the ﬂux is given by

In all the MCNP scattering distributions and in the standard sources, we assume azimuthal

symmetry. Therefore,

pµϕ,()dΩ

dΩµϕ,() ϕ

pµϕ,()dΩe

Σts()sd

∫

–

⋅

dΩµϕ,()

Σts()s

∫

–[]exp

dΩdA R2

⁄=

pµϕ,()

-------e

Σts()sd

∫

–

pµϕ,()

------------------ eΣts()sd

∫

–

2-86 April 10, 2000

CHAPTER 2

TALLIES

and is sampled uniformly on (0,2π). That is, .

If is substituted in the expression for the ﬂux, the expression used in

MCNP is arrived at:

when

W=particle weight;

λ= total number of mean free paths integrated over the trajectory

from the source or collision point to the detector;

R = distance from source or collision event to detector; and

p(µ)= value of probability density function at µ, the cosine of the angle between the

particle trajectory and the direction to the detector.

A point detector is known as a “next-event estimator” because it is a tally of the ﬂux at a point

if the next event is a trajectory without further collision directly to the point detector.

A contribution to the point detector is made at every source or collision event. The e−λ term

accounts for attenuation between the present event and the detector point. The 1/2π R2 term

accounts for the solid angle effect. The p(µ) term accounts for the probability of scattering

toward the detector instead of the direction selected in the random walk. For an isotropic source

or scatter, p(µ)= 0.5 and the solid angle terms reduce to the expected 1/4π R2. (Note that p(µ)

can be larger than unity, because it is the value of a density function and not a probability.) Each

contribution to the detector can be thought of as the transport of a pseudoparticle to the detector.

The R2 term in the denominator of the point detector causes a singularity that makes the

theoretical variance of this estimator inﬁnite. That is, if a source or collision event occurs near

the detector point, Rapproaches zero and the ﬂux approaches inﬁnity. The technique is still valid

and unbiased, but convergence is slower and often impractical. If the detector is not in a source

or scattering medium, a source or collision close to the detector is impossible. For problems

where there are many scattering events near the detector, a cell or surface estimator should be

used instead of a point detector tally. If there are so few scattering events near the detector that

cell and surface tallies are impossible, a point detector can still be used with a speciﬁed average

ﬂux region close to the detector. This region is deﬁned by a ﬁctitious sphere of radius Ro

surrounding the point detector. Rocan be speciﬁed either in centimeters or in mean free paths. If

Rois speciﬁed in centimeters and if R<Ro, the point detector estimation inside Rois assumed to

be the average ﬂux uniformly distributed in volume.

pµ() pµϕ,()ϕd

2π

∫

ϕpµϕ,()pµ()2π⁄=

pµϕ,()pµ()2π⁄=

ΦrEtµ,,,()Wp µ()eλ–2πR2

()⁄=

Σts()sd

∫=

April 10, 2000 2-87

CHAPTER 2

TALLIES

If Σt = 0, the detector is not in a scattering medium, no collision can occur, and

If the ﬁctitious sphere radius is speciﬁed in mean free paths , then = Σt Ro and

The choice of Romay require some experimentation. For a detector in a void region or a region

with very few collisions (such as air), Rocan be set to zero. For a typical problem, setting Roto

a mean free path or some fraction thereof is usually adequate. If Rois in centimeters, it should

correspond to the mean free path for some average energy in the sphere. Be certain when

deﬁning Ro that the sphere it deﬁnes does not encompass more than one material unless you

understand the consequences. This is especially true when deﬁning Roin terms of mean free path

because Ro becomes a function of energy and can vary widely. In particular, if Ro is deﬁned in

terms of mean free paths and if a detector is on a surface that bounds a void on one side and a

material on the other, the contribution to the detector from the direction of the void will be zero

even though the importance of the void is nonzero. The reason is simply that the volume of the

artiﬁcial sphere is inﬁnite in a void. Contributions to the detector from the other direction (that

is, across the material) will be accounted for.

Detectors differing only in Ro are coincident detectors (see page 2–94), and there is little cost

incurred by experimenting with several detectors that differ only by Roin a single problem.

ΦRR

<()

ΦVd

∫

---------------=

Wp µ() eΣtr–()

4πr2rd

∫4

---πRo

----------------------------------------

Wp µ()1eΣtRo

–

–()

---πRo

3Σt

---------------------------------------------=

ΦRR

oΣt0=,<()

Wp µ()Ro

---πRo

-----------------------=

λ0λ0

Φλ λ

<()

Wp µ()1eλ0

–

–()Σ

---πλ0

-----------------------------------------------=

2-88 April 10, 2000

CHAPTER 2

TALLIES

2. Ring Detector

A ring detector84 tally is a point detector tally in which the point detector location is not ﬁxed

but rather sampled from some location on a ring. Most of the previous section on point detectors

applies to ring detectors as well. In MCNP three ring detector tallies FX, FY, and FZ correspond

to rings located rotationally symmetric about the x,y, and z coordinate axes. A ring detector

usually enhances the efﬁciency of point detectors for problems that are rotationally symmetric

about a coordinate axis. Ring detectors also can be used for problems where the user is interested

in the average ﬂux at a point on a ring about a coordinate axis.

Although the ring detector is based on the point detector that has a 1/R2 singularity and an

unbounded variance, the ring detector has a ﬁnite variance and only a 1/Rmin singularity, where

Rmin is the minimum distance between the contributing point and the detector ring.85

In a cylindrically symmetric system, the ﬂux is constant on a ring about the axis of symmetry.

Hence, one can sample uniformly for positions on the ring to determine the ﬂux at any point on

the ring. The ring detector efﬁciency is improved by biasing the selection of point detector

locations to favor those near the contributing collision or source point. This bias results in the

same total number of detector contributions, but the large contributions are sampled more

frequently, reducing the relative error.

For isotropic scattering in the lab system, experience has shown that a good biasing function is

proportional to e−PR−2, where Pis the number of mean free paths and Ris the distance from the

collision point to the detector point. For most practical applications, using a biasing function

involving Ppresents prohibitive computational complexity except for homogeneous medium

problems. For air transport problems, a biasing function resembling e−Phas been used with good

results. A biasing function was desired that would be applicable to problems involving dissimilar

scattering media and would be effective in reducing variance. The function R−2 meets these

requirements.

In Fig. 2-7, consider a collision point, (xo,yo,zo) at a distance R from a point detector location

(x,y,z). The point (x,y,z) is to be selected from points on a ring of radius rthat is symmetric about

the y-axis in this case.

April 10, 2000 2-89

CHAPTER 2

TALLIES

Figure 2-7.

To sample a position (x,y,z) on the ring with a 1/R2 bias, we pick from the density function

, where C is a normalization constant. To pick from , let ξ be a

random number on the unit interval. Then

where

b=−2rxo

c=−2rzo

C = (a2− b2 − c2)1/2.

(x ,y ,z )

ooo

(x,y,z)

pϕ() C2πR2

()⁄=ϕpϕ()

ξC

2π

------ ϕ′d

--------

π–

∫

2π

------ ϕ′d

xorϕ′cos–()

2yoy–()

2zorϕ′sin–()

--------------------------------------------------------------------------------------------------------

π–

∫

2π

------ ϕ′d

ab ϕ′ cϕ′sin+cos+

--------------------------------------------------

π–

∫

---1– 1

----ab–()

--- c+tan







---+tan=

r2xo

2yy

–()

2zo

++ +

2-90 April 10, 2000

CHAPTER 2

TALLIES

The above expression is valid if a2> b2+ c2, which is true except for collisions exactly on the

ring.

Solving for tan ,

Letting ,

then x=r cos

y=y (ﬁxed)

z=r sin .

For ring detectors, the 1/R2biasing has been supplemented when it is weak to include a biasing

based on angle to select the point on the ring. This angle is in the plane of the ring and is relative

to the shortest line from the collision point to the detector ring. The angle that would most likely

be selected would pick the same point on the ring as a straight line through the axis of the

problem, the collision point, and the ring. The angle least likely to be picked would choose the

point on the opposite side of the ring. This approach will thus make scores with smaller

attenuations more often. This supplemental biasing is achieved by requiring that

in the above equation.

If the radius of the ring is very large compared to the dimensions of the scattering media (such

that the detector sees essentially a point source in a vacuum), the ring detector is still more

efﬁcient than a point detector. The reason for this unexpected behavior is that the individual

scores to the ring detector for a speciﬁc history have a mean closer to the true mean than to the

regular point detector contributions. That is, the point detector contributions from one history

will tend to cluster about the wrong mean because the history will not have collisions uniformly

in volume throughout the problem, whereas the ring detector will sample many paths through

the problem geometry to get to different points on the ring.

3. General Considerations of Point Detector Estimators

a. Pseudoparticles and detector reliability: Point and ring detectors are Monte Carlo

methods wherein the simulation of particle transport from one place to another is

deterministically short-circuited. Transport from the source or collision point to the detector is

replaced by a deterministic estimate of the potential contribution to the detector. This transport

between the source or collision point and the detector can be thought of as being via

---

tan 1

ab–

------------Cπξ 1

---–





c–tan







tϕ2⁄tan=

ϕr1t2

–()1t2

+()⁄=

ϕ2rt 1t2

+()⁄=

a32b2c2

+()⁄12⁄

≤

April 10, 2000 2-91

CHAPTER 2

TALLIES

“pseudoparticles.” Pseudoparticles undergo no further collisions. These particles do not reduce

the weight or otherwise affect the random walk of the particles that produced them. They are

merely estimates of a potential contribution. The only resemblance to Monte Carlo particles is

that the quantity they estimate requires an attenuation term that must be summed over the

trajectory from the source or collision to the detector. Thus most of the machinery for

transporting particles can also be used for the pseudoparticles. No records (for example, tracks

entering) are kept about pseudoparticle passage.

Because detectors rely on pseudoparticles rather than particle simulation by random walk, they

should be considered only as a very useful last resort. Detectors are unbiased estimators, but their

use can be tricky, misleading, and occasionally unreliable. Consider the problem illustrated in

Fig. 2-8.

The monoenergetic isotropic point source always will make the same contribution to the point

detector, so the variance of that contribution will be zero. If no particles have yet collided in the

scattering region, the detector tally will be converged to the source contribution, which is wrong

and misleading. But as soon as a particle collides in the scattering region, the detector tally and

its variance will jump. Then the detector tally and variance will steadily decrease until the next

particle collides in the scattering region, at which time there will be another jump.

These jumps in the detector score and variance are characteristic of undersampling important

regions. Next event estimators are prone to undersampling as already described on page 2–62

for the p(µ) term of photon coherent scattering. The jump discussed here is from the sudden

change in the R and possibly λ terms. Jumps in the tally caused by undersampling can be

eliminated only by better sampling of the undersampled scattering region that caused them.

Biasing Monte Carlo particles toward the tally region would cause the scattering region to be

sampled better, thus eliminating the jump problem. It is recommended that detectors be used

with caution and with a complete understanding of the nature of next event estimators. When

detectors are used, the tally ﬂuctuation charts printed in the output ﬁle should be examined

closely to see the degree of the ﬂuctuations. Also the detector diagnostic print tables should be

examined to see if any one pseudoparticle trajectory made an unusually large contribution to the

Monoenergetic

isotropic source Detector

Scattering

region

Figure 2-8.

2-92 April 10, 2000

CHAPTER 2

TALLIES

tally. Detector results should be viewed suspiciously if the relative error is greater than 5%.

Close attention should be paid to the tally statistical analysis and the ten statistical checks

described on page 2–121.

b. Detectors and reﬂecting, white or periodic surfaces: Detectors used with reﬂecting,

white, or periodic surfaces give wrong answers because pseudoparticles travel only in straight

lines. Consider Fig. 2-9, with a point detector and eight source cells. The imaginary cells and

point detector are also shown on the other side of the mirror. The solid line shows the source

contribution from the indicated cell. MCNP does not allow for the dashed-line contribution on

the other side of the reﬂecting surface. The result is that contributions to the detector will always

be from the solid path instead of from a mixture of solid and dashed contributions. This same

situation occurs at every collision. Therefore, the detector tally will be lower (with the same

starting weight) than the correct answer and should not be used with reﬂecting, white, or periodic

surfaces. The effect is even worse for problems with multiple reﬂecting, white or periodic

surfaces.

c. Variance reduction schemes for detectors: Pseudoparticles of point detectors are not

subject to the variance reduction schemes applied to particles of the random walk. They do not

split according to importances, weight windows, etc., although they are terminated by entering

zero importance cells. However, two Russian roulette games are available speciﬁcally for

detector pseudoparticles.

The PD card can be used to specify the pseudoparticle generation probability for each cell. The

entry for each cell i is pi where . Pseudoparticles are created with probability piand

weight 1/pi. If pi= 1, which is the default, every source or collision event produces a

pseudoparticle. If pi=0, no pseudoparticle is produced. Setting pi=0in a cell that can actually

contribute to a detector erroneously biases the detector tally by eliminating such contributions.

Thus pi=0should be used only if the true probability of scoring is zero or if the score from cell

i is unwanted for some legitimate reason such as problem diagnostics. Fractional entries of pi

should be used with caution because the PD card applies equally to all pseudoparticles. The DD

card can be used to Russian roulette just the unimportant pseudoparticles. However, the DD card

Detector

Reflecting plane

Source cells

Figure 2-9.

0pi1≤≤

April 10, 2000 2-93

CHAPTER 2

TALLIES

roulette game often requires particles to travel some distance along their trajectory before being

killed. When cells are many mean free paths from the detector, the PD card may be preferable.

The DD card controls both the detector diagnostic printing and a Russian roulette game played

on pseudoparticles in transit to detectors. The Russian roulette game is governed by the input

parameter kthat controls a comparison weight wc internal to MCNP, such that

wc=−k if k < 0;

wc= 0 if k = 0;

wc= 0 if k > 0 and ;

wc= ifk > 0 and N > 200,

where N=number of histories run so far,

I=number of pseudoparticles started so far,

=Wp(µ)e−λ/(2πR2),

I=contribution of the ith pseudoparticle to the detector tally.

When each pseudoparticle is generated, W,p(µ), and Rare already known before the expensive

tracking process is undertaken to determine λ. If Wp(µ)/(2πR2)< wc, the pseudoparticle

contribution to the detector will be less than the comparison weight. Playing Russian roulette

on all pseudoparticles with < wcavoids the expensive tracking of unimportant

pseudoparticles. Most are never started. Some are started but are rouletted as soon as λ has

increased to the point where Wp(µ)e−λ/(2/πR2)<w

c. Rouletting pseudoparticles whose expected

detector contribution is small also has the added beneﬁt that those pseudoparticles surviving

Russian roulette now have larger weights, so the disparity in particle weights reaching the

detector is reduced. Typically, using the DD card will increase the efﬁciency of detector

problems by a factor of ten. This Russian roulette is so powerful that it is one of two MCNP

variance reduction options that is turned on by default. The default value of k is 0.1. The other

default variance reduction option is implicit capture.

The DD card Russian roulette game is almost foolproof. Performance is relatively insensitive to

the input value of k. For most applications the default value of k = 0.1 is adequate. Usually,

choose kso that there are 1–5 transmissions (pseudoparticle contributions) per source history. If

k is too large, too few pseudoparticles are sampled; thus is a fatal error.

Because a random number is used for the Russian roulette game invoked by k>0, the addition

of a detector tally affects the random walk tracking processes. Detectors are the only tallies that

affect results. If any other tally type is added to a problem, the original problem tallies remain

unchanged. Because detectors use the default DD card Russian roulette game, and that game

affects the random number sequence, the whole problem will track differently and the original

tallies will agree only to within statistics. Because of this tracking difference, it is recommended

N200≤

kN⁄()Σ

Iϕi

ϕi

k1≥

2-94 April 10, 2000

CHAPTER 2

TALLIES

that k<0be used once a good guess at wccan be made. This is especially important if a problem

needs to be debugged by starting at some history past the ﬁrst one. Also, k < 0 makes the ﬁrst

200 histories run faster.

There are two cases when it is beneﬁcial to turn off the DD card Russian roulette game by setting

k=0. First, when looking at the tail of a spectrum or some other low probability event, the DD

card roulette game will preferentially eliminate small scores and thus eliminate the very

phenomenon of interest. For example, if energy bias is used to preferentially produce high

energy particles, these biased particles will have a lower weight and thus preferentially will be

rouletted by the DD card game. Second, in very deep penetration problems, pseudoparticles will

sometimes go a long way before being rouletted. In this rare case it is wasteful to roulette a

pseudoparticle after a great deal of time has been spent following it and perhaps a fractional PD

card should be used or, if possible, a cell or surface tally.

d. Coincident detectors: Because tracking pseudoparticles is very expensive, MCNP uses

a single pseudoparticle for multiple detectors, known as coincident detectors, that must be

identical in: geometric location,

particle type (that is, neutron or photon),

upper time bin limit,

DD card Russian Roulette control parameter, k, and

PD card entries, if any.

Energy bins, time bins, tally multipliers, response functions, ﬁctitious sphere radii, user-supplied

modiﬁcations (TALLYX), etc., can all be different. Coincident detectors require little additional

computational effort because most detector time is spent in tracking a pseudoparticle. Multiple

detectors using the same pseudoparticle are almost “free.”

e. Direct vs. total contribution: Unless speciﬁcally turned off by the user, MCNP

automatically prints out both the direct and total detector contribution. Recall that

pseudoparticles are generated at source and collision events. The direct contribution is that

portion of the tally from pseudoparticles born at source events. The total contribution is the total

tally from both source and collision events. For Mode N P problems with photon detectors, the

direct contribution is from pseudophotons born in neutron collisions. The direct contributions

for detailed photon physics will be smaller than the simple physics direct results because

coherent scattering is included in the detailed physics total cross section and omitted in the

simple physics treatment.

f. Angular distribution functions for point detectors: All detector estimates require

knowledge of the p(µ) term, the value of the probability density function at an angle , where µ

= cos . This quantity is available to MCNP for the standard source and for all kinds of

collisions. For user-supplied source subroutines, MCNP assumes an isotropic distribution

April 10, 2000 2-95

CHAPTER 2

TALLIES

Therefore, the variable PSC=p(µ) = 1/2. If the source distribution is not isotropic in a user-

supplied source subroutine, the user must also supply a subroutine SRCDX if there are any

detectors or DXTRAN spheres in the problem. In subroutine SRCDX, the variable PSC must be

set for each detector and DXTRAN sphere. An example of how this is done and also a

description of several other source angular distribution functions is in Chapter 4.

g. Detectors and the S(α,β) thermal treatment: The S(α,β) thermal treatment poses

special challenges to next event estimators because the probability density function for angle has

discrete lines to model Bragg scattering and other molecular effects. Therefore, MCNP has an

approximate model42 that, for the PSC calculation (not the transport calculation), replaces the

discrete lines with ﬁnite histograms of width µ< .1

This approximation has been demonstrated to accurately model the discrete line S(α,β) data. In

cases where continuous data is approximated with discrete lines, the approximate scheme

cancels the errors and models the scattering better than the random walk.43 Thus the S(α,β)

thermal treatment can be used with conﬁdence with next event estimators like detectors and

DXTRAN.

F. Additional Tally Features

The standard MCNP tally types can be controlled, modiﬁed, and beautiﬁed by other tally cards.

These cards are described in detail in Chapter 3; an overview is given here.

1. Bin limit control

The integration limits of the various tally types are controlled by E, T, C, and FS cards. The E

card establishes energy bin ranges; the T card establishes time bin ranges; the C card establishes

cosine bin ranges; and the FS card segments the surface or cell of a tally into subsurface or

subcell bins.

2. Flagging

Cell and surface ﬂagging cards, CF and SF, determine what portion of a tally comes from where.

Example: F4 1

CF4 2 3 4

The ﬂux tally for cell 1 is output twice: ﬁrst, the total ﬂux in cell 1; and second, the ﬂagged tally,

or that portion of the ﬂux caused by particles having passed through cells 2, 3, or 4.

pµ()dµdΩ

4π

------- µϕdd

4π

-------------

2π

∫1

---µd== =

2-96 April 10, 2000

CHAPTER 2

TALLIES

3. Multipliers and modification

MCNP tallies can be modiﬁed in many different ways. The EM, TM, and CM cards multiply the

quantities in each energy, time, or cosine bin by a different constant. This capability is useful for

modeling response functions or changing units. For example, a surface current tally can have its

units changed to per steradian by entering the inverse steradian bin sizes on the CM card.

The DE and DF cards allow modeling of an energy-dependent dose function that is a continuous

function of energy from a table whose data points need not coincide with the tally energy bin

structure (E card). An example of such a dose function is the ﬂux-to-radiation dose conversion

factor given in Appendix H.

The FM card multiplies the F1, F2, F4, and F5 tallies by any continuous-energy quantity

available in the data libraries. For example, average heating numbers Havg(E) and total cross

section σT(E) are stored on the MCNP data libraries. An F4 tally multiplied by σTHavg(E)ρa/ρg

converts it to an F6 tally, or an F5 detector tally multiplied by the same quantity calculates

heating at a point (see page 2–82). The FM card can modify any ﬂux or current tally of the form

into , where R(E) is any combination of sums and products of energy-

dependent quantities known to MCNP.

The FM card can also model attenuation. Here the tally is converted to ,

where xis the thickness of the attenuator, ρais its atom density, and σtis its total cross section.

Double parentheses allow the calculation of . More complex

expressions of σt(E)ρaxare allowed so that many attenuators may be stacked. This is useful for

calculating attenuation in line-of-sight pipes and through thin foils and detector coatings,

particularly when done in conjunction with point and ring detector tallies. Beware, however, that

attenuation assumes that the attenuated portion of the tally is lost from the system by capture or

escape and cannot be scattered back in.

Two special FM card options are available. The ﬁrst option sets R(E) =1/ϕ(E) to score tracks or

collisions. The second option sets R(E) = 1/velocity to score population or prompt removal

lifetime.

4. Special Treatments

A number of special tally treatments are available using the FT tally card. A brief description of

each one follows.

a. Change current tally reference vector: F1 current tallies measure bin angles relative to

the surface normal. They can be binned relative to any arbitrary vector with the FRV option.

ϕE()Ed

∫RE()ϕE()Ed

∫

ϕE()eσtE()ρ

ax–Ed

∫

ϕE()eσtE()ρ

ax–RE()Ed

∫

April 10, 2000 2-97

CHAPTER 2

TALLIES

b. Gaussian energy broadening: The GEB option can be used to better simulate a

physical radiation detector in which energy peaks exhibit Gaussian energy broadening. The

tallied energy is broadened by sampling from the Gaussian:

where E= the broadened energy;

Eo= the unbroadened energy of the tally;

C= a normalization constant; and

A= the Gaussian width.

The Gaussian width is related to the full width half maximum (FWHM) by

=.60056120439322 ∗FWHM

The desired FWHM is speciﬁed by the user–provided constants, a, b, and c, where

The FWHM is deﬁned as FWHM = 2(EFWHM – Eo),

where EFWHM is such that f(EFWHM) = f(Eo)

and f(Eo) is the maximum value of f(E).

c. Time convolution: Because the geometry and material compositions are independent

of time, except in the case of time-dependent temperatures, the expected tally T(t,t + τ)at time

t+τfrom a source particle emitted at time tis identical to the expected tally T(0,τ)from a source

particle emitted at time 0. Thus, if a calculation is performed with all source particles started at

t=0, one has an estimate of T(0,τ)and the tallies from a number of time-distributed sources.

Qi(t) can be calculated at time as

by sampling tfrom Qi(t) and recording each particle’s tally (shifted by t), or after the calculation

by integrating Qi(t) multiplied by the histogram estimate of . The latter method is

used in MCNP to simulate a source as a square pulse starting at time a and ending at time b,

where a and b are supplied by the TMC option.

fE() Ce

–

---------------





–

AFWHM

22ln

-------------------=

FWHM a b E cE2

++=

---

TQi

TQiη() Qit()Ttη,()td

∫Qit()T0ηt–,()td

∫

T0ηt–,()

2-98 April 10, 2000

CHAPTER 2

TALLIES

d. Binning by the number of collisions: Tallies can be binned by the number of collisions

that caused them with the INC option and an FU card. A current tally, for example, can be

subdivided into the portions of the total current coming from particles that have undergone zero,

one, two, three, ... collisions before crossing the surface. In a point detector tally, the user can

determine what portion of the score came from particles having their 1st, 2nd, 3rd, ... collision.

Collision binning is particularly useful with the exponential transform because the transform

reduces variance by reducing the number of collisions. If particles undergoing many collisions

are the major contributor to a tally, then the exponential transform is ill-advised. When the

exponential transform is used, the portion of the tally coming from particles having undergone

many collisions should be small.

e. Binning by detector cell: The ICD option with an FU card is used to determine what

portion of a detector tally comes from what cells. This information is similar to the detector

diagnostics print, but the FT card can be combined with energy and other binning cards. The

contribution to the normalized rather than unnormalized tally is printed.

f. Binning by source distribution: The SCX and SCD options are used to bin a tally score

according to what source distribution caused it.

g. Binning by multigroup particle type: The PTT option with an FU card is used to bin

multigroup tallies by particle type. The MCNP multigroup treatment is available for neutron,

coupled neutron/photon, and photon problems. However, charged particles or any other

combinations of particles can be run with the various particles masquerading as neutrons and are

printed out in the OUTP ﬁle as if they were neutrons. With the PTT option, the tallies can be

segregated into particle types by entering atomic weights in units of MeV on the FU card. The

FU atomic weights must be speciﬁed to within 0.1% of the true atomic weight in MeV units:

thus FU .511 speciﬁes an electron, but .510 is not recognized.

h. Binning by particle charge: The ELC option allows binning F1 current tallies by

particle charge. There are three ELC options:

1. cause negative electrons to make negative scores and positrons to make positive

scores. Note that by tallying positive and negative numbers the relative error is

unbounded and this tally may be difficult to converge;

2. segregate electrons and positrons into separate bins plus a total bin. There will be three

bins (positron, electron, and total) all with positive scores. The total bin will be the

same as the single tally bin without the ELC option.

3. segregate electrons and positrons into separate bins plus a total bin, with the electron

bin scores being all negative to reflect their charge. The bins will be for positrons

(positive scores), electrons (negative scores), and total. The total bin will be the same

as the single bin with the first ELC option above (usually with negative scores because

there are more electrons than positrons).

April 10, 2000 2-99

CHAPTER 2

ESTIMATION OF THE MONTE CARLO PRECISION

5. User modification

If the above capabilities do not provide exactly what is desired, tallies can be modiﬁed by a user-

supplied TALLYX subroutine (FU card). As with a user-supplied SOURCE subroutine, which

lets the user provide his own specialized source, the TALLYX subroutine lets the user modify

any tally, with all the programming changes conveniently located in a single subroutine.

6. Tally output format

Not only can users change the contents of MCNP tallies, the output format can be modiﬁed as

well. Any desired descriptive comment can be added to the tally title by the tally comment (FC)

card. The printing order can be changed (FQ card) so that instead of, for instance, getting the

default output blocks in terms of time vs. energy, they could be printed in blocks of segment vs.

cosine. The tally bin that is monitored for the tally ﬂuctuation chart printed at the problem end

and used in the statistical analysis of the tally can be selected (TF card). Detector tally diagnostic

prints are controlled with the DD card. Finally, the PRINT card controls what optional tables are

displayed in the output ﬁle.

VI. ESTIMATION OF THE MONTE CARLO PRECISION

Monte Carlo results represent an average of the contributions from many histories sampled

during the course of the problem. An important quantity equal in stature to the Monte Carlo

answer (or tally) itself is the statistical error or uncertainty associated with the result. The

importance of this error and its behavior vs. the number of histories cannot be overemphasized

because the user not only gains insight into the quality of the result, but also can determine if a

tally appears statistically well behaved. If a tally is not well behaved, the estimated error

associated with the result generally will not reﬂect the true conﬁdence interval of the result and,

thus, the answer could be completely erroneous. MCNP contains several quantities that aid the

user in assessing the quality of the conﬁdence interval.86

The purpose of this section is to educate MCNP users about the proper interpretation of the

MCNP estimated mean, relative error, variance of the variance, and history score probability

density function. Carefully check tally results and the associated tables in the tally ﬂuctuation

charts to ensure a well-behaved and properly converged tally.

A. Monte Carlo Means, Variances, and Standard Deviations

Monte Carlo results are obtained by sampling possible random walks and assigning a score xi

(for example, xi= energy deposited by the ith random walk) to each random walk. Random walks

typically will produce a range of scores depending on the tally selected and the variance

reduction chosen.

2-100 April 10, 2000

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ESTIMATION OF THE MONTE CARLO PRECISION

Suppose f(x) is the history score probability density function for selecting a random walk that

scores xto the tally being estimated. The true answer (or mean) is the expected value of x,E(x),

where

= true mean.

The function f(x) is seldom explicitly known; thus, f(x) is implicitly sampled by the Monte Carlo

random walk process. The true mean then is estimated by the sample mean where

, (2.15)

where xi is the value of x selected from f(x) for the ith history and Nis the number of histories

calculated in the problem. The Monte Carlo mean is the average value of the scores xifor all

the histories calculated in the problem. The relationship between E(x) and is given by the

Strong Law of Large Numbers1 that states that if E(x) is ﬁnite, tends to the limit E(x) as N

approaches inﬁnity.

The variance of the population of xvalues is a measure of the spread in these values and is given

by1

The square root of the variance is σ, which is called the standard deviation of the population of

scores. As with E(x),σis seldom known but can be estimated by Monte Carlo as S, given by (for

large N)

(2.16a)

and

. (2.16b)

The quantity Sis the estimated standard deviation of the population of xbased on the values of

xi that were actually sampled.

The estimated variance of is given by

Ex() xf x()xd

∫

----xi

i1=

∑

σ2xEx()–()

2fx()xd

∫Ex

() Ex()()

–==

S2Σi1=

Nxix–()

N1–

--------------------------------- x2x2

–∼=

x21

----xi

i1=

∑

April 10, 2000 2-101

CHAPTER 2

ESTIMATION OF THE MONTE CARLO PRECISION

. (2.17)

These formulae do not depend on any restriction on the distribution of xor (such as normality)

beyond requiring that E(x) and σ2 exist and are ﬁnite. The estimated standard deviation of the

mean is given by .

It is important to note that is proportional to 1/ , which is the inherent drawback to the

Monte Carlo method. To halve , four times the original number of histories must be

calculated, a calculation that can be computationally expensive. The quantity can also be

reduced for a speciﬁed Nby making Ssmaller, reducing the inherent spread of the tally results.

This can be accomplished by using variance reduction techniques such as those discussed in

section VII of this chapter.

B. Precision and Accuracy

There is an extremely important difference between precision and accuracy of a Monte Carlo

calculation. As illustrated in Fig. 2-10, precision is the uncertainty in caused by the statistical

ﬂuctuations of the xi’s for the portion of physical phase space sampled by the Monte Carlo

process. Important portions of physical phase space might not be sampled because of problem

cutoffs in time or energy, inappropriate use of variance reduction techniques, or an insufﬁcient

sampling of important low-probability events. Accuracy is a measure of how close the expected

value of , E(x), is to the true physical quantity being estimated. The difference between this

true value and E(x) is called the systematic error, which is seldom known. Error or uncertainty

estimates for the results of Monte Carlo calculations refer only to the precision of the result and

not to the accuracy. It is quite possible to calculate a highly precise result that is far from the

physical truth because nature has not been modeled faithfully.

2S2

-----=

x Sx

SxN

SxSx

Figure 2-10.

2-102 April 10, 2000

CHAPTER 2

ESTIMATION OF THE MONTE CARLO PRECISION

1. Factors Affecting Problem Accuracy

Three factors affect the accuracy of a Monte Carlo result: (1) the code, (2) problem modeling,

and (3) the user. Code factors encompass: the physics features included in a calculation as well

as the mathematical models used; uncertainties in the data, such as the transport and reaction

cross sections, Avogadro's number, atomic weights, etc.; the quality of the representation of the

differential cross sections in energy and angle; and coding errors (bugs). All of the applicable

physics must be included in a calculation to produce accurate results. Even though the

evaluations are not perfect, more faithful representation of the evaluator's data should produce

more accurate results. The descending order of preference for Monte Carlo data for calculations

is continuous energy, thinned continuous energy, discrete reaction, and multigroup. Coding

errors can always be a problem because no large code is bug-free. MCNP, however, is a very

mature, heavily used production code. With steadily increasing use over the years, the likelihood

of a serious coding error continues to diminish.

The second area, problem-modeling factors, can quite often contribute to a decrease in the

accuracy of a calculation. Many calculations produce seemingly poor results because the model

of the energy and angular distribution of the radiation source is not adequate. Two other

problem-modeling factors affecting accuracy are the geometrical description and the physical

characteristics of the materials in the problem.

The third general area affecting calculational accuracy involves user errors in the problem input

or in user-supplied subroutines and patches to MCNP. The user can also abuse variance

reduction techniques such that portions of the physical phase space are not allowed to contribute

to the results. Checking the input and output carefully can help alleviate these difﬁculties. A last

item that is often overlooked is a user's thorough understanding of the relationship of the Monte

Carlo tallies to any measured quantities being calculated. Factors such as detector efﬁciencies,

data reduction and interpretation, etc., must be completely understood and included in the

calculation, or the comparison is not meaningful.

2. Factors Affecting Problem Precision

The precision of a Monte Carlo result is affected by four user-controlled choices: (1) forward vs.

adjoint calculation, (2) tally type, (3) variance reduction techniques, and (4) number of histories

run.

The choice of a forward vs. adjoint calculation depends mostly on the relative sizes of the source

and detector regions. Starting particles from a small region is easy to do, whereas transporting

particles to a small region is generally hard to do. Because forward calculations transport

particles from source to detector regions, forward calculations are preferable when the detector

(or tally) region is large and the source region is small. Conversely, because adjoint calculations

transport particles backward from the detector region to the source region, adjoint calculations

April 10, 2000 2-103

CHAPTER 2

ESTIMATION OF THE MONTE CARLO PRECISION

are preferable when the source (or tally) region is large and the detector region is small. MCNP

can be run in multigroup adjoint mode. There is no continuous-energy adjoint capability.

As alluded to above, the smaller the tally region, the harder it becomes to get good tally

estimates. An efﬁcient tally will average over as large a region of phase space as practical. In this

connection, tally dimensionality is extremely important. A one-dimensional tally is typically 10

to 100 times easier to estimate than a two-dimensional tally, which is 10 to 100 times easier than

a three-dimensional tally. This fact is illustrated in Fig. 2-15 later in this section.

Variance reduction techniques can be used to improve the precision of a given tally by increasing

the nonzero tallying efﬁciency and by decreasing the spread of the nonzero history scores. These

two components are depicted in a hypothetical f(x) shown in Fig. 2-11. See page 2–113 for more

discussion about the empirical f(x) for each tally ﬂuctuation chart bin. A calculation will be more

precise when the history-scoring efﬁciency is high and the variance of the nonzero scores is low.

The user should strive for these conditions in difﬁcult Monte Carlo calculations. Examples of

these two components of precision are given on page 2–109.

More histories can be run to improve precision (see section Cfollowing). Because the precision

is proportional to 1/ , running more particles is often costly in computer time and therefore

is viewed as the method of last resort for difﬁcult problems.

C. The Central Limit Theorem and Monte Carlo Conﬁdence Intervals

To deﬁne conﬁdence intervals for the precision of a Monte Carlo result, the Central Limit

Theorem1 of probability theory is used, stating that

Figure 2-11.

N∞–

lim Pr E x() ασ

-------- xEx() βσ

--------

+<<+1

2π

------ et22⁄–td

∫

2-104 April 10, 2000

CHAPTER 2

ESTIMATION OF THE MONTE CARLO PRECISION

where αand βcan be any arbitrary values and Pr[Z] means the probability of Z. In terms of the

estimated standard deviation of , , this may be rewritten in the following approximation for

large N:

This crucial theorem states that for large values of N (that is, as Ntends to inﬁnity) and

identically distributed independent random variables xiwith ﬁnite means and variances, the

distribution of the ’s approaches a normal distribution. Therefore, for any distribution of tallies

(an example is shown in Fig. 2-11), the distribution of resulting ’s will be approximately

normally distributed, as shown in Fig. 2-10, with a mean of E(x).IfSis approximately equal to

σ, which is valid for a statistically signiﬁcant sampling of a tally (i.e, Nhas tended to inﬁnity),

then

~ 68% of the time and (2.18a)

, ~ 95% of the time (2.18b)

from standard tables for the normal distribution function. Eq. (2.18a) is a 68% conﬁdence

interval and Eq. (2.18b) is a 95% conﬁdence interval.

The key point about the validity of these conﬁdence intervals is that the physical phase space

must be adequately sampled by the Monte Carlo process. If an important path in the geometry

or a window in the cross sections, for example, has not been well sampled, both and will

be unknowingly incorrect and the results will be wrong, usually tending to be too small. The user

must take great care to be certain that adequate sampling of the source, transport, and any tally

response functions have indeed taken place. Additional statistical quantities to aid in the

assessment of proper conﬁdence intervals are described in later portions of section VI.

D. Estimated Relative Errors in MCNP

All standard MCNP tallies are normalized to be per starting particle history (except for some

criticality calculations) and are printed in the output with a second number, which is the

estimated relative error deﬁned as

(2.19a)

The relative error is a convenient number because it represents statistical precision as a fractional

result with respect to the estimated mean.

xSx

Pr αSxxEx()–

σN

-------------------- βSx

<<1

2π

---------- et22⁄–td

∫

∼

x2SxEx() xS

x,+<<–

x2SxEx() x2Sx

+<<–

x Sx

xx⁄≡

April 10, 2000 2-105

CHAPTER 2

ESTIMATION OF THE MONTE CARLO PRECISION

Combining Eqs. (2.15), (2.16), and (2.17), R can be written (for large N) as

. (2.19b)

Several important observations about the relative error can be made from Eq. (2.19b). First, if

all the xi’s are nonzero and equal, Ris zero. Thus, low-variance solutions should strive to reduce

the spread in the xi’s. If the xi’s are all zero, Ris deﬁned to be zero. If only one nonzero score is

made, Rapproaches unity as Nbecomes large. Therefore, for xi’s of the same sign, can never

be greater than because R never exceeds unity. For positive and negative xi’s, R can exceed

unity. The range of R values for xi’s of the same sign is therefore between zero and unity.

To determine what values of Rlead to results that can be stated with conﬁdence using Eqs. (2.6),

consider Eq. (2.19b) for a difﬁcult problem in which nonzero scores occur very infrequently. In

this case,

. (2.20a)

For clarity, assume that there are nout of nonzero scores that are identical and equal

to x. With these two assumptions, R for “difﬁcult problems” becomes

RD.P. ~. (2.20b)

This result is expected because the limiting form of a binomial distribution with infrequent

nonzero scores and large Nis the Poisson distribution, which is the form in Eq. (2.20b) used in

detector “counting statistics.”

Through use of Eqs. (2.8), a table of Rvalues versus the number of tallies or “counts” can be

generated as shown in Table 2.2. A relative error of 0.5 is the equivalent of four counts, which

is hardly adequate for a statistically signiﬁcant answer. Sixteen counts is an improvement,

TABLE 2.2:

Estimated Relative Error R vs. Number of Identical Tallies nfor Large N

n1 4 16 25 100 400

R1.0 0.5 0.25 0.20 0.10 0.05

----x2

-----1–







12⁄Σi1=

Nxi

Σi1=

Nxi

()

------------------------ 1

----–

12⁄

----Σi1=

Nxi

Σi1=

Nxi

()

------------------------«

Nn N«()

nx2

n2x2

-----------

12⁄1

-------nN«,=

2-106 April 10, 2000

CHAPTER 2

ESTIMATION OF THE MONTE CARLO PRECISION

reducing R to 0.25, but still is not a large number of tallies. The same is true for nequals 25.

When nis 100, Ris 0.10, so the results should be much improved. With 400 tallies, an Rof 0.05

should be quite good indeed.

Based on this qualitative analysis and the experience of Monte Carlo practitioners, Table 2.3

presents the recommended interpretation of the estimated 1σconﬁdence interval for

various values of R associated with an MCNP tally. These guidelines were determined

empirically, based on years of experience using MCNP on a wide variety of problems. Just

before the tally ﬂuctuation charts, a “Status of Statistical Checks” table prints how many tally

bins of each tally have values of R exceeding these recommended guidelines.

Point detector tallies generally require a smaller value of R for valid conﬁdence interval

statements because some contributions, such as those near the detector point, are usually

extremely important and may be difﬁcult to sample well. Experience has shown that for R less

than 0.05, point detector results are generally reliable. For an Rof 0.10, point detector tallies may

only be known within a factor of a few and sometimes not that well (see the pathological

example on page 2–123.)

MCNP calculates the relative error for each tally bin in the problem using Eq. (2.19b). Each xi

is deﬁned as the total contribution from the ith starting particle and all resulting progeny. This

deﬁnition is important in many variance reduction methods, multiplying physical processes such

as ﬁssion or (n,xn) neutron reactions that create additional neutrons, and coupled neutron/

photon/electron problems. The ith source particle and its offspring may thus contribute many

times to a tally and all of these contributions are correlated because they are from the same

source particle.

Figure 2.12 represents the MCNP process of calculating the ﬁrst and second moments of each

tally bin and relevant totals using three tally storage blocks of equal length for each tally bin. The

TABLE 2.3:

Guidelines for Interpreting the Relative Error Ra

aand represents the estimated statistical relative error at the 1σlevel. These in-

terpretations of Rassume that all portions of the problem phase space have been well sam-

pled by the Monte Carlo process.

Range of RQuality of the Tally

0.5 to 1 Garbage

0.2 to 0.5 Factor of a few

0.1 to 0.2 Questionable

<0.10 Generally reliable except for point detector

<0.05 Generally reliable for point detector

x1R±()

xx⁄=

April 10, 2000 2-107

CHAPTER 2

ESTIMATION OF THE MONTE CARLO PRECISION

hypothetical grid of tally bins in the bottom half of Fig. 2-12 has 24 tally bins including the time

and energy totals. During the course of the ith history, sums are performed in the ﬁrst MCNP tally

storage block. Some of the tally bins receive no contributions and others receive one or more

contributions. At the conclusion of the ith history, the sums are added to the second MCNP tally

storage block. The sums in the ﬁrst MCNP tally storage block are squared and added to the third

tally storage block. The ﬁrst tally storage block is then ﬁlled with zeros and history i+1begins.

After the last history N, the estimated tally means are computed using the second MCNP tally

storage block and Eq. (2.15). The estimated relative errors are calculated using the second and

third MCNP tally storage blocks and Eq. (2.19b). This method of estimating the statistical

uncertainty of the result produces the best estimate because the batch size is one, which

minimizes the variance of the variance.87,88

Note that there is no guarantee that the estimated relative error will decrease inversely

proportional to the as required by the Central Limit Theorem because of the statistical

nature of the tallies. Early in the problem, R will generally have large statistical ﬂuctuations.

Later, infrequent large contributions may cause ﬂuctuations in and to a lesser extent in and

therefore in R. MCNP calculates a FOM for one bin of each numbered tally to aid the user in

determining the statistical behavior as a function of N and the efﬁciency of the tally.

Sxx

Time

Energy

Time

Total Grand

Total

XXX X X XXXXX

XXXX

XX XX

Energy

Total

HYPOTHETICAL TALLY GRID

X=Score from the present history

Particle batch size is one

MCNP TALLY BLOCKS

{

}

Running

History

Scores

Sums performed

after each history

Figure 2-12.

2-108 April 10, 2000

CHAPTER 2

ESTIMATION OF THE MONTE CARLO PRECISION

E. MCNP Figure of Merit

The estimated relative error squared R2should be proportional to 1/N, as shown by Eq. (2.19a).

The computer time Tused in an MCNP problem should be directly proportional to N; therefore,

R2T should be approximately a constant within any one Monte Carlo run. It is convenient to

deﬁne a ﬁgure of merit (FOM) of a tally to be

. (2.21a)

MCNP prints the FOM for one bin of each numbered tally as a function of N, where the unit of

computer time T is minutes The table is printed in particle increments of 1000 up to 20,000

histories. Between 20,000 and 40,000 histories, the increment is doubled to 2000. This trend

continues, producing a table of up to 20 entries. The default increment can be changed by the

5th entry on the PRDMP card.

The FOM is a very important statistic about a tally bin and should be studied by the user. It is a

tally reliability indicator in the sense that if the tally is well behaved, the FOM should be

approximately a constant with the possible exception of statistical ﬂuctuations very early in the

problem. An order-of-magnitude estimate of the expected fractional statistical ﬂuctuations in the

FOM is 2R. This result assumes that both the relative statistical uncertainty in the relative error

is of the order of the relative error itself and the relative error is small compared to unity. The

user should always examine the tally ﬂuctuation charts at the end of the problem to check that

the FOMs are approximately constant as a function of the number of histories for each tally.

The numerical value of the FOM can be better appreciated by considering the relation

(2.21b)

Table 2.4 shows the expected value of Rthat would be produced in a one-minute problem (T=1)

as a function of the value of the FOM. It is clearly advantageous to have a large FOM for a

problem because the computer time required to reach a desired level of precision is

proportionally reduced. Examination of Eq. (2.21b) shows that doubling the FOM for a problem

will reduce the computer time required to achieve the same R by a factor of two.

TABLE 2.4:

R Values as a Function of the FOM for T = 1 Minute

FOM 1 10 100 1000 10000

R 1.0 0.32 0.10 0.032 0.010

FOM 1

R2T

----------

≡

R1FOM T⁄=∗

April 10, 2000 2-109

CHAPTER 2

ESTIMATION OF THE MONTE CARLO PRECISION

In summary, the FOM has three uses. The most important use is as a tally reliability indicator.

If the FOM is not approximately a constant (except for statistical ﬂuctuations early in the

problem), the conﬁdence intervals may not overlap the expected score value, E(x), the expected

fraction of the time. A second use for the FOM is to optimize the efﬁciency of the Monte Carlo

calculation by making several short test runs with different variance reduction parameters and

then selecting the problem with the largest FOM. Remember that the statistical behavior of the

FOM (i.e., R) for a small number of histories may cloud the selection of techniques competing

at the same level of efﬁciency. A third use for the FOM is to estimate the computer time required

to reach a desired value of R by using T ~ 1/R2FOM.

F. Separation of Relative Error into Two Components

Three factors that affect the efﬁciency of a Monte Carlo problem are (1) history-scoring

efﬁciency, (2) dispersions in nonzero history scores, and (3) computer time per history. All three

factors are included in the FOM. The ﬁrst two factors control the value of R; the third is T.

The relative error can be separated into two components: the nonzero history-scoring efﬁciency

component and the intrinsic spread of the nonzero xi scores . Deﬁning qto be the

fraction of histories producing nonzero xi’s, Eq. 2.19b can be rewritten as

. (2.22a)

Note by Eq. 2.19b that the ﬁrst two terms are the relative error of the qN nonzero scores. Thus

deﬁning,

and (2.22b)

yields (2.22c)

. (2.22d)

For identical nonzero xi’s, is zero and for a 100% scoring efﬁciency, is zero. It is

usually possible to increase qfor most problems using one or more of the MCNP variance

reduction techniques. These techniques alter the random walk sampling to favor those particles

that produce a nonzero tally. The particle weights are then adjusted appropriately so that the

expected tally is preserved. This topic is described in Sec. VII (Variance Reduction) beginning

Reff

2Rint

Σi1=

Nxi

Σi1=

Nxi

()

------------------------ 1

----– Σxi0≠x2

Σxi0≠xi

()

------------------------- 1

----– Σxi0≠x2

Σxi0≠xi

()

--------------------------- 1

-------–1q–

------------+== =

Rint

2Σxi0≠x2

Σxi0≠xi

()

--------------------------- 1

-------–=

Reff

21q–()qN()⁄=

R2Reff

2Rint

Rint

2Reff

2-110 April 10, 2000

CHAPTER 2

ESTIMATION OF THE MONTE CARLO PRECISION

on page 2–127 . The sum of the two terms of Eq. (2.22d) produces the same result as Eq. (2.19b).

Both and are printed for the tally ﬂuctuation chart bin of each tally so that the

dominant component of R can be identiﬁed as an aid to making the calculation more efﬁcient.

These equations can be used to better understand the effects of scoring inefﬁciency; that is, those

histories that do not contribute to a tally. Table 2.5 shows the expected values of as a

function of qand the number of histories N. This table is appropriate for identical nonzero scores

and represents the theoretical minimum relative error possible for a speciﬁed q and N. It is no

surprise that small values of qrequire a compensatingly large number of particles to produce

precise results.

A practical example of scoring inefﬁciency is the case of infrequent high-energy particles in a

down-scattering-only problem. If only a small fraction of all source particles has an energy in

the highest energy tally bin, the dominant component of the relative error will probably be the

scoring efﬁciency because only the high-energy source particles have a nonzero probability of

contributing to the highest energy bin. For problems of this kind, it is often useful to run a

separate problem starting only high-energy particles from the source and to raise the energy

cutoff. The much-improved scoring efﬁciency will result in a much larger FOM for the high-

energy tally bins.

To further illustrate the components of the relative error, consider the ﬁve examples of selected

discrete probability density functions shown in Fig. 2-13. Cases I and II have no dispersion in

the nonzero scores, cases III and IV have 100% scoring efﬁciency, and case V contains both

elements contributing to R. The most efﬁcient problem is case III. Note that the scoring

inefﬁciency contributes 75% to R in case V, the second worst case of the ﬁve.

TABLE 2.5:

Expected Values of Reff as a Function of q and N

N0.001 0.01 0.1 0.5

1030.999 0.315 0.095 0.032

1040.316 0.099 0.030 0.010

1050.100 0.031 0.009 0.003

1060.032 0.010 0.003 0.001

Rint

2Reff

Reff

April 10, 2000 2-111

CHAPTER 2

ESTIMATION OF THE MONTE CARLO PRECISION

G. Variance of the Variance

Previous sections have discussed the relative error R and ﬁgure of merit FOM as measures of the

quality of the mean. A quantity called the relative variance of the variance (VOV) is another

useful tool that can assist the user in establishing more reliable conﬁdence intervals. The VOV

is the estimated relative variance of the estimated R. The VOV involves the estimated third and

fourth moments of the empirical history score probability density function (PDF) f(x) and is

much more sensitive to large history score ﬂuctuations than is R. The magnitude and NPS

behavior of the VOV are indicators of tally ﬂuctuation chart (TFC) bin convergence. Early work

was done by Estes and Cashwell87 and Pederson89 later reinvestigated this statistic to determine

its usefulness.

The VOV is a quantity that is analogous to the square of the R of the mean, except it is for R

instead of the mean. The estimated relative VOV of the mean is deﬁned as

1/3

0 0.5 1

E[x]=0x1/3+1/3x1/2+1/3x1=0.5

R=0.82/sqrt(N)

R =0.41/sqrt(N) 25%

R =0.71/sqrt(N) 75%

eff

int

IV f

0.5

0 1/4 3/4 1

E[x]=1/2x1/4+1/2x3/4=0.5

R=R =0.5/sqrt(N)

R =0

eff

int

III f

0.5

0 1/3 2/3 1

E[x]=1/2x1/3+1/2x2/3=0.5

R=R =0.33/sqrt(N)

R =0

eff

int

II f

0.75

0.25

02/3 1

E[x]=0x1/4+2/3x3/4=0.5

R=R =0.58/sqrt(N)

R =0

int

eff

0.5

E[x]=0.5(0+1)=0.5

R=R =1/sqrt(N)

R =0

int

eff

FIVE CASES WITH A MEAN OF 0.5

Figure 2-13.

2-112 April 10, 2000

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ESTIMATION OF THE MONTE CARLO PRECISION

where is the estimated variance of and is the estimated variance in . The VOV

is a measure of the relative statistical uncertainty in the estimated R and is important because S

must be a good approximation of σ to use the Central Limit Theorem to form conﬁdence

intervals.

The VOV for a tally bin89 is

. (2.23)

This is the fourth central moment minus the second central moment squared normed by the

product of Nand the second central moment squared.

When Eq. (2.23) is expanded in terms of sums of powers of xi, it becomes

(2.24)

Now consider the truncated Cauchy formula for the following analysis. The truncated Cauchy is

similar in shape to some difﬁcult Monte Carlo tallies. After numerous statistical experiments on

sampling a truncated positive Cauchy distribution

, (2.25)

it is concluded that the VOV should be below 0.1 to improve the probability of forming a reliable

conﬁdence interval. The quantity 0.1 is a convenient value and is why the VOV is used for the

statistical check and not the square root of the VOV (R of the R). Multiplying numerator and

denominator of Eq. (2.24) by 1/N converts the terms into averages and shows that the VOV

is expected to decrease as 1/N.

It is interesting to examine the VOV for the nidentical history scores xthat were used

to analyze R in Table 2.2, page 2–105. The VOV behaves as 1/n in this limit. Therefore, ten

VOV S2Sx

()Sx

⁄=

2x S2Sx

() Sx

VOV Σxix–()

=Σxix–()

()⁄21N⁄–

VOV Σxi

44ΣxiΣxi

3N6Σxi

2Σxi

()

2N23Σxi

()

4N3

⁄–⁄+⁄–

Σxi

2Σxi

()

2N⁄–()

------------------------------------------------------------------------------------------------------------------------------ 1

----–=

VOV Σxi

44ΣxiΣxi

3N8Σxi

2Σxi

()

2N⁄24Σxi

()

4N⁄3

–Σxi

()

2N⁄–+⁄–

Σxi

2Σxi

()

2N⁄–()

-------------------------------------------------------------------------------------------------------------------------------------------------------------=

Cauchy f x() 2π1x2

+()0xx

max

≤≤,⁄=

nN«()

April 10, 2000 2-113

CHAPTER 2

ESTIMATION OF THE MONTE CARLO PRECISION

identical history scores would be enough to satisfy the VOV criterion, a factor of at least ten less

than the R criterion. There are two reasons for this phenomenon: 1) it is more important to know

R well than the VOV in forming conﬁdence intervals; and 2) the history scores will ordinarily

not be identical and thus the fourth moment terms in the VOV will increase rapidly over the

second moment terms in R.

The behavior of the VOV as a function of Nfor the TFC bin is printed in the OUTP ﬁle. Because

the VOV involves third and fourth moments, the VOV is a much more sensitive indicator to large

history scores than the R, which is based on ﬁrst and second moments. The desired VOV

behavior is to decrease inversely with N. This criterion is deemed to be a necessary, but not

sufﬁcient, condition for a statistically well-behaved tally result. A tally with a VOV that matches

this criteria is NOT guaranteed to produce a high quality conﬁdence interval because

undersampling of high scores will also underestimate the higher score moments.

To calculate the VOV of every tally bin, put a nonzero 15th entry on the DBCN card. This option

creates two additional history score moment tables each of length MXF in the TAL array to sum

and (see Fig. 2-12). This option is not the default because the amount of tally storage will

increase by 2/5, which could be prohibitive for a problem with many tally bins. The magnitude

of the VOV in each tally bin is reported in the “Status of Statistical Checks” table. History–

dependent checks of the VOV of all tally bins can be done by printing the tallies to the output

ﬁle at some frequency using the PRDMP card.

H. Empirical History Score Probability Density Function f(x)

1. Introduction

This section discusses another statistic that is useful in assessing the quality of conﬁdence

intervals from Monte Carlo calculations. Consider a generic Monte Carlo problem with difﬁcult

to sample, but extremely important, large history scores. This type of problem produces three

possible scenarios.86

The ﬁrst, and obviously desired, case is a correctly converged result that produces a statistically

correct conﬁdence interval. The second case is the sampling of an infrequent, but very large,

history score that causes the mean and R to increase and the FOM to decrease signiﬁcantly. This

case is easily detectable by observing the behavior of the FOM and the R in the TFCs.

The third and most troublesome case yields an answer that appears statistically converged based

on the accepted guidelines described previously, but in fact may be substantially smaller than the

correct result because the large history tallies were not well sampled. This situation of too few

large history tallies is difﬁcult to detect. The following sections discuss the use of the empirical

history score probability density function (PDF) f(x) to gain insight into the TFC bin result. A

pathological example to illustrate the third case follows.

3xi

2-114 April 10, 2000

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2. The History Score Probability Density Function f(x)

A history score posted to a tally bin can be thought of as having been sampled from an

underlying and generally unknown history score PDF f(x), where the random variable x is the

score from one complete particle history to a tally bin. The history score can be either positive

or negative. The quantity f(x)dx is the probability of selecting a history score between x and

x+dx for the tally bin. Each tally bin will have its own f(x).

The most general form for expressing f(x) mathematically is

where fc(x) is the continuous nonzero part and represents the n different

discrete components occurring at xiwith probability pi.Anf(x) could be composed of either or

both parts of the distribution. A history score of zero is included in f(x) as the discrete component

δ(x − 0).

By the deﬁnition of a PDF,

As discussed on page 2–99, f(x) is used to estimate the mean, variance, and higher moment

quantities such as the VOV.

3. The Central Limit Theorem and f(x)

As discussed on page 2–103, the Central Limit Theorem (CLT) states that the estimated mean

will appear to be sampled from a normal distribution with a known standard deviation

when N approaches inﬁnity. In practice, σ is NOT known and must be approximated by the

estimated standard deviation S. The major difﬁculty in applying the CLT correctly to a Monte

Carlo result to form a conﬁdence interval is knowing when N has approached inﬁnity.

The CLT requires the ﬁrst two moments of f(x) to exist. Nearly all MCNP tally estimators

(except point detectors with zero neighborhoods in a scattering material and some exponential

transform problems) satisfy this requirement. Therefore, the history score PDF f(x) also exists.

One can also examine the behavior of f(x) for large history scores to assess if f(x) appears to have

been “completely” sampled. If “complete” sampling has occurred, the largest values of the

sampled x’s should have reached the upper bound (if such a bound exists) or should decrease

faster than 1/x3so that exists (σis assumed to be ﬁnite in the CLT).

fx() fcx() piδxx

–()

i1=

∑

Σi1=

npiδxx

–()

fx()x1≡d

∞–

∞

∫

σN()⁄

() x2fx()xd

∞–

∞

∫

April 10, 2000 2-115

CHAPTER 2

ESTIMATION OF THE MONTE CARLO PRECISION

Otherwise, Nis assumed not to have approached inﬁnity in the sense of the CLT. This is the basis

for the use of the empirical f(x) to assess Monte Carlo tally convergence.

The argument should be made that since S must be a good estimate of σ, the expected value of

the fourth history score moment should exist. It will be assumed that

only the second moment needs to exist so that the f(x) convergence criterion will be relaxed

somewhat. Nevertheless, this point should be kept in mind.

4. Analytic Study of f(x) for Two-State Monte Carlo Problems

Booth90,91 examined the distribution of history scores analytically for both an analog two-state

splitting problem and two exponential transform problems. This work provided the theoretical

foundation for statistical studies,92 on relevant analytic functions to increase understanding of

conﬁdence interval coverage rates for Monte Carlo calculations.

It was found that the two–state splitting problem f(x) decreases geometrically as the score

increases by a constant increment. This is equivalent to a negative exponential behavior for a

continuous f(x). The f(x) for the exponential transform problem decreases geometrically with

geometrically increasing x. Therefore, the splitting problem produces a linearly decreasing f(x)

for the history score on a lin-log plot of the score probability versus score. The exponential

transform problem generates a linearly decreasing score behavior (with high score negative

exponential roll off) on a log-log plot of the score probability versus score plot. In general, the

exponential transform problem is the more difﬁcult to sample because of the larger impact of the

low probability high scores.

The analytic shapes were compared with a comparable problem calculated with a modiﬁed

version of MCNP. These shapes of the analytic and empirical f(x)s were in excellent

agreement.92

5. Proposed Uses for the Empirical f(x) in Each TFC Bin

Few papers discuss the underlying or empirical f(x) for Monte Carlo transport problems.93,86

MCNP provides a visual inspection and analysis of the empirical f(x) for the TFC bin of each

tally. This analysis helps to determine if there are any unsampled regions (holes) or spikes in the

empirical history score PDF f(x) at the largest history scores.

The most important use for the empirical f(x) is to help determine if Nhas approached inﬁnity

in the sense of the CLT so that valid conﬁdence intervals can be formed. It is assumed that the

underlying f(x) satisﬁes the CLT requirements; therefore, so should the empirical f(x). Unless

there is a largest possible history score, the empirical f(x) must eventually decrease more steeply

than x−3 for the second moment to exist. It is postulated94 that if such

() x4fx()xd

∞–

∞

∫

x2fx()xd

∞–

∞

∫





2-116 April 10, 2000

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ESTIMATION OF THE MONTE CARLO PRECISION

decreasing behavior in the empirical f(x) with no upper bound has not been observed, then Nis

not large enough to satisfy the CLT because f(x) has not been completely sampled. Therefore, a

larger N is required before a conﬁdence interval can be formed. It is important to note that this

convergence criterion is NOT affected by any correlations that may exist between the estimated

mean and the estimated R. In principle, this lack of correlation should make the f(x) diagnostic

robust in assessing “complete” sampling.

Both the analytic and empirical history score distributions suggest that large score ﬁll-in and one

or more extrapolation schemes for the high score tail of the f(x) could provide an estimate of

scores not yet sampled to help assess the impact of the unsampled tail on the mean. The

magnitude of the unsampled tail will surely affect the quality of the tally conﬁdence interval.

6. Creation of f(x) for TFC Bins

The creation of the empirical f(x) in MCNP automatically covers nearly all TFC bin tallies that

a user might reasonably be expected to make, including the effect of large and small tally

multipliers. A logarithmically spaced grid is used for accumulating the empirical f(x) because

the tail behavior is assumed to be of the form 1/xn,n>3(unless an upper bound for the history

scores exists). This grid produces an equal width histogram straight line for f(x) on a log-log plot

that decreases n decades in f(x) per decade increase in x.

Ten bins per x decade are used and cover the unnormalized tally range from 10−30 to 1030. The

term “unnormalized” indicates that normalizations that are not performed until the end of the

problem, such as cell volume or surface area, are not included in f(x). The user can multiply this

range at the start of the problem by the 16th entry on the DBCN card when the range is not

sufﬁcient. Both history score number and history score for the TFC bin are tallied in the x grid.

With this x grid in place, the average empirical between xi and xi+1 is deﬁned to be

= (number of history scores in ith score bin)/N(xi+1 −xi)) ,

where xi+1 = 1.2589 xi.The quantity 1.2589 is 100.1 and comes from 10 equally spaced log bins

per decade. The calculated s are available on printed plots or by using the “z” plot option

(MCPLOT) with the TFC command mnemonics. Any history scores that are outside the xgrid

are counted as either above or below to provide this information to the user.

Negative history scores can occur for some electron charge deposition tallies. The MCNP default

is that any negative history score will be lumped into one bin below the lowest history score in

the built-in grid (the default is ). If DBCN(16) is negative, f(−x) will be created from

the negative scores and the absolute DBCN(16) value will be used as the score grid multiplier.

Positive history scores then will be lumped into the lowest bin because of the sign change.

()

110

30–

April 10, 2000 2-117

CHAPTER 2

ESTIMATION OF THE MONTE CARLO PRECISION

Figures 2.14 and 2.15 show two simple examples of empirical f(x)s from MCNP for 10 million

histories each. Figure 2.14 is from an energy leakage tally directly from a source that is uniform

in energy from 0 to 10 MeV. The analytic f(x) is a constant 0.1 between 0 and 10 MeV. The

empirical f(x) shows the sampling, which is 0.1 with statistical noise at the lower x bins where

fewer samples are made in the smaller bins.

Figure 2-14.

Figure 2-15.

2-118 April 10, 2000

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ESTIMATION OF THE MONTE CARLO PRECISION

Figure 2.15 shows the sampled distance to ﬁrst collision in a material that has a macroscopic

cross section of about 0.1 cm−1. This analytic function is a negative exponential given by

f(x) = Σexp−Σ x(see page 2–27) with a mean of 10. The empirical f(x) transitions from a constant

0.1 at values of xless than unity to the expected negative exponential behavior.

7. Pareto Fit to the Largest History Scores for the TFC Bin

The slope nin 1/xnof the largest history tallies xmust be estimated to determine if and when the

largest history scores decrease faster than 1/x3. The 201 largest history scores for each TFC bin

are continuously updated and saved during the calculation. A generalized Pareto function95

Pareto f(x) = a−1(1 + kx/a)−(1/k)−1

is used to ﬁt the largest x’s. This function ﬁts a number of extreme value distributions including

1/xn,exponential (k=0), and constant (k=−1). The large history score tail ﬁtting technique uses

the robust “simplex” algorithm,96 which ﬁnds the values of aand kthat best ﬁt the largest history

scores by maximum likelihood estimation.

The number of history score tail points used for the Pareto ﬁt is a maximum of 201 points

because this provides about 10% precision95 in the slope estimator at n=3. The precision

increases for smaller values of nand vice versa. The number of points actually used in the ﬁt is

the lesser of 5% of the nonzero history scores or 201. The minimum number of points used for

a Pareto ﬁt is 25 with at least two different values, which requires 500 nonzero history scores

with the 5% criterion. If less than 500 history scores are made in the TFC bin, no Pareto ﬁt is

made.

From the Pareto ﬁt, the slope of f(xlarge) is deﬁned to be

A SLOPE value of zero is deﬁned to indicate that not enough f(xlarge)tail information exists for

a SLOPE estimate. The SLOPE is not allowed to exceed a value of 10 (a “perfect score”), which

would indicate an essentially negative exponential decrease. If the 100 largest history scores all

have values with a spread of less than 1%, an upper limit is assumed to have been reached and

the SLOPE is set to 10. The SLOPE should be greater than 3 to satisfy the second moment

existence requirement of the CLT. Then, f(x) will appear to be “completely” sampled and hence

N will appear to have approached inﬁnity.

A printed plot of f(x) is automatically generated in the OUTP ﬁle if the SLOPE is less than 3 (or

if any of the other statistical checks described in the next section do not pass). If

0 < SLOPE < 10, several “S’s” appear on the printed plot to indicate the Pareto ﬁt, allowing the

quality of the ﬁt to the largest history scores to be assessed visually. If the largest scores are not

Pareto in shape, the SLOPE value may not reﬂect the best estimate of the largest history score

SLOPE 1k⁄()1+≡

April 10, 2000 2-119

CHAPTER 2

ESTIMATION OF THE MONTE CARLO PRECISION

decrease. A new SLOPE can be estimated graphically. A blank or 162 on the PRINT card also

will cause printed plots of the ﬁrst two cumulative moments of the empirical f(x) to be made.

Graphical plots of various f(x) quantities can be made using the “z” plot option (MCPLOT) with

the TFC plot command. These plots should be examined for unusual behavior in the empirical

f(x), including holes or spikes in the tail. MCNP tries to assess both conditions and prints a

message if either condition is found.

I. Forming Statistically Valid Conﬁdence Intervals

The ultimate goal of a Monte Carlo calculation is to produce a valid conﬁdence interval for each

tally bin. Section VI has described different statistical quantities and the recommended criteria

to form a valid conﬁdence interval. Detailed descriptions of the information available in the

output for all tally bins and the TFC bins are now discussed.

1. Information Available for Forming Statistically Valid Confidence

The R is calculated for every user-speciﬁed tally bin in the problem. The VOV and the shifted

conﬁdence interval center, discussed below, can be obtained for all bins with a nonzero entry for

the 15th entry on the DBCN card at problem initiation.

a. R Magnitude Comparisons With MCNP Guidelines: The quality of MCNP Monte

Carlo tallies historically has been associated with two statistical checks that have been the

responsibility of the user: 1) for all tally bins, the estimated relative error magnitude rules–of–

thumb that are shown in Fig. 2-3 (i.e., R<0.1 for nonpoint detector tallies and R<0.05 for point

detector tallies); and 2) a statistically constant FOM in the user-selectable (TFn card) TFC bin

so that the estimated R is decreasing by as required by the CLT.

In an attempt to make the user more aware of the seriousness of checking these criteria, MCNP

provides checks of the R magnitude for all tally bins. A summary of the checks is printed in the

“Status of Statistical Checks” table. Messages are provided to the user giving the results of these

checks.

b. Asymmetric Conﬁdence Intervals: A correlation exists between the estimated mean

and the estimated uncertainty in the mean.89 If the estimated mean is below the expected value,

the estimated uncertainty in the mean will most likely be below its expected value. This

correlation is also true for higher moment quantities such as the VOV. The worst situation for

forming valid conﬁdence intervals is when the estimated mean is much smaller than the expected

value, resulting in smaller than predicted coverage rates. To correct for this correlation and

improve coverage rates, one can estimate a statistic shift in the midpoint of the conﬁdence

interval to a higher value. The estimated mean is unchanged.

1N⁄

2-120 April 10, 2000

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ESTIMATION OF THE MONTE CARLO PRECISION

The shifted conﬁdence interval midpoint is the estimated mean plus a term proportional to the

third central moment. The term arises from an Edgeworth expansion89 to attempt to correct the

conﬁdence interval for non-normality effects in the estimate of the mean. The adjustment term

is given by

Substituting for the estimated mean and expanding produces

The SHIFT should decrease as 1/N. This term is added to the estimated mean to produce the

midpoint of the now asymmetric conﬁdence interval about the mean. This value of the

conﬁdence interval midpoint can be used to form the conﬁdence interval about the estimated

mean to improve coverage rates of the true, but unknown, mean E(x). The estimated mean plus

the SHIFT is printed automatically for the TFC bin for all tallies. A nonzero entry for the 15th

DBCN card entry produces the shifted value for all tally bins.

This correction approaches zero as Napproaches inﬁnity, which is the condition required for the

CLT to be valid. Kalos97 uses a slightly modiﬁed form of this correction to determine if the

requirements of the CLT are “substantially satisﬁed.” His relation is

which is equivalent to

The user is responsible for applying this check.

c. Forming Valid Conﬁdence Intervals for Non–TFC Bins: The amount of statistical

information available for non–TFC bins is limited to the mean and R. The VOV and the center

of the asymmetric conﬁdence can be obtained for all tally bins with a nonzero 15th entry on the

DBCN card in the initial problem. The magnitude criteria for R (and the VOV, if available)

should be met before forming a conﬁdence interval. If the shifted conﬁdence interval center is

available, it should be used to form asymmetric conﬁdence intervals about the estimated mean.

History dependent information about R (and the VOV, if available) for non–TFC bins can be

obtained by printing out the tallies periodically during a calculation using the PRDMP card. The

N–dependent behavior of R can then be assessed. The complete statistical information available

can be obtained by creating a new tally and selecting the desired tally bin with the TFn card.

SHIFT Σxix–()

32S2N()⁄=

SHIFT Σxi

33Σxi

2ΣxiN2Σxi

()

3N2

⁄+⁄–()2NΣxi

2Σxi

()

–()()⁄=

Σxix–()

3S3N«

SHIFT Sx2⁄«

April 10, 2000 2-121

CHAPTER 2

ESTIMATION OF THE MONTE CARLO PRECISION

2. Information Available for Forming Statistically Valid Confidence Intervals for TFC Bins

Additional information about the statistical behavior of each TFC bin result is available. A TFC

bin table is produced by MCNP after each tally to provide the user with detailed information

about the apparent quality of the TFC bin result. The contents of the table are discussed in the

following subsections, along with recommendations for forming valid conﬁdence intervals using

this information.

a. TFC Bin Tally Information: The ﬁrst part of the TFC bin table contains information

about the TFC bin result including the mean, R, scoring efﬁciency, the zero and nonzero history

score components of R (see page 2–109), and the shifted conﬁdence interval center. The two

components of R can be used to improve the problem efﬁciency by either improving the history

scoring efﬁciency or reducing the range of nonzero history scores.

b. The Largest TFC Bin History Score Occurs on the Next History: There are occasions

when the user needs to make a conservative estimate of a tally result. Conservative is deﬁned so

that the results will not be less than the expected result. One reasonable way to make such an

estimate is to assume that the largest observed history score would occur again on the very next

history, N + 1.

MCNP calculates new estimated values for the mean, R, VOV, FOM, and shifted conﬁdence

interval center for the TFC bin result for this assumption. The results of this proposed occurrence

are summarized in the TFC bin information table. The user can assess the impact of this

hypothetical happening and act accordingly.

c. Description of the 10 Statistical Checks for the TFC Bin: MCNP prints the results of

ten statistical checks of the tally in the TFC bin at each print. In a “Status of Statistical Checks”

table, the results of these ten checks are summarized at the end of the output for all TFC bin

tallies. The quantities involved in these checks are the estimated mean, R, VOV, FOM, and the

large history score behavior of f(x). Passing all of the checks should provide additional assurance

that any conﬁdence intervals formed for a TFC bin result will cover the expected result the

correct fraction of the time. At a minimum, the results of these checks provide the user with more

information about the statistical behavior of the result in the TFC bin of each tally.

The following 10 statistical checks are made on the TFCs printed at the end of the output for

desirable statistical properties of Monte Carlo solutions:

MEAN

(1) a nonmonotonic behavior (no up or down trend) in the estimated mean as a

function of the number histories N for the last half of the problem;

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(2) an acceptable magnitude of the estimated R of the estimated mean (< 0.05 for a

point detector tally or < 0.10 for a non-point detector tally);

(3) a monotonically decreasing R as a function of the number histories Nfor the last

half of the problem;

(4) a decrease in the R as a function of N for the last half of the problem;

VOV

(5) the magnitude of the estimated VOV should be less than 0.10 for all types of

tallies;

(6) a monotonically decreasing VOV as a function of N for the last half of the

problem;

(7) a 1/N decrease in the VOV as a function of Nfor the last half of the problem;

FOM

(8) a statistically constant value of the FOM as a function of Nfor the last half of the

problem;

(9) a nonmonotonic behavior in the FOM as a function of N for the last half of the

problem; and

f(x)

(10) the SLOPE (see page 2–118) of the 25 to 201 largest positive (negative with a

negative DBCN(16) entry) history scores xshould be greater than 3.0 so that the

second moment will exist if the SLOPE is extrapolated to inﬁnity.

The seven N-dependent checks for the TFC bin are for the last half of the problem. The last half

of the problem should be well behaved in the sense of the CLT to form the most valid conﬁdence

intervals. “Monotonically decreasing” in checks 3 and 5 allows for some increases in both R and

the VOV. Such increases in adjacent TFC entries are acceptable and usually do not, by

themselves, cause poor conﬁdence intervals. A TFC bin R that does not pass check 3, by

deﬁnition in MCNP, does not pass check 4. Similarly, a TFC bin VOV that does not pass check

6, by deﬁnition, does not pass check 7.

A table is printed after each tally for the TFC bin result that summarizes the results and the pass

or no-pass status of the checks. Both asymmetric and symmetric conﬁdence intervals are printed

for the one, two, and three σlevels when all of the statistical checks are passed. These intervals

can be expected to be correct with improved probability over historical rules of thumb. This is

NOT A GUARANTEE, however; there is always a possibility that some as–yet–unsampled

portion of the problem would change the conﬁdence interval if more histories were calculated.

1N⁄

x2fx()xd

∞–

∞

∫

April 10, 2000 2-123

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ESTIMATION OF THE MONTE CARLO PRECISION

A WARNING is printed if one or more of these ten statistical checks is not passed, and one page

of printed plot information about f(x) is produced for the user to examine.

An additional information-only check is made on the largest ﬁve f(x) score grid bins to determine

if there are bins that have no samples or if there is a spike in an f(x) that does not appear to have

an upper limit. The result of the check is included in the TFC summary table for the user to

consider. This check is not a pass or no-pass test because a hole in the tail may be appropriate

for a discrete f(x) or an exceptional sample occurred with so little impact that none of the ten

checks was affected. The empirical f(x) should be examined to assess the likelihood of

“complete” sampling.

d. Forming Valid TFC Bin Conﬁdence Intervals: For TFC bin results, the highest

probability of creating a valid conﬁdence interval occurs when all of the statistical checks are

passed. Not passing several of the checks is an indication that the conﬁdence interval is less

likely to be correct. A monotonic trend in the mean for the last half of the problem is a strong

indicator that the conﬁdence interval is likely to produce incorrect coverage rates. The

magnitudes of R and the VOV should be less than the recommended values to increase the

likelihood of a valid conﬁdence interval. Small jumps in the R, VOV, and/or the FOM as a

function of Nare not threatening to the quality of a result. The slope of f(x) is an especially strong

indicator that Nhas not approached inﬁnity in the sense of the CLT. If the slope appears too

shallow (< 3), check the printed plot of f(x) to see that the estimated Pareto ﬁt is adequate. The

use of the shifted conﬁdence interval is recommended, although it will be a small effect for a

well–converged problem.

The last half of the problem is determined from the TFC. The more information available about

the last half of the problem, the better the N-dependent checks will be. Therefore, a problem that

has run 40,000 histories will have 20 TFC N entries, which is more Nentries than a 50,000

history problem with 13 entries. It is possible that a problem that passes all tests at 40,000 may

not pass all the tests at 40,001. As is always the case, the user is responsible for deciding when

a conﬁdence interval is valid. These statistical diagnostics are designed to aid in making this

decision.

J. A Statistically Pathological Output Example

A statistically pathological test problem is discussed in this section. The problem calculates the

surface neutron leakage ﬂux above 12 MeV from an isotropic 14 MeV neutron point source of

unit strength at the center of a 30 cm thick concrete shell with an outer radius of 390 cm. Point

and ring detectors were deliberately used to estimate the surface neutron leakage ﬂux with highly

inefﬁcient, long-tailed f(x)s. The input is shown on page 5–50.

The variance reduction methods used were implicit capture with weight cutoff, low-score point

detector Russian roulette, and a 0.5 mean free path (4 cm) neighborhood around the detectors to

2-124 April 10, 2000

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ESTIMATION OF THE MONTE CARLO PRECISION

produce large, but ﬁnite, higher moments. Other tallies or variance reduction methods could be

used to make this calculation much more efﬁcient, but that is not the object of this example. A

surface ﬂux estimator would have been over a factor of 150 to 30,000 times more efﬁcient than

ring and point detectors, respectively.

Figure 2.16 shows MCNP plots of the estimated mean, R, VOV and slope of the history score

PDF as a function of N values of 20,000 (left column) and 5 million (right column). The ring

detector results are shown as the solid line and the point detector result is the dashed line.

Column 1 shows the results as a function of N for 20,000 histories. The point detector result at

14,000 histories (not shown) was (R=0.041). The FOM varied somewhat

randomly between about 800 and 1160 for the last half of the problem. With no other

information, this result could be accepted by even a careful Monte Carlo practitioner. However,

the VOV never gets close to the required 0.1 value and the slope of the unbounded f(x) is less

than 1.4. This slope could not continue indeﬁnitely because even the mean of f(x) would not

exist. Therefore, a conﬁdence interval should not be formed for this tally. At 20,000 histories, R

increases substantially and the FOM crashes, indicating serious problems with the result.

The ring detector result is having problems of its own. The ring detector result for 14,000

histories was (R=0.17, VOV=0.35, slope=2.1, FOM=67). None of the

plotted quantities satisﬁes the required convergence criteria.The correct detector result, obtained

from a 5 million history ring detector tally, is (R=0.0169, VOV=0.023,

slope=4.6, FOM=19). The apparently converged 14,000 history point detector result is a factor

of four below the correct result!

If you were to run 200,000 histories, you would see the point detector result increasing to

(R=0.20, VOV=0.30, slope=1.6, FOM=1.8). The magnitudes of R and the

VOV are much too large for the point detector result to be accepted. The slope of f(x) is slowly

increasing, but has only reached a value of 1.6. This slope is still far too shallow compared to the

required value of 3.0.

The ring detector result of (R=0.0579, VOV=0.122, slope=2.8, FOM=22)

at 192,000 histories is interesting. All of these values are close to being acceptable, but just miss

the requirements. The ring detector result is more than two estimated standard deviations below

the correct result.

Column 2 shows the results as a function of Nfor 5 million histories. The ring detector result of

(R=0.0169, VOV=0.023, slope=4.6, FOM=19) now appears very well

behaved in all categories. This tally passed all 10 statistical checks. There appears to be no

reason to question the validity of this tally. The point detector result is

(R=0.11, VOV=0.28, slope=2.1, FOM=0.45). The result is clearly improving, but does not meet

the acceptable criteria for convergence. This tally did not pass 3 out 10 statistical checks.

1.41 10 8–

×ncm

2s⁄⁄

4.60 10 8–

×ncm

2s⁄⁄

5.72 10 8–

×ncm

2s⁄⁄

3.68 10 8–

×ncm

2s⁄⁄

5.06 10 8–

×ncm

2s⁄⁄

5.72 10 8–

×ncm

2s⁄⁄

4.72 10 8–

×ncm

2s⁄⁄

April 10, 2000 2-125

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ESTIMATION OF THE MONTE CARLO PRECISION

Figure 2-16.

Mean

VOV

Slope

2-126 April 10, 2000

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When you compare the empirical point detector f(x)s for 14,000 and 200 million histories you

see that the 14,000 history f(x) clearly has unsampled regions in the tail, indicating incomplete

f(x) sampling.94 For the point detector, seven decades of x have been sampled by 200 million

histories compared to only three decades for 14,000 histories. The largest x’s occur from the

extremely difﬁcult to sample histories that have multiple small energy loss collisions close to the

detector. The 200 million history point detector result is (R=0.035,

VOV=0.60, slope=2.4, FOM=0.060). The point detector f(x) slope is increasing, but still is not

yet completely sampled. This tally did not pass 6 of 10 checks with 200 million histories. The

result is about 1.5 estimated standard deviations below the correct answer. It is important to note

that calculating a large number of histories DOES NOT guarantee a precise result. The more

compact empirical ring f(x) for 20 million histories appears to be completely sampled because

of the large slope. The results for 1 billion histories are shown in Ref. 86.

For difﬁcult to sample problems such as this example, it is possible that an even larger history

score could occur that would cause the VOV and possibly the slope to have unacceptable values.

The mean and RE will be much less affected than the VOV. The additional running time required

to reach acceptable values for the VOV and the slope could be prohibitive. The large history

score should NEVER be discarded from the tally result. It is important that the cause for the large

history score be completely understood. If the score was created by a poorly sampled region of

phase space, the problem should be modiﬁed to provide improved phase space sampling. It is

also possible that the large score was created by an extremely unlikely set of circumstances that

occurred “early” in the calculation. In this situation, if the RE is within the guidelines, the

empirical f(x) appears to be otherwise completely sampled, and the largest history score appears

to be a once in a lifetime occurrence, a good conﬁdence interval can still be formed. If a

conservative (large) answer is required, the printed result that assumes the largest history score

occurs on the very next history can be used.

Comparing several empirical f(x)s for the above problem with 200 million histories that have

been normalized so that the mean of each f(x) is unity, you see that the point detector at 390 cm

clearly is quite Cauchy–like (see Eq. (2.25) for many decades.93 The point detector at 4000 cm

is a much easier tally (by a factor of 10,000) as exhibited by the much more compact empirical

f(x). The large–score tail decreases in a manner similar to the negative exponential f(x). The

surface ﬂux estimator is the most compact f(x) of all. The blip on the high–score tail is caused

by the average cosine approximation of 0.05 between cosines of 0 and 0.1 (see page 2–80). This

tally is 30,000 times more efﬁcient than the point detector tally.

5.41 10 8–

×ncm

2s⁄⁄

April 10, 2000 2-127

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VARIANCE REDUCTION

VII. VARIANCE REDUCTION

A. General Considerations

1. Variance Reduction and Accuracy

Variance-reducing techniques in Monte Carlo calculations reduce the computer time required to

obtain results of sufﬁcient precision. Note that precision is only one requirement for a good

Monte Carlo calculation. Even a zero variance calculation cannot accurately predict natural

behavior if other sources of error are not minimized. Factors affecting accuracy were discussed

in Section VI beginning on page 2–99.

2. Two Choices That Affect Efficiency

The efﬁciency of a Monte Carlo calculation is affected by two choices, tally type and random

walk sampling. The tally choice (for example, point detector ﬂux tally vs. surface crossing ﬂux

tally) amounts to trying to obtain the best results from the random walks sampled. The chosen

random walk sampling amounts to preferentially sampling “important” random walks at the

expense of “unimportant” random walks. (A random walk is important if it has a large affect on

a tally.) These two choices usually affect the time per history and the history variance as

described in Sec. 3 below. MCNP estimates tallies of the form

< T >

by sampling particle histories that statistically produce the correct particle density .

The tally function is zero except where a tally is required. For example, for a surface

crossing tally (F1),Twill be one on the surface and zero elsewhere. MCNP variance reduction

techniques allow the user to try to produce better statistical estimates of N where T is large,

usually at the expense of poorer estimates where T is zero or small.

There are many ways to statistically produce . Analog Monte Carlo simply samples

the events according to their natural physical probabilities. In this way, an analog Monte Carlo

calculation estimates the number of physical particles executing any given random walk.

Nonanalog techniques do not directly simulate nature. Instead, nonanalog techniques are free to

do anything if N, hence <T>, is preserved. This preservation is accomplished by adjusting the

weight of the particles. The weight can be thought of as the number of physical particles

represented by the MCNP particle (see page 2–26). Every time a decision is made, the nonanalog

techniques require that the expected weight associated with each outcome be the same as in the

analog game. In this way, the expected number of physical particles executing any given random

walk is the same as in the analog game.

rvtNrvt,,()Tr vt,,()d

∫

Nr vt,,()

Trvt,,()

Nr vt,,()

2-128 April 10, 2000

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For example, if an outcome “A” is made qtimes as likely as in the analog game, when a particle

chooses outcome “A,” its weight must be multiplied by q−1to preserve the expected weight for

outcome “A.” Let p be the analog probability for outcome “A”; then pq is the nonanalog

probability for outcome “A.” If w0is the current weight of the particle, then the expected weight

for outcome “A” in the analog game is w0∗pand the expected weight for outcome “A” in the

nonanalog game is (w0/q)∗pq.

MCNP uses three basic types of nonanalog games: (1) splitting, (2) Russian roulette, and

(3) sampling from nonanalog probability density functions. The previous paragraph discusses

type 3. Splitting refers to dividing the particle's weight among two or more daughter particles

and following the daughter particles independently. Usually the weight is simply divided evenly

among kidentical daughter particles whose characteristics are identical to the parent except for

a factor 1/k in weight (for example, splitting in the weight window). In this case the expected

weight is clearly conserved because the analog technique has one particle of weight w0 at

, whereas the splitting results in k particles of weight w0/k at . In both cases

the outcome is weight w0 at .

Other splitting techniques split the parent particle into k, typically two, differing daughter

particles. The weight of the jth daughter represents the expected number of physical particles that

would select outcome jfrom a set of k mutually exclusive outcomes. For example, the MCNP

forced collision technique considers two outcomes: (1) the particle reaches a cell boundary

before collision, or (2) the particle collides before reaching a cell boundary. The forced collision

technique divides the parent particle representing w0physical particles into two daughter

particles, representing w1physical particles that are uncollided and w2 physical particles that

collide. The uncollided particle of weight w1is then put on the cell boundary. The collision site

of the collided particle of weight w2 is selected from a conditional distance-to-collision

probability density, the condition being that the particle must collide in the cell. This technique

preserves the expected weight colliding at any point in the cell as well as the expected weight

not colliding. A little simple mathematics is required to demonstrate this technique.

Russian roulette takes a particle at of weight w0 and turns it into a particle of weight

w1>w0with probability w0/w1and kills it (that is, weight=0) with probability (1−(w0/w1)). The

expected weight at is w1∗(w0/w1)+(1−(w0/w1)) ∗0=w0, the same as in the analog

game.

Some techniques use a combination of these basic games and DXTRAN uses all three.

3. Efficiency, Time per History, and History Variance

Recall from page 2–108 that the measure of efﬁciency for MCNP calculations is the

FOM: , where

rvt,,() rvt,,()

rvt,,()

FOM 1R2T()⁄≡

April 10, 2000 2-129

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VARIANCE REDUCTION

R2= sample relative standard deviation of the mean and

T= computer time for the calculation (in minutes).

Recall from Eqns. 2.17 and 2.19a that , where

S2=sample history variance,

N=number of particles, and

=sample mean.

Generally we are interested in obtaining the smallest R in a given time T. The equation above

indicates that to decrease Rit is desirable to: 1) decrease S and 2) increase N; that is, decrease

the time per particle history. Unfortunately, these two goals usually conﬂict. Decreasing S

normally requires more time because better information is required. Increasing Nnormally

increases Sbecause there is less time per history to obtain information. However, the situation

is not hopeless. It is often possible either to decrease S substantially without decreasing Ntoo

much or to increase N substantially without increasing Stoo much, so that Rdecreases.

Many variance reduction techniques in MCNP attempt to decrease Rby either producing or

destroying particles. Some techniques do both. In general, techniques that produce tracks work

by decreasing S(we hope much faster than Ndecreases) and techniques that destroy tracks work

by increasing N (we hope much faster than S increases).

4. Strategy

Successful use of MCNP variance reduction techniques is often difﬁcult, tending to be more art

than science. The introduction of the weight window generator has improved things, but the user

is still fundamentally responsible for the choice and proper use of variance reducing techniques.

Each variance reduction technique has its own advantages, problems, and peculiarities.

However, there are some general principles to keep in mind while developing a variance

reduction strategy.

Not surprisingly, the general principles all have to do with understanding both the physical

problem and the variance reduction techniques available to solve the problem. If an analog

calculation will not sufﬁce to calculate the tally, there must be something special about the

particles that tally. The user should understand the special nature of those particles that tally.

Perhaps, for example, only particles that scatter in particular directions can tally. After the user

understands why the tallying particles are special, MCNP techniques can be selected (or

developed by the user) that will increase the number of special particles followed.

After the MCNP techniques are selected the user typically has to supply appropriate parameters

to the variance reduction techniques. This is probably more difﬁcult than is the selection of

RSN⁄()x⁄=

2-130 April 10, 2000

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VARIANCE REDUCTION

techniques. The ﬁrst guess at appropriate parameters typically comes either from experience

with similar problems or from experience with an analog calculation of the current problem. It

is usually better to err on the conservative side; that is, too little biasing rather than too much

biasing. After the user has supplied parameters for the variance reduction techniques, a short

Monte Carlo run is done so that the effectiveness of the techniques and parameters can be

monitored with the MCNP output.

The MCNP output contains much information to help the user understand the sampling. This

information should be examined to ensure that

(1) the variance reduction techniques are improving the sampling of the particles that

tally;

(2) the variance reduction techniques are working cooperatively; that is, one is not

destructively interfering with another;

(3) the FOM table is not erratic, which would indicate poor sampling; and

(4) there is nothing that looks obviously ridiculous.

Unfortunately, analyzing the output information requires considerable thought and experience.

Reference 98 shows in detail strategies and analysis for a particular problem.

After ascertaining that the techniques are improving the calculation, the user makes a few more

short runs to reﬁne the parameters until the sampling no longer improves. The weight window

generator can also be turned on to supply information about the importance function in different

regions of the phase space. This rather complex subject is described on page 2–139.

5. Erratic Error Estimates

Erratic error estimates are sometimes observed in MCNP calculations. In fact, the primary

reason for the Tally Fluctuation Chart (TFC) table in the MCNP output is to allow the user to

monitor the FOM and the relative error as a function of the number of histories. With few

exceptions, such as an analog point detector embedded in a scattering medium with Ro= 0 (a

practice highly discouraged), MCNP tallies are ﬁnite variance tallies. For ﬁnite variance tallies

the relative error should decrease roughly as so the FOM should be roughly constant and

the ten statistical checks of the tallies (see page 2–121) should all be passed. If the statistical

checks are not passed, the error estimates should be considered erratic and unreliable, no matter

how small the relative error estimate is.

Erratic error estimates occur typically because a high-weight particle tallies from an important

region of phase space that has not been well sampled. A high-weight particle in a given region

of phase space is a particle whose weight is some nontrivial fraction of \underbar{all} the weight

that has tallied from that region because of all previous histories. A good example is a particle

that collides very close to a point or ring detector. If not much particle weight has previously

April 10, 2000 2-131

CHAPTER 2

VARIANCE REDUCTION

collided that close to the detector, the relative error estimate will exhibit a jump for that history.

Another example is coherent photon scattering towards a point detector (see page 2–62).

To avoid high-weight particles in important regions, the user should try to ensure that these

regions are well sampled by many particles and try to minimize the weight ﬂuctuation among

these particles. Thus the user should try to use biasing techniques that preferentially push

particles into important regions without introducing large weight ﬂuctuations in these regions.

The weight window can often be very useful in minimizing weight ﬂuctuations caused by other

variance reduction techniques.

If, despite a user's efforts, an erratic error estimate occurs, the user should obtain event logs for

those particles causing the estimate to be erratic. The event logs should be studied to learn what

is special about these particles. When the special nature of these particles is understood, the user

can adjust the variance reduction techniques to sample these particles more often. Thus their

weight will be smaller and they will not be as likely to cause erratic estimates. Under absolutely

no circumstances should these particles be discarded or ignored! The fact that these particles

contribute very heavily to the tally indicates that they are important to the calculation and the

user should try to sample more of them.

6. Biasing Against Random Walks of Presumed Low Importance

It was mentioned earlier that one should be cautious and conservative when applying variance

reduction techniques. Many more people get into trouble by overbiasing than by underbiasing.

Note that preferentially sampling some random walks means that some walks will be sampled

(for a given computer time) less frequently than they would have been in an analog calculation.

Sometimes these random walks are so heavily biased against that very few, or even none, are ever

sampled in an actual calculation because not enough particles are run.

Suppose that (on average) for every million histories only one track enters cell 23. Further

suppose that a typical run is 100,000 histories. On any given run it is unlikely that a track enters

cell~23. Now suppose that tracks entering cell 23 turn out to be much more important than a

user thought. Maybe 10% of the answer should come from tracks entering cell 23. The user

could run 100,000 particles and get 90% of the true tally with an estimated error of 1%, with

absolutely no indication that anything is amiss. However, suppose the biasing had been set such

that (on average) for every 10,000 particles, one track entered cell 23, about 10 tracks total. The

tally probably will be severely affected by at least one high weight particle and will hover closer

to the true tally with a larger and perhaps erratic error estimate. The essential point is this:

following ten tracks into cell 23 does not cost much computer time and it helps ensure that the

estimated error cannot be low when the tally is seriously in error. Always make sure that all

regions of the problem are sampled enough to be certain that they are unimportant.

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B. Variance Reduction Techniques

There are four classes of variance reduction techniques99 that range from the trivial to the

esoteric.

Truncation Methods are the simplest of variance reduction methods. They speed up calculations

by truncating parts of phase space that do not contribute signiﬁcantly to the solution. The

simplest example is geometry truncation in which unimportant parts of the geometry are simply

not modeled. Speciﬁc truncation methods available in MCNP are energy cutoff and time cutoff.

Population Control Methods use particle splitting and Russian roulette to control the number of

samples taken in various regions of phase space. In important regions many samples of low

weight are tracked, while in unimportant regions few samples of high weight are tracked. A

weight adjustment is made to ensure that the problem solution remains unbiased. Speciﬁc

population control methods available in MCNP are geometry splitting and Russian roulette,

energy splitting/roulette, weight cutoff, and weight windows.

Modiﬁed Sampling Methods alter the statistical sampling of a problem to increase the number of

tallies per particle. For any Monte Carlo event it is possible to sample from any arbitrary

distribution rather than the physical probability as long as the particle weights are then adjusted

to compensate. Thus with modiﬁed sampling methods, sampling is done from distributions that

send particles in desired directions or into other desired regions of phase space such as time or

energy, or change the location or type of collisions. Modiﬁed sampling methods in MCNP

include the exponential transform, implicit capture, forced collisions, source biasing, and

neutron-induced photon production biasing.

Partially-Deterministic Methods are the most complicated class of variance reduction methods.

They circumvent the normal random walk process by using deterministic-like techniques, such

as next event estimators, or by controlling of the random number sequence. In MCNP these

methods include point detectors, DXTRAN, and correlated sampling.

The available MCNP variance reduction techniques now are described.

1. Energy Cutoff

The energy cutoff in MCNP is either a single user-supplied, problem-wide energy level or a cell-

dependent energy level. Particles are terminated when their energy falls below the energy cutoff.

The energy cutoff terminates tracks and thus decreases the time per history. The energy cutoff

should be used only when it is known that low-energy particles are either of zero or almost zero

importance. An energy cutoff is like a Russian roulette game with zero survival probability. A

number of pitfalls exist.

April 10, 2000 2-133

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1. Remember that low-energy particles can often produce high-energy particles (for

example, fission or low-energy neutrons inducing high-energy photons). Thus, even

if a detector is not sensitive to low-energy particles, the low-energy particles may be

important to the tally.

2. The CUT card energy cutoff is the same throughout the entire problem. Often low-

energy particles have zero importance in some regions and high importance in others,

and so a cell-dependent energy cutoff is also available with the ELPT card.

3. The answer will be biased (low) if the energy cutoff is killing particles that might

otherwise have contributed. Furthermore, as the apparent error will go to zero

and therefore mislead the unwary. Serious consideration should be given to two

techniques discussed later, energy roulette and space-energy weight window, that are

always unbiased.

The energy cutoff has one advantage not directly related to variance reduction. A lower energy

cutoff requires more cross sections so that computer memory requirements go up and interactive

computing with a timesharing system is degraded.

2. Time Cutoff

The time cutoff in MCNP is a single user-supplied, problem-wide time value. Particles are

terminated when their time exceeds the time cutoff. The time cutoff terminates tracks and thus

decreases the computer time per history. A time cutoff is like a Russian roulette game with zero

survival probability. The time cutoff should only be used in time-dependent problems where the

last time bin will be earlier than the cutoff.

Although the energy and time cutoffs are similar, more caution must be exercised with the

energy cutoff because low energy particles can produce high energy particles, whereas a late

time particle cannot produce an early time particle.

3. Geometry Splitting with Russian Roulette

Geometry splitting/Russian roulette is one of the oldest and most widely used variance-reducing

techniques in Monte Carlo codes. When used judiciously, it can save substantial computer time.

As particles migrate in an important direction, they are increased in number to provide better

sampling, but if they head in an unimportant direction, they are killed in an unbiased manner to

avoid wasting time on them. Oversplitting, however, can substantially waste computer time.

Splitting generally decreases the history variance but increases the time per history, whereas

Russian roulette generally increases the history variance but decreases the time per history.

Each cell in the problem geometry setup is assigned an importance Iby the user on the IMP input

card. The number I should be proportional to the estimated value that particles in the cell have

for the quantity being scored. When a particle of weight Wpasses from a cell of importance Ito

N∞→

2-134 April 10, 2000

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VARIANCE REDUCTION

one of higher importance , the particle is split into a number of identical particles of lower

weight according to the following recipe. If is an integer , the particle is split into

nidentical particles, each weighing W/n. Weight is preserved in the integer splitting process. If

is not an integer but still greater than 1, splitting is done probabilistically so that the

expected number of splits is equal to the importance ratio. Denoting to be the

largest integer in , is deﬁned. Then with probability p,n+ 1 particles are

used, and with probability 1−p,n particles are used. For example, if is 2.75, 75% of the

time split 3 for 1 and 25% of the time split 2 for 1. The weight assigned to each particle is

, which is the expected weight, to minimize dispersion of weights.

On the other hand, if a particle of weight Wpasses from a cell of importance Ito one of lower

importance I', so that I'/I < 1, Russian roulette is played and the particle is killed with

probability 1−(I'/I), or followed further with probability I'/I and weight .

Geometry splitting with Russian roulette is very reliable. It can be shown that the weights of all

particle tracks are the same in a cell no matter which geometrical path the tracks have taken to

get to the cell, assuming that no other biasing techniques, e.g. implicit capture, are used. The

variance of any tally is reduced when the possible contributors all have the same weight.

The assigned cell importances can have any value—they are not limited to integers. However,

adjacent cells with greatly different importances place a greater burden on reliable sampling.

Once a sample track population has deteriorated and lost some of its information, large splitting

ratios (like 20 to 1) can build the population back up, but nothing can regain the lost information.

It is generally better to keep the ratio of adjacent importances small (for example, a factor of a

few) and have cells with optical thicknesses in the penetration direction less than about two mean

free paths. MCNP prints a warning message if adjacent importances or weight windows have a

ratio greater than 4. PRINT TABLE 120 in the OUTP ﬁle lists the affected cells and ratios.

Generally, in a deep penetration shielding problem the sample size (number of particles)

diminishes to almost nothing in an analog simulation, but splitting helps keep the size built up.

A good rule is to keep the population of tracks traveling in the desired direction more or less

constant—that is, approximately equal to the number of particles started from the source. A

good initial approach is to split the particles 2 for 1 wherever the track population drops by a

factor of 2. Near-optimum splitting usually can be achieved with only a few iterations and

additional iterations show strongly diminishing returns. Note that in a combined neutron/photon

problem, importances will probably have to be set individually for neutrons and for photons.

MCNP never splits into a void, although Russian roulette can be played entering a void.

Splitting into a void accomplishes nothing except extra tracking because all the split particles

must be tracked across the void and they all make it to the next surface. The split should be done

according to the importance ratio of the last nonvoid cell departed and the ﬁrst nonvoid cell

entered.Note four more items:

I′I′I⁄nn 2≥()

I′I⁄nI′I⁄[]=

I′I⁄pI′In–⁄=I′I⁄

WII′⁄⋅

April 10, 2000 2-135

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1. Geometry splitting/Russian roulette works well only in problems that do not have

extreme angular dependence. In the extreme case, splitting/Russian roulette can be

useless if no particles ever enter an important cell where the particles can be split.

2. Geometry splitting/Russian roulette will preserve weight variations. The technique is

“dumb” in that it never looks at the particle weight before deciding appropriate action.

An example is geometry splitting/Russian roulette used with source biasing.

3. Geometry splitting/Russian roulette are turned on or off together.

4. Particles are killed immediately upon entering a zero importance cell, acting as a

geometry cutoff.

4. Energy Splitting/Roulette

Energy splitting and roulette typically are used together, but the user can specify only one if

desired. Energy splitting/roulette is independent of spatial cell. If the problem has a space-

energy dependence, the space-energy dependent weight window is normally a better choice.

1. Splitting: In some cases, particles are more important in some energy ranges than in

others. For example, it may be difficult to calculate the number of 235U fissions

because the thermal neutrons are also being captured and not enough thermal neutrons

are available for a reliable sample. In this case, once a neutron falls below a certain

energy level it can be split into several neutrons with an appropriate weight

adjustment. A second example involves the effect of fluorescent emission after

photoelectric absorption. With energy splitting, the low-energy photon track

population can be built up rather than rapidly depleted, as would occur naturally with

the high photoelectric absorption cross section. Particles can be split as they move up

or down in energy at up to five different energy levels.

Energy splitting can increase as well as decrease tally variances. Currently, the MCNP

weight cutoff game does not take into account whether a particle has undergone

energy splitting or not. Consequently, particles undergoing energy splitting may then

be rouletted by the weight cutoff game, defeating any advantages of the energy

splitting.

With only a minor modiﬁcation to MCNP, the mechanics for energy splitting can be

used for time splitting.

2. Russian roulette: In many cases the number of tracks increases with decreasing

energy, especially neutrons near the thermal energy range. These tracks can have

many collisions requiring appreciable computer time. They may be important to the

problem and cannot be completely eliminated with an energy cutoff, but their number

can be reduced by playing a Russian roulette game to reduce their number and

computer time.

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If a track's energy drops through a prescribed energy level, the roulette game is played,

based on the input value of the survival probability. If the game is won, the track's

history is continued, but its weight is increased by the reciprocal of the survival

probability to conserve weight.

5. Weight Cutoff

In weight cutoff, Russian roulette is played if a particle's weight drops below a user-speciﬁed

weight cutoff. The particle is either killed or its weight is increased to a user-speciﬁed level. The

weight cutoff was originally envisioned for use with geometry splitting/Russian roulette and

implicit capture, see page 2–144. Because of this intent,

1. The weight cutoffs in cell j depend not only on WC1 and WC2 on the CUT card, but

also on the cell importances.

2. Implicit capture is always turned on (except in detailed photon physics) whenever a

nonzero WC1 is specified.

Referring to item 1 above, the weight cutoff is applied when the particle’s weight falls below

Rj∗WC2, where Rjis the ratio of the source cell importance (IMP card) to cell j’s importance.

With probability W/(WC1 ∗Rj) the particle survives with new weight WC1 ∗Rj; otherwise the

particle is killed. When WC1 and WC2 on the CUT card are negative, the weight cutoff is scaled

to the minimum source weight of a particle so that source particles are not immediately killed

by falling below the cutoff.

As mentioned earlier, the weight cutoff game was originally envisioned for use with geometry

splitting and implicit capture. To illustrate the need for a weight cutoff when using implicit

capture, consider what can happen without a weight cutoff. Suppose a particle is in the interior

of a very large medium and there are neither time nor energy cutoffs. The particle will go from

collision to collision, losing a fraction of its weight at each collision. Without a weight cutoff, a

particle's weight would eventually be too small to be representable in the computer, at which

time an error would occur. If there are other loss mechanisms (for example, escape, time cutoff,

or energy cutoff), the particle’s weight will not decrease indeﬁnitely, but the particle may take

an unduly long time to terminate.

Weight cutoff's dependence on the importance ratio can be easily understood if one remembers

that the weight cutoff game was originally designed to solve the low-weight problem sometimes

produced by implicit capture. In a high-importance region, the weights are low by design, so it

makes no sense to play the same weight cutoff game in high- and low-importance regions.

Comments: Many techniques in MCNP cause weight change. The weight cutoff was really

designed with geometry splitting and implicit capture in mind. Care should be taken in the use

of other techniques.

April 10, 2000 2-137

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VARIANCE REDUCTION

Weight cutoff games are unlike time and energy cutoffs. In time and energy cutoffs, the random

walk is always terminated when the threshold is crossed. Potential bias may result if the particle's

importance was not zero. A weight cutoff (weight roulette would be a better name) does not bias

the game because the weight is increased for those particles that survive.

Setting the weight cutoff is not typically an easy task and requires thought and experimentation.

Essentially the user must guess what weight is worth following and start experimenting with

weight cutoffs in that vicinity.

6. Weight Window

The weight window (Fig. 2-17) is a space-energy-dependent splitting and Russian roulette

technique. For each space-energy phase space cell, the user supplies a lower weight bound. The

upper weight bound is a user-speciﬁed multiple of the lower weight bound. These weight bounds

deﬁne a window of acceptable weights. If a particle is below the lower weight bound, Russian

roulette is played and the particle's weight is either increased to a value within the window or the

particle is terminated. If a particle is above the upper weight bound, it is split so that all the split

particles are within the window. No action is taken for particles within the window.

Figure 2.18 is a more detailed picture of the weight window. Three important weights deﬁne the

weight window in a space-energy cell

poof W

Particles here: play

roulette, kill,

or move to W

Particles within

window: do

nothing

Particles here: split

Increasing

Weight Lower weight bound

specified for each

space-energy cell

Survival weight

specified as a constant

C times W

Upper weight bound

specified as a constant

C times W

The constants C

and C are for

the entire problem

Figure 2-17. Figure 2-18.

2-138 April 10, 2000

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VARIANCE REDUCTION

1. WL, the lower weight bound,

2. WS, the survival weight for particles playing roulette, and

3. WU, the upper weight bound.

The user speciﬁes WL for each space-energy cell on WWN cards. WS and WU are calculated

using two problem-wide constants, CS and CU (entries on the WWP card), as WS=CSWLand

WU=CUWL. Thus all cells have an upper weight bound CUtimes the lower weight bound and a

survival weight CS times the lower weight bound.

Although the weight window can be effective when used alone, it was designed for use with

other biasing techniques that introduce a large variation in particle weight. In particular, a

particle may have several “unpreferred” samplings, each of which will cause the particle weight

to be multiplied by a weight factor substantially larger than one. Any of these weight

multiplications by itself is usually not serious, but the cumulative weight multiplications can

seriously degrade calculational efﬁciency. Worse, the error estimates may be misleading until

enough extremely high-weight particles have been sampled. Monte Carlo novices are prone to

be mislead because they do not have enough experience reading and interpreting the summary

information on the sampling supplied by MCNP. Hence, a novice may put more faith in an

answer than is justiﬁed.

Although it is impossible to eliminate all pathologies in Monte Carlo calculations, a properly

speciﬁed weight window goes far toward eliminating the pathology referred to in the preceding

paragraph. As soon as the weight gets above the weight window, the particle is split and

subsequent weight multiplications will thus be multiplying only a fraction of the particle’s

weight (before splitting). Thus, it is hard for the tally to be severely perturbed by a particle of

extremely large weight. In addition, low-weight particles are rouletted, so time is not wasted

following particles of trivial weight.

One cannot ensure that every history contributes the same score (a zero variance solution), but

by using a window inversely proportional to the importance, one can ensure that the mean score

from any track in the problem isroughly constant. (A weight window generator exists to estimate

these importance reciprocals; see page 2–139.) In other words, the window is chosen so that the

track weight times the mean score (for unit track weight) is approximately constant. Under these

conditions, the variance is due mostly to the variation in the number of contributing tracks rather

than the variation in track score.

Thus far, two things remain unspeciﬁed about the weight window: the constant of inverse

proportionality and the width of the window. It has been observed empirically that an upper

weight bound ﬁve times the lower weight bound works well, but the results are reasonably

insensitive to this choice anyway. The constant of inverse proportionality is chosen so that the

April 10, 2000 2-139

CHAPTER 2

VARIANCE REDUCTION

lower weight bound in some reference cell is chosen appropriately. In most instances the

constant should be chosen so that the source particles start within the window.

1. Weight Window Compared to Geometry Splitting: Although both techniques use

splitting and Russian roulette, there are some important differences.

a. The weight window is space-energy dependent. Geometry splitting is only space

dependent.

b. The weight window discriminates on particle weight before deciding appropriate

action. Geometry splitting is done regardless of particle weight.

c. The weight window works with absolute weight bounds. Geometry splitting is

done on the ratio of the importance across a surface.

d. The weight window can be applied at surfaces, collision sites, or both. Geometry

splitting is applied only at surfaces.

e. The weight window can control weight ﬂuctuations introduced by other biasing

techniques by requiring all particles in a cell to have weight WL<W<WU. The

geometry splitting will preserve any weight ﬂuctuations because it is weight

independent.

f. In the rare case where no other weight modiﬁcation schemes are present,

importances will cause all particles in a given cell to have the same weight.

Weight windows will merely bound the weight.

g. The weight windows can be turned off for a given cell or energy regime by

specifying a zero lower bound. This is useful in long or large regions where no

single importance function applies. Care should be used because when the weight

window is turned off at collisions, the weight cutoff game is turned on,

sometimes causing too many particles to be killed.

2. The Weight Window Generator: The generator is a method that automatically

generates weight window importance functions.100 The task of choosing importances

by guessing, intuition, experience, or trial and error is simplified and insight into the

Monte Carlo calculation is provided.

Although the window generator has proved very useful, two caveats are appropriate.

The generator is by no means a panacea for all importance sampling problems and

certainly is not a substitute for thinking on the user's part. In fact, in most instances,

the user will have to decide when the generator's results look reasonable and when

they do not. After these disclaimers, one might wonder what use to make of a

2-140 April 10, 2000

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VARIANCE REDUCTION

generator that produces both good and bad results. To use the generator effectively, it

is necessary to remember that the generated parameters are only statistical estimates

and that these estimates can be subject to considerable error. Nonetheless, practical

experience indicates that a user can learn to use the generator effectively to solve some

very difﬁcult transport problems.

Examples of the weight window generator are given in Ref. 98 and Ref. 100 and

should be examined before using the generator. Note that this importance estimation

scheme works regardless of what other variance reduction techniques are used in a

calculation.

3. Theory: The importance of a particle at a point Pin phase space equals the expected

score a unit weight particle will generate. Imagine dividing the phase space into a

number of phase space “cells” or regions. The importance of a cell then can be defined

as the expected score generated by a unit weight particle after entering the cell. Thus,

with a little bookkeeping, the cell's importance can be estimated as

After the importances have been generated, MCNP assigns weight windows inversely

proportional to the importances. Then MCNP supplies the weight windows in an

output ﬁle suitable for use as an input ﬁle in a subsequent calculation. The spatial

portion of the phase space is divided using either standard MCNP cells or a

superimposed mesh grid, which can be either rectangular or cylindrical. The energy

portion of the phase space is divided using the WWGE card. The time portion of the

phase space can be divided also. The constant of proportionality is speciﬁed on the

WWG card.

4. Limitations of the Weight-Window Generator: The principal problem encountered

when using the generator is bad estimates of the importance function because of the

statistical nature of the generator. In particular, unless a phase space region is sampled

adequately, there will be either no generator importance estimate or an unreliable one.

The generator often needs a very crude importance guess just to get any tally; that is,

the generator needs an initial importance function to estimate a (we hope) better one

for subsequent calculations.

Fortunately, in most problems the user can guess some crude importance function

sufﬁcient to get enough tallies for the generator to estimate a new set of weight

windows. Because the weight windows are statistical, several iterations usually are

Importance

(expected score)

total score because of particles (and

their progeny) entering the cell

total weight weight entering the cell

April 10, 2000 2-141

CHAPTER 2

VARIANCE REDUCTION

required before the optimum importance function is found for a given tally. The ﬁrst

set of generated weight windows should be used in a subsequent calculation, which

generates a better set of windows, etc.

In addition to iterating on the generated weight windows, the user must exercise some

degree of judgment. Speciﬁcally, in a typical generator calculation, some generated

windows will look suspicious and will have to be reset. In MCNP, this task is

simpliﬁed by an algorithm that automatically scrutinizes cell-based importance

functions, either input by the user or generated by a generator. By ﬂagging the

generated windows that are more than a factor of 4 different from those in adjacent

spatial regions, often it is easy to determine which generated weight windows are

likely to be statistical ﬂukes that should be revised before the next generator iteration.

For example, suppose the lower weight bounds in adjacent cells were 0.5, 0.3, 0.9,

0.05, 0.03, 0.02, etc.; here the user would probably want to change the 0.9 to

something like 0.1 to ﬁt the pattern, reducing the 18:1 ratio between cells 3 and 4.

The weight window generator also will fail when phase space is not sufﬁciently

subdivided and no single set of weight window bounds is representative of the whole

region. It is necessary to turn off the weight windows (by setting a lower bound of

zero) or to further subdivide the geometry or energy phase space. Use of a

superimposed importance mesh grid for weight window generation is a good way to

subdivide the spatial portion of the phase space without complicating the MCNP cell

geometry.

On the other hand, the weight window generator will also fail if the phase space is too

ﬁnely subdivided and subdivisions are not adequately sampled. Adequate sampling of

the important regions of phase space is always key to accurate Monte Carlo

calculations and the weight window generator is a tool to help the user determine the

important phase space regions. When using the mesh-based weight window generator,

resist the temptation to create mesh cells that are too small.

7. Exponential Transform

The exponential transform samples the distance to collision from a nonanalog probability

density function. Although many impressive results are claimed for the exponential transform,

it should be remembered that these results are usually obtained for one-dimensional geometries

and quite often for energy-independent problems. A review article by Clark101 gives theoretical

background and sample results for the exponential transform. Sarkar and Prasad102 have done a

purely analytical analysis for the optimum transform parameter for an inﬁnite slab and one

energy group. The exponential transform allows particle walks to move in a preferred direction

by artiﬁcially reducing the macroscopic cross section in the preferred direction and increasing

the cross section in the opposite direction according to

2-142 April 10, 2000

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VARIANCE REDUCTION

where Σt*=ﬁctitious transformed cross section,

Σt= true total cross section,

Σa= absorption cross section,

Σs= scattering cross section,

p= the exponential transform parameter used to vary the degree

of biasing |p|< 1. Can be a constant or p= Σa/Σt, in which case

Σt*= Σs, and

µ= cosine of the angle between the preferred direction and the

particle's direction. . The preferred direction can be

speciﬁed on a VECT card.

At a collision a particle's weight is multiplied by a factor wc(derived below) so that the expected

weight colliding at any point is preserved. The particle's weight is adjusted such that the weight

multiplied by the probability that the next collision is in ds about s remains constant.

The probability of colliding in ds about s is

where Σ is either Σtor Σt*, so that preserving the expected collided weight requires

If the particle reaches a cell surface, time cutoff, DXTRAN sphere, or tally segment instead of

colliding, the particle's weight is adjusted so that the weight, multiplied by the probability that

the particle travels a distance s to the surface, remains constant. The probability of traveling a

distance s without collision is

so that preserving the expected uncollided weight requires

, or

ΣtΣt1pµ–()=

µ1≤

ΣeΣs–sd

ΣteΣts–sdw

cΣteΣts–sd=*

ΣteΣts–

---------------- eρΣtµs–

1pµ–

----------------==

eΣs–

eΣts–wseΣts–

April 10, 2000 2-143

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VARIANCE REDUCTION

For one–dimensional deep penetration through highly absorbing media, the variance typically

will decrease as p goes from zero to some p', and then increase as p goes from p' to one. For

p<p', the solution is “underbiased” and for p>p', the solution is “overbiased.”

Choosing p' is usually a matter of experience, although some insight may be gleaned by

understanding what happens in severely underbiased and severely overbiased calculations. For

illustration, apply the variance analysis of page 2–109 to a deep penetration problem when the

exponential transform is the only nonanalog technique used. In a severely underbiased

calculation , very few particles will score, but those that do will all contribute unity.

Thus the variance in an underbiased system is caused by a low scoring efﬁciency rather than a

large dispersion in the weights of the penetrating particles. In a severely overbiased system

particles will score, but there will be a large dispersion in the weights of the penetrating

particles with a resulting increase in variance.

Comments: MCNP gives a warning message if the exponential transform is used without a

weight window. There are numerous examples where an exponential transform without a weight

window gives unreliable means and error estimates. However, with a good weight window both

the means and errors are well behaved. The exponential transform works best on highly

absorbing media and very poorly on highly scattering media. For neutron penetration of concrete

or earth, experience indicates that a transform parameter p= 0.7 is about optimal. For photon

penetration of high-Z material, even higher values such as p= 0.9 are justiﬁed.

The following explains what happens with an exponential transform without a weight window.

For simplicity consider a slab of thickness T with constant Σt. Let the tally be a simple count

(F1 tally) of the weight penetrating the slab and let the exponential transform be the only

nonanalog technique used. Suppose for a given penetrating history that there are kﬂights, mthat

collide and n that do not collide. The penetrating weight is thus:

However, the particle's penetration of the slab means that

and hence

wseΣts–

eΣts–

-----------eρΣtµs–

p0→()

p1→()

wpeρΣtµisi

–

1pµi

–()

---------------------- eρΣtµjsj

–

jm1+=

∏

i1=

∏

µlsl

l1=

∑T=

2-144 April 10, 2000

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VARIANCE REDUCTION

The only variation in wpis because of the (1−pµ)−1 factors that arise only from collisions. For

a perfectly absorbing medium, every particle that penetrates scores exactly . If a particle

has only a few collisions, the weight variation will be small compared to a particle that has many

collisions. The weight window splits the particle whenever the weight gets too large, depriving

the particle of getting a whole series of weight multiplications upon collision that are

substantially greater than one.

By setting p=Σ

a/Σtand µ=1so that Σ*=Σ

s, we sample distance to scatter rather than distance

to collision. It is preferable to sample distance to scatter in highly absorbing media — in fact,

this is the standard procedure for astrophysics problems. Sampling distance to scatter is also

equivalent to implicit capture along a ﬂight path (see page 2–35). However, in such highly

absorbing media there is usually a more optimal choice of transform parameter, p, and it is

usually preferable to take advantage of the directional component by not ﬁxing µ = 1.

8. Implicit Capture

“Implicit capture,” “survival biasing,” and “absorption by weight reduction” are synonymous.

Implicit capture is a variance reduction technique applied in MCNP \underbar{after} the

collision nuclide has been selected. Let

σti = total microscopic cross section for nuclide i and

σai = microscopic absorption cross section for nuclide i.

When implicit capture is used rather than sampling for absorption with probability σai/σti, the

particle always survives the collision and is followed with new weight: W∗(1 − σai}/σti).

Implicit capture is a splitting process where the particle is split into absorbed weight (which need

not be followed further) and surviving weight.

Implicit capture can also be done along a ﬂight path rather than at collisions when a special form

of the exponential transform is used. See page 2–35 for details.

Two advantages of implicit capture are

1. a particle that has finally, against considerable odds, reached the tally region is not

absorbed just before a tally is made, and

2. the history variance, in general, decreases when the surviving weight (that is, 0 or W)

is not sampled, but an expected surviving weight is used instead (see weight cutoff,

page 2–136).

peρΣtT–1pµi

–()

1–

i1=

∏

epΣt

–T

April 10, 2000 2-145

CHAPTER 2

VARIANCE REDUCTION

Two disadvantages are

1. a fluctuation in particle weight is introduced, and

2. the time per history is increased (see weight cutoff, page 2–136).

9. Forced Collisions

The forced collision method is a variance reduction scheme that increases sampling of collisions

in speciﬁed cells. Because detector contributions and DXTRAN particles arise only from

collisions and at the source, it is often useful in certain cells to increase the number of collisions

that can produce large detector contributions or large weight DXTRAN particles. Sometimes

we want to sample collisions in a relatively thin cell (a fraction of a mean free path) to improve

the estimate of quantities like a reaction rate or energy deposition or to cause collisions that are

important to some other part of the problem.

The forced collision method splits particles into collided and uncollided parts. The collided part

is forced to collide within the current cell. The uncollided part exits the current cell without

collision and is stored in the bank until later when its track is continued at the cell boundary. Its

weight is

where Wo= current particle weight before forced collision,

d= distance to cell surface in the particle's direction, and

Σt= macroscopic total cross section of the cell material.

That is, the uncollided part is the current particle weight multiplied by the probability of exiting

the cell without collision.

The collided part has weight , which is the current particle weight

multiplied by the probability of colliding in the cell. The uncollided part is always produced. The

collided part may be produced only a fraction fof the time, in which case the collided weight is

. This is useful when several forced collision cells are adjacent or when too

much time is spent producing and following forced collision particles.

The collision distance is sampled as follows. If P(x) is the unconditional probability of colliding

within a distance x,P(x)/P(d) is the conditional probability of colliding within a distance xgiven

that a collision is known to occur within a distance d. Thus the position xof the collision must

be sampled on the interval 0<x<d within the cell according to ξ=P(x)/P(d), where

and ξ is a random number. Solving for x, one obtains

oeΣtd–

01eΣtd–

–()=

Wo1eΣtd–

–()f⁄

Px() 1exΣt

–

–=

2-146 April 10, 2000

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VARIANCE REDUCTION

Because a forced collision usually yields a collided particle having a relatively small weight,

care must be taken with the weight-cutoff game (page 2–136), the weight-window game

(page 2–137), and subsequent collisions of the particle within the cell. The weight window game

is not played on the surface of a forced collision cell that the particle is entering. For collisions

inside the cell the user has two options.

Option 1: (negative entry for the cell on the forced collision card.) After the forced collision,

subsequent collisions of the particle are sampled normally. The weight cutoff game is turned off

and detector contributions and DXTRAN particles are made before the weight window game is

played. If weight windows are used, they should be set to the weight of the collided particle

weight or set to zero if detector contributions or DXTRAN particles are desired.

Option 2: (positive entry for the cell on the forced collision card.) After the forced collision,

detector contributions or DXTRAN particles are made and either the weight cutoff or weight

window game is played. Surviving collided particles undergo subsequent forced collisions. If

weight windows are used, they should bracket the weight of particles entering the cell.

10. Source Variable Biasing

Provision is made for biasing the MCNP sources in any or all of the source variables speciﬁed.

MCNP's source biasing, although not completely general, allows the production of more source

particles, with suitably reduced weights, in the more important regimes of each variable. For

example, one may start more “tracks” at high energies and in strategic directions in a shielding

problem while correcting the distribution by altering the weights assigned to these tracks.

Sizable variance reductions may result from such biasing of the source. Source biasing samples

from a nonanalog probability density function.

If negative weight cutoff values are used on the CUT card, the weight cutoff is made relative to

the lowest value of source particle weight generated by the biasing schemes.

Source biasing is the only variance reduction scheme allowed with F8 tallies having energy

binning (see page 2–83).

1. Biasing by Specifying Explicit Sampling Frequencies: The SB input card determines

source biasing for a particular variable by specifying the frequency at which source

particles will be produced in the variable regime. If this fictitious frequency does not

correspond to the fraction of actual source particles in a variable bin, the corrected

weight of the source particles in a particular bin is determined by the ratio of the actual

frequency (defined on the SP card) divided by the fictitious frequency (defined on the

Σt

-----1ξ1edΣt

–

–()–[]ln–=

April 10, 2000 2-147

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VARIANCE REDUCTION

SB card) except for the lin-lin interpolation where it is defined to be the ratio of the

actual to fictitious frequency evaluated at the exact value of the interpolated variable.

The total weight of particles started in a given SI bin interval is thus conserved.

2. Biasing by Standard Prescription: Source biasing can use certain built-in prescriptions

similar in principle to built-in analytic source distributions. These biasing options are

detailed in the sections below for the appropriate source variables. The SB card input

is analogous to that of an SP card for an analytic source distribution; that is, the first

entry is a negative prescription number for the type of biasing required, followed by

one or more optional user-specified parameters, which are discussed in the following

sections.

a. Direction Biasing: The source direction can be biased by sampling from a continuous

exponential function or by using cones of ﬁxed size and starting a ﬁxed fraction of particles

within each cone. The user can bias particles in any arbitrary direction or combination of

directions.

In general, continuous biasing is preferable to ﬁxed cone biasing because cone biasing can cause

problems from the discontinuities of source track weight at the cone boundaries. However, if the

cone parameters (cone size and fraction of particles starting in the cone) are optimized through

a parameter study and the paths that tracks take to contribute to tallies are understood, ﬁxed cone

biasing sometimes can outperform continuous biasing. Unfortunately, it is usually time

consuming (both human and computer) and difﬁcult to arrive at the necessary optimization.

Source directional biasing can be sampled from an exponential probability density function

p(µ)=CeKµ, where Cis a norming constant equal to K/(eK−e−K) and , where is an

angle relative to the biasing direction. Kis typically about 1; K= 3.5 deﬁnes the ratio of weight

of tracks starting in the biasing direction to tracks starting in the opposite direction to be 1/1097.

This ratio is equal to e−2K.

Table 2.6 may help to give you a feel for the biasing parameter K.r

TABLE 2.6:

Exponential Biasing Parameter

KCumulative

Probability Theta Weight K Cumulative

Probability Theta Weight

.01 0 0 0.990 2.0 0 0.245

.25 60 0.995 .25 31 .325

.50 90 1.000 .50 48 .482

.75 120 1.005 .75 70 .931

1.00 180 1.010 1.00 180 13.40

µθcos=θ

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From this table for K=1, we see that half the tracks start in a cone of 64oopening about the axis,

and the weight of tracks at 64ois 0.762 times the unbiased weight of source particles. K= 0.01

is almost equivalent to no biasing, and K= 3.5 is very strong.

Cone directional biasing can be invoked by specifying cone cosines on the SI card, the true

distribution on the SP card, and the desired biasing probabilities on the SB card. Both histogram

and linear interpolation can be used. For example, consider the following case in which the true

distribution is isotropic:

The direction cosine relative to the reference direction, say v, is sampled uniformly within the

cone ν<v<1with probability p2and within −1 < v<νwith the complementary probability p1.

The weights assigned are W(1 − ν)/(2p2) and W(1 + ν)/(2p1), respectively. Note that for a very

small cone deﬁned by νand a high probability p2>> p1for being within the cone, the few source

particles generated outside the cone will have a very high weight that can severely perturb a tally.

The sampling of the direction cosines azimuthal to the reference axis is not biased.

b. Covering Cylinder Extent Biasing: This biasing prescription for the SDEF EXT

variable allows the automatic spatial biasing of source particles in a cylindrical-source-covering-

volume along the axis of the cylinder. Such biasing can aid in the escape of source particles from

optically thick source regions and thus represents a variance reduction technique.

c. Covering Cylinder or Sphere Radial Biasing: This biasing prescription for the SDEF

RAD variable allows for the radial spatial biasing of source particles in either a spherical or

cylindrical source covering volume. Like the previous example of extent biasing, this biasing can

be used to aid in the escape of source particles from optically thick source regions.

1.0 0 0 .432 3.5 0 0 .143

.25 42 .552 .25 23 .190

.50 64 .762 .50 37 .285

.75 93 1.230 .75 53 .569

1.00 180 3.195 1.00 180 156.5

TABLE 2.6:

Exponential Biasing Parameter

SIn 1v1–

SPn 01v+

------------1v–

---------

SBn0p1p2

April 10, 2000 2-149

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VARIANCE REDUCTION

3. Biasing Standard Analytic Source Functions:103 The preceding examples discuss the

biasing of source variables by either input of specific sampling frequencies

corresponding to SP card entries or by standard analytic biasing functions. A third

biasing category can be used in conjunction with standard analytic source probability

functions (for example, a Watt fission spectrum).

A negative entry on an SP card, that is,

SPn −iab

causes MCNP to sample source distribution n from probability function i with input

variables a,b,... . Sampling schemes are typically unbiasable. For example, for

SPn −5a

the evaporation spectrum f(E) =CEexp(−E/a) is sampled according to the sampling

prescription E=−alog (\ξ1∗ξ2), where ξi1and ξi2are random numbers. Biasing this

sampling scheme is usually very difﬁcult or impossible. Fortunately, there is an

approximate method available in MCNP for biasing any arbitrary probability

function.103 The code approximates the function as a table, then uses the usual SB card

biasing scheme to bias this approximate table function. The user inputs a coarse bin

structure to govern the bias and the code adds up to 300 additional equiprobable bins

to assure accuracy. For example, suppose we wish to sample the function

f(E) = C E exp(−E/a)

and suppose that we want half the source to be in the range .005 < E<.1and the other

half to be in the range .1 < E< 20. Then the input is

SPn -5 a

SIn .005 .1 20

SBn C 0 .5 1 .

MCNP breaks up the function into 150 equiprobable bins below E=.1and 150 more

equiprobable bins above E= .1. Half the time E is chosen from the upper set of bins

and half the time it is chosen from the lower set. Particles starting from the upper bins

have a different weight than that of particles starting from the lower bins to adjust for

the bias, and a detailed summarys provided when the PRINT option is used.

Note that in the above example the probability distribution function is truncated below

E= .005 and above E= 20. MCNP prints out how much of the distribution is lost in

this manner and reduces the weight accordingly.

It is possible for the user to choose a foolish biasing scheme. For example,

SPn -5 a

SIn .005 297I .1 20

SBn 0 1 298R

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VARIANCE REDUCTION

causes each of the 299 bins to be chosen with equal probability. This would be all right

except that since there are never more than 300 equiprobable bins, this allocates only

1 equiprobable bin per user-supplied bin. The single equiprobable bin for .1 < E<20

is inadequate to describe the distribution function over this range. Thus the table no

longer approximates the function and the source will be sampled erroneously. MCNP

issues an error message whenever too much of the source distribution is allocated to a

single equiprobable bin, alerting users to a poor choice of binning which might

inadequately represent the function. The coarse bins used for biasing should be chosen

so that the probability function is roughly equally distributed among them.

11. Point Detector Tally

The point detector is a tally and does not bias random walk sampling. Recall from Section VI,

however, that the tally choice affects the efﬁciency of a Monte Carlo calculation. Thus, a little

will be said here in addition to the discussion in the tally section.

Although ﬂux is a point quantity, ﬂux at a point cannot be estimated by either a track-length tally

(F4) or a surface ﬂux tally (F2) because the probability of a track entering the volume or crossing

the surface of a point is zero. For very small volumes, a point detector tally can provide a good

estimate of the ﬂux where it would be almost impossible to get either a track-length or surface-

crossing estimate because of the low probability of crossing into the small volume.

It is interesting that a DXTRAN sphere of vanishingly small size with a surface-crossing tally

across the diameter normal to the particle's trajectory is equivalent to a point detector. Thus,

many of the comments on DXTRAN are appropriate and the DXC cards essentially are identical

to the PD cards.

For a complete discussion of point detectors, see page 2–75.

12. DXTRAN

DXTRAN typically is used when a small region is being inadequately sampled because particles

have a very small probability of scattering toward that region. To ameliorate this situation, the

user can specify in the input ﬁle a DXTRAN sphere that encloses the small region. Upon

collision (or exiting the source) outside the sphere, DXTRAN creates a special “DXTRAN

particle” and deterministically scatters it toward the DXTRAN sphere and deterministically

transports it, without collision, to the surface of the DXTRAN sphere. The collision itself is

otherwise treated normally, producing a non-DXTRAN particle that is sampled in the normal

way, with no reduction in weight. However, the non-DXTRAN particle is killed if it tries to enter

the DXTRAN sphere. DXTRAN uses a combination of splitting, Russian roulette, and sampling

from a nonanalog probability density function.

April 10, 2000 2-151

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VARIANCE REDUCTION

The subtlety about DXTRAN is how the extra weight created for the DXTRAN particles is

balanced by the weight killed as non-DXTRAN particles cross the DXTRAN sphere. The non-

DXTRAN particle is followed without any weight correction, so if the DXTRAN technique is

to be unbiased, the extra weight put on the DXTRAN sphere by DXTRAN particles must

somehow (on average) balance the weight of non-DXTRAN particles killed on the sphere.

1. DXTRAN Viewpoint 1: One can view DXTRAN as a splitting process (much like the

forced collision technique) wherein each particle is split upon departing a collision (or

source point) into two distinct pieces:

a. the weight that does not enter the DXTRAN sphere on the next ﬂight, either

because the particle is not pointed toward the DXTRAN sphere or because

the particle collides before reaching the DXTRAN sphere, and

b. the weight that enters the DXTRAN sphere on the next ﬂight.

Let wobe the weight of the particle before exiting the collision, let p1be the analog probability

that the particle does not enter the DXTRAN sphere on its next ﬂight, and let p2be the analog

probability that the particle does enter the DXTRAN sphere on its next ﬂight. The particle must

undergo one of these mutually exclusive events, thus p1+p2= 1. The expected weight not

entering the DXTRAN sphere is w1=wop1, and the expected weight entering the DXTRAN

sphere is w2=wop2. Think of DXTRAN as deterministically splitting the original particle with

weight wointo two particles, a non-DXTRAN (particle 1) particle of weight w1and a DXTRAN

(particle 2) particle of weight w2. Unfortunately, things are not quite that simple.

Recall that the non-DXTRAN particle is followed with unreduced weight worather than weight

w1=wop1. The reason for this apparent discrepancy is that the non-DXTRAN particle (#1) plays

a Russian roulette game. Particle 1’s weight is increased from w1 to woby playing a Russian

roulette game with survival probability p1=w1/wo. The reason for playing this Russian roulette

game is simply that p1is not known, so assigning weight w1=p1woto particle 1 is impossible.

However, it is possible to play the Russian roulette game without explicitly knowing p1. It is not

magic, just slightly subtle.

The Russian roulette game is played by sampling particle 1 normally and keeping it only if it

does not enter (on its next ﬂight) the DXTRAN sphere; that is, particle 1 survives (by deﬁnition

of p1) with probability p1. Similarly, the Russian roulette game is lost if particle 1 enters (on its

next ﬂight) the DXTRAN sphere; that is, particle 1 loses the roulette with probability p2. To

restate this idea, with probability p1, particle 1 has weight woand does not enter the DXTRAN

sphere and with probability p2, the particle enters the DXTRAN sphere and is killed. Thus, the

expected weight not entering the DXTRAN sphere is wop1+0∗p2=w1, as desired.

So far, this discussion has concentrated on the non-DXTRAN particle and ignored exactly what

happens to the DXTRAN particle. The sampling of the DXTRAN particle will be discussed after

a second viewpoint on the non-DXTRAN particle.

2-152 April 10, 2000

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VARIANCE REDUCTION

2. DXTRAN Viewpoint 2: This second way of viewing DXTRAN does not see

DXTRAN as a splitting process but as an accounting process in which weight is both

created and destroyed on the surface of the DXTRAN sphere. In this view, DXTRAN

estimates the weight that should go to the DXTRAN sphere upon collision and creates

this weight on the sphere as DXTRAN particles. If the non-DXTRAN particle does

not enter the sphere, its next flight will proceed exactly as it would have without

DXTRAN, producing the same tally contributions and so forth. However, if the non-

DXTRAN particle's next flight attempts to enter the sphere, the particle must be killed

or there would be (on average) twice as much weight crossing the DXTRAN sphere

as there should be because the weight crossing the sphere has already been accounted

for by the DXTRAN particle.

3. The DXTRAN Particle: Although the DXTRAN particle does not confuse people

nearly as much as the non-DXTRAN particle, the DXTRAN particle is nonetheless

subtle.

The most natural approach for scattering particles toward the DXTRAN sphere would

be to sample the scattering angle proportional to the analog density. This approach

is not used because it is too much work to sample proportional to the analog density

and because it is sometimes useful to bias the sampling.

To sample in an unbiased fashion when it is known that points to the DXTRAN

sphere, one samples the conditional density

Pcon}( ) =P( )/ (the set S( ) points toward the sphere)

and multiplies the weight by , the probability of scattering into the

cone (see Fig. 2-19). However, it is too much work to calculate the above integral for

each collision. Instead, an arbitrary density function Parb( ) is sampled and the

weight is multiplied by

The total weight multiplication is the product of the fraction of the weight scattering

into the cone, , and the weight correction for sampling Parb( ) instead

of Pcon( ). Thus, the weight correction on scattering is

P()Parb().

Ω

ΩΩ

ΩΩ PΩ()Ωd

SΩ()

∫Ω

PΩ()Ω()d

SΩ()

∫

Ω

Pcon Ω()

Parb Ω()

-------------------- PΩ()

Parb Ω() PΩ()Ω()d

SΩ()

∫

------------------------------------------------------------=

PΩ()Ωd

SΩ()

∫Ω

Ω

ΩΩ

April 10, 2000 2-153

CHAPTER 2

VARIANCE REDUCTION

If µis the cosine of the angle between the scattering direction and the particle’s

incoming direction, then P()=P(µ)/(2π) because the scattering is symmetric in the

azimuthal angle. If is the cosine of the angle with respect to the cone axis (see

Fig. 2-19) and if the azimuthal angle about the cone axis is uniformly sampled, then

Parb()=Parb( )/(2π). Thus

= weight multiplier for DXTRAN particle.

This result can be obtained more directly, but the other derivation does not explain why

Pcon( ) is not sampled.

Because Parb( ) is arbitrary, MCNP can choose a scheme that samples from a two-

step density that favors particles within the larger interval. In fact, the inner

DXTRAN sphere has to do only with this arbitrary density and is not essential to the

DXTRAN concept. The DXTRAN particles are always created on the outside

DXTRAN sphere, with the inner DXTRAN sphere deﬁning only the boundary

between the two steps in the density function.

After = cos has been chosen, the azimuthal angle is sampled uniformly on

[0,2π]; this completes the scattering. Recall, however, that the DXTRAN particle

arrives at the DXTRAN sphere without collision. Thus the DXTRAN particle also has

its weight multiplied by the negative exponential of the optical path between the

collision site and the sphere.

4. Inside the DXTRAN Sphere: So far, only collisions outside the DXTRAN sphere have

been discussed. At collisions inside the DXTRAN sphere, the DXTRAN game is not

played because first, the particle is already in the desired region, and second, it is

impossible to define the angular cone of Fig. 2-19. If there are several DXTRAN

spheres and the collision occurs in sphere i, DXTRAN will be played for all spheres

except sphere i.

5. Terminology—Real particle and Pseudoparticle: Sometimes the\break DXTRAN

particle is called a pseudoparticle and the non-DXTRAN particle is called the original

or real particle. The terms “real particle” and “pseudoparticle” are potentially

misleading. Both particles are equally real: both execute random walks, both carry

nonzero weight, and both contribute to tallies. The only sense in which the DXTRAN

particle should be considered “pseudo” or “not real” is during creation. A DXTRAN

particle is created on the DXTRAN sphere, but creation involves determining what

weight the DXTRAN particle should have upon creation. Part of this weight

determination requires calculating the optical path between the collision site and the

DXTRAN sphere. This is done in the same way as point detectors (see point detector

pseudoparticles on page 2–90.) MCNP determines the optical path by tracking a

Ω

Ωη

Pµ()

Parb η()

-------------------

Ω

ηη

ηθ ϕ

2-154 April 10, 2000

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VARIANCE REDUCTION

pseudoparticle from the collision site to the DXTRAN sphere. This pseudoparticle is

deterministically tracked to the DXTRAN sphere simply to determine the optical path.

No distance to collision is sampled, no tallies are made, and no records of the

pseudoparticle's passage are kept (for example, tracks entering). In contrast, once the

DXTRAN particle is created at the sphere's surface, the particle is no longer a

pseudoparticle. The particle has real weight, executes random walks, and contributes

to tallies.

6. DXTRAN Details: To explain how the scheme works, consider the neighborhood of

interest to be a spherical region surrounding a designated point in space. In fact,

consider two spheres of arbitrary radii about the point Po=(xo,yo,zo). Further, assume

that the particle having direction (u,v,w) collides at the point P1=(x,y,z), as shown in

Fig. 2-19.

The quantities I,O,I,O,RI, and Roare deﬁned in the ﬁgure. Thus L, the

distance between the collision point and center of the spheres, is

On collision, a DXTRAN particle is placed at a point on the outer sphere of radius Ro

as described below. Provision is made for biasing the contributions of these DXTRAN

particles on the outer sphere within the cone deﬁned by the inner sphere. The weight

of the DXTRAN particle is adjusted to account for the probability of scattering in the

direction of the point on the outer sphere and traversing the distance with no further

collision.

The steps in sampling the DXTRAN particles are outlined:

(u,v,w)

P1P0

θθθ

ηθ

I= cos I

0= cos 0

0LI

Figure 2-19.

θθ ηη

Lxx

–()

2yy

–()

2zz

–()

++=

April 10, 2000 2-155

CHAPTER 2

VARIANCE REDUCTION

Sample =I+ξ(1 −I) uniformly in ( I,1) with probability

Q(1 − I)/[Q(1 − I) + I−O]

and with probability

(I−O)/[Q(1 − I) + I−O]

sample =O+ξ(I – O) uniformly in ( O,I). The quantity Q (equal to 5 in

MCNP) is a factor that measures the importance assigned to scattering in the inner

cone relative to the outer cone. Therefore, Q is also the ratio of weights for particles

put in the two different cones.

With = cos chosen, a new direction is computed by considering the

rotation through the polar angle (and a uniform azimuthal angle ) from the

reference direction

The particle is advanced in the direction to the surface of the sphere of

radius Ro. The new DXTRAN particle with appropriate direction and coordinates is

banked. The weight of the DXTRAN particle is determined by multiplying the weight

of the particle at collision by

and

where

µ=uu' +vv' +ww',

P(µ) = scattering probability density function for scattering through the angle

cos−1 µ in the lab system for the event sampled at (x,y,z),

ηIθI

cos L2RI

–()

12⁄L⁄==

ηOθO

cos L2Ro

–()

12⁄L⁄==

ηη η η

ηηηη

ηη η ηη

ηη η η ηη

ηθ u′v′w′,,()

θϕ

xox–

----------- yoy–

--------------zoz–

-------------





u′v′w′,,()

νPµ()Q1ηI

–()η

IηO

–+{}e

σts()sd

∫

–

---------------------------------------------------------------------------------------------- ηIη1≤≤,⋅

νPµ()Q1ηI

–()η

IηO

–+{}e

σts()sd

∫

–

ηOηη

≤≤,⋅

2-156 April 10, 2000

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VARIANCE REDUCTION

ν= number of particles emitted from the event, and

=the attenuation along the line between P1(x,y,z) and Ps, the point

on the sphere where the particle is placed.

In arriving at the weight factor, note that the density function for sampling is given

Thus the weight of the DXTRAN particle is the weight of the incoming particle at P1

modiﬁed by the ratio of the probability density function for actually scattering from

P1 and arriving at Pswithout collision to the density function actually sampled in

choosing Ps. Therefore, particles in the outer cone have weights Q=5times higher

than the weights of similar particles in the inner cone.

The attenuation is calculated at the energy obtained by scattering through the angle µ.

The energy is uniquely determined from µ in elastic scattering (and also in level

scattering), whereas for other nonelastic events, the energy is sampled from the

corresponding probability density function for energy, and may not depend on µ.

7. Auxiliary Games for DXTRAN: The major disadvantage to DXTRAN is the extra

time consumed following DXTRAN particles with low weights. Three special games

can control this problem:

1. DXTRAN weight cutoffs,

2. DXC games, and

3. DD game.

Particles inside a DXTRAN sphere are not subject to the normal MCNP weight cutoff

or weight window game. Instead DXTRAN spheres have their own weight cutoffs,

allowing the user to roulette DXTRAN particles that, for one reason or another, do not

have enough weight to be worth following.

Sometimes low-weighted DXTRAN particles occur because of collisions many free

paths from the DXTRAN sphere. The exponential attenuation causes these particles

to have extremely small weights. The DXTRAN weight cutoff will roulette these

particles only after much effort has been spent producing them. The DXC cards are

cell dependent and allow DXTRAN contributions to be taken only some fraction of

the time. They work just like the PD cards for detectors (see page 2–92). The user

Σts()sd

∫

–

QQ1ηI

–()η

IηO

–+[]⁄η

Iη1≤<,

1Q1ηI

–()η

IηO

–+[](),⁄η

Oηη

≤≤

April 10, 2000 2-157

CHAPTER 2

VARIANCE REDUCTION

speciﬁes a probability pithat a DXTRAN particle will be produced at a given collision

or source sampling in cell i. The DXTRAN result remains unbiased because when a

DXTRAN particle is produced its weight is multiplied by . (The non-DXTRAN

particle is treated exactly as before, unaffected unless it enters the DXTRAN sphere,

whereupon it is killed.) To see the utility, suppose that the DXTRAN weight cutoff was

immediately killing 99% of the DXTRAN particles from cell i. Only 1% of the

DXTRAN particles survive anyway, so it might be appropriate to produce only 1%

(pi= .01) and have these not be killed immediately by the DXTRAN weight cutoff. Or

the pi’s can often be set such that all DXTRAN particles from all cells are created on

the DXTRAN sphere with roughly the same weight. Choosing the pi’s is often difﬁcult

and the method works well typically when the material exponential attenuation is the

major source of the weight ﬂuctuation.

Often the weight ﬂuctuation arises because the probability P(µ) of scattering toward

the DXTRAN sphere varies greatly, depending on what nuclide is hit and what the

collision orientation is with respect to the DXTRAN sphere. For example, consider a

highly forward-peaked scattering probability density. If the DXTRAN sphere were

close to the particle’s precollision direction, P(µ) will be large; if the DXTRAN sphere

were at 105οto the precollision direction, P(µ) will be small. The DD game can be

used to reduce the weight ﬂuctuation on the DXTRAN sphere caused by these

geometry effects, as well as the material exponential attenuation effects.

The DD game selectively roulettes the DXTRAN pseudoparticles during creation,

depending on the DXTRAN particles’ weight compared to some reference weight.

This is the same game that is played on detector contributions, and is described on

page 2–92 The reference weight can be either a fraction of the average of previous

DXTRAN particle weights or a user input reference weight. Recall that a DXTRAN

particle's weight is computed by multiplying the exit weight of the non-DXTRAN

particle by a weight factor having to do with the scattering probability and the negative

exponential of the optical path between collision site and DXTRAN sphere. The

optical path is computed by tracking a pseudoparticle from collision to DXTRAN

sphere. The weight of the pseudoparticle is monotonically decreasing, so the DD game

compares the pseudoparticle's weight at the collision site and, upon exiting each cell,

against the reference weight. A roulette game is played when the pseudoparticle's

weight falls below the reference weight. The DD card stops tracking a pseudoparticle

as soon as the weight becomes inconsequential, saving time by eliminating subsequent

tracking.

8. Final Comments:

a. DXTRAN should be used carefully in optically thick problems. Do not rely

on DXTRAN to do penetration.

pi1–

2-158 April 10, 2000

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VARIANCE REDUCTION

b. If the source is user supplied, some provision must be made for obtaining

the source contribution to particles on the DXTRAN sphere.

c. Extreme care must be taken when more than one DXTRAN sphere is in a

problem. Cross-talk between spheres can result in extremely low weights

and an excessive growth in the number of particle tracks.

d. Never put a zero on the DXC card. A zero will bias the calculation by not

creating DXTRAN particles but still killing the non-DXTRAN particle if it

enters the DXTRAN sphere.

e. Usually there should be a rough balance in the summary table of weight

created and lost by DXTRAN.

f. DXTRAN cannot be used with reﬂecting surfaces for the same reasons that

point detectors cannot be used with reﬂecting surfaces. See page 2–92 for

further explanation.

g. Both DXTRAN and point detectors track pseudoparticles to a point.

Therefore, most of the discussion about detectors applies to DXTRAN.

Refer to the section on detectors, page 2–85, for more information.

13. Correlated Sampling

Correlated sampling estimates the change in a quantity resulting from a small perturbation of

any type in the problem. This technique enables the evaluation of small quantities that would

otherwise be masked by the statistical errors of uncorrelated calculations. MCNP correlates a

pair of runs by providing each new history in the unperturbed and perturbed problems with the

same initial pseudorandom number. The same sequence of subsequent numbers is used, until a

perturbation causes the sequences to diverge. This sequencing is done by incrementing the

random number generator at the beginning of each history by a stride Sof random numbers from

the beginning of the previous history. The default value of S is 152,917. The stride should be a

quantity greater than would be needed by most histories (see page 2–187).

MCNP does not provide an estimate of the error in the difference. Reference 98 shows how the

error in the difference between two correlated runs can be estimated. A postprocessor code

would have to be written to do this.

Correlated sampling should not be confused with more elaborate Monte Carlo perturbation

schemes that calculate differences and their variances directly. MCNP has no such scheme at

present.

April 10, 2000 2-159

CHAPTER 2

CRITICALITY CALCULATIONS

VIII.CRITICALITY CALCULATIONS

Nuclear criticality, the ability to sustain a chain reaction by ﬁssion neutrons, is characterized by

keff, the eigenvalue to the neutron transport equation. In reactor theory, keff is thought of as the

ratio between the number of neutrons in successive generations, with the ﬁssion process

regarded as the birth event that separates generations of neutrons.104 For critical systems, keff =1

and the chain reaction will just sustain itself. For subcritical systems, keff <1and the chain

reaction will not sustain itself. For supercritical systems, keff >1 and the number of ﬁssions in

the chain reaction will increase with time. In addition to the geometry description and material

cards, all that is required to run a criticality problem is a KCODE card, described below, and an

initial spatial distribution of ﬁssion points using either the KSRC card, the SDEF card, or an

SRCTP ﬁle.

Calculating keff consists of estimating the mean number of ﬁssion neutrons produced in one

generation per ﬁssion neutron started. A generation is the life of a neutron from birth in ﬁssion

to death by escape, parasitic capture, or absorption leading to ﬁssion. In MCNP, the

computational equivalent of a ﬁssion generation is a keff cycle; i.e., a cycle is a computed

estimate of an actual ﬁssion generation. Processes such as (n,2n) and (n,3n) are considered

internal to a cycle and do not act as termination. Because ﬁssion neutrons are terminated in each

cycle to provide the ﬁssion source for the next cycle, a single history can be viewed as continuing

from cycle to cycle. The effect of the delayed neutrons is included by using the total . The

spectrum of delayed neutrons is assumed to be the same as neutrons from prompt ﬁssion. In a

Mode N,P problem, secondary photon production from neutrons is turned off during inactive

cycles. MCNP uses three different estimators for keff. We recommend using, for the ﬁnal keff

result, the statistical combination of all three.105

It is extremely important to emphasize that the result from a criticality calculation is a conﬁdence

interval for keff that is formed using the ﬁnal estimated keff and the estimated standard deviation.

A properly formed conﬁdence interval from a valid calculation should include the true answer

the fraction of time used to deﬁne the conﬁdence interval. There will always be some probability

that the true answer lies outside of a conﬁdence interval.

Reference 106 is an introduction to using MCNP for criticality calculations, focusing on the

unique aspects of setting up and running a criticality problem and interpreting the results. A

quickstart chapter gets the new MCNP user on the computer running a simple criticality problem

as quickly as possible.

A. Criticality Program Flow

Because the calculation of keff entails running successive ﬁssion cycles, criticality calculations

have a different program ﬂow than MCNP ﬁxed source problems. They require a special

2-160 April 10, 2000

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criticality source that is incompatible with the surface source and user-supplied sources. Unlike

ﬁxed source problems, where the source being sampled throughout the problem never changes,

the criticality source changes from cycle to cycle.

1. Criticality Problem Definition

To set up a criticality calculation, the user initially supplies an INP ﬁle that includes the KCODE

card with the following information:

1. the nominal number of source histories, N, per keff cycle;

2. an initial guess of keff;

3. the number of source cycles, Ic, to skip before keff accumulation;

4. the total number of cycles, It, in the problem.

Other KCODE entries are discussed in Chapter 3, page 3–70. The initial spatial distribution of

ﬁssion neutrons can be entered by using (1) the KSRC card with sets of x,y,z point locations,

(2) the SDEF card to deﬁne points uniformly in volume, or (3) a ﬁle (SRCTP) from a previous

MCNP criticality calculation. If the SDEF card is used, the default WGT value should not be

changed. Any KSRC points in geometric cells that are void or have zero importance are rejected.

The remaining KSRC points are duplicated or rejected enough times so the total number of

points M in the source spatial distribution is approximately the nominal source size N. The

energy of each source particle for the ﬁrst keff cycle is selected from a generic Watt thermal

ﬁssion distribution if it is not available from the SRCTP ﬁle.

2. Particle Transport for Each keff Cycle

In each keff cycle, M (varying with cycle) source particles are started isotropically. For the ﬁrst

cycle, these Mpoints come from one of three user–selected source possibilities. For subsequent

cycles, these points are the ones written at collision sites from neutron transport in the previous

cycle. The total source weight of each cycle is a constant N. That is, the weight of each source

particle is N/M, so all normalizations occur as if Nrather than Mparticles started in each cycle.

Source particles are transported through the geometry by the standard random walk process,

except that ﬁssion is treated as capture, either analog or implicit as deﬁned on the PHYS:N or

CUT:N card. At each collision point the following four steps are performed for the cycle:

1. the three prompt neutron lifetime estimates are accumulated;

2. if fission is possible, the three keff estimates are accumulated; and

3. if fission is possible, fission sites (including the sampled outgoing energy of the

fission neutron) at each collision are stored for use as source points in the next cycle,

n0≥

April 10, 2000 2-161

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where n=+ random number;

W= particle weight (before implicit capture weight reduction or

analog capture);

= average number of neutrons produced by ﬁssion at the

incident energy of this collision, with either prompt or

total (default) used;

σf= microscopic material ﬁssion cross section;

σt= microscopic material total cross section; and

keff = estimated collision keff from previous cycle.

For ﬁrst cycle, the second KCODE card entry.

M=Σn= number of ﬁssion source points to be used in next cycle. The number of

ﬁssion sites n stored at each collision is rounded up or down to an integer (including

zero) with a probability proportional to its closeness to that integer. If the initial guess

of keff is too low or too high, the number of ﬁssion sites written as source points for the

next cycle will be, respectively, too high or too low relative to the desired nominal

number N. A bad initial guess of keff causes only this consequence.

Avery} poor initial guess for the spatial distribution of ﬁssions can cause the ﬁrst

cycle estimate of keff to be extremely low. This situation can occur when only a fraction

of the ﬁssion source points enter a cell with a ﬁssionable material. As a result, one of

two error messages can be printed: (1) no new source points were generated, or (2)

the new source has overrun the old source. The second message occurs when the

MCNP storage for the ﬁssion source points is exceeded because the small keff that

results from a poor initial source causes n to become very large.

The ﬁssion energy of the next–cycle neutron is sampled separately for each source

point and stored for the next cycle. It is sampled from the same distributions as ﬁssions

would be sampled in the random walk based on the incident neutron energy and

ﬁssionable isotope. The geometric coordinates and cell of the ﬁssion site are also

stored.

4. The collision nuclide and reaction are sampled (after steps 1, 2, and 3) but the fission

reaction is not allowed to occur because fission is treated as capture. The fission

neutrons that would have been created are accrued in three different ways to estimate

keff for this cycle.

3. keff Cycle Termination

At the end of each keff cycle, a new set of M source particles has been written from ﬁssions in

that cycle. The number Mvaries from cycle to cycle but the total starting weight in each cycle is

a constant N. These M particles are written to the SRCTP ﬁle at certain cycle intervals. The

Wνσ

fσt

⁄()1keff

⁄()

νν

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SRCTP ﬁle can be used as the initial source in a subsequent criticality calculation with a similar,

though not identical, geometry. Also, keff quantities are accumulated, as is described below.

The ﬁrst Iccycles in a criticality calculation are inactive cycles, where the spatial source changes

from the initial deﬁnition to the correct distribution for the problem. No keff accumulation,

summary table, activity table, or tally information is accrued for inactive cycles. Photon

production, perturbations, and DXTRAN are turned off during inactive cycles. Ic is an input

parameter on the KCODE card for the number of keff cycles to be skipped before keff and tally

accumulation. After the ﬁrst Iccycles, the ﬁssion source spatial distribution is assumed to have

achieved equilibrium, active cycles begin, and keff and tallies are accumulated. Cycles are run

until either a time limit is reached or the total cycles on the KCODE card have been completed.

B. Estimation of keff Conﬁdence Intervals and Prompt Neutron Lifetimes

The criticality eigenvalue keff and various prompt neutron lifetimes, along with their standard

deviations, are automatically estimated in every criticality calculation in addition to any user-

requested tallies. keff and the lifetimes are estimated for every active cycle, as well as averaged

over all active cycles. keff and the lifetimes are estimated in three different ways. These estimates

are combined105 using observed statistical correlations to provide the optimum ﬁnal estimate of

keff and its standard deviation.

It is known107 that the power iteration method with a ﬁxed source size produces a very small

negative bias ∆keff in keff that is proportional to 1/N. This bias is negligible107 for all practical

problems where N is greater than about 200 neutrons per cycle and as long as too many active

cycles are not used. It has been shown107 that this bias is less, probably much less, than one-half

of one standard deviation for 400 active cycles when the ratio of the true keff standard deviation

to keff is 0.0025 at the problem end.

In MCNP the deﬁnition of keff is:

where the phase-space variables are t, E, and for time, energy, direction, and implicitly rfor

position with incremental volume dV around r. The denominator is the loss rate, which is the sum

of leakage, capture (n,0n), ﬁssion, and multiplicity (n,xn) terms. By particle balance, the loss rate

is also the source rate, which is unity in a criticality calculation. If the number of ﬁssion neutrons

keff ﬁssion neutrons in generation i1+

fission neutrons in generation i

-------------------------------------------------------------------------------------=

ρaνσfΦVdtdEΩdd

Ω

∫

∞

∫

∇J•VtddEΩρ

aσcσfσm

++()ΦVtEΩdddd

Ω

∫

∞

∫

+dd

Ω

∫

∞

∫

-------------------------------------------------------------------------------------------------------------------------------------------------------------------------------=

Ω

April 10, 2000 2-163

CHAPTER 2

CRITICALITY CALCULATIONS

produced in one generation is equal to the number in the previous generation, then the system is

critical. If it is greater, the system is supercritical. If it is less, then the system is subcritical. The

multiplicity term is:

The above deﬁnition of keff comes directly from the time-integrated Boltzmann transport

equation (without external sources):

which may be rewritten to look more like the deﬁnition of keff as:

The loss rate is on the left and the production rate is on the right.

The neutron prompt removal lifetime is the average time from the emission of a prompt neutron

in ﬁssion to the removal of the neutron by some physical process such as escape, capture, or

ﬁssion. In MCNP “absorption” and “capture” are used interchangeably, both meaning (n,0n),

and σc and σa are used interchangeably. Also, even with the TOTNU card to produce delayed

neutrons as well as prompt neutrons (KCODE default), the neutrons are all born at time zero, so

ρaσmΦVtdd()EΩdd

Ω

∫

∞

∫

ρaσn2n,ΦVdtdEΩ2ρaσn2n,ΦVtEΩdddd

Ω

∫

∞

∫

–dd

Ω

∫

∞

∫

aσn3n,ΦVdtdEΩ3ρaσn3n,ΦVtEΩ…+dddd

Ω

∫

∞

∫

–dd

Ω

∫

∞

∫

∇JVtEΩρ

aσTΦVtEΩdddd

Ω

∫

∞

∫

+dddd•

Ω

∫

∞

∫

keff

-------- ρaνσfΦVtEΩρ

aσsΦ′ E′VtEΩddddd

E′

∫

Ω

∫

∞

∫

+dddd

Ω

∫

∞

∫

=‘

∇JVtEΩdddd•

Ω

∫

∞

∫

ρaσcσfσn2n,σn3n,…++ + +()ΦVtEΩdddd

Ω

∫

∞

∫

keff

-------- ρaνσfΦVtEΩdddd

Ω

∫

∞

∫

ρa2σn2n,3σn3n,…++()ΦVtEΩdddd

Ω

∫

∞

∫

2-164 April 10, 2000

CHAPTER 2

CRITICALITY CALCULATIONS

the removal lifetimes calculated in MCNP are prompt removal lifetimes, even if there are

delayed neutrons.

The deﬁnition of the prompt removal lifetime108 is

where is the population per unit volume per unit energy per unit solid angle. In a multiplying

system in which the population is increasing or decreasing on an asymptotic period, the

population changes in accordance with

where τr is the adjoint–weighted removal lifetime. MCNP calculates the nonadjoint–weighted

prompt removal lifetime τr that can be signiﬁcantly different in a multiplying system. In a

nonmultiplying system, keff = 0 and , the population decays as

where the nonadjoint–weighted removal lifetime τris also the relaxation time.

Noting that the ﬂux is deﬁned as

where v is the speed, the MCNP nonadjoint–weighted prompt removal lifetime τr is deﬁned as

The prompt removal lifetime is a fundamental quantity in the nuclear engineering point kinetics

equation. It is also useful in nuclear well-logging calculations and other pulsed source problems

because it gives the population time-decay constant.

1. Collision Estimators

The collision estimate for keff for any active cycle is:

τr

ηVtEΩdddd

Ω

∫

∞

∫

∇JVtEΩρ

aσcσfσm

++()ΦVtEΩdddd

Ω

∫

∞

∫

+dddd•

Ω

∫

∞

∫

-------------------------------------------------------------------------------------------------------------------------------------------------------------------------------=

ηη

0ekeff 1–()tτr

⁄

τrτr

→+

ηη

0etτr

⁄–

Φηv=

τr

---- VtEΩdddd

Ω

∫

∞

∫

∇JVtEΩρ

aσcσfσm

++()ΦVtEΩdddd

Ω

∫

∞

∫

+dddd•

Ω

∫

∞

∫

-------------------------------------------------------------------------------------------------------------------------------------------------------------------------------=

April 10, 2000 2-165

CHAPTER 2

CRITICALITY CALCULATIONS

where iis summed over all collisions in a cycle where ﬁssion is possible;

kis summed over all nuclides of the material involved in the ith collision;

= total microscopic cross section;

= microscopic ﬁssion cross section;

= average number of prompt or total neutrons produced per ﬁssion by the

collision nuclide at the incident energy;

fk= atomic fraction for nuclide k;

N= nominal source size for cycle; and

Wi= weight of particle entering collision.

Because Wi represents the number of neutrons entering the ith collision,

is the expected number of neutrons to be produced from all ﬁssion processes in the collision.

Thus is the mean number of ﬁssion neutrons produced per cycle. The collision estimator

tends to be best, sometimes only marginally so, in very large systems.

The collision estimate of the prompt removal lifetime for any active cycle is the average time

required for a ﬁssion source neutron to be removed from the system by either escape, capture

(n,0n), or ﬁssion.

where Te and Tx are the times from the birth of the neutron until escape or collision. We is the

weight lost at each escape. Wc+Wf is the weight lost to (n,0n) and ﬁssion at each collision,

where is the microscopic capture (n,0n) cross section, and Wi is the weight entering the

collision.

keff

----Wi

Σkfkvkσfk

ΣkfkσTk

-------------------------

∑

σTk

σfk

νk

WiΣkfkνkσfk

ΣkfkσTk

--------------------------

keff

τr

CΣWeTeΣWcWf

+()Tx

ΣWeΣWcWf

+()+

-------------------------------------------------------------=

WcWf

+Wi

Σkfkσckσfk

+()

ΣkfkσTk

--------------------------------------

σck

2-166 April 10, 2000

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2. Absorption Estimators

The absorption estimator for keff for any active cycle is made when a neutron interacts with a

ﬁssionable nuclide. The estimator differs for analog and implicit capture. For analog capture,

where iis summed over each analog capture event in the kth nuclide. Note that in analog capture,

the weight is the same both before and after the collision. Because analog capture includes

ﬁssion in criticality calculations, the frequency of analog capture at each collision with nuclide

k is . The analog absorption keff estimate is very similar to the collision

estimator of keff except that only the kth absorbing nuclide, as sampled in the collision, is used

rather than averaging over all nuclides.

For implicit capture, the following is accumulated:

where iis summed over all collisions in which ﬁssion is possible and

is the weight absorbed in the implicit capture. The difference between the implicit absorption

estimator and the collision estimator is that only the nuclide involved in the collision

is used for the absorption keff estimate rather than an average of all nuclides in the material for

the collision keff estimator.

The absorption estimator with analog capture is likely to produce the smallest statistical

uncertainty of the three for systems where the ratio is nearly constant. Such

would be the case for a thermal system with a dominant ﬁssile nuclide such that the 1/velocity

cross section variation would tend to cancel.

The absorption estimate differs from the collision estimate in that the collision estimate is based

upon the expected value at each collision, whereas the absorption estimate is based upon the

events actually sampled at a collision. Thus all collisions will contribute to the collision estimate

of and by the probability of ﬁssion (or capture for ) in the material. Contributions to

the absorption estimator will only occur if an actual ﬁssion (or capture for ) event occurs for

the sampled nuclide in the case of analog capture. For implicit capture, the contribution to the

absorption estimate will only be made for the nuclide sampled.

The absorption estimate of the prompt removal lifetime for any active cycle is again the average

time required for a ﬁssion source neutron to be removed from the system by either escape,

capture (n,0n), or ﬁssion.

keff

----Wiνk

σfk

σckσfk

---------------------

∑

σckσfk

+()σ

⁄

keff

----Wi′νk

σfk

σckσfk

---------------------

∑

Wi′Wiσckσfk

+()σ

⁄=

keff

Akeff

νkσfkσckσfk

+()⁄

keff

Cτr

τr

April 10, 2000 2-167

CHAPTER 2

CRITICALITY CALCULATIONS

For implicit capture,

where

For analog capture,

where Te,Tc,Tf, and Txare the times from the birth of the neutron until escape, capture (n,0n),

ﬁssion, or collision. Weis the weight lost at each escape. Wcand Wfare the weights lost to capture

(n,0n) and ﬁssion at each capture (n,0n) or ﬁssion event with the nuclide sampled for the

collision.

3. Track Length Estimators

The track length estimator of keff is accumulated every time the neutron traverses a distance din

a ﬁssionable material cell:

where iis summed over all neutron trajectories,

ρis the atomic density in the cell, and

dis the trajectory track length from the last event.

Because is the expected number of ﬁssion neutrons produced along trajectory d,

is a third estimate of the mean number of ﬁssion neutrons produced in a cycle per nominal

ﬁssion source neutron.

The track length estimator tends to display the lowest variance for optically thin fuel cells (e.g.,

plates) and fast systems where large cross–section variations because of resonances may cause

high variances in the other two estimators.

AWeTeWcTcWfTf

+()T

∑

WeWcWf

∑

-------------------------------------------------------------------------------

WcWf

+Wiσckσfk

σTk

-----------------------------=

τr

AWeTeWcTcWfT

∑

WeWcWf

∑

---------------------------------------------------------------------------

keff

TL 1

----Wiρdf

kνkσfk

∑

ρdΣkfkνkσfk

keff

2-168 April 10, 2000

CHAPTER 2

CRITICALITY CALCULATIONS

The track length estimator for the prompt removal lifetime for each cycle is accumulated every

time the neutron traverses a distance d in any material in any cell:

where Wsis the source weight summed over all histories in the cycle and vis the velocity. Note

that d/v is the time span of the track. Note further that:

and in criticality problems:

These relationships show how is related to the deﬁnition of τron page 2–164.

4. Other Lifetime Estimators

In addition to the collision, absorption, and track length estimators of the prompt removal

lifetime τr, MCNP provides the escape, capture (n,0n), and ﬁssion prompt lifespans and lifetimes

for all KCODE problems having a sufﬁcient number of settle cycles. Further, the “average time

of” printed in the problem summary table is related to the lifespans, and track-length estimates

of many lifetimes can be computed using the 1/v tally multiplier option on the FM card for track-

length tallies.

In KCODE problems, MCNP calculates the lifespan of escape le, capture (n,0n) lc, ﬁssion lf, and

removal lr:

, and

τr

TL ΣiWidv⁄

----------------------=

Widv⁄

∑ρaΦ

---- VdtdEdΩd

Ω

∫

∞

∫

Ws1

keff

-------- ρaνσfΦVtEΩdddd

Ω

∫

∞

∫

∇J•VtEΩρ

aσcσfσm

++()ΦVtEΩdddd

Ω

∫

∞

∫

+dddd

Ω

∫

∞

∫

τr

ΣWeTe

ΣWe

-----------------=

ΣWcTc

ΣWc

-----------------=

ΣWfTf

ΣWf

-------------------=

April 10, 2000 2-169

CHAPTER 2

CRITICALITY CALCULATIONS

These sums are taken over all the active histories in the calculation. (If KC8 = 0 on the KCODE

card, then the sums are over both active and inactive cycle histories, but KC8 = 1, the default, is

assumed for the remainder of this discussion.) The capture (n,0n) and ﬁssion contributions are

accumulated at each collision with a nuclide, so these are absorption estimates. Thus,

The difference is that is the average of the for each cycle and lr is the average over all

histories. lr= if there is precisely one active cycle, but then neither nor lris printed out

because there are too few cycles. The cycle average does not precisely equal the history

average lr because they are ratios.

le and lc are the “average time to” escape and capture (n,0n) that is printed in the problem

summary table for all neutron and photon problems.

, , and are the weight lost to escape, capture (n,0n), and ﬁssion in the

problem summary table.

The “fractions” Fx printed out below the lifespan in the KCODE summary table are, for

x=e,c,f,orr,

The prompt lifetimes108 for the various reactions τx are then

Both and the covariance-weighted combined estimator are used. Note again that

the slight differences between similar quantities are because lxand Fxare averaged over all active

histories whereas and are averaged within each active cycle, and then the ﬁnal

values are the averages of the cycle values, i.e., history–averages vs. batch–averages.

The prompt removal lifetime can also be calculated using the F4 track-length tally with the 1/v

multiplier option on the FM card and using the volume divided by the average source weight Ws

as the multiplicative constant. The standard track length tally is then converted from

ΣWeTeΣWcTcΣWfTf

ΣWeΣWcΣWf

-------------------------------------------------------------------=

lrτr

≈

τr

Aτr

τr

Aτr

τr

----ΣWe1

----ΣWc1

----ΣWf

ΣWeΣWcΣWf

------------------------------------------------=

τx

τr

------ρa

---- Vtdd

∞

∫

σxΦVtdd

∞

∫

------------------------------------

τr

AτrCAT⁄⁄()

τr

AτrCAT⁄⁄()

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Remember to multiply by volume, either by setting the FM card constant to the volume or

overriding the F4 volume divide by using segment divisors of unity on the SD card. Ws should

be unity for KCODE calculations. The only difference between and the modiﬁed F4 tally

will be any variations from unity in Ws and the error estimation, which will be batch-averaged

for and history-averaged for the F4 tally.

Lifetimes for all other processes also can be estimated by using the FM multiplier to calculate

reaction rates as well (the numerator and denominator are separate tallies that must be divided

by the user — see the examples in Chapter 4 and 5):

Note that the lifetimes are inversely additive:

5. Combined keff and τr Estimators

MCNP provides a number of combined keff and τrestimators that are combinations of the three

individual keff and τrestimators using two at a time or all three. The combined keff's and τr's are

computed by using a maximum likelihood estimate, as outlined by Halperin109 and discussed

further by Urbatsch.105 This technique, which is a generalization of the inverse variance

weighting for uncorrelated estimators, produces the maximum likelihood estimate for the

combined average keff and τr, which, for multivariate normality, is the almost–minimum variance

estimate. It is “almost” because the covariance matrix is not known exactly and must be

estimated. The three-combined keff and τrestimators are the best ﬁnal estimates from an MCNP

calculation.105

This method of combining estimators can exhibit one feature that is disconcerting: sometimes

(usually with highly positively correlated estimators) the combined estimate will lie outside the

interval deﬁned by the two or three individual average estimates. Statisticians at Los Alamos

have shown105 that this is the best estimate to use for a ﬁnal keff and τr value. Reference 105

shows the results of one study of 500 samples from three highly positively correlated normal

F4Φtd

∫

F4V

-------Φ

---- td

∫

τr

τx

TL 1v multiplier⁄()

reaction rate multiplier

------------------------------------------------------------------ ρa

---- Vtdd

∞

∫

σxΦVtdd

∞

∫

------------------------------------

τr

---- 1

τe

---- 1

τc

---- 1

τf

-----++=

April 10, 2000 2-171

CHAPTER 2

CRITICALITY CALCULATIONS

distributions, all with a mean of zero. In 319 samples, all three estimators fell on the same side

of the expected value. This type of behavior occurs with high positive correlation because if one

estimator is above or below the expected value, the others have a good probability of being on

the same side of the expected value. The advantage of the three–combined estimator is that the

Halperin algorithm correctly predicts that the true value will lie outside of the range.

6. Error Estimation and Estimator Combination

After the ﬁrst Icinactive cycles, during which the ﬁssion source spatial distribution is allowed to

come into spatial equilibrium, MCNP begins to accumulate the estimates of keff and τrwith those

estimates from previous active (after the inactive) cycles. The relative error Rof each quantity is

estimated in the usual way as

where M= the number of active cycles,

where xm = a quantity, such as , from cycle m. This assumes that the cycle–to–cycle

estimates of each keff are uncorrelated. This assumption generally is good for keff, but not for the

eigenfunction (ﬂuxes) of optically large systems.110

MCNP also combines the three estimators in all possible ways and determines the covariance

and correlations. The simple average of two estimators is deﬁned as xij =(1/2)(xi+xj), where,

for example, ximay be the collision estimator and xjmay be the absorption estimator .

The “combined average” of two estimators is weighted by the covariances as

where the covariance Cij is

Note that for estimator i.

---x2x2

–

M1–

----------------=

-----xmand x2

∑1

-----xm

∑

keff

Ckeff

xij xixixj

–()Cii Cij

–()

Cii Cjj 2Cij

–+()

---------------------------------------------–Cjj Cij

–()xiCii Cij

–()xj

Cii Cjj 2Cij

–+()

-------------------------------------------------------------------==

Cij 1

----xm

ixm

-----xm

∑





-----xm

∑





–

∑

Cii x2x2

–=

2-172 April 10, 2000

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The “correlation” between two estimators is a function of their covariances and is given by

correlation = .

The correlation will be between unity (perfect positive correlation) and minus one (perfect anti

or negative correlation). If the correlation is one, no new information has been gained by the

second estimator. If the correlation is zero, the two estimators appear statistically independent

and the combined estimated standard deviation should be signiﬁcantly less than either. If the

correlation is negative one, even more information is available because the second estimator will

tend to be low, relative to the expected value, when the ﬁrst estimator is high and vice versa. Even

larger improvements in the combined standard deviation should occur.

The combined average estimator (keff or τr) and the estimated standard deviation of all three

estimators are based on the method of Halperin109 and is much more complicated than the two-

combination case. The improvements to the standard deviation of the three-combined estimator

will depend on the magnitude and sign of the correlations as discussed above. The details and

analysis of this method are given in Ref. 105.

For many problems, all three estimators are positively correlated. The correlation will depend on

what variance reduction (e.g., implicit or analog capture) is used. Occasionally, the absorption

estimator may be only weakly correlated with either the collision or track length estimator. It is

possible for the absorption estimator to be signiﬁcantly anticorrelated with the other two

estimators for some fast reactor compositions and large thermal systems. Except in the most

heterogeneous systems, the collision and track length estimators are likely to be strongly

positively correlated.

There may be a negative bias107 in the estimated standard deviation of keff for systems with

dominance ratios (second largest to largest eigenvalue) close to unity. These systems are

typically large with small neutron leakage. The magnitude of this effect can be estimated by

batching the cycle keff values in batch sizes much greater than one cycle,107 which MCNP

provides automatically. For problems where there is a reason to suspect the results, a more

accurate calculation of this effect can be done by making several independent calculations of the

same problem (using different random number sequences) and observing the variance of the

population of independent keff’s. The larger the number of independent calculations that can be

made, the better the distribution of keff values can be assessed.

7. Creating and Interpreting keff Confidence Intervals

The result of a Monte Carlo criticality calculation (or any other type of Monte Carlo calculation)

is a conﬁdence interval. For criticality, this means that the result is not just keff,butkeff plus and

minus some number of estimated standard deviations to form a conﬁdence interval (based on the

Cij

CiiCjj

----------------------

April 10, 2000 2-173

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Central Limit Theorem) in which the true answer is expected to lie a certain fraction of the time.

The number of standard deviations used (e.g., from a Student's t Table) determines the fraction

of the time that the conﬁdence interval will include the true answer, for a selected conﬁdence

level. For example, a valid 99% conﬁdence interval should include the true result 99% of the

time. There is always some probability (in this example, 1%) that the true result will lie outside

of the conﬁdence interval. To reduce this probability to an acceptable level, either the conﬁdence

interval must be increased according to the desired Student's t percentile, or more histories need

to be run to get a smaller estimated standard deviation.

MCNP uses three different estimators for keff. The advantages of each estimator vary with the

problem: no one estimator will be the best for all problems. All estimators and their estimated

standard deviations are valid under the assumption that they are unbiased and consistent,

therefore representative of the true parameters of the population. This statement has been

validated empirically105 for all MCNP estimators for small dominance ratios. The batched keff

results table should be used to estimate if the calculated batch-size-of-one keff standard deviation

appears to be adequate.

The conﬁdence interval based on the three-statistically-combined keff estimator is the

recommended result to use for all ﬁnal keff conﬁdence interval quotations because all of the

available information has been used in the ﬁnal result. This estimator often has a lower estimated

standard deviation than any of the three individual estimators and therefore provides the smallest

valid conﬁdence interval as well. The ﬁnal estimated keff value, estimated standard deviation, and

the estimated 68%, 95%, and 99% conﬁdence intervals (using the correct number of degrees of

freedom) are presented in the box on the keff results summary page of the output. If other

conﬁdence intervals are wanted, they can be formed from the estimated standard deviation of

keff. At least 30 active cycles need to be run for the ﬁnal keff results box to appear. Thirty cycles

are required so that there are enough degrees of freedom to form conﬁdence intervals using the

well-known estimated standard deviation multipliers. (When constructing a conﬁdence interval

using any single keff estimator, its standard deviation, and a Student’s t Table, there are It−Ic−1

degrees of freedom. For the two- and three-combined keff estimators, there are It−Ic− 2 and

It−Ic−3 degrees of freedom, respectively.)

All of the keff estimators and combinations by two or three are provided in MCNP so that the user

can make an alternate choice of conﬁdence interval if desired. Based on statistical studies, using

the individual keff estimator with the smallest estimated standard deviation is not recommended.

Its use can lead to conﬁdence intervals that do not include the true result the correct fraction of

the time.105 The studies have shown that the standard deviation of the three-combined keff

estimator provides the correct coverage rates, assuming that the estimated standard deviations in

the individual keff estimators are accurate. This accuracy can be veriﬁed by checking the batched

keff results table. When signiﬁcant anti-correlations occur among the estimators, the resultant

much smaller estimated standard deviation of the three-combined average has been veriﬁed105

by analyzing a number of independent criticality calculations.

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8. Analysis to Assess the Validity of a Criticality Calculation

The two most important requirements for producing a valid criticality calculation for a speciﬁed

geometry are sampling all of the ﬁssionable material well and ensuring that the fundamental

spatial mode was achieved before and maintained during the active keff cycles. MCNP has checks

to assess the fulﬁllment of both of these conditions.

MCNP veriﬁes that at least one ﬁssion source point was generated in each cell containing

ﬁssionable material. A WARNING message is printed on the keff results summary page that

includes a list of cells that did not have any particles entering, and/or no collisions, and/or no

ﬁssion source points. For repeated structures geometries, a source point in any one cell that is

repeated will satisfy this test. For example, assume a problem with a cylinder and a cube that are

both ﬁlled with the same universe, namely a sphere of uranium and the space outside the sphere.

If a source point is placed in the sphere inside the cylinder but not in the sphere inside the cube,

the test will be satisﬁed.

One basic assumption that is made for a good criticality calculation is that the normal spatial

mode for the ﬁssion source has been achieved after Ic cycles were skipped. MCNP attempts to

assess this condition in several ways. The estimated combined keff and its estimated standard

deviation for the ﬁrst and second active cycle halves of the problem are compared. A WARNING

message is issued if either the difference of the two values of combined col/abs/track-length keff

does not appear to be zero or the ratio of the larger-to-the-smaller estimated standard deviations

of the two col/abs/track-length keff is larger than expected. Failure of either or both checks

implies that the two active halves of the problem do not appear to be the same and the output

from the calculation should be inspected carefully.

MCNP checks to determine which number of cycles skipped produces the minimum estimated

standard deviation for the combined keff estimator. If this number is larger than Ic, it may indicate

that not enough inactive cycles were skipped. The table of combined keff–by–number–of–cycles

skipped should be examined to determine if enough inactive cycles were skipped.

It is assumed that N is large enough so that the collection of active cycle keff estimates for each

estimator will be normally distributed if the fundamental spatial mode has been achieved in Ic

cycles and maintained for the rest of the calculation. To test this assumption, MCNP performs

normality checks111,112 on each of the three keff estimator cycle data at the 95% and 99%

conﬁdence levels. A WARNING message is issued if an individual keff data set does not appear

to be normally distributed at the 99% conﬁdence level. This condition will happen to good data

about 1% of the time. Unless there is a high positive correlation among the three estimators, it

is expected to be rare that all three keff estimators will not appear normally distributed at the 99%

conﬁdence level when the normal spatial mode has been achieved and maintained. When the

condition that all three sets of keff estimators do not appear to be normal at the 99% conﬁdence

level occurs, the box with the ﬁnal keff will not be printed. The ﬁnal conﬁdence interval results

April 10, 2000 2-175

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CRITICALITY CALCULATIONS

are available elsewhere in the output. Examine the calculation carefully to see if the normal

mode was achieved before the active cycles began. The normality checks are also made for the

batched-keff and keff-by- cycles-skipped tables so that normality behavior can be studied by batch

size and Ic.

These normality checks test the assumption that the individual cycle keff values behave in the

assumed way. Even if the underlying individual cycle keff values are not normally distributed, the

three average keff values and the combined keff estimator will be normally distributed if the

conditions required by the Central Limit Theorem are met for the average. If required, this

assumption can be tested by making several independent calculations to verify empirically that

the population of average keff’s appears to be normally distributed with the same population

variance as estimated by MCNP.

MCNP tests for a monotonic trend of the three-combined keff estimator over the last ten active

cycles. This type of behavior is not expected in a well converged solution for keff and could

indicate a problem with achieving or maintaining the normal spatial mode. A WARNING

message is printed if such a monotonic trend is observed.

9. Normalization of Standard Tallies in a Criticality Calculation

Track length ﬂuxes, surface currents, surface ﬂuxes, heating and detectors–all the standard

MCNP tallies—can be made during a criticality calculation. The tallies are for one ﬁssion

neutron generation. Biases may exist in these criticality results, but appear to be smaller than

statistical uncertainties.107 These tallied quantities are accumulated only after the Ic inactive

cycles are ﬁnished. The tally normalization is per active source weight w, where

w=N∗(It−Ic), and Nis the nominal source size (from the KCODE card); Itis the total number

of cycles in the problem; and Ic is the number of inactive cycles (from KCODE card). The

number w is appropriately adjusted if the last cycle is only partially completed. If the tally

normalization ﬂag (on the KCODE card) is turned on, the tally normalization is the actual

number of starting particles during the active cycles rather than the nominal weight above. Bear

in mind, however, that the source particle weights are all set to W = N/M so that the source

normalization is based upon the nominal source size N for each cycle.

An MCNP tally in a criticality calculation is for one ﬁssion neutron being born in the system at

the start of a cycle. The tally results must be scaled either by the total number of neutrons in a

burst or by the neutron birth rate to produce, respectively, either the total result or the result per

unit time of the source. The scaling factor is entered on the Fm card.

The statistical errors that are calculated for the tallies assume that all the neutron histories are

independent. They are not independent because of the cycle–to–cycle correlations that become

worse as the dominance ratio approaches one. In this limit, each keff cycle effectively provides

no new source information. For extremely large systems (dominance ratio > 0.995), the

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estimated standard deviation for a tally that involves only a portion of the problem could be

underestimated by a factor of ﬁve or more (see Ref. 110, page 42–44). This value also is a

function of the size of the tally region. In the Ref. 110 slab reactor example, the entire problem

(i.e., keff) standard deviation was not underestimated at all. An MCNP study113 of the FFTF fast

reactor with a smaller dominance ratio indicates that 90% coverage rates for ﬂux tallies are good,

but that 2 out of 300 tallies were beyond four estimated standard deviations. Independent runs

can be made to study the real eigenfunction distribution (i.e., tallies) and the estimated standard

deviations for difﬁcult criticality calculations. This method is the only way to determine

accurately these conﬁdence intervals for large dominance ratio problems.

10. Neutron Tallies and the MCNP Net Multiplication Factor

The MCNP net multiplication factor M printed out on the problem summary page differs from

the keff from the criticality code. We will examine a simple model to illustrate the approximate

relationship between these quantities and compare the tallies between standard and criticality

calculations.

Assume we run a standard MCNP calculation using a ﬁxed neutron source distribution identical

in space and energy to the source distribution obtained from the solution of an eigenvalue

problem with keff <1. Each generation will have the same space and energy distribution as the

source. The contribution to an estimate of any quantity from one generation is reduced by a

factor of keff from the contribution in the preceding generation. The estimate Ekof a tally quantity

obtained in a criticality eigenvalue calculation is the contribution for one generation produced

by a unit source of ﬁssion neutrons. An estimate for a standard MCNP ﬁxed source calculation,

Es, is the sum of contributions for all generations starting from a unit source.

. (2.26)

Note that 1/(1 −keff) is the true system multiplication. The above result depends on our

assumptions about the unit ﬁssion source used in the standard MCNP run. Usually, Eswill vary

considerably from the above result, depending on the difference between the ﬁxed source and

the eigenmode source generated in the eigenvalue problem. Es will be a fairly good estimate if

the ﬁxed source is a distributed source roughly approximating the eigenmode source. Tallies

from a criticality calculation are appropriate only for a critical system and the tally results can

be scaled to a desired ﬁssion neutron source (power) level or total neutron pulse strength.

In a ﬁxed source MCNP problem, the net multiplication M is deﬁned to be unity plus the gain

Gf in neutrons from ﬁssion plus the gain Gx from nonﬁssion multiplicative reactions. Using

neutron weight balance (creation equals loss),

M=1 + Gf+ Gx= We + Wc, (2.27)

EsEkkeff Ekkeff

2Ekkeff

3Ek…++++Ek1keff

–()⁄==

April 10, 2000 2-177

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CRITICALITY CALCULATIONS

where Weis the weight of neutrons escaped per source neutron and Wcis the weight of neutrons

captured per source neutron. In a criticality calculation, ﬁssion is treated as an absorptive

process; the corresponding relationship for the net multiplication is then

, (2.28)

where the superscript odesignates results from the criticality calculation and is the weight

of neutrons causing ﬁssion per source neutron. Because keff is the number of ﬁssion neutrons

produced in a generation per source neutron, we can also write

, (2.29)

where is the average number of neutrons emitted per ﬁssion for the entire problem. Making

the same assumptions as above for the ﬁxed source used in the standard MCNP calculation and

using equations (2.26), (2.27), and (2.28), we obtain

or, by using (2.28) and (2.29),

Often, the nonﬁssion multiplicative reactions . This implies that keff can be approximated

by (from an appropriate Fixed Source calculation)

, (2.30)

when the two ﬁssion neutron source distributions are nearly the same. The average value of

in a problem can be calculated by dividing the ﬁssion neutrons gained by the ﬁssion neutrons

lost as given in the totals of the neutron weight balance for physical events. Note, however, that

the above estimate is subject to the same limitations as described in Eq. 2.26.

Mo1Gx

+We

oWc

oWf

++==

keff νWf

eWc

+We

oWc

1keff

–

----------------------MoWf

–

1keff

–

----------------------===

Mokeff

--------–

1keff

–

----------------------

1keff

--------–Gx

1keff

–

-------------------------------==

o1«

keff

keff keff

≈M1–

---–

--------------=

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C. Recommendations for Making a Good Criticality Calculation

1. Problem Set-Up

As with any calculation, the geometry must be adequately and correctly speciﬁed to represent

the true physical situation. Plot the geometry and check cells, materials, and masses for

correctness. Specify the appropriate nuclear data, including S(α,β) thermal data, at the correct

material temperatures. Do as good a job as possible to put initial ﬁssion source points in every

cell with ﬁssionable material. Try running short problems with both analog and implicit capture

(see the PHYS:N card) to improve the ﬁgure of merit for the combined keff and any tallies being

made. Follow the tips for good calculations listed at the end of Chapter 1.

2. Number of Neutrons per Cycle and Number of Cycles

Criticality calculations can suffer from two potential problems. The ﬁrst is the failure to

sufﬁciently converge the spatial distribution of the ﬁssion source from its initial guess to a

distribution ﬂuctuating around the fundamental eigenmode solution. It is recommended that you

make an initial run with a relatively small number of source particles per generation (perhaps

500) and generously allow a large enough number of cycles so that the eigenvalue appears to be

ﬂuctuating about a constant value. You should examine the results and continue the calculation

if any trends in the eigenvalue are noticeable. The SRCTP ﬁle from the last keff cycle of the initial

run can then be used as the source for the ﬁnal production run to be made with a larger number

of histories per cycle.

This convergence procedure can be extended for very slowly convergent problems–typically

large, thermal, low-leakage systems, where a convergence run might be made with 500 histories

per cycle. Then a second convergence run would be made with 1000 histories per cycle, using

the SRCTP ﬁle from the ﬁrst run as an initial ﬁssion source guess. If the results from the second

run appear satisfactory, then a ﬁnal run might be made using 4000 particles per cycle with the

SRCTP ﬁle from the second run as an initial ﬁssion source guess. In the ﬁnal run, only a few

cycles should need to be skipped. The bottom line is this: skip enough cycles so that the normal

spatial mode is achieved.

The second potential problem arises from the fact that the criticality algorithm produces a very

small negative bias in the estimated eigenvalue. The bias depends upon 1/N, where N is the

number of source particles per generation. Thus it is desirable to make Nas large as possible.

Any value of N > 200 should be sufﬁcient to reduce the bias to a small level.The eigenvalue bias

∆keff has been shown107 to be

, (2.31)∆keff

–ItIc

–()

2keff

-------------------σkeff

2σapprox

–()=

April 10, 2000 2-179

CHAPTER 2

CRITICALITY CALCULATIONS

where is the true standard deviation for the ﬁnal keff,

σapprox is the approximate standard deviation computed assuming

the individual keff values are statistically independent, and

The standard deviations are computed at the end of the problem. Because the σ2s decrease as

1/(It−Ic),∆keff is independent of the number of active cycles. Recall that ∆keff is proportional to

1/N, the number of neutrons per keff cycle.

Eqn. (2.31) can be written107 as the following inequality:

. (2.32)

This inequality is useful for determining an upper limit to the number of active cycles that should

be used for a calculation without having ∆keff dominate . If is 0.0025, which is a

reasonable value for criticality calculations, and It−Icis 400, then and ∆keff

will not dominate the keff conﬁdence interval. If is reasonably well approximated by

MCNP's estimated standard deviation, this ratio will be much less than 0.5.

The total running time for the active cycles is proportional to N(It−Ic), and the standard

deviation in the estimated eigenvalue is proportional to . From the results of the

convergence run, the total number of histories needed to achieve the desired standard deviation

can be estimated.

It is recommended that 200 to 400 active cycles be used, assuming that the above

is much less than unity in doing so. This large number of cycles will provide large batch sizes of

keff cycles (e.g., 40 batches of 10 cycles each for 400 active cycles) to compare estimated

standard deviations with those obtained for a batch size of one keff cycle. For example, for 400

active cycles, 40 batches of 10 keffs are created and analyzed for a new average keff and a new

estimated standard deviation. The behavior of the average keff by a larger number of cycles can

also be observed to ensure a good normal spatial mode. Fewer than 30 active cycles is not

recommended because trends in the average keff may not have enough cycles to develop.

3. Analysis of Criticality Problem Results

The goal of the calculation is to produce a keff conﬁdence interval that includes the true result the

desired fraction of the time. Check all WARNING messages. Understand their signiﬁcance to

σkeff

2σapprox

∆keff

σkeff

--------------- ItIc

–()σ

keff

2keff

-----------------------------

σkeff σkeff keff

⁄

∆keff σkeff

⁄0.5<

σkeff

1NI

tIc

–()⁄

∆keff σkeff

⁄

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the calculation. Study the results of the checks that MCNP makes that were described starting

on page 2–174.

The criticality problem output contains a lot of useful information. Study it to make sure that:

1) the problem terminated properly; 2) enough cycles were skipped to ensure that the normal

spatial mode for ﬁssion sources was achieved; 3) all cells with ﬁssionable material were

sampled; 4) the average combined keff appears to be varying randomly about the average value

for the active cycles; 5) the average combined keff–by–cycles–skipped does not exhibit a trend

during the latter stages of the calculation; 6) the conﬁdence intervals for the batched (with at

least 30 batch values) combined keff do not differ signiﬁcantly from the ﬁnal result; 7) the impact

of having the largest of each of the three keff estimators occurring on the next cycle is not too

great on the ﬁnal conﬁdence interval; and 8) the combined keff ﬁgure of merit should be stable.

The combined keff ﬁgure of merit should be reasonably stable, but not as stable as a tally ﬁgure

of merit because the number of histories for each cycle is not exactly the same and combined keff

relative error may experience some changes because of changes in the estimated covariance

matrix for the three individual estimators.

Plots (using the z option) can be made of the three individual and average keff estimators by cycle,

as well as the three-estimator-combined keff. Use these plots to better understand the results.

If there is concern about a calculation, the keff–by–cycles–skipped table presents the results that

would be obtained in the ﬁnal result box for differing numbers of cycles skipped. This

information can provide insight into ﬁssion source spatial convergence, normality of the keff data

sets, and changes in the 95% and 99% conﬁdence intervals. If concern persists, a problem could

be run that tallies the track length estimator keff using an F4:n tally and an FM card using the −6

and −7 reaction multipliers (see Chapter 4 for an example). In the most drastic cases, several

independent calculations can be made and the variance of the keff values (and any other tallies)

could be computed from the individual values.

If a conservative (too large) keff conﬁdence interval is desired, the results from the largest keff

occurring on the next cycle table can be used. This situation could occur with a maximum

probability of 1/(It−Ic) for highly positively correlated keff’s to 1/(It−Ic)3 for no correlation.

Finally, keep in mind the discussion in starting on page 2–175. For large systems with a

dominance ratio close to one, the estimated standard deviations for tallies could be much smaller

than the true standard deviation. The cycle–to–cycle correlations in the ﬁssion sources are not

taken into account, especially for any tallies that are not made over the entire problem. The only

way to obtain the correct statistical errors in this situation is to run a series of independent

problems using different random number sequences and analyze the sampled tally results to

estimate the statistical uncertainties.

April 10, 2000 2-181

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VOLUMES AND AREAS114

IX. VOLUMES AND AREAS114

The particle ﬂux in Monte Carlo transport problems often is estimated as the track length per

unit volume or the number of particles crossing a surface per unit area. Therefore, knowing the

volumes and surface areas of the geometric regions in a Monte Carlo problem is essential.

Knowing volumes is useful in calculating the masses and densities of cells and thus in

calculating volumetric or mass heating. Furthermore, calculation of the mass of a geometry is

frequently a good check on the accuracy of the geometry setup when the mass is known by other

means.

Calculating volumes and surface areas in modern Monte Carlo transport codes is nontrivial.

MCNP allows the construction of cells from unions and/or intersections of regions deﬁned by

an arbitrary combination of second-degree surfaces, toroidal fourth-degree surfaces, or both.

These surfaces can have different orientations or be segmented for tallying purposes. The cells

they form even can consist of several disjoint subcells. Cells can be constructed from

quadralateral or hexagonal lattices or can be embedded in repeated structures universes.

Although such generality greatly increases the ﬂexibility of MCNP, computing cell volumes and

surface areas understandably requires increasingly elaborate computational methods.

MCNP automatically calculates volumes and areas of polyhedral cells and of cells or surfaces

generated by surfaces of revolution about any axis, even a skew axis. If a tally is segmented, the

segment volumes or areas are computed. For nonrotationally symmetric or nonpolyhedral cells,

a stochastic volume and surface area method that uses ray tracing is available. See page 2–182.

A. Rotationally Symmetric Volumes and Areas

The procedure for computing volumes and surface areas of rotationally symmetric bodies

follows:

1. Determine the common axis of symmetry of the cell.114 If there is none and if the cell

is not a polyhedron, MCNP cannot compute the volume (except stochastically) and the

area of each bounding surface cannot be computed on the side of the asymmetric cell.

2. Convert the bounding surfaces to q-form:

ar2+br +cs2+ds +e=0 ,

where s is the axis of rotational symmetry in the r-s coordinate system. All MCNP

surfaces except tori are quadratic surfaces and therefore can be put into q-form.

3. Determine all intersections of the bounding surfaces with each other in the r-s

coordinate system. This procedure generally requires the solution of a quartic

equation.22 For spheres, ellipses, and tori, extra intersection points are added so that

2-182 April 10, 2000

CHAPTER 2

VOLUMES AND AREAS114

these surfaces are not infinite. The list of intersections are put in order of increasing

s-coordinate. If no intersection is found, the surface is infinite; its volume and area on

one side cannot be computed.

4. Integrate over each bounding surface segment between intersections:

for volumes;

for surface areas.

A bounding surface segment lies between two intersections that bound the cell of interest.

A numerical integration is required for the area of a torroidal surface; all other integrals are

directly solved by integration formulas. The sense of a bounding surface to a cell determines the

sign of V. The area of each surface is determined cell-by-cell twice, once for each side of the

surface. An area will be calculated unless bounded on both sides by asymmetric or inﬁnite cells.

B. Polyhedron Volumes and Areas

A polyhedron is a body bounded only by planes that can have an arbitrary orientation. The

procedure for calculating the volumes and surface areas of polyhedra is as follows:

1. For each facet side (planar surface), determine the intersections (ri,si) of the other

bounding planes in the r-scoordinate system. The r-scoordinate system is redefined

for each facet to be an arbitrary coordinate system in the plane of the facet.

2. Determine the area of the facet:

and the coordinates of its centroid, rc,sc:

The sums are over all bounding edges of the facet where i and i + 1 are the ends of

the bounding edge such that, in going from ito i+1, the facet is on the right side. As

with rotationally symmetric cells, the area of a surface is determined cell-by-cell

twice, once for each side. The area of a surface on one side is the sum over all facets

on that side.

Vπr2sd

∫

A2πr1rd

-----





+sd

∫

---si1+ si

–()

∑ri1+ ri

+()=

rc16a()⁄si1+ si

–()ri1+

2ri1+ riri

++()

∑

sc16a()⁄ri1+ ri

–()si1+

2si1+ sisi

++()

∑

April 10, 2000 2-183

CHAPTER 2

PLOTTER

3. The volume of a polyhedron is computed by using an arbitrary reference plane.

Prisms are projected from each facet normal to the reference plane, and the volume of

each prism is V = da cos where

d= distance from reference plane to facet centroid;

a= facet area; and

= angle between the external normal of the facet and the positive

normal of the reference plane.

The sum of the prism volumes is the polyhedron cell volume.

C. Stochastic Volume and Area Calculation

MCNP cannot calculate the volumes and areas of asymmetric, nonpolyhedral, or inﬁnite cells.

Also, in very rare cases, the volume and area calculation can fail because of roundoff errors. For

these cases a stochastic estimation is possible by ray tracing. The procedure is as follows:

1. Void out all materials in the problem (VOID card).

2. Set all nonzero importances to one and all positive weight windows to zero.

3. Use a planar source with a source weight equal to the surface area to flood the

geometry with particles. This will cause the particle flux throughout the geometry to

statistically approach unity. Perhaps the best way to do a stochastic volume estimation

is to use an inward-directed, biased cosine source on a spherical surface with weight

equal to πr2.

4. Use the cell flux tally (F4) to tabulate volumes and the surface flux tally (F2) to

tabulate areas. The cell flux tally is inversely proportional to cell volume. Thus in cells

whose volumes are known, the unit flux will result in a tally of unity and in cells whose

volume is uncalculated, the unit flux will result in a tally of volumes. Similarly, the

surface flux tally is inversely proportional to area so that the unit flux will result in a

tally of unity wherever the area is known and a tally of area wherever it is unknown.

X. PLOTTER

The MCNP plotter draws cross-sectional views of the problem geometry according to

commands entered by the user. See Appendix B for the command vocabulary and examples of

use. The pictures can be drawn on the screen of a terminal or on some local or remote hard copy

graphics device, as directed by the user. The pictures are drawn in a square viewport on the

graphics device. The mapping between the viewport and the portion of the problem space to be

plotted, called the window, is user–deﬁned. A plane in problem space, the plot plane, is deﬁned

by specifying an origin and two perpendicular basis vectors and . The size of the window

in the plot plane is deﬁned by specifying two extents. The picture appears in the viewport with

roa b

2-184 April 10, 2000

CHAPTER 2

PLOTTER

the origin at the center, the ﬁrst basis vector pointing to the right and the second basis vector

pointing up. The width of the picture is twice the ﬁrst extent and the height is twice the second

extent. If the extents are unequal, the picture is distorted. The central task of the plotter is to plot

curves representing the intersections of the surfaces of the geometry with the plot plane within

the window.

All plotted curves are conics, deﬁned here to include straight lines. The intersection of a plane

with any MCNP surface that is not a torus is always a conic. A torus is plotted only if the plot

plane contains the torus axis or is perpendicular to it, in which cases the intersection curves are

conics. The ﬁrst step in plotting the curves is to ﬁnd equations for them, starting from the

equations for the surfaces of the problem. Equations are needed in two forms for each curve: a

quadratic equation and a pair of parametric equations. The quadratic equations are needed to

solve for the intersections of the curves. The parametric equations are needed for deﬁning the

points on the portions of the curves that are actually plotted.

The equation of a conic is

As2 + 2Hst + Bt2+ 2Gs + 2Ft + C = 0 ,

where sand tare coordinates in the plot plane. They are related to problem coordinates (x,y,z) by

or in matrix form

In matrix form the conic equation is

Thus, ﬁnding the equation of a curve to be plotted is a matter of ﬁnding the QM matrix, given

the PL matrix and the coefﬁcients of the surface.

Any surface in MCNP, if it is not a torus, can be readily written as

osa tb++=

100

xoaxbx

yoayby

zoazbz

PL 1

1st[]

CGF

GAH

FHB

0or 1st[]QM 1

April 10, 2000 2-185

CHAPTER 2

PLOTTER

Ax2 + By2 + Cz2 + Dxy + Eyz + Fzx + Gx + Hy + Jz + K = 0 ,

or in matrix form as

The transpose of the transformation between (s,t) and (x,y,z) is

where PLT is the transpose of the PL matrix. Substitution in the surface equation gives

Therefore, QM = PLT AM PL.

A convenient set of parametric equations for conics is

straight line s = C1 + C2p

t=C4 + C5p

parabola s = C1 + C2p + C3p2

t=C4 + C5p + C6p2

ellipse s = C1+ C2 sin p+C

3 cos p

t=C4+ C5 sin p + C6 cos p

hyperbola s = C1+ C2 sinh p + C3 cosh p

t=C4+ C5 sinh p + C6 cosh p.

1xyz[]

KG2⁄H2⁄J2⁄

G2⁄AD2⁄F2⁄

H2⁄D2⁄BE2⁄

J2⁄F2⁄E2⁄C

1xyz[]AM

1xyz[]1st[]PLT

1st[]PLTAM PL 1

2-186 April 10, 2000

CHAPTER 2

PLOTTER

The type of a conic is determined by examination of the conic invariants,115 which are simple

functions of the elements of QM. Some of the surfaces produce two curves, such as the two

branches of a hyperbola or two straight lines. A separate set of parametric coefﬁcients, C1

through C6, is needed for each curve in such cases. The parametric coefﬁcients are found by

transforming QM into yet another coordinate system where most of its elements are zero. The

parametric coefﬁcients are then simple functions115 of the remaining elements. Finally, the

coefﬁcients are transformed from that coordinate system back to the (s,t) system.

For a plottable torus, the curves are either a pair of identical ellipses or a pair of concentric

circles. The parametric coefﬁcients are readily calculated from the surface coefﬁcients and the

elements of QM are simple functions of the parametric coefﬁcients.

The next step is to reject all curves that lie entirely outside the window by ﬁnding the

intersections of each curve with the straight line segments that bound the window, taking into

account the possibility that an ellipse may lie entirely inside the window.

The remaining curves are plotted one at a time. The intersections of the current curve with all of

the other remaining curves and with the boundaries of the window are found by solving the

simultaneous equations

where i = 1 is the current curve and i = 2 is one of the other curves. This process generally

requires ﬁnding the roots of a quartic. False roots and roots outside the window are rejected and

the value of the parameter pfor each remaining intersection is found. The intersections then are

arranged in order of increasing values of p.

Each segment of the curve–the portion of the curve between two adjacent intersections–is

examined to see whether and how it should be plotted. A point near the center of the segment is

transformed back to the (x,y,z) coordinate system. All cells immediately adjacent to the surface

at that point are found. If there is exactly one cell on each side of the surface and those cells are

the same, the segment is not plotted. If there is exactly one cell on each side and those cells are

different, the segment is plotted as a solid line. If anything else is found, the segment is plotted

as a dotted line, which indicates either that there is an error in the problem geometry or that some

other surface of the problem also intersects the plot plane along the segment.

If a curve to be plotted is not a straight line, it is plotted as a sequence of short straight lines

between selected points on the curve. The points are selected according to the criterion that the

middle of the line drawn between points must not lie farther from the nearest point on the true

1st[]QMi

April 10, 2000 2-187

CHAPTER 2

PSEUDORANDOM NUMBERS

curve than the nominal resolution of the picture. The nominal resolution is ﬁxed at 1/3000 of a

side of the viewport. This is a bit coarse for the best plotting devices and is quite a bit too ﬁne

for the worst ones, but it produces adequate pictures at reasonable cost.

XI. PSEUDORANDOM NUMBERS

Like any other Monte Carlo program, MCNP uses a sequence of pseudorandom numbers to

sample from probability distributions. MCNP has always used the congruential scheme of

Lehmer,15 though the mechanics of implementation have been modiﬁed for portability to

different computer platforms. In particular, a method has been devised that multiplies two

64-bit words to get a 128-bit word without using more than 64-bit words for the 128-bit word.116

A pseudorandom sequence of integers In is generated by

In+1 = mod(M In, 248),

where Mis the random number multiplier, and 48-bit integers and 48-bit ﬂoating point mantissas

are assumed. The default value of M, which can be changed with the DBCN card, is

M = 519 = 19,073,486,328,125 .

The pseudorandom number is then

Rn= 2−48In.

The starting pseudorandom number of each history is

In+S = mod(MSIn,248),

where S is the pseudorandom number stride. Because each pseudorandom number is the least

signiﬁcant (lower) 48 bits of Mmultiplied by the previous random number, the lower 48 bits of

In+S are the same as the lower 48 bits of MSIn. The default value of S, which can be changed

with the DBCN card, is

S = 15291710 = 4525258= 1001010101010101012.

The 01010101 pattern ensures that the bit pattern will change when the stride is multiplied by

almost anything.

The period P of the MCNP algorithm is

P246 7.04 1013

×≈=

2-188 April 10, 2000

CHAPTER 2

PERTURBATIONS

because the last two binary bits of the lower 48 bits of Mkare 012for all values of k. The period

could be increased from 246 to

by adding 1 as follows:

In= mod(M In-1, 248)+ 1 .

MCNP prints a WARNING and counts the number of histories for which the stride Sis exceeded.

MCNP also prints a WARNING if the period Pis exceeded. Exceeding the stride or the period

does not result in wrong answers but does result in an underestimate of the variance. However,

because the random numbers are used for very different purposes, MCNP seems quite

insensitive to overrunning either the stride or the period.116

Sometimes users wish to know how much of the variation between problems is purely statistical

and the variance is insufﬁcient to provide this information. In correlated sampling (see

page 2–158) and criticality problems, the variances can be underestimated because of correlation

between histories. In this case, rerun the problems with a different random number sequence,

either by starting with a new random number or by changing the random number stride or

multiplier on the DBCN card. MCNP checks for and does not allow invalid choices, such as an

even numbered initial random number that, after a few pseudorandom numbers, would result in

all subsequent random numbers being zero.

XII. PERTURBATIONS

The evaluation of response or tally sensitivities to cross–section data involves ﬁnding the ratio

of the change in a tally to the inﬁnitesimal change in the data, as given by the Taylor series

expansion. In deterministic methods, this ratio is approximated by performing two calculations,

one with the original data and one with the perturbed data. This approach is useful even when

the magnitude of the perturbation becomes very small. In Monte Carlo methods, however, this

approach fails as the magnitude of the perturbation becomes small because of the uncertainty

associated with the response. For this reason, the differential operator technique was developed.

The differential operator perturbation technique as applied in the Monte Carlo method was

introduced by Olhoeft117 in the early 1960’s. Nearly a decade after its introduction, this

technique was applied to geometric perturbations by Takahashi.118 A decade later, the method

was generalized for perturbations in cross–section data by Hall119,120 and later Rief.121 A

rudimentary implementation into MCNP followed shortly thereafter.122 With an enhancement of

the user interface and the addition of second order effects, this implementation has evolved into

a standard MCNP feature.

P248 2.81 1014

×≈=

April 10, 2000 2-189

CHAPTER 2

PERTURBATIONS

A. Derivation of the Operator

In the differential operator approach, a change in the Monte Carlo response c, due to changes in

a related data set (represented by the parameter v), is given by a Taylor series expansion

. . . + + . . . ,

where the nth order coefﬁcient is

This can be written as

for the data set

where Kb(h) is some constant, Brepresents a set of macroscopic cross sections, and Hrepresents

a set of energies or an energy interval.

For a track-based response estimator

where tj is the response estimator and qj is the probability of path segment j (path segment j is

comprised of segment j− 1 plus the current track.) This gives

∆cdc

------∆v1

-----d2c

dv2

-------- ∆v2+⋅⋅+⋅=1

-----dnc

dvn

-------- ∆vn

⋅⋅

un1

-----dnc

dvn

--------

⋅=

un1

-----xb

nh() ∂nc

∂xb

nh()

----------------







hH∈

∑

bB∈

∑

bh() Kkbh()evbBhH∈,∈;⋅=

jqj

∑

un1

-----xb

nh() ∂n

∂xb

nh()

---------------- tjqj

()







hH∈

∑

bB∈

∑

2-190 April 10, 2000

CHAPTER 2

PERTURBATIONS

where

With some manipulations presented in Ref. 123, the path segment estimator γnjtj can be

converted to a particle history estimator of the form

where piis the probability of the ith history and Vni is the nth order coefﬁcient estimator for history

i, given by

Note that this sum involves only those path segments j' in particle history i. The Monte Carlo

expected value of un becomes

for a sample of N particle histories.

The probability of path segment j is the product of the track probabilities,

where rk is the probability of track k and segment j contains m+ 1 tracks. If the kth track starts

with a neutron undergoing reaction type “a” at energy E' and is scattered from angle θ'to angle

θ and E, continues for a length λk, and collides, then

un1

-----γnjtjqj

∑

γnj xb

nh() ∂n

∂xb

nh()

---------------- tjqj

()







tjqj

---------





hH∈

∑

bB∈

∑

≡

unVnipi

∑

Vni 1

-----γnj′tj′

∑

≡

〈〉 1

----Vni

∑

Nn!

--------- γnj′tj′

j′

∑





∑

qjrk

k0=

∏

April 10, 2000 2-191

CHAPTER 2

PERTURBATIONS

where xa(E') is the macroscopic reaction cross section at energy E',xT(E') is the total cross

section at energy E', and is the probability distribution function in

phase space of the emerging neutron. If the track starts with a collision and ends in a boundary

crossing

If the track starts with a boundary crossing and ends with a collision,

And ﬁnally, if the track starts and ends with boundary crossings

1. First Order

For a ﬁrst order perturbation, the differential operator becomes

whereas,

then

xaE′()

xTE′()

----------------





PaE′Eθ′ θ→;→()dEdτθ exTE()λ

–

()xTE()d

PaE′Eθ′ θ→;→()dEdθ

xae′()

xTE'()

----------------





PaE'Eθ′ θ→;→()dEdθexTE()λ

–

()=

rkexTE()λ

–

()xTE()dλ=

rkexTE()λ

–

γ1j′xbh() xbh()∂∂tj′qj′

()





tj′qj′

-----------





hH∈

∑

bB∈

∑

≡

xbh()

qj′

------------- qj′

∂

xbh()∂

---------------- xbh()

tj′

------------- tj′

∂

xbh()∂

----------------





hH∈

∑

bB∈

∑

qj′

------qj′

∂

xbh()∂

---------------- 1

---- rk

∂

xbh()∂

----------------

k0=

∑

λ1j'βj'kR1j′

k0=

∑

2-192 April 10, 2000

CHAPTER 2

PERTURBATIONS

where

for a track segment k that starts with a particle undergoing reaction type “a” at energy E' and is

scattered to energy E and collides after a distance λk. Note that δhE and δba are unity if h=E and

b=a; otherwise they vanish. For other types of tracks (for which the various expressions for rk

were given in the previous section), i.e., collision to boundary, boundary to collision, and

boundary to boundary, derivatives of rkcan be taken leading to one or more of these four terms

for βj'k.

The second term of γ1j'is

where the tally response is a linear function of some combination of reaction cross sections, or

where cis an element of the tally cross sections, , and may be an element of the perturbed

cross sections, . Then,

R1j'is the fraction of the reaction rate tally involved in the perturbation. If none of the nuclides

participating in the tally is involved in the perturbation, then R1j' = 0, which is always the case

for F1, F2, and F4 tallies without FM cards. For F4 tallies with an FM card, if the FM card

multiplicative constant is positive (no ﬂag to multiply by atom density) it is assumed that the FM

βj′k

xbh()

-------------





∂

xbh()∂

----------------





hH∈

∑

bB∈

∑

≡

δhEδba

δhE'xbE'()

xTE'()

-------------------------– δhExbE()λ

–δhExbE()

xTE()

-----------------------+





hH∈

∑

bB∈

∑

R1j'

xbh()

tj'

------------- ∂tj'

∂xbh()

----------------

hH∈

∑

bB∈

∑

tj'λkxcE()

cC∈

∑

cC∈

cB∈

R1j′

xbh()

xch()

cC∈

∑





----------------------------- ∂

∂xbh()

---------------- xch()

cC∈

∑





hH∈

∑

bB∈

∑

xcE()

EH∈

∑

cB∈

∑

xcE()

cC∈

∑

-------------------------------------=

April 10, 2000 2-193

CHAPTER 2

PERTURBATIONS

tally cross sections are unaffected by the perturbation and R1j' =0. For KCODE keff track length

estimates, F6 and F7 heating tallies, and F4 tallies with FM cards with negative multipliers

(multiply by atom density to get macroscopic cross sections), if the tally cross section is affected

by the perturbation, then R1j' > 0. For keff and F6 and F7 tallies in perturbed cells where all

nuclides are perturbed, generally R1j' = 1.

Finally, the expected value of the ﬁrst order coefﬁcient is

2. Second Order

For a second order perturbation, the differential operator becomes

Whereas tj' is a linear function of xb(h), then

and by taking ﬁrst and second derivatives of the rkterms of qj' as for the ﬁrst order perturbation,

where

The expected value of the second order coefﬁcient is

〈〉 1

----βj′kR1j′

k0=

∑







tj′

j′

∑

γ2j′x2

bh() ∂2

∂x2

bh()

-------------------tj′qj′

()







tj′qj′

-----------





hH∈

∑

bB∈

∑

≡

bh()

tj′qj′

----------------tj′

∂2qj′

∂x2

bh()

-------------------







2∂qj′

∂xbh()

---------------- ∂tj′

∂xbh()

---------------- qj′

∂2tj′

∂xbh()

----------------

hH∈

∑

bB∈

∑

∂2tj′

∂

xbh()

--------------- 0=

γ2j′αj′kβj′k

–()R1j′

–βj′kR1j′

k0=

∑







k0=

∑

αj′k

2δhE′xb

2E′()

----------------------------- 2δhE′δbaxbE′()

xTE′()

-------------------------------------–δhExb

2λk

22δhExb

2E()λ

xTE()

--------------------------------–+







hH∈

∑

bB∈

∑

2-194 April 10, 2000

CHAPTER 2

PERTURBATIONS

where βj'k and αj'k are given by one or more terms as described above for track kand R1j' is again

the fraction of the perturbation with nuclides participating in the tally.

3. Implementation in MCNP

The total perturbation printed in the MCNP output ﬁle is

For each history i and path j',

Let the ﬁrst order perturbation with R1j' = 0 be

and let the second order perturbation with R1j' = 0 be

Then the Taylor series expansion for R1j' = 0 is

If then

〈〉 1

------- αj′kβj′k

–()R2

ij′

–βj′kR+1j′

k0=

∑







k0=

∑







tj′

j′

∑

∆c〈〉 1

----∆cj′

j′

∑

∆cj′

dcj′

--------- ∆v1

---d2cj′

dv2

------------∆v2

⋅⋅+⋅=

P1j′βj′k

k0=

∑







tj′

j′

∑

P2j′αj′kβj′k

–()

k0=

∑







tj′

j′

∑

∆cj′P1j′∆v1

---P2j′P1j′

+()∆v2





tj′

R1j′0≠

∆cj′P1j′R1j′

+()∆v1

---P2j′R1j′

–P1j′R1j′

+()

+()∆v2

+tj′

April 10, 2000 2-195

CHAPTER 2

PERTURBATIONS

That is, the case is just a correction to the case.

In MCNP, P1j' and P2j' are accumulated along every track length through a perturbed cell. All

perturbed tallies are multiplied by

and then if the tally is further corrected by

R1j' ∆v+P1j' R1j' ∆v2.

R1j' is the fraction of the reaction rate tally involved in the perturbation. R1j' = 0 for F1, F2, F4

tallies without FM cards, and F4 tallies with FM cards with positive multiplicative constants.

B. Limitations

Although it is always a high priority to minimize the limitations of any MCNP feature, the

perturbation technique has the limitations given below. Chapter 3, page 3–144, has examples you

can refer to.

1. A fatal error is generated if a PERT card attempts to unvoid a region. The simple

solution is to include the material in the unperturbed problem and void the region of

interest with the PERT card. See Appendix B of Ref. 124.

2. A fatal error is generated if a PERT card attempts to alter a material composition in

such a way as to introduce a new nuclide. The solution is to set up the unperturbed

problem with a mixture of both materials and introduce PERT cards to remove each.

See Appendix B of Ref. 124.

3. The track length estimate of keff in KCODE criticality calculations assumes the

fundamental eigenvector (fission distribution) is unchanged in the perturbed

configuration.

4. DXTRAN, point detector tallies, and pulse height tallies are not currently compatible

with the PERT card.

5. While there is no limit to the number of perturbations, they should be kept to a

minimum, as each perturbation can degrade performance by 10–20%.

6. The METHOD keywork can indicate if a perturbation is so large that higher than

second order terms are needed to prevent inaccurate tallies.

P1j′∆v1

---P2j′P1j′

+( )∆v2R1j′∆vP

1j′R1j′∆v2

+++tj′

R1j′0≠R1j′0=

P1j′∆v1

---P2j′P2

1j′

+()∆v

R1j′0≠

2-196 April 10, 2000

CHAPTER 2

PERTURBATIONS

C. Accuracy

Analyzing the ﬁrst and second order perturbation results presented in Ref. 124 leads to the

following rules of thumb. The ﬁrst order perturbation estimator typically provides sufﬁcient

accuracy for response or tally changes that are less than 5%. The default ﬁrst and second order

estimator offers acceptable accuracy for response changes that are less than 20–30%. This upper

bound depends on the behavior of the response as a function of the perturbed parameter. The

magnitude of the second order estimator is a good measure of the range of applicability. If this

magnitude exceeds 30% of the ﬁrst order estimator, it is likely that higher order terms are needed

for an accurate prediction. The METHOD keyword on the PERT card allows one to tally the

second order term separate from the ﬁrst. See Chapter 3, page 3–142.

April 10, 2000 2-197

CHAPTER 2

REFERENCES

XIII.REFERENCES

1. L. L. Carter and E. D. Cashwell, Particle Transport Simulation with the Monte Carlo

Method, ERDA Critical Review Series, TID-26607 (1975).

2. Ivan Lux and Laszlo Koblinger, Monte Carlo Particle Transport Methods: Neutron and

Photon Calculations, CRC Press, Boca Raton (1991).

3. C. J. Everett and E. D. Cashwell, “A Third Monte Carlo Sampler,” Los Alamos National

Laboratory Report, LA-9721-MS, (March 1983).

4. G. Compte de Buffon, “Essai d'arithmetique morale,” Supplement a la Naturelle, Vol. 4,

1777.

5. A Hall, “On an Experimental Determination of Pi,” Messeng. Math., 2, 113-114 (1873).

6. J. M. Hammersley and D. C. Handscomb, Monte Carlo Methods, John Wiley & Sons, New

York (1964).

7. Marquis Pierre-Simon de Laplace, Theorie Analytique des Probabilities, Livre 2

pp. 356-366 contained in Oeuvres Completes de Laplace, de L'Aca\-demie des Sciences,

Paris, Vol. 7, part 2, 1786.

8. Lord Kelvin, “Nineteenth Century Clouds Over the Dynamical Theory of Heat and Light,”

Philosophical Magazine, series 6, 2, 1 (1901).

9. W. W. Wood, “Early History of Computer Simulations in Statistical Mechanics and

Molecular Dynamics,” International School of Physics “Enrico Fermi,” Varenna, Italy,

1985, Molecular-Dynamics Simulation of Statistical Mechanical Systems, XCVII Corso

(Soc. Italiana di Fisica, Bologna) (1986).

10. Necia Grant Cooper, Ed., From Cardinals to Chaos — Reflections on the Life and Legacy

of Stanislaw Ulam, Cambridge University Press, New York (1989).

11. “Fermi Invention Rediscovered at LASL,” The Atom, Los Alamos Scientific Laboratory

(October 1966).

12. N. Metropolis and S. Ulam, “The Monte Carlo Method,” J. Amer. Stat. Assoc., 44, 335

(1949).

13. Herman Kahn, “Modifications of the Monte Carlo Method,” Proceeding, Seminar on

Scientific Computation, Nov. 1949, IBM, New York, 20-27 (1950).

14. A. S. Householder, G. E. Forsythe, and H. H. Germond, Ed., Monte Carlo Methods, NBS

Applied Mathematics Series, Vol. 12, 6, (1951).

15. D. H. Lehmer, “Mathematical Methods in Large-Scale Computing Units,” Ann. Comp.

Lab., Harvard Univ. 26, 141-146 (1951).

16. Herman Kahn, “Applications of Monte Carlo,” AECU-3259 Report, Rand Corporation,

Santa Monica, CA, (1954).

17. E. D. Cashwell and C. J. Everett, A Practical Manual on the Monte Carlo Method for

Random Walk Problems, Pergamon Press, Inc., New York (1959).

18. Robert R. Johnston, “A General Monte Carlo Neutronics Code,” Los Alamos Scientific

Laboratory Report, LAMS–2856 (March 1963).

19. E. D. Cashwell, J. R. Neergaard, W. M. Taylor, and G. D. Turner, “MCN: A Neutron

Monte Carlo Code,” Los Alamos Scientific Laboratory Report, LA–4751 (January 1972).

2-198 April 10, 2000

CHAPTER 2

REFERENCES

20. E. D. Cashwell, J. R. Neergaard, C. J. Everett, R. G. Schrandt, W. M. Taylor, and

G. D. Turner, “Monte Carlo Photon Codes: MCG and MCP,” Los Alamos National

Laboratory Report, LA–5157–MS (March 1973).

21. J. A. Halblieb and T. A. Mehlhorn, “ITS: The Integrated TIGER Series of Coupled

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u˙˙

o˙˙ φλ() 1

2πi

-------- eµµln λµ+µd

σi∞–

σi∞+

∫

u˙˙

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84. J. E. Stewart, “A General Point-on-a-Ring Detector,” Transactions of the American

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90. T. E. Booth, “Analytic Comparison of Monte Carlo Geometry Splitting and Exponential

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91. T. E. Booth, “A Caution on Reliability Using “Optimal” Variance Reduction Parameters,”

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92. T. E. Booth, “Analytic Monte Carlo Score Distributions for Future Statistical Confidence

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April 10, 2000 3-1

CHAPTER 3

INP FILE

CHAPTER 3

DESCRIPTION OF MCNP INPUT

Input to MCNP consists of several ﬁles, but the main one supplied by the user is the INP (the

default name) ﬁle, which contains the input information necessary to describe the problem. Only a

small subset of all available input cards will be needed in any particular problem. The input cards

are summarized by card type on page 3–146. The word “card” is used throughout this manual to

describe a single line of input up to 80 characters.

Maximum dimensions exist for some MCNP input items; they are summarized on page 3–150. The

user can increase any of these maximum values by altering the code and recompiling.

All features of MCNP should be used with caution and knowledge. This is especially true of

detectors and variance reduction schemes; you are encouraged to read the appropriate sections of

Chapter 2 before using them.

The units used throughout MCNP are given in Chapter 1 on page 1–20.

I. INP FILE

The INP ﬁle can have two forms, initiate-run and continue-run. Either can contain an optional

message block that replaces or supplements the MCNP execution line information.

A. Message Block

A user has the option to use a message block before the problem identiﬁcation title card in the INP

ﬁle. In computer environments where there are no execution line messages, the message block is

the only means for giving MCNP an execution message. Less crucially, it is a convenient way to

avoid retyping an often-repeated message. The message block starts with the string MESSAGE:

and is limited to columns 1−80. Alphabetic characters can be upper, lower, or mixed case. The

message block ends with a blank line delimiter before the title card.All cards before the blank line

delimiter are continuation cards. A $ and & in the message block are end−of−line markers. The

syntax and components of the message are the same as for the regular execution line message

discussed on page 1–32. Any ﬁlename substitution, program module execution option or keyword

entry on the execution line takes precedence over conﬂicting information in the message block.

INP = ﬁlename is not a legitimate entry in the message block. The name INP can be changed on

the execution line only.

3-2 April 10, 2000

CHAPTER 3

INP FILE

B. Initiate-Run

This form is used to set up a Monte Carlo problem (describe geometry, materials, tallies, etc.) and

run if message block is present. The initiate-run ﬁle has the following form:

The ﬁrst card in the ﬁle after the optional message block is the required problem title card. It is

limited to one 80−column line and is used as a title in various places in the MCNP output.It can

contain any information the user desires (or can even be blank) and often contains information

describing the particular problem. Note that a blank card elsewhere is used as a delimiter or as a

terminator. Alphabetic characters can be upper, lower, or mixed case.

With a valid set of data cards MCNP will run with or without the blank line terminator. With the

terminator MCNP will stop reading the input ﬁle there even if additional lines are in the ﬁle. Some

users like to keep additional material, such as alternative versions of the problem or textual

information, associated with the input ﬁle itself. The terminator will prevent such additional lines

from being read.

C. Continue−Run

Continue-run is used to continue running histories in a problem that was terminated earlier−for

example, to run the job 2 hours and then to run it an additional hour later. It can also be used to

reconstruct the output of a previous run. A continue-run must contain C or CN in the MCNP

execution line or message block to indicate a continue-run. It will start with the last dump unless

C m is used to start with the mth dump.

Message Block

Blank Line Delimiter

Title Card

Cell Cards

⋅

Blank Line Delimiter

Surface Cards

⋅

Blank Line Delimiter

Data Cards

⋅

Blank Line Terminator

Anything Else Recommended

Optional

}Optional

April 10, 2000 3-3

CHAPTER 3

INP FILE

In addition to the C or CN option on the MCNP execution line, two ﬁles can be important for this

procedure: (1) the restart ﬁle (default name RUNTPE), and (2) an optional continue-run input ﬁle

(default name INP).

The run ﬁle, generated by MCNP in the initiate-run sequence, contains the geometry, cross

sections, problem parameters, tallies, and all other information necessary to restart the job. In

addition the problem results at various stages of the run are recorded in a series of dumps. See the

PRDMP card (page 3–127) for a discussion of the selection of dump times. As discussed below,

the run may be restarted from any of these dumps.

The CN execution message option differs from the C option only in that the dumps produced during

the continue-run are written immediately after the ﬁxed data portion of the RUNTPE ﬁle rather

than after the dump from which the continue-run started. The new dumps overwrite the old dumps,

providing a way for the user to prevent unmanageable growth of RUNTPE ﬁles. RUNTPE growth

also can be controlled by the NDMP entry on the PRDMP card.

The optional continue-run input ﬁle must have the word CONTINUE as the ﬁrst entry on the ﬁrst

line (title card), or after the optional Message Block and its blank line delimiter. Alphabetic

characters can be upper, lower, or mixed case. This ﬁle has the following form:

The data cards allowed in the continue-run input ﬁle are a subset of the data cards available for an

initiate-run ﬁle. The allowed continue-run data cards are FQ, DD, NPS, CTME, IDUM, RDUM,

PRDMP, LOST, DBCN, PRINT, KCODE, MPLOT, ZA, ZB, and ZC.

A very convenient feature is that if none of the above items is to be changed (and if the computing

environment allows execution line messages), the continue-run input ﬁle is not required; only the

run ﬁle RUNTPE and the C option on the MCNP execution line are necessary. For example, if you

run a job for a minute but you want more particles run, execute with the C or CN message on the

execute line, and the job will pick up where it stopped and continue until another time limit or

particle cutoff is reached or until you stop it manually. This example assumes that a restart ﬁle

called RUNTPE from the initial run is in your current directory.

Message Block

Blank Line Delimiter

CONTINUE

Data Cards

Blank Line Terminator

Anything else

⋅

}Optional

Recommended

Optional

3-4 April 10, 2000

CHAPTER 3

INP FILE

The complete continue-run execution line option is C m or CN m, where m speciﬁes which dump

to pick up from the RUNTPE and to continue with. If m is not speciﬁed, the last dump is taken by

default. If the initial run producing the RUNTPE was stopped because of particle cutoff (NPS card,

page 3–125), NPS must be increased for a continue-run. The NPS card refers to total histories to

be run, including preceding continue-runs and the initial run. CTME in a continue−run is the

number of minutes more to run, not cumulative total time. To run more KCODE cycles, only the

fourth entry KCT matters. Like NPS, KCT refers to total cycles to be run, including previous ones.

In a continue-run, a negative number entered on the NPS card produces a print output ﬁle at the

time of the requested dump. No more histories will be run. This can be useful when the printed

output has been lost or you want to alter the content of the output with the PRINT or FQ cards.

Be cautious if you use a FILES card in the initial run. See page 3–133.

D. Card Format

All input lines are limited to 80 columns. Alphabetic characters can be upper, lower, or mixed case.

Most input is entered in horizontal form; however, a vertical input format is allowed for data cards.

A comment can be added to any input card. A $ (dollar sign) terminates data entry and anything

that follows the $ is interpreted as a comment. Blank lines are used as delimiters and terminators.

Data entries are separated by one or more blanks.

Comment cards can be used anywhere in the INP ﬁle after the problem title card and before the last

blank terminator card. These cards must have a C anywhere in columns 1−5 followed by at least

one blank. Comment cards are printed only with the input ﬁle listing and not anywhere else in the

MCNP output ﬁle. The FCn input card is available for user comments and is printed as a heading

for tally n (as a tally title, for example). The SCn card is available for user comments and is printed

as a heading for source probability distribution n.

1. Horizontal Input Format

Cell, surface, and data cards all must begin within the ﬁrst ﬁve columns. The card name or number

and particle designator is followed by data entries separated by one or more blanks. Blanks in the

ﬁrst ﬁve columns indicate a continuation of the data from the last named card. An & (ampersand)

preceded by at least one blank ending a line indicates data will continue on the following card. Data

on the continuation card can be in columns 1−80. Completely blank cards are reserved as delimiters

between major sections of the input ﬁle. An individual entry must be entirely on one line. There

can be only one card of any given type for a given particle designation (see page 3–7). Integers must

be entered where integer input is required. Other numerical data can be entered as integer or

ﬂoating point and will be read properly by MCNP. (In fact noninteger numerical data can be entered

in any form acceptable to a FORTRAN E-edit descriptor.)

April 10, 2000 3-5

CHAPTER 3

INP FILE

Four features incorporated in the code facilitate input card preparation:

1. nR means repeat the immediately preceding entry on the card n times. For example, 2

4R is the same as 2 2 2 2 2.

2. nI means insert n linear interpolates between the entries immediately preceding and

following this feature. For example, 1.5 2I 3.0 on a card is the same as 1.5 2.0 2.5 3. In

the construct X nI Y, if X and Y are integers, and if Y − X is an exact multiple of n+1,

correct integer interpolates will be created. Otherwise only real interpolates will be

created, but Y will be stored directly in all cases. In the above example, the 2.0 may not

be exact, but in the example 1 4I 6 = 1 2 3 4 5 6, all interpolates are exact.

3. xM is a multiply feature and when used on an input card, it is replaced by the value of

the previous entry on the card multiplied by the factor x. For example, 1 1 2M 2M 2M

2M 4M 2M 2M is equivalent to 1 1 2 4 8 16 64 128 256.

4. nJ can be used on an input card to jump over the entry where used and take the default

value. As an example, the following two cards are identical in their effect:

DD .1 1000

DD J 1000

JJJisalso equivalent to 3J. You can jump to a particular entry on a card without having

to explicitly specify prior items on the card. This feature is convenient if you know you

want to use a default value but can’t remember it. DBCN 7J 5082 is another example.

These features apply to both integer and ﬂoating point quantities. If n (an integer) is omitted in the

constructs nR, nI, and nJ, then n is assumed to be 1. If x (integer or ﬂoating point) is omitted in xM,

it is a fatal error. The rules for dealing with adjacent special input items are as follows:

1. nR must be preceded by a number or by an item created by R or M.

2. nI must be preceded by a number or by an item created by R or M, and must be followed

by a number.

3. xM must be preceded by a number or by an item created by R or M.

4. nJ may be preceded by anything except I and may begin the card input list.

Examples: 1 3M 2R = 1 3 3 3

1 3M I 4 = 1 3 3.5 4

1 3M 3M = 1 3 9

1 2R 2I 2.5 = 1 1 1 1.5 2.0 2.5

1 R 2M = 1 1 2

1 R R = 1 1 1

1 2I 4 3M = 1 2 3 4 12

1 2I 4 2I 10 = 1 2 3 4 6 8 10

3-6 April 10, 2000

CHAPTER 3

INP FILE

3J 4R is illegal.

1 4I 3M is illegal.

1 4I J is illegal.

2. Vertical Input Format

Column input is particularly useful for cell parameters and source distributions. Cell importances

or volumes strung out on horizontal input lines are not very readable and often cause errors when

users add or delete cells. In column format, all the cell parameters for one cell can be on a single

line, labeled with the name of the cell. If a cell is deleted, the user deletes just one line of cell

parameters instead of hunting for the data item that belongs to the cell in each of several multiline

cell parameter cards. For source distributions, corresponding SI, SP, and SB values are side by side.

Source options, other than defaults, are on the next line and must all be entered explicitly. The &

continuation symbol is not needed, and if present, is ignored.

In column format, card names are put side by side on one input line and the data values are listed

in columns under the card names. A # is put somewhere in columns 1−5 on the line with the card

names. The card names must be all cell parameters, all surface parameters, or all something else.

If a card name appears on a # card, there must not be a regular horizontal card by that name in the

same input ﬁle. If there are more entries on data value lines than card names on the # line, the ﬁrst

data entry is a cell or surface number. If any cell names are entered, all must be entered. If cell

names are entered, the cells don’t have to be in the same order as they are in the cell cards block.

If cell names are omitted, the default order is the order of the cells in the cell card block. The same

rules apply to surface parameters, but because we presently have only one surface parameter

(AREA), column input of surface parameters is less useful.

There can be more than one block of column data in an input ﬁle. Typically, there would be one

block for cell parameters and one for each source distribution. If a lot of cell parameter options are

being used, additional blocks of column data would be needed.

The entries in each column do not need to be precisely under the card name at the top of the column,

but you might want the columns to be reasonably neat for readability. The column format is

intended for input data that naturally ﬁt into columns of equal length, but less tidy data are not

prohibited. If a longer column is to the right of a shorter column, the shorter column must be ﬁlled

with enough J entries to eliminate any ambiguity about which columns the data items are in.

Special syntax items (R, M, I, and J) are not as appropriate in column format as they are on

horizontal lines, but they are not prohibited. They are, of course, interpreted vertically instead of

horizontally. Multiple special syntax items, such as 9R, are not allowed if cell or surface names are

present.

The form of a column input block is

April 10, 2000 3-7

CHAPTER 3

INP FILE

1S2Sm

K1D11 D12 D1m

K2D21 D22 D2m

.. .. .

.....

KnDn1 Dn2 Dnm

1. The # is somewhere in columns 1−5.

2. Each line can be only 80 columns wide.

3. Each column, Sithrough Dli, where lmay be less than n, represents a regular input card.

4. The Si must be valid MCNP card names. They must be all cell parameters, all surface

parameters, or all something else.

5. D1i through Dni must be valid entries for an Sicard, except that Dl+1,i through Dni may

be some J’s possibly followed by some blanks.

6. If Dji is nonblank, Dj,i-1 must also be nonblank. A J may be used if necessary to make

Dj,i-1 nonblank.

7. The Si must not appear anywhere else in the input file.

8. The Kj are optional integers. If any are nonblank, all must be nonblank.

9. If the Siare cell parameter card names, the Kj, if present, must be valid cell names. The

same is true with surface parameters.

10. If the Kj are present, the Dji must not be multiple special syntax items, such as 9R.

E. Particle Designators

Several of the input cards require a particle designator to distinguish between input data for

neutrons, for photons and for electrons. These cards are IMP, EXT, FCL, WWN, WWE, WWP,

WWGE, DXT, DXC, F, F5X, F5Y, F5Z, PHYS, ELPT, ESPLT, CUT and PERT. The particle

designator consists of the symbol : (colon) and the letter N, P or E immediately after the name of

the card. At least one blank must follow the particle designator. For example, to enter neutron

importances, use an IMP:N card; enter photon importances on an IMP:P card. To specify the same

value for more than one kind of particle, a single card can be used instead of several.

Example: IMP:E,P,N 1 1 0. With a tally card, the particle designator follows the card name

including tally number. For example, ∗F5:N indicates a neutron point detector energy tally. In the

heating tally case, both particle designators may appear. The syntax F6:N,P indicates the combined

heating tally for both neutrons and photons.

…

3-8 April 10, 2000

CHAPTER 3

INP FILE

F. Default Values

Many MCNP input parameters have default values that are summarized on page 3–146. Therefore

you do not always have to specify explicitly every input parameter every time if the defaults match

your needs. If an input card is left out, the default values for all parameters on the card are used.

However, if you want to change a particular default parameter on a card but that parameter is

preceded by others, you have to specify the others or use the nJ jump feature to jump over the

parameters for which you still want the defaults. CUT:P 3J −.10 is a convenient way to use the

defaults for the ﬁrst three parameters on the photon cutoff card but change the fourth.

G. Input Error Messages

MCNP makes extensive checks (over 400) of the input ﬁle for user errors. A fatal error message is

printed, both at the terminal and in the OUTP ﬁle, if the user violates a basic constraint of the input

speciﬁcation, and MCNP will terminate before running any particles. The ﬁrst fatal error is real;

subsequent error messages may or may not be real because of the nature of the ﬁrst fatal message.

The FATAL option on the MCNP execution line instructs MCNP to ignore fatal errors and run

particles, but the user should be extremely cautious about doing this.

Most MCNP error messages are warnings and are not fatal. The user should not ignore these

messages but should understand their signiﬁcance before making important calculations.

In addition to FATAL and WARNING messages, MCNP issues BAD TROUBLE messages

immediately before any impending catastrophe, such as a divide by zero, which would otherwise

cause the program to “crash.” MCNP terminates as soon as the BAD TROUBLE message is issued.

User input errors in the INP ﬁle are the most common reason for issuing a BAD TROUBLE

message. These error messages indicate what corrective action is required.

H. Geometry Errors

There is one important kind of input error that MCNP will not detect while processing data from

the INP ﬁle. MCNP cannot detect overlapping cells or gaps between cells until a particle track

actually gets lost. Even then the precise nature of the error may remain unclear. However, there is

much that you can and should do to check your geometry before starting a long computer run.

Use the geometry-plotting feature of MCNP to look at the system from several directions and at

various scales. Be sure that what you see is what you intend. Any gaps or overlaps in the geometry

will probably show up as dashed lines. The intersection of a surface with the plot plane is drawn as

a dashed line if there is not exactly one cell on each side of the surface at each point. Dashed lines

can also appear if the plot plane happens to coincide with a plane of the problem, if there are any

cookie-cutter cells in the source, or if there are DXTRAN spheres in the problem.

April 10, 2000 3-9

CHAPTER 3

INP FILE

Set up and run a short problem in which your system is ﬂooded with particle tracks from an external

source. The necessary changes in the INP ﬁle are as follows:

1. Add a VOID card to override some of the other specifications in the problem and make

all the cells voids, turn heating tallies into flux tallies, and turn off any FM cards.

2. Add another cell and a large spherical surface to the problem such that the surface

surrounds the system and the old outside world cell is split by the new surface into two

cells: the space between the system and the new surface, which is the new cell, and the

space outside the new surface, which is now the outside world cell. Be sure that the new

cell has nonzero importance. Actually, it is best to make all nonzero importances equal.

If the system is infinite in one or two dimensions, use one or more planes instead of a

sphere.

3. Replace the source specifications by an inward directed surface source to flood the

geometry with particles:

SDEF SUR=m NRM = −1

where m is the number of the new spherical surface added in Step 2. If the new surface

is a plane, you must specify the portion to be used by means of POS and RAD or possibly

X, Y, and Z source distributions.

Because there are no collisions, a short run will generate a great many tracks through your system.

If there are any geometry errors, they should cause some of the particles to get lost.

When a particle ﬁrst gets lost, whether in a special run with the VOID card or in a regular

production run, the history is rerun to produce some special output on the OUTP ﬁle. Event-log

printing is turned on during the rerun. The event log will show all surface crossings and will tell

you the path the particle took to the bad spot in the geometry. When the particle again gets lost, a

description of the situation at that point is printed. You can usually deduce the cause of the lost

particle from this output. It is not possible to rerun lost particles in a multitasking run.

If the cause of the lost particle is still obscure, try plotting the geometry with the origin of the plot

at the point where the particle got lost and with the horizontal axis of the plot plane along the

direction the particle was moving. The cause of the trouble is likely to appear as a dashed line

somewhere in the plot or as some discrepancy between the plot and your idea of what it should look

like.

3-10 April 10, 2000

CHAPTER 3

CELL CARDS

II. CELL CARDS

Form: j m d geom params

or: j LIKE n BUT list

j=cell number; .

If cell has transformation, . See page 3–27.

m=0 if the cell is a void.

=material number if the cell is not a void. This indicates that the cell is to

contain material m, which is speciﬁed on the Mm card. See page 3–108.

d=absent if the cell is a void.

=cell material density. A positive entry is interpreted as the atomic density

in units of 1024 atoms/cm3. A negative entry is interpreted as the mass

density in units of g/cm3.

geom =speciﬁcation of the geometry of the cell. It consists of signed surface

numbers and Boolean operators that specify how the regions bounded by

the surfaces are to be combined.

params =optional speciﬁcation of cell parameters by entries in the

keyword = value form.

n=name of another cell

list =set of keyword =value speciﬁcations that deﬁne the attributes that differ

between cell n and j.

In the geometry speciﬁcation, a signed surface number stands for the region on the side of the

surface where points have the indicated sense. The plus sign for positive sense is optional. The

regions are combined by Boolean operators: intersection (no symbol—implicit, like multiplication

in algebra); union, :; and complement, #. Parentheses can be used to control the order of the

operations. Parentheses and operator symbols also function as delimiters. Where they are present,

blank delimiters are not necessary. The default order of operations is complement ﬁrst, intersection

second, and union last. A number immediately after a complement operator, without parentheses,

is interpreted as a cell number and is shorthand for the geometry speciﬁcation of that cell number.

Example: 3 0 -1 2 -4 $ deﬁnition of cell 3

#3 $ equivalent to next line

#(-1 2 -4)

For a simple cell (no union or complement operators), the geometry speciﬁcation is just a blank-

delimited list of the bounding surfaces and ambiguity surfaces of the cell with signs determined by

the sense of the cell with respect to each surface. See the Geometry sections of Chapters 1, 2, and

4 for complete explanations of how to specify the geometry of cells in MCNP.

Cell parameters can be deﬁned on cell cards instead of in the data card section of the INP ﬁle. A

blank is equivalent to the equal sign. If a cell parameter is entered on any cell card, a cell-parameter

1j99999≤≤ 1j999≤≤

April 10, 2000 3-11

CHAPTER 3

CELL CARDS

card with that name can not be present, nor can the mnemonic appear on any column-format input

card. Some cell parameters can be speciﬁed on cell cards and a different subset on cell-parameter

or column-format cards. The form is keyword=value, where the allowed keywords are IMP, VOL,

PWT, EXT, FCL, WWN, DXC, NONU, PD, and TMP, with particle designators where necessary.

Four cell parameter cards associated with the repeated structures capability are U, TRCL, LAT and

FILL. Like any cell parameter card, these four cards can be placed in the data card section of the

INP ﬁle. Our recommendation is that the mnemonic and entry for each cell be placed on the cell

card line after the cell description. The entries on the TRCL card and the FILL card, in particular,

can be quite long and involved and it seems to be conceptually simpler when they are placed on the

cell card line.

The LIKE n BUT feature uses keywords for the cell material number and density. The mnemonics

are MAT and RHO, respectively. These keywords only can be used following the LIKE n BUT

construct. In a normal cell description, material number and density are still the second and third

entries on the cell card.

TMP and WWN data can be entered on cell cards in two ways. The keyword=value form

TMP1=value TMP2=value etc. can be used or a special syntax is available where the single

keyword TMP is followed by all the temperatures of the cell in an order corresponding to the times

on the THTME card. The form for the WWN card is analogous: WWN1:n=value or WWN:n

followed by all the lower weight bounds for the energy intervals of the cell.

Example: 10 16 −4.2 1 −2 3 IMP:N=4 IMP:P=8 EXT:N=−.4X

This says that cell 10 is to be ﬁlled with material 16 at a density of 4.2 g/cm3. The cell consists of

the intersections of the regions on the positive side of surface 1, the negative side of surface 2, and

the positive side of surface 3. The neutron importance in cell 10 is 4 and the photon importance is

8. Neutrons in cell 10 are subject to an exponential transform in the minus X direction with

stretching parameter 0.4.

Here are some precautions when you are preparing cell cards:

1. Avoid excessively complicated cells. MCNP runs faster when the problem geometry is

made up of many simpler cells rather than fewer more complicated cells.

2. Avoid adding unneeded surfaces to the geometry description of a cell through poor use

of the complement operator. The extra surfaces make the problem run slower and may

destroy the necessary conditions for volume and area calculations. See page 4–15.

3. Always use the geometry-plotting feature of MCNP to check the geometry of a problem.

See Appendix B.

4. Flood the system with particles from an outside source to find errors in the geometry.

See page 3–8.

3-12 April 10, 2000

CHAPTER 3

SURFACE CARDS

5. If you add or remove cells, change all your cell parameter cards accordingly. The

difficulty of this can be reduced if you use vertical format for your cell parameter cards.

See page 3–6. Alternatively, define the values of cell parameters on cell cards and

eliminate cell parameter cards entirely.

A. Shorthand Cell Speciﬁcation

The LIKE n BUT feature is very useful in problems with a lot of repeated structures. Cell j inherits

from cell n the values of all attributes that are not speciﬁed in the list. The cell card for cell n must

be before the cell card for cell j in the INP ﬁle. Any card name that appears after the BUT is a cell

parameter on a cell card and, therefore, must appear on cell cards only, not on any cards in the data

block of the INP ﬁle.

Example: 2 3 −3.7 −1 IMP:N=2 IMP:P=4

3 LIKE 2 BUT TRCL=1 IMP:N=10

This says that cell 3 is the same as cell 2 in every respect except that cell 3 has a different location

(TRCL=1) and a different neutron importance. The material in cell 3, the density and the deﬁnition

are the same as cell 2 and the photon importance is the same.

III. SURFACE CARDS

A. Surfaces Deﬁned by Equations

Form: j n a list

j=surface number: , with asterisk for a reﬂecting surface

or plus for a white boundary.

If surface deﬁnes a cell that is transformed with TRCL, .

See page 3–27.

n=absent or 0 for no coordinate transformation.

=>0, speciﬁes number of a TRn card.

=< 0, speciﬁes surface j is periodic with surface n.

a=equation mnemonic from Table 3.1

list =one to ten entries, as required.

The surface types, equations, mnemonics, and the order of the card entries are given in Table 3.1.

To specify a surface by this method, ﬁnd the surface in Table 3.1 and determine the coefﬁcients for

the equation (you may need to consult a book on analytical geometry). The information is entered

on a surface card according to the above form. Under certain conditions a surface can be deﬁned

by specifying geometrical points, as discussed in sections B and C. Surfaces also can be produced

by combinatorial–geometry–like macrobodies, described in section D.

1j99999≤≤

1j999≤≤

April 10, 2000 3-13

CHAPTER 3

SURFACE CARDS

A point (x,y,z) is deﬁned as having positive sense with respect to a surface when the expression for

that surface evaluated at (x,y,z) is positive. The expression for a surface is the left side of the

equation for the surface in Table 3.1. With the sphere, cylinder, cone, and torus, this deﬁnition is

identical to deﬁning the sense to be positive outside the ﬁgure. With planes normal to axes (PX,

PY, or PZ), the deﬁnition gives positive sense for points with x,y, or zvalues exceeding the

intercept of the plane. For the P, SQ and GQ surfaces, the user supplies all of the coefﬁcients for

the expression and thus can determine the sense of the surface at will. This is different from the

other cases where the sense, though arbitrary, is uniquely determined by the form of the expression.

Therefore, in a surface transformation (see the TRn card on page 3–30) a PX, PY, or PZ surface

will sometimes be replaced by a P surface just to prevent the sense of the surface from getting

reversed.

If the surface number is preceded by an asterisk, a reﬂecting surface is deﬁned. A particle track that

hits a reﬂecting surface is reﬂected specularly. If the surface number is preceded by a plus, a white

boundary is deﬁned. Detectors and DXTRAN (next–event estimators) usually should not be used

in problems that have reﬂecting surfaces or white boundaries. See page 2–92. Tallies in problems

with reﬂecting surfaces will need to be normalized differently. See page 2–14.

A negative second entry n speciﬁes that surface j is periodic with surface k. The following

restrictions apply:

1. Surfaces j and k must be planes.

2. No surface transformation is allowed for the periodic planes.

3. The periodic cell(s) can be infinite or bounded by planes on the top and bottom that can

be reflecting or white, but cannot be periodic.

4. Periodic planes can only bound other periodic planes or top and bottom planes.

5. A single zero–importance cell must be on one side of each periodic plane.

6. All periodic planes must have a common rotational vector normal to the geometry top

and bottom.

7. Next–event estimators such as detectors and DXTRAN should not be used.

3-14 April 10, 2000

CHAPTER 3

SURFACE CARDS

TABLE 3.1: MCNP Surface Cards

Mnemonic Type Description Equation Card Entires

Plane General

Normal to X–axis

Normal to Y–axis

Normal to Z–axis

Ax + By + Cz – D = 0

x – D = 0

y – D = 0

z – D = 0

ABCD

Sphere Centered at Origin

General

Centered on X–axis

Centered on Y–axis

Centered on Z–axis

C/X

C/Y

C/Z

Cylinder Parallel to X–axis

Parallel to Y–axis

Parallel to Z–axis

On X–axis

On Y–axis

On Z–axis

K/X

K/Y

K/Z

Cone Parallel to X–axis

Parallel to Y–axis

Parallel to Z–axis

On X–axis

On Y–axis

On Z–axis

used only

for 1 sheet cone

SQ Ellipsoid

Hyperboloid

Paraboloid

Axis not parallel

to X–, Y–, or Z–axis

A B C D E

F G

GQ Cylinder

Cone

Ellipsoid

Hyperboloid

Paraboloid

Axes not parallel

to X–, Y–, or Z–axis +

A B C D E

F G H J K

Elliptical or

circular torus.

Axis is

Parallel to

X–,Y–, or Z– axis

A B C

XYZP Surfaces deﬁned by points See pages 3–16 and 3–18

x2y2z2R2

–++ 0=

xx–()

2yy–()

2zz–()

2R2

–++ 0=

xx–()

2y2z2R2

–++ 0=

x2yy–()

2z2R2

–++ 0=

y2y2zz–()

2R2

–++ 0=

xy z R

y R

yy–()

2zz–()

2R2

–+0=

xx–()

2zz–()

2R2

–+0=

xx–()

2yy–()

2R2

–+0=

y2z2R2

–+0=

x2z2R2

–+0=

x2y2R2

–+0=

yz R

xz R

xy R

yy–()

2zz–()

+tx x–()–0=

xx–()

2zz–()

+ty y–()–0=

xx–()

2yy–()

+tz z–()–0=

y2z2

+tx x–()–0=

x2z2

+ty y–()–0=

x2y2

+tz z–()–0=

x y z t21±

xt21±

y t21±

z t21±

1±

Ax x–()

2By y–()

2Cz z–()

2Dx x–()2Ey y–()++

2Fz z–()G++0=

x y z

Ax2By2Cz2Dxy Eyz+++ +

Fzx Gz Hy Jz K++++ 0=

xx–()

2B2

⁄yy–()

2zz–()

+A–()

2C21–⁄+0=

yy–()

2B2

⁄xx–()

2zz–()

+A–()

2C2

⁄1–+0=

zz–()

2B2

⁄xx–()

2yy–()

+A–()

2C21–()⁄+0=

x y z

April 10, 2000 3-15

CHAPTER 3

SURFACE CARDS

Example 1: j PY 3

This describes a plane normal to the y–axis at y = 3 with positive sense for all points with y> 3.

Example 2: j K/Y 0 0 2 .25 1

This speciﬁes a cone whose vertex is at (x,y,z) = (0,0,2) and whose axis is parallel to the y–axis.

The tangent tof the opening angle of the cone is 0.5 (note that t2is entered) and only the positive

(right hand) sheet of the cone is used. Points outside the cone have a positive sense.

Example 3: j GQ 1 .25 .75 0 –.866

0 –12 –2 3.464 39

This is a cylinder of radius 1 cm whose axis is in a plane normal to the x–axis at x=6, displaced 2

cm from the x–axis and rotated about the x–axis off the y–axis toward the z–axis. The sense is

positive for points outside the cylinder. Such a cylinder would be much easier to specify by ﬁrst

deﬁning it in an auxiliary coordinate system where it is symmetric about a coordinate axis and then

using the TRn input card (see page 3–30) to deﬁne the relation between the basic and auxiliary

coordinate systems. The input would then be

j 7 CX 1

*TR7 6 1 –1.732 0 30 60

See Chapter 4 for additional examples of the TRn card.

The TX, TY, and TZ input cards represent elliptical tori (fourth degree surfaces) rotationally

symmetric about axes parallel to the x,y, and z axes, respectively. A TY torus is illustrated in

Figure 3.1a. Note that the input parameters a b c specify the ellipse

rotated about the s–axis in the (r,s) cylindrical coordinate system (Figure 3.1b) whose origin is at

in the x,y,z system. In the case of a TY torus,

and

A torus is degenerate if |a|<c where 0< a < c produces the outer surface (Figure 3.1c), and

−c< a < 0 produces the inner surface (Figure 3.1d).

30°

xy z

-----ra–()

------------------+1=

xy z

syy–()=

rxx–()

2zz–()

3-16 April 10, 2000

CHAPTER 3

SURFACE CARDS

Figure 3-1. Torus

Coordinate transformations for tori are limited to those in which each axis of the auxiliary

coordinate system is parallel to an axis of the main system.

B. Axisymmetric Surfaces Deﬁned by Points

Form: j n a list

j=surface number: . If surface deﬁnes a cell that is

transformed with TRCL, . See page 3–27.

n=absent for no coordinate transformation, or number of TRn card.

a=the letter X, Y, or Z

list =one to three coordinate pairs.

Surface cards of type X, Y, and Z can be used to describe surfaces by coordinate points rather than

by equation coefﬁcients as in the previous section. The surfaces described by these cards must be

xy s

Fig. a c

Fig. b

Fig. c

cb0< a< c

outer surface

Fig. d

cba< 0< c

inner surface

1j99999≤≤

1j999≤≤

April 10, 2000 3-17

CHAPTER 3

SURFACE CARDS

symmetric about the x−,y−, or z−axis, respectively, and, if the surface consists of more than one

sheet, the speciﬁed coordinate points must all be on the same sheet.

Each of the coordinate pairs deﬁnes a geometrical point on the surface. On the Y card, for example,

the entries may be

jYy1r1y2r2

where and yi is the coordinate of point i.

If one coordinate pair is used, a plane (PX, PY, or PZ) is deﬁned.

If two coordinate pairs are used, a linear surface (PX, PY, PZ, CX, CY, CZ, KX, KY, or KZ) is

deﬁned.

If three coordinate pairs are used, a quadratic surface (PX, PY, PZ, SO, SX, SY, SZ, CX, CY, CZ,

KX, KY, KZ, or SQ) is deﬁned.

When a cone is speciﬁed by two points, a cone of only one sheet is generated.

The senses of these surfaces (except SQ) are determined by the code to be identical to the senses

one would obtain by specifying the surface by equations. For SQ, the sense is deﬁned so that points

sufﬁciently far from the axis of symmetry have positive sense. Note that this is different from the

equation-deﬁned SQ, where the user could choose the sense freely.

Example 1: j X 7 5 3 2 4 3

This describes a surface symmetric about the x–axis, which passes through the three (x,r) points

(7,5), (3,2), and (4,3). This surface is a hyperboloid of two sheets, converted in MCNP to its

equivalent

jSQ−.083333333 1 1 0 0 0 68.52083 −26.5 0 0.

Example 2: j Y 1 2 1 3 3 4

This describes two parallel planes at Y = 1 and Y = 3 and is a fatal error because the requirement

that all points be on the same sheet is not met.

Example 3: j Y 3 0 4 1 5 0

This describes a sphere of radius 1 with center at (x,y,z) = (0,4,0).

rixi

2zi

+()=

3-18 April 10, 2000

CHAPTER 3

SURFACE CARDS

Example 4: j Z 1 0 2 1 3 4

This surface is rejected because the points are on two different sheets of the hyperboloid

However, the surface

j Z 2 1 3 4 5 9.380832

which has the same surface equation as above is accepted because all coordinates lie on a single

surface, the right sheet of the hyperboloid.

Example 5: 1 0 1 −2 3 $ cell 1

1Y −32 21

2 Y 23 33 42

3 Y 21 31 42

This ﬁnal example deﬁnes a cell bounded by a cone, hyperboloid, and an ellipsoid. The three

surfaces deﬁne the donut-like cell that is symmetric about the y−axis. A cross section of this cell is

seen in Figure 3.2. To plot this view, type PX = 0EX= 5. One surface goes through the points (-

3,2) and (2,1). The second surface goes through (2,3), (3,3), and (4,2). The last surface is deﬁned

by the points (2,1), (3,1), and (4,2). These coordinate points are in the form (y,r). Using these cards,

MCNP indicates that surface 1 is a cone of one sheet, surface 2 is an ellipsoid, and surface 3 is a

hyperboloid of one sheet. The equation coefﬁcients for the standard surface equations are printed

out for the various surfaces when the PRINT input card or execution option is used. For example,

an SQ card deﬁning surface 3 is

3 SQ 1 -1.5 1000−.625 0 2.5 0

Figure 3-2.

x2y27z2

–20z13–++ 0=

April 10, 2000 3-19

CHAPTER 3

SURFACE CARDS

C. General Plane Deﬁned by Three Points

Form: j n P X1Y1Z1X2Y2Z2X3Y3Z3

j=surface number: or if repeated structure.

n=absent or 0 for no coordinate transformation.

=> 0, speciﬁes number of a TRn card.

=< 0, speciﬁes surface j is periodic with surface n.

(Xi,Yi,Zi)=coordinates of points to deﬁne the plane.

If there are four entries on a P card, they are assumed to be the general plane equation coefﬁcients

as in Table 3.1. If there are more than four entries, they give the coordinates of three points lying

in the desired plane. The code converts them to the required surface coefﬁcients to produce the

plane

The sense of the plane is determined by requiring the origin to have negative sense. If the plane

passes through the origin (D = 0), the point has positive sense. If this fails (D =C=0),

the point has positive sense. If this fails (D =C=B=0), the point has positive

sense. If this fails, the three points lie in a line and a fatal error is issued.

D. Surfaces Deﬁned by Macrobodies

Using a combinatorial–geometry–like macrobody capability is an alternative method of deﬁning

cells and surfaces. The combinatorial geometry bodies available are similar to those in the

Integrated Tiger Series (ACCEPT) codes. The macrobodies can be mixed with the standard cells

and surfaces. The macrobody surface is decomposed internally into surface equations and the

facets are assigned individual numbers according to a predetermined sequence. The assigned

numbers are the number selected by the user followed by a decimal point and 1, 2, .... The facets

can be used for tallying, tally segmentation, other cell deﬁnitions, SDEF sources, etc. They cannot

be used on the SSR/SSW cards, the surface ﬂagging card, PTRAC, or MCTAL ﬁles.

The space inside a body has a negative sense with respect to the macrobody surface and all its

facets. The space outside a body has a positive sense. The sense of a facet is the sense assigned to

it by the macrobody “master” cell and the facet retains that assigned sense if it appears in other cell

descriptions and must be properly annotated. See an example at the end of this section for an

illustration.

The following geometry bodies are available and their complete descriptions follow.

BOX Arbitrarily oriented orthogonal box

RPP Rectangular ParallelePiped

1j99999≤≤ 999≤

Ax By Cz D–++ 0=

00∞,,()

0∞0,,() ∞00,,()

3-20 April 10, 2000

CHAPTER 3

SURFACE CARDS

SPH Sphere

RCC Right Circular Cylinder

RHP or HEX Right Hexagonal Prism

BOX: Arbitrarily oriented orthogonal box (all corners are 90˚.)

BOX Vx Vy Vz A1x A1y A1z A2x A2y A2z A3x A3y A3z

where Vx Vy Vz = x,y,z coordinates of corner

A1x A1y A1z = vector of 1st side

A2x A2y A2z = vector of 2nd side

A2x A3y A3z = vector of 3rd side

Example: BOX –1 –1 –1 2 0 0 0 2 0 0 0 2

a cube centered at the origin, 2 cm on a side, sides parallel to the major axes.

RPP: Rectangular ParallelePiped, surfaces normal to major axes, x,y,z values relative to origin.

RPP Xmin Xmax Ymin Ymax Zmin Zmax

Example: RPP –1 1 –1 1 –1 1

equivalent to BOX above.

SPH: Sphere. Equivalent to surface equation for general sphere.

SPH Vx Vy Vz R

where Vx Vy Vz = x,y,z coordinates of center

R = radius

RCC: Right Circular Cylinder, can

RCC Vx Vy Vz Hx Hy Hz R

where Vx Vy Vz = center of base

Hx Hy Hz = cylinder axis vector

R = radius

Example: RCC 0 –5 0 0 10 0 4

a 10-cm high can about the y-axis, base plane at y=–5 with radius of 4 cm.

RHP or HEX: Right Hexagonal Prism. Differs from ITS (ACCEPT) format.

RHP v1 v2 v3 h2 h2 h3 r1 r2 r3 s1 s2 s3 t1 t2 t3

where v1 v2 v3 = x,y,z coordinates of the bottom of the hex

h1 h2 h3 = vector from the bottom to the top

for a z-hex with height h, h1,h2,h3 = 0 0 h

r1 r2 r3 = vector from the axis to the middle of the ﬁrst facet

for a pitch 2p facet normal to y-axis, r1,r2,r3 = 0 p 0

s1 s2 s3 = vector to center of the 2nd facet

April 10, 2000 3-21

CHAPTER 3

SURFACE CARDS

t1 t2 t3 = vector to center of the 3rd facet

Example: RHP 0 0 –4 0 0 8 0 2 0

a hexagonal prism about the z-axis whose base plane is at z=–4 with a height

of 8-cm and whose ﬁrst facet is normal to the y-axis at y=2.

The facets of the bodies are sequentially numbered and can be used on other MCNP cards. BOX

and RPP can be inﬁnite in a dimension, in which case those two facets are skipped and the numbers

of the remaining facets are decreased by two. RHP can be inﬁnite in the axial dimension in which

case facets 7 and 8 do not exist. The order of the facet numbering follows for each geometry body.

Facet numbering can be displayed graphically with MBODY=OFF in the geometry plotter.

BOX: 1 plane normal to end of A1x A1y A1z

2 plane normal to beginning of A1x A1y A1z

3 plane normal to end of A2x A2y A2z

4 plane normal to beginning of A2x A2y A2z

5 plane normal to end of A3x A3y A3z

6 plane normal to beginning of A3x A3y A3z

RPP: 1 Plane Xmax

2 Plane Xmin

3 Plane Ymax

4 Plane Ymin

5 Plane Zmax

6 Plane Zmin

SPH: treated as a regular surface so no facet

RCC: 1 Cylindrical surface of radius R

2 Plane normal to end of Hx Hy Hz

3 Plane normal to beginning of Hx Hy Hz

RHP or HEX 1 Plane normal to end of r1 r2 r3

2 Plane opposite facet 1

3 Plane normal to end of s1 s2 s3

4 Plane opposite facet 3

5 Plane normal to end of t1 t2 t3

6 Plane opposite facet 5

7 Plane normal to end of h1 h2 h3

8 Plane normal to beginning of h1 h2 h3

3-22 April 10, 2000

CHAPTER 3

DATA CARDS

The following input ﬁle describes ﬁve cells and illustrates a combination of the various body and

cell/surface descriptions. Surface numbers are in italics alongside the planes they deﬁne. Note that

the cell and surface numbers do not have to start with 1 or be consecutive.

IV. DATA CARDS

All MCNP input cards other than those for cells and surfaces are entered after the blank card

delimiter following the surface card block. The mnemonic must begin within the ﬁrst ﬁve columns.

These cards fall into the following categories:

Category Page

(A) Problem type 3–23

(B) Geometry cards 3–23

(D) Source speciﬁcation 3–49

(E) Tally speciﬁcation 3–73

(F) Material and cross section speciﬁcation 3–107

(G) Energy and thermal treatment 3–116

(H) Problem cutoffs 3–123

(I) User data arrays 3–126

(J) Peripheral cards 3–127

10–1

2 like 1 but trcl = (2 0 0)

4 0 1.1 –2001.1 –5.3 –5.5 –5.6 –5.4

9 0 (–5.1 : 1.3 : 2001.1 : –99 : 5.5 : 5.6) #5 cell 1 cell 2

cell 3 cell 4

cell 9

1.15.1

1.3

5.3

2001.1

5 rpp –2 0 –2 0 –1 1

99 py –2

1 rpp 0 2 0 2 –1 1

3 0 –1.2 –1.1 1.4 –1.5 –1.6 99

50 –5

cell 5

5.2

3 0 5.1 –1.1 1.4 –5.5 –5.6 –5.4

3 0 –1.2 –1.1 –5.3 –5.5 –5.6 –5.4

3 0 5.1 –1.1 –5.3 –5.5 –5.6 99

alternative descriptions of cell 3:

April 10, 2000 3-23

CHAPTER 3

DATA CARDS

These card categories are described below. Only the cards listed on page 3–3 are allowed in a

continue-run input ﬁle. No data card can be used more than once with the same number or particle

type designations. For example, M1 and M2 are acceptable, as are CUT:N and CUT:P, but two M1

cards or two CUT:N cards are disallowed.

A. Problem Type (MODE) Card

Form: MODE

xi=N for neutron transport

P for photon transport

E for electron transport

Default: If the MODE card is omitted, MODE N is assumed.

Use: A MODE card is required unless MODE=N. The entries are space delineated.

B. Geometry Cards

1. VOL Cell Volume Card

Form: VOL x1x2xi

or: VOL NO x1x2xi

xi= volume of cell i where i=1, 2, ... number of cells in the problem.

NO = no volumes or areas are calculated.

Default: MCNP attempts to calculate the volume of all cells unless “NO” appears on

the VOL card. If no value is entered for a cell on the VOL card, the calculated

volume is used.

Use: Optional card used to input cell volumes.

Mnemonic Card Type Page

VOL Cell volumes 3–23

AREA Surface areas 3–24

U Universes 3–26

TRCL Cell transformations 3–27

LAT Lattices 3–28

FILL Fill card 3–29

TR Coordinate transformation 3–30

x1…xi

…

3-24 April 10, 2000

CHAPTER 3

DATA CARDS

With the VOL card, if the number of entries does not equal the number of cells in the problem, it

is a fatal error. Use the nJ feature to skip over cells for which you do not want to enter values. The

entry NO on the VOL card will bypass the volume calculation altogether. The xientries following

NO are optional. If present, xientries are the volume values the code will use. For some problems

the NO option saves considerable computer time.

Volumes or masses of cells are required for some tallies. MCNP calculates the volumes of all cells

that are rotationally symmetric (generated by surfaces of revolution) about any axis, even a skew

axis. It will also calculate the volumes of polyhedral cells. As a byproduct of the volume

calculation, areas and masses are also calculated. These volumes, areas, and masses can be printed

in the OUTP ﬁle by using the PRINT card. The user can enter values on the VOL card for the

volume of any cell and these values, instead of the calculated values, will be used for tally purposes.

If a cell volume required for a tally cannot be calculated and is not entered on the VOL or SDn

cards, a fatal error message is printed.

The VOL card provides an alternative way to enter volumes required by tallies. Normally the SDn

card would be used. The VOL card can be used only for cell volumes, whereas the SDn card can

be used for cell and segment volumes or masses.

Volumes of cells or segments that cannot be calculated by MCNP or by the user can be obtained in

a separate MCNP run using the ray-tracing technique described on page 2–183.

2. AREA Surface Area Card

Form: AREA

xi= area of surface i where i=1, 2, ... number of surfaces in the problem.

Default: MCNP attempts to calculate the area of all surfaces. If no value is entered for a surface

on the AREA card, the calculated area, if any, is used.

Use: Optional card used to input surface areas.

This card is analogous to the VOL card. MCNP calculates the area of surfaces as a byproduct of

the volume calculation. If the volume of all cells on either side of the surface can be calculated, the

area of the surface will be calculated. Otherwise the area calculation will fail. A fatal error occurs

if an area is required for tallying purposes and is not available either from the MCNP calculation

or from an AREA or SDn card.

The AREA card provides an alternative way to enter areas required by tallies. Normally the SDn

card would be used. The AREA card can be used only for areas of whole surfaces, whereas the SDn

card can be used for areas of surface segments as well as whole surfaces.

x1…xi…xn

April 10, 2000 3-25

CHAPTER 3

DATA CARDS

3–6. Repeated Structures Cards

The primary goal of the repeated-structures capability is to make it possible to describe only once

the cells and surfaces of any structure that appears more than once in a geometry. The amount of

input data the user has to provide and the amount of computer memory needed by problems that

have a lot of geometrical repetition is reduced. Problems that would be impractical because they

take an unreasonable amount of work to set up or they use too much memory can be run. One

example of such a problem is a reactor core that has dozens of nearly identical fuel modules.

Another example is a room containing some complicated nearly identical objects arranged in some

not necessarily regular order. This feature reduces input and memory use but problems won’t run

any faster than with any other description. Examples of the use of repeated structures cards are in

Chapter 4.

The repeated structures capability extends the concept of an MCNP cell. The user can specify that

a cell is to be ﬁlled with something called a universe. A universe is either a lattice or an arbitrary

collection of cells. A single universe, described only once, can be designated to ﬁll each of any

number of cells in the geometry. Some or all of the cells in a universe may themselves be ﬁlled with

universes. Several concepts and cards combine in order to use this capability.

• Remember that cell parameters can be deﬁned on cell cards.

• The “LIKE m BUT” feature is a shorthand making it possible to make one cell equivalent to

another except for assorted attributes that can be speciﬁed with keyword=value entries. See

page 3–11.

• The universe card, the U card, is used to specify to what universe the cell belongs.

• The ﬁll card is used to specify with which universe a cell is to be ﬁlled.

• The TRCL card makes it possible to deﬁne only once the surfaces that bound several cells

identical in size and shape but located at different places in the geometry. It follows the

transformation rules established for the TR card. See page 3–30.

• The lattice card, the LAT card, is used to deﬁne an inﬁnite array of hexahedra or hexagonal

prisms. The order of speciﬁcation of the surfaces of a lattice cell identiﬁes which lattice

element lies beyond each surface.

• A general source description can be deﬁned in a repeated structures part of the geometry.

Surface source surfaces must be regular MCNP surfaces, not surfaces associated with a

repeated structures part of the geometry. No check is made that this requirement is met. The

user must remember and this notiﬁcation is your only warning.

• An importance in a cell that is in a universe is interpreted as a multiplier of the importance of

the ﬁlled cell. Weight–window lower bounds are handled the same way.

3-26 April 10, 2000

CHAPTER 3

DATA CARDS

Chapter 4 contains several examples that illustrate the repeated structures input and logic. The

reader is strongly encouraged to become familiar with these examples and to use them as teaching

aids to help understand the card descriptions that follow.

3. U Universe Card

As mentioned earlier, a universe can be either a lattice or a collection of ordinary cells. A nonzero

entry on the U card is the number of the universe that the corresponding cell belongs to. Lack of

a U card or a zero entry means that the cell does not belong to any universe. Universe numbers are

arbitrary integers chosen by the user. The FILL card, page 3–29, indicates that a cell is ﬁlled by all

the cells having a corresponding integer entry on the U card. The cells of a universe may be ﬁnite

or inﬁnite, but they must ﬁll all of the space inside any cell that the universe is speciﬁed to ﬁll.

One way to think about the connection between a ﬁlled cell and the ﬁlling universe is that the ﬁlled

cell is a “window” that looks into a second level, like a window in a wall provides a view of the

outdoors. Cells in the second level can be inﬁnite because they will be “ended” when they bump

into or intersect the surfaces of the “window.” The second level can have its own origin, in a primed

coordinate system, unrelated to the upper level origin. However, if the ﬁlled cell and ﬁlling universe

have all their surfaces in the same coordinate system, one TRCL card, explained on page 3–27, will

deﬁne the coordinate system of both ﬁlled and ﬁlling cells. The ﬁrst repeated structures example

in Chapter 4 illustrates this fact.

A cell in a universe can be ﬁlled by another universe, in which case a third level is introduced.

There is a maximum of 10 levels, more than most problems will need. To clarify some jargon about

hierarchies, the highest to lowest level is in inverse order to the associated numerical value. The

highest level is level zero, lower is level one, lower still is level two, etc.

Planar surfaces of a ﬁlled cell and those in a ﬁlling universe CAN be coincident. In other words,

the cells of a universe can ﬁt exactly into the ﬁlled cell. The following cell and surface cards

illustrate this feature. They represent a –cm box ﬁlled with a lattice of –

cm cubes, each of which is ﬁlled with a sphere.

A problem will run faster by preceding the U card entry with a minus sign for any cell that is not

truncated by the boundary of any higher level cell. The minus sign indicates that calculating

distances to boundary in higher level cells can be omitted. In the problem below, cell 3 has a

negative universe number. It is a ﬁnite cell and is not truncated by any other cell. Cell 4 cannot have

a negative universe number because it is an inﬁnite region that is truncated by cell 2.

CAUTION: Use this capability AT YOUR OWN RISK. MCNP cannot detect errors in this feature

as the logic that enables detection is omitted by the presence of the negative universe. Extremely

wrong answers can be quietly calculated. Use this feature with EXTREME caution. Plot several

views of the geometry or run with the VOID card (see page 3–8) to check for errors.

50 20 10×× 10 10 10××

April 10, 2000 3-27

CHAPTER 3

DATA CARDS

10 1−2 –3 4 –5 6 ﬁll=1

2 0 –7 1 –3 8 u=1 ﬁll=2 lat=1

3 0 –11 u=−2

4 0 11 u=2

5 0 –1:2:3:–4:5:–6

1px 0

2px50

3py10

4 py –10

5pz 5

6pz –5

7px10

8py0

10 py 10

11 s 5504

Every cell in the problem is either part of the real world (universe level 0) or part of some universe,

but the surfaces of a problem are less restricted. A single planar surface can be used to describe

cells in more than one universe. Coincident surfaces can not be reﬂecting or periodic, source

surfaces, or tally surfaces. Materials are normally put into the cells of the lowest level universe, not

in the higher level but there is an exception in the case of a lattice.

The above example can be described with macrobodies as follows:

4. TRCL Cell Transformation Card

The TRCL card makes it possible to describe just once the surfaces that bound several cells

identical in size and shape but located at different places in the geometry. It is especially valuable

when these cells are ﬁlled with the same universe. If the surfaces of these ﬁlled cells and the

surfaces of the cells in the universe that ﬁlls them are all described in the same auxiliary coordinate

system, a single transformation will completely deﬁne the interior of all these ﬁlled cells because

the cells of the universe will inherit the transformation of the cells they ﬁll.TRCL is intended to be

1 0 –20 ﬁll=1

2 0 –30 u=1 ﬁll=2 lat=1

3 0 –11 u=–2

4 0 11 u=2

50 20

20 rpp 0 50 –10 10 –5 5

30 rpp 0 10 0 10

11 s 5504

3-28 April 10, 2000

CHAPTER 3

DATA CARDS

used with LIKE BUT, LAT, etc. With a regular cell description, it is suggested the TR on the surface

cards be used.

The basic form of an entry is an integer that is interpreted as the number of a TR card that contains

a transformation for all of the surfaces of the cell and is located in the data card section of the INP

ﬁle. The absence of the TRCL card or zero means there is no transformation, the default. The actual

transformation can be entered following the TRCL mnemonic, enclosed by parentheses. If the

actual transformation is entered, all the rules applying to the TR card

(page 3–30) are valid. If the symbol ∗TRCL is used, the rotation matrix entries are angles in

degrees instead of cosines, the same as the ∗TR card.

If a cell has a transformation, a set of new surfaces with unique names is generated from the

original surfaces. The name of the generated surface is equal to the name of the original surface

plus 1000 times the name of the cell. This formula gives generated names that are predictable and

can be used on other cell cards and on tally cards. This method limits cell names and original

surface names to no more than three digits, however. These generated surfaces are only the

bounding surfaces of the transformed cell, not the surfaces of any universe that ﬁlls it. MCNP

requires only one full description of each universe, no matter how many times that universe is

referenced in the problem.

5. LAT Lattice Card

LAT=1 means the lattice is made of hexahedra, solids with six faces. LAT=2 means the lattice is

made of hexagonal prisms, solids with eight faces.A nonzero entry on the LAT card means that the

corresponding cell is the (0,0,0) element of a lattice. The cell description of a lattice cell has two

main purposes. It is a standard MCNP cell description and the order of speciﬁcation of the surfaces

of the cell identiﬁes which lattice element lies beyond each surface.

After you have designed your lattice, decide which element you want to be the (0,0,0) element and

in which directions in the lattice you want the three lattice indices to increase. In the case of a

hexagonal prism lattice you have two constraints: the ﬁrst and second indices must increase across

adjacent surfaces and the third index must increase in one or the other direction along the length of

the prism. You will then enter the bounding surfaces of the (0,0,0) element on the cell card in the

right order, in accordance with the following conventions. For a hexahedral lattice cell, beyond the

ﬁrst surface listed is the (1,0,0) element, beyond the second surface listed is the (-1,0,0) element,

then the (0,1,0), (0,-1,0), (0,0,1) and (0,0,-1) lattice elements in that order. This method provides

the order of arrangement of the lattice to the code so that when you specify element (7,9,3), the

code knows which one that is. For a hexagonal prism lattice cell, on the opposite side of the ﬁrst

surface listed is element (1,0,0), opposite the second listed surface is

(-1,0,0), then (0,1,0), (0,-1,0), (-1,1,0), (1,-1,0), (0,0,1), and (0,0,-1). These last two surfaces must

be the base surfaces of the prism. Example 7, page 4–34, illustrates a hexagonal prism lattice cell.

April 10, 2000 3-29

CHAPTER 3

DATA CARDS

The hexahedra need not be rectangular and the hexagonal prisms need not be regular, but the

lattices made out of them must ﬁll space exactly. This means that opposite sides have to be identical

and parallel. A hexahedral lattice cell may be inﬁnite in one or two of its dimensions. A hexagonal

prism lattice cell may be inﬁnite in the direction along the length of the prism. The cross section

must be convex (no butterﬂies). It does not matter whether the lattice is left-handed or right-handed.

A lattice must be the only thing in its universe. The real world (universe level 0) itself can be a

lattice. If a particle leaves the last cell of a real-world, limited-extent lattice (see the FILL card for

how the extent of a lattice can be limited), it is killed (escapes).

6. FILL Fill Card

A nonzero entry on the FILL card indicates the number of the universe that ﬁlls the corresponding

cell. The same number on the U card identiﬁes the cells making up the ﬁlling universe. The FILL

entry may optionally be followed by, in parentheses, either a transformation number or the

transformation itself. This transformation is between the coordinate systems of the ﬁlled cell and

the ﬁlling universe, with the universe considered to be in the auxiliary coordinate system. If no

transformation is speciﬁed, the universe inherits the transformation, if any, of the ﬁlled cell. A

∗FILL may be used if the rotation matrix entries are angles in degrees rather than cosines. In the

data card section of the INP ﬁle you cannot have both a FILL and a ∗FILL entry. If you want to

enter some angles by degrees (∗FILL) and some angles by cosines (FILL), all FILL and ∗FILL data

must be placed on the cell cards of the INP ﬁle.

If the ﬁlled cell is a lattice, the FILL speciﬁcation can be either a single entry, as described above,

or an array. If it is a single entry, every cell of the lattice is ﬁlled by the same universe. If it is an

array, the portion of the lattice covered by the array is ﬁlled and the rest of the lattice does not exist.

It is possible to ﬁll various elements of the lattice with different universes, as shown below and in

examples in Chapter 4, section III,

The array speciﬁcation for a cell ﬁlled by a lattice has three dimension declarators followed by the

array values themselves. The dimension declarators deﬁne the ranges of the three lattice indices.

They are in the same form as in FORTRAN, but both lower and upper bounds must be explicitly

stated with positive, negative, or zero integers, separated by a colon. The indices of each lattice

element are determined by its location with respect to the (0,0,0) element. Reread the LAT card

section, if needed, with particular emphasis on how the order of speciﬁcation of the surfaces of the

cell identiﬁes the ordering of the lattice elements. The ﬁrst two surfaces listed on the cell card

deﬁne the direction the ﬁrst lattice index must cover. The numerical range of the indices depends

on where in the lattice the (0,0,0) element is located. For example, −5:5, 0:10, and −10:0 all deﬁne

a range of 11 elements. The third and fourth surfaces listed in the cell description deﬁne the

direction of the second lattice index.

The array values follow the dimension declarators. Each element in the array corresponds to an

element in the lattice. Only those elements of the lattice that correspond to elements in the array

3-30 April 10, 2000

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actually exist. The value of each array element is the number of the universe that is to ﬁll the

corresponding lattice. There are two values that can be used in the array that have special meanings.

A zero in a real world (level zero) lattice means that the lattice element does not exist, making it

possible, in effect, to specify a nonrectangular array. If the array value is the same as the number

of the universe of the lattice, that element is not ﬁlled with any universe but with the material

speciﬁed on the cell card for the lattice cell. A real world (level zero) lattice, by default, is universe

zero and only can be universe zero. Therefore, using the universe number of the lattice as an array

value to ﬁll that element with the cell material is not possible. As with a single entry FILL

speciﬁcation, any value in the array optionally can be followed by, in parentheses, a transformation

number or the transformation itself.

Example: FILL=0:2 1:2 0:1 4 42040433040

Only eight elements of this lattice exist. Elements (0,1,0), (1,1,0), (1,2,0), (0,1,1) and (1,2,1) are

ﬁlled with universe 4. Element (2,1,0) is ﬁlled with universe 2. Elements (1,1,1) and (2,1,1) are

ﬁlled with universe 3.

7. TRn Coordinate Transformation Card

Form: TRn O1O2O3B1B2B3B4B5B6B7B8B9M

n=number of the transformation: 1< n < 999.∗TRn means that the Bi

are angles in degrees rather than being the cosines of the angles.

O1O2O3=displacement vector of the transformation.

B1to B9=rotation matrix of the transformation.

M=1 (the default) means that the displacement vector is the location of

the origin of the auxiliary coordinate system, deﬁned in the main

system.

=−1 means that the displacement vector is the location of the origin of

the main coordinate system, deﬁned in the auxiliary system.

Default: TRn 0 0 0 1 0 0 0 1 0 0 0 1 1

The maximum number of transformations in a single problem is 999. A cone of one sheet can be

rotated only from being on or parallel to one coordinate axis to being on or parallel to another

coordinate axis (multiples of ). A cone of one sheet can have any origin displacement vector

appropriate to the problem. A cone of two sheets can be transformed anywhere. A cone of two

sheets with an ambiguity surface in the cell description to cut off one half (the cell looks like one

sheet) can be transformed. The ambiguity surface must have the same transformation number as

the cone of two sheets. Ambiguity surfaces are described on page 2–12.

90°

April 10, 2000 3-31

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The B matrix speciﬁes the relationship between the directions of the axes of the two coordinate

systems. Bi is the cosine of the angle (or the angle itself, in degrees in the range from 0 to 180, if

the optional asterisk is used) between an axis of the main coordinate system (x,y,z) and an axis of

the auxiliary coordinate system as follows:

Element B1B2B3B4B5B6B7B8B9

Axes x,x' y,x' z,x' x,y' y,y' z,y' x,z' y,z' z,z'

The meanings of the Bido not depend on M. It is usually not necessary to enter all of the elements

of the B matrix. These patterns are acceptable:

1. All nine elements.

2. Two of the three vectors either way in the matrix (6 values). MCNP will create the third

vector by cross product.

3. One vector each way in the matrix (5 values). The component in common must be less

than 1. MCNP will fill out the matrix by the Eulerian angles scheme.

4. One vector (3 values). MCNP will create the other two vectors in some arbitrary way.

5. None. MCNP will create the identity matrix.

A vector consists of the three elements in either a row or a column in the matrix. In all cases MCNP

cleans up any small nonorthogonality and normalizes the matrix. In this process, exact vectors like

(1,0,0) are left unchanged. A warning message is issued if the nonorthogonality is more than about

0.001 radian.

Pattern #5 is appropriate when the transformation is a pure translation. Pattern #4 is appropriate

when the auxiliary coordinate system is being used to describe a set of surfaces that are all surfaces

of rotation about a common skew axis. Patterns 2 and 3 are about equally useful in more general

cases. Pattern #1 is required if one of the systems is right handed and the other is left handed.

Coordinate transformations in MCNP are used to simplify the geometrical description of surfaces

and to relate the coordinate system of a surface source problem to the coordinate system of the

problem that wrote the surface source ﬁle. See the surface source SSR card on page 3–66. Periodic

boundary surfaces cannot have surface transformations.

To use a transformation to simplify the description of a surface, choose an auxiliary coordinate

system in which the description of the surface is easy, include a transformation number n on the

surface card, and specify the transformation on a TRn card. See page 4–16 for an example showing

how much easier it is to specify a skewed cylinder this way than as a GQ surface. Often a whole

cluster of cells will have a common natural coordinate system. All of their surfaces can be

described in that system, which can then be speciﬁed by a single TRn card.

x′y′z′

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Example: 17 4 PX 5

TR4 7 .9 1.3 0 −10001−100

Surface 17 is set up in an auxiliary coordinate system that is related to the main coordinate system

by transformation number 4. (Presumably there are many other surfaces in this problem that are

using the same transformation, probably because they came from the input ﬁle of an earlier

problem. Otherwise there would be no reason to use a transformation to set up a surface as simple

as a plane perpendicular to a coordinate axis.) MCNP will produce coefﬁcients in the main

coordinate system as if surface 17 had been entered as

17 P 0 −1 0 4.1

It will not produce

17 PY 4.1

that is located at the same place in space, because this PY surface has the wrong sense. More

examples of the transformation are in Chapter 4.

C. Variance Reduction

The following cards deﬁne parameters for variance reduction cards.

Mnemonic Card Type Page

IMP Cell importances 3–33

ESPLT Energy splitting and roulette 3–34

PWT Photon production weights 3–35

EXT Exponential transform 3–36

VECT Vector input 3–38

FCL Forced collision 3–38

WWE Weight window energies 3–40

WWN Weight window bounds 3–40

WWP Weight window parameter 3–41

WWG Weight window generation 3–43

WWGE Weight window generation energies 3–44

MESH Superimposed importance mesh for 3–44

mesh-based weight window generator

PD Detector contribution to tally 3–47

DXC DXTRAN cell contributions 3–48

BBREM Bremsstrahlung biasing 3–48

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Either an IMP or WWN card is required; most of the other cards are for optional variance reduction

techniques.

Entries on a cell or surface parameter card correspond in order to the cell or surface cards that

appear earlier in the INP ﬁle. To get to the particular cell(s) or surface(s) on a card, you must supply

the appropriate default values on the cards as spacers (the nR repeat or nJ jump features may help).

The number of entries on a cell or surface parameter card should always equal the number of cells

or surfaces in the problem or a FATAL error will result.

Many of these cards require a knowledge of both the Monte Carlo method and the particular

variance reduction technique being used. Chapter 2 and some of the references listed at the end of

the manual may provide some of this knowledge.

1. IMP Cell Importance Cards

Form: IMP:n x1x2xixI

n=N for neutrons, P for photons, E for electrons. N,P or P,E or N,P,E

is allowed if importances are the same for different particle types.

xi=importance for cell i

I=number of cells in the problem

Default: If an IMP:P card is omitted in a MODE N P problem, all photon cell importances

are set to unity unless the neutron importance is 0. Then the photon importance

is 0 also.

Use: An IMP:n card is required with an entry for every cell unless a WWN weight

window bound card is used.

The importance of a cell is used to terminate the particle’s history if the importance is zero, for

geometry splitting and Russian roulette as described on page 2–135 to help particles move to more

important regions of the geometry, and in the weight cutoff game described on page 3–124. An

importance in a cell that is in a universe is interpreted as a multiplier of the importance of the ﬁlled

cell.

Neutrons, photons, and electrons can be split differently by having separate IMP:N, IMP:P, and

IMP:E cards. It is a fatal error if the number of entries on any IMP:n card is not equal to the number

of cells in the problem. The nJ feature is allowed and provides the default importance of zero. The

nR repeat and nM multiply features are especially useful with this card.

Example: IMP:N 1 2 2M 0 1 20R

……

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The neutron importance of cell 1 is 1, cell 2 is 2, cell 3 is 4, cell 4 is 0, and cells 5 through 25 is 1.

A track will be split 2 for 1 going from cell 2 into cell 3, each new track having half the weight of

the original track before splitting. A track moving in the opposite direction will be terminated in

about half (that is, probability=0.5) the cases but followed in the remaining cases with twice the

weight.

Remember that both tracks and contributions to detectors and DXTRAN spheres are killed in cells

of zero importance.

A track will neither be split nor rouletted when it enters a void cell even if the importance ratio of

the adjacent cells would normally call for a split or roulette. However, the importance of the

nonvoid cell it left is remembered and splitting or Russian roulette will be played when the particle

next enters a nonvoid cell. As an example of the beneﬁt of not splitting into a void, consider a long,

void pipe surrounded by a material like concrete where the importances are decreasing radially

away from the pipe. Considerable computer time can be wasted by tracks bouncing back and forth

across the pipe and doing nothing but splitting then immediately undergoing roulette. Splitting into

a void increases the time per history but has no counterbalancing effect on the expected history

variance. Thus the FOM is reduced by the increased time per history.

If a superimposed weight window mesh is used, the IMP card is required but splitting/Russian

roulette is not done at surfaces. Cell importances are only used for the weight cutoff game in zero–

window meshes.

2. ESPLT Energy Splitting and Roulette Card

Form: ESPLT:n N1E1... N5E5

n=N for neutrons, P for photons, E for electrons.

Ni= number of tracks into which a particle will be split.

Ei= energy (MeV) at which particles are to undergo splitting.

Default: Omission of this card means that energy splitting will not take place for those

particles for which the card is omitted.

Use: Optional; use energy-dependent weight windows instead.

The ESPLT card allows for splitting and Russian roulette in energy, as the IMP card allows for

splitting and Russian roulette as a function of geometry. Energy splitting can result in low weight

particles that are inadvertently killed by the weight cutoff game (CUT card). Because energy

dependent weight windows perform the same function as the ESPLT card, are not limited to ﬁve

energy groups, can have spatial dependence, and are more compatible with other variance

reduction features, use of the ESPLT card is discouraged.

April 10, 2000 3-35

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The entries on this card consist of pairs of energy-biasing parameters, Niand Ei, with a maximum

of ﬁve pairs allowed. Nican be noninteger and also can be between 0 and 1, in which case Russian

roulette on energy is played. For Nibetween 0 and 1 the quantity becomes the survival probability

in the roulette game.

If the particle’s energy falls below Ei, the speciﬁed splitting or roulette always occurs. If the

particle’s energy increases above Ei, the inverse game is normally played. For example, suppose

roulette is speciﬁed at 1 eV with survival probability 0.5; if a particle’s energy increases above 1

eV, it is split 2 for 1.

A neutron’s energy may increase by ﬁssion or from thermal up–scattering. There are cases when it

may not be desirable to have the inverse splitting or roulette game played on energy increases

(particularly in a ﬁssion-dominated problem). If N1<0, then splitting or roulette will be played

only for energy decreases and not for energy increases.

Example: ESPLT:N 2 .1 2 .01 .25 .001

This example speciﬁes a 2 for 1 split when the neutron energy falls below 0.1 MeV, another 2 for

1 split when the energy falls below 0.01 MeV, and Russian roulette when the energy falls below

0.001 MeV with a 25% chance of surviving.

3. PWT Photon Weight Card

Form: PWT W1W2... Wi... WI

Wi=relative threshold weight of photons produced at neutron collisions in cell i

I=number of cells in the problem.

Use: Recommended for MODE N P and MODE N P E problems without weight windows.

The PWT card is used in Mode N P or ModeNPEproblems. Its purpose is to control the number

and weight of neutron-induced photons produced at neutron collisions. Only prompt photons are

produced from neutron collisions. Delayed gammas are neglected by MCNP. The PWT card

application is further discussed on page 2–33.

For each cell with a positive Wi entry, only neutron-induced photons with weights greater than

Wi∗Is/Iiare produced, where Isand Iiare the neutron importances of the collision and source cells,

respectively. Russian roulette is played to determine if a neutron-induced photon with a weight

below this value survives.

For each cell with a negative Wi entry, only neutron-induced photons with weights greater than −

Wi∗Ws∗Is/Ii are produced, where Ws is the starting weight of the neutron for the history being

3-36 April 10, 2000

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DATA CARDS

followed, and Isand Iiare the neutron importances of the collision and source cells, respectively.

Russian roulette is played to determine if a neutron-induced photon with weight below this value

survives.

If Wi= 0, exactly one photon will be generated at each neutron collision in cell i, provided that

photon production is possible. If Wi= −1.0E6, photon production in cell i is turned off.

The PWT card controls the production of neutron-induced photons by comparing the total weight

of photons produced with a relative threshold weight speciﬁed on the PWT card. This threshold

weight is relative to the neutron cell importance and, if Wi<0, to the source neutron weight. If more

neutron-induced photons are desired, the absolute value of Wi should be lowered to reduce the

weight and therefore increase the number of photons. If fewer neutron-induced photons are

desired, the absolute value of Wi should be increased.

For problems using photon cell importances (IMP:P), rather than photon weight windows

(WWNn:P), a good ﬁrst guess for PWT card entries is either the default value, Wi= −1, or set Wi

in every cell to the average source weight.

For problems with photon weight windows, the PWT card is ignored and the correct number of

photons are produced to be born within the weight windows.

4. EXT Exponential Transform Card

Form: EXT:n A1A2... Ai... AI

n=N for neutrons, P for photons, not available for electrons.

Ai=entry for cell i. Each entry Ai is of the form A= QVm, where Q describes

the amount of stretching and Vm deﬁnes the stretching direction.

I=number of cells in the problem.

Default: No transform, Ai= 0.

Use: Optional. Use cautiously. Weight windows strongly recommended.

The exponential transform should not be used in the same cell as forced collisions or without good

weight control, such as the weight window. The transform works well only when the particle ﬂux

has an exponential distribution, such as in highly absorbing problems.

The exponential transform method stretches the path length between collisions in a preferred

direction by adjusting the total cross section as follows:

, whereΣtΣt1pµ–()=

April 10, 2000 3-37

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DATA CARDS

=artiﬁcially adjusted total cross section;

Σt=true total cross section;

p=the stretching parameter; and

µ=cosine of angle between particle direction and stretching direction.

The stretching parameter p can be speciﬁed by the stretching entry Qthree ways:

Q= 0 ;p= 0 exponential transform not used

Q= p ;0< p < 1 constant stretching parameter

Q= S ; p=Σ

aΣt, where Σa is the capture cross section.

Letting p=Σ

a/Σt can be used for implicit capture along a ﬂight path, as described on page 2–36.

The stretching direction is deﬁned by the Vm part of each Ai entry on the EXT card with three

options.

1. Omit the Vm part of the Aientry; that is, enter only the stretching entry Ai=Qfor a given

cell. This causes the stretching to be in the particle direction (µ=1), independent of the

particle direction and is not recommended unless you want to do implicit capture along

a flight path, in which case Ai=Q=Sand the distance to scatter rather than the distance

to collision is sampled.

2. Specify the stretching direction as Vm, the line from the collision point to the point

(xm,ym,zm), where (xm,ym,zm)is specified on the VECT card (see next section). The

direction cosine µ is now the cosine of the angle between the particle direction and the

line drawn from the collision point to point (xm,ym,zm). The sign of Ai governs whether

stretching is toward or away from (xm,ym,zm).

3. Specify the stretching direction as Vm =X,Y,orZ, so the direction cosine µis the cosine

of the angle between the particle direction and the X−,Y−, or Z−axis, respectively. The

sign of Aigoverns whether stretching is toward or away from the

X−,Y−, or Z−axis.

Example: EXT:N 0 0 .7V2 S −SV2 −.6V9 0 .5V9 SZ −.4X

VECT V9 0 0 0 V2 1 1 1

The 10 entries are for the 10 cells in this problem. Path length stretching is not turned on for

photons or for cells 1, 2, and 7. Following is a summary of path length stretching in the other cells.

Σt

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5. VECT Vector Input Card

Form: VECT Vm xmymzm... Vn xnynzn...

m,n =any numbers to uniquely identify vectors Vm,Vn ...

xm ym zm= coordinate triplets to deﬁne vector Vm.

Default: None.

Use: Optional.

The entries on the VECT card are quadruplets which deﬁne any number of vectors for either the

exponential transform or user patches. See the EXT card (page 3–36) for a usage example.

6. FCL Forced Collision Card

Form: FCL:nx

1x2... xi... xI

n=N for neutrons, P for photons, not available for electrons.

xi= forced collision control for cell i.

I=number of cells in the problem.

Default: xi= 0, no forced collisions.

Use: Optional. Exercise caution.

The FCL card controls the forcing of neutron or photon collisions in each cell. This is particularly

useful for generating contributions to point detectors or DXTRAN spheres. The weight window

game at surfaces is not played when entering forced collision cells.

cell AiQVm stretching

parameter direction

3 .7V2 .7 V2 p= .7 toward point (1,1,1)

4SS p =Σ

a/Σtparticle direction

5−SV2 S −V2 p=Σ

a/Σtaway from point (1,1,1)

6−.6V9 .6 −V9 p= .6 away from origin

8 .5V9 .5 V9 p=.5 toward origin

9SZS Zp=Σ

a/Σtalong +Z-axis

10 −.4X .4 −Xp= .4 along −X-axis

1xi1≤≤–

April 10, 2000 3-39

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If , all particles entering cell iare split into collided and uncollided parts with the appropriate

weight adjustment (see page 2–147). If |xi|<1, Russian roulette is played on the collided parts with

survival probability |xi|to keep the number of collided histories from getting too large. Fractional

xi entries are recommended if a number of forced collision cells are adjacent to each other.

If xi< 0, the forced collision process applies only to particles entering the cell. After the forced

collision, the weight cutoff is ignored and all subsequent collisions are handled in the usual analog

manner. Weight windows are not ignored and are applied after contributions are made to detectors

and DXTRAN spheres.

If xi> 0, the forced collision process applies both to particles entering cell i and to the collided

particles surviving the weight cutoff or weight window games. Particles will continue to be split

into uncollided and (with probability |xi|) collided parts until killed by either weight cutoff or

weight windows.

Usage tips:

Let xi=1or−1 unless a number of forced collision cells are adjacent to each other or the number

of forced collision particles produced is higher than desired. Then fractional values are usually

needed.

When cell–based weight window bounds bracket the typical weight entering the cell, choose

xi> 0. When cell–based weight window bounds bracket the weight typical of forced collision

particles, choose xi< 0. For mesh–based windows, xi> 0 usually is recommended.

When using importances, use xi> 0 because xi< 0 turns off the weight cutoff game.

7–9. Weight Window Cards

Weight windows can be either cell–based or mesh–based. Mesh–based windows eliminate the need

to subdivide geometries ﬁnely enough for importance functions.

Weight windows provide an alternative means to importances (IMP:n cards) and energy splitting

(ESPLT:n cards) for specifying space and energy importance functions. They also can provide

time–dependent importance functions. The advantages of weight windows are that they (1) provide

an importance function in space and time or space and energy; (2) control particle weights; (3) are

more compatible with other variance reduction features such as the exponential transform (EXT:n

card); (4) can be applied at surface crossings, collisions, or both; (5) the severity of splitting or

Russian roulette can be controlled; (6) can be turned off in selected space or energy regions; and

(7) can be automatically generated by the weight window generator. The disadvantages are that (1)

weight windows are not as straightforward as importances; and (2) when the source weight is

xi0≠

3-40 April 10, 2000

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DATA CARDS

changed, the weight windows may have to be renormalized. You are strongly advised to read the

section on weight windows in Chapter 2.

A cell–based weight-window lower bound in a cell that is in a universe is interpreted as a multiplier

of the weight-window lower bound of the ﬁlled cell. Mesh–based windows are recommended in

repeated structures.

7. WWE Weight Window Energies or Times

Form: WWE:n E1E2... Ei... Ej; j 99

n=N for neutrons, P for photons, E for electrons

Ei=upper energy or time bound of ith window

Ei-1 =lower energy or time bound of ith window

E0= 0, by deﬁnition

Default: If this card is omitted and the weight window is used, a single energy or time

interval is established corresponding to the energy or time limits of the problem

being run.

Use: Optional. Use only with WWN card.

The WWE card deﬁnes the energy or time intervals for which weight window bounds will be

speciﬁed on the WWN card. The minimum energy, which is not entered on the WWE card, is zero.

The minimum time is –∞. Whether energy or time is speciﬁed is determined by the 6th entry on

the WWP card.

8. WWN Cell–Based Weight Window Bounds

Form: WWNi:n wi1 wi2 ... wij ... wiJ

n=N for neutrons, P for photons, E for electrons

wij =lower weight bound in cell j and energy or time interval Ei-1 < E < Ei,E

0= 0,

as deﬁned on the WWE card. If no WWE card, i= 1.

J=number of cells in the problem.

Default: None.

Use: Weight windows (WWN and WWP cards) are required unless importances (IMP

card) or mesh–based windows are used.

The WWN card speciﬁes the lower weight bound of the space and energy dependent weight

windows in cells. It must be used with the WWP card, and, if the weight windows are energy or

≤

April 10, 2000 3-41

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DATA CARDS

time dependent, with the WWE card. The IMP:n card should not be used if a WWN:n card, where

n is the same particle type, is used.

If wij < 0, any particle entering cell j is killed. That is, negative entries correspond to zero

importance. If negative entries are used for one energy group, they should be used for all the other

energy groups in the same cell.

If wij > 0, particles entering or colliding in cell j are split or rouletted according to the options on

the WWP card, described in the next section.

If wij =0, the weight window game is turned off in cell jfor energy bin iand the weight cutoff game

is turned on with a 1–for–2 roulette limit. Sometimes it is useful to specify the weight cutoffs on

the CUT card as the lowest permissible weights desired in the problem. Otherwise, too many

particles entering cells with wij =0 may be killed by the weight cutoff. Usually the

1–for–2 roulette limitation is sufﬁcient to use the default weight cutoffs, but caution is needed and

the problem output ﬁle should be examined carefully. The capability to turn the weight window

game off in various energy and spatial regions is useful when these regions cannot be characterized

by a single importance function or set of weight window bounds.

In terms of the weight window, particle weight bounds are always absolute and not relative; you

have to explicitly account for weight changes from any other variance reduction techniques such

as source biasing. You must specify one lower weight bound per cell per energy interval. There

must be no holes in the speciﬁcation; that is, if WWNi is speciﬁed, WWNj for 1< j < i must also

be speciﬁed.

Example 1: WWE:N E1E2E3

WWN1:N w11 w12 w13 w14

WWN2:N w21 w22 w23 w24

WWN3:N w31 w32 w33 w34

These cards deﬁne three energy or time intervals and the weight window bounds for a four-cell

neutron problem.

Example 2: WWN1:P w11 w12 w13

This card, without an accompanying WWE card, deﬁnes an energy or time independent photon

weight window for a three-cell problem.

9. WWP Weight Window Parameter Card

Form: WWP:n WUPN WSURVN MXSPLN MWHERE SWITCHN MTIME

n=N for neutrons, P for photons, E for electrons

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WUPN = If the particle weight goes above WUPN times the lower weight bound,

the particle will be split.

Required: WUPN .

WSURVN = If the particle survives the Russian roulette game, its weight becomes

MIN(WSURVN times the lower weight bound,WGT∗MXSPLN).

Required: 1 < WSURVN < WUPN.

MXSPLN = No particle will ever be split more than MXSPLN-for-one or be rouletted

more harshly than one-in-MXSPLN.

MXSPLN=2 in zero window cells or meshes.

Required: MXSPLN > 1.

MWHERE = decides where to check a particle’s weight.

−1 means check the weight at collisions only

0 means check the weight at surfaces and collisions

1 means check the weight at surfaces only

SWITCHN = decides where to get the lower weight window bounds.

< 0 means get them from an external WWINP ﬁle.

0 means get them from WWNi cards.

> 0 means set the lower weight window bounds equal to SWITCHN

divided by the cell importances from the IMP card.

MTIME = 0 energy dependent windows (WWE card)

1 time dependent windows (WWE card)

Defaults: WUPN=5; WSURVN=0.6∗WUPN; MXSPLN=5; MWHERE=0;

SWITCHN=0, MTIME=0

Use: Weight windows are required unless importances are used.

The WWP card contains parameters that control use of the weight window lower bounds speciﬁed

on the WWN cards, the IMP cards, or an external ﬁle, depending on the value of SWITCHN.

Having SWITCHN > 0 and also having WWNi cards is a fatal error. If SWITCHN is zero, the

lower weight window bounds must be speciﬁed with the WWNi cards. If SWITCHN < 0, an

external WWINP ﬁle with either cell- or mesh-based lower weight window bounds must exist.

This ﬁle name can be changed on the MCNP execution line using “WWINP = ﬁlename.” The

different formats of the WWINP ﬁle will indicate to the code whether the weight windows are cell

or mesh based. For mesh-based weight windows, the mesh geometry will also be read from the

WWINP ﬁle. The WWINP ﬁle format is provided in Appendix J.

Using Existing Cell Importances to Specify the Lower Weight Bound

An energy-independent weight window can be speciﬁed using existing importances from the IMP

card and setting the ﬁfth entry (SWITCHN) on the WWP card to a positive constant C. If this

option is selected, the lower weight bounds for the cells become C/I, where I is the cell importance.

The remaining entries on the WWP card are entered as described above. A suggested value for C

2≥

April 10, 2000 3-43

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DATA CARDS

is one in which source particles start within the weight window such as .25 times the source weight.

If that is not possible, your window is probably too narrow or you need to respecify your source.

10–12. Weight Window Generation Cards

The weight window generator estimates the importances of the space-energy regions of phase

space speciﬁed by the user. The space-energy weight window parameters are then calculated

inversely proportional to the importances.

Recall that the cell–based generator estimates the average importance of a phase-space cell. If the

cells are too large, the importance variation inside the cell will be large and the average importance

will not represent the cell. Inadequate geometry speciﬁcation also occurs with large importance

differences between adjacent cells. Fortunately, the generator provides information about whether

the geometry speciﬁcation is adequate for sampling purposes. If geometries are inadequately

subdivided for importances, mesh–based weight windows should be used.

The user is advised to become familiar with the section on the weight window in Chapter 2 before

trying to use the weight window generator.

10. WWG Weight Window Generation

Form: WWG ItIcWgJJJJI

It= problem tally number (n of the Fn card). The particular tally bin for which

the weight window generator is optimized is deﬁned by the TFn card.

Ic= invokes cell- or mesh-based weight window generator .

> 0 means use the cell-based weight window generator with Ic as the

reference cell (typically a source cell).

0 means use the mesh-based weight window generator. (MESH card.)

Wg= value of the generated lower weight window bound for cell Ic or for the

reference mesh (see MESH card).

0 means lower bound will be half the average source weight.

J= unused

IEtoggles energy- or time-dependent weight windows.

0 means interpret WWGE card as energy bins.

1 means interpret WWGE card as time bins.

Default: No weight window values are generated.

Use: Optional.

The WWG card causes the optimum importance function for tally Itto be generated. For the cell-

based weight window generator, the importance function is written on WWE and WWNi cards that

3-44 April 10, 2000

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DATA CARDS

are printed, evaluated, and summarized in the OUTP ﬁle and are also printed on the weight window

generator output ﬁle WWOUT. For the mesh-based weight window generator, the importance

function and the mesh description are written only on the WWOUT ﬁle. (The format of the mesh-

based WWOUT ﬁle is provided in Appendix J.) In either case, the generated weight window

importance function easily can be used in subsequent runs using SWITCHN < 0 on the WWP card.

For many problems, this importance function is superior to anything an experienced user can guess

on an IMP card. To generate energy- or time-dependent weight-windows, use the WWGE card

described below.

11. WWGE Weight Window Generation Energies or Times

Form: WWGE:n E1 E2... Ei ... Ej; j 15

n=N for neutrons, P for photons, E for electrons

Ei= upper energy or time bound for weight window group to be

generated, Ei+1 > Ei.

Default: If this card is omitted and the weight window is used, a single energy or time

interval will be established corresponding to the energy/time limits of the problem

being run. If the card is present but has no entries, ten energy/time bins will be

generated with energies/times of Ei=10

i-8 MeV/shake and j=10. Both the single

time/energy and the energy/time–dependent windows are generated.

Use: Optional.

If this card is present, time/energy-dependent weight windows are generated and written on the

WWOUT ﬁle and, for cell-based weight windows, on the OUTP ﬁle. If IE= 1 on the WWG card,

time-dependent windows are generated. In addition, single-group energy- or time-independent

weight windows are written on a separate output ﬁle, WWONE. Energy- and time-independent

weight windows are useful for trouble-shooting the energy- and time-dependent weight windows

on the WWOUT ﬁle. The WWONE ﬁle format is the same as that of the WWOUT ﬁle provided

in Appendix J.

12. MESH Superimposed Importance Mesh for Mesh-Based Weight Window Generator

Form: MESH mesh variable=speciﬁcation

Use: Required if mesh-based weight windows are used or generated.

The equal sign is optional. Keywords can be entered in any order. Special input features I, M, and

R can be used except with GEOM. Table 3.2 summarizes the superimposed mesh variables and lists

their defaults. The default geometry is rectangular and the default ORIGIN point is (0,0,0). For a

cylindrical mesh, the default cylindrical axis is parallel to the MCNP geometry zaxis and the half-

≤

April 10, 2000 3-45

CHAPTER 3

DATA CARDS

plane deﬁning θ=0 is the MCNP geometry positive x axis. The reference point must always be

speciﬁed.

TABLE 3.2: Superimposed Mesh Variables

The location of the n’th coarse mesh in the udirection (runin what follows) is given in terms of the

most positive surface in the udirection. For a rectangular mesh, the coarse mesh locations rxn,ryn,

and rzn are given as planes perpendicular to the x,y, and z axes, respectively, in the MCNP cell

coordinate system; thus, the ORIGIN point (x0,y0,z0) is the most negative point of the mesh. For

a cylindrical mesh, (r0,z0,θ0) = (0., 0., 0.), corresponds to the bottom center point, which is the

cylindrical ORIGIN point in MCNP cell geometry. The coarse mesh locations must increase

monotonically (beginning with the ORIGIN point for a rectangular mesh).

The ﬁne meshes are evenly distributed within the n’th coarse mesh in the udirection. The mesh in

which the reference point lies becomes the reference mesh for the mesh-based weight window

Variable Meaning Default

GEOM Mesh geometry; either Cartesian (“xyz” or “rec”) or

cylindrical (“rzt” or “cyl”). xyz

REF x,y, and z coordinates of the reference point None (variable must be

present)

ORIGIN x,y, and z coordinates in MCNP cell geometry of the

origin (bottom center for cylindrical or bottom, left,

behind for rectangular) of the superimposed mesh

0., 0., 0.

AXS vector giving the direction of the axis of the

cylindrical mesh 0., 0., 1.

VEC vector deﬁning, along with AXS, the plane for θ= 0 1., 0., 0.

IMESH locations of the coarse meshes in the x direction for

rectangular geometry or in the r direction for

cylindrical geometry

None

IINTS number of ﬁne meshes within corresponding coarse

meshes in the xdirection for rectangular geometry or

in the r direction for cylindrical geometry

10 in each coarse mesh

JMESH locations of the coarse meshes in the y direction for

rectangular geometry or in the z direction for

cylindrical geometry

None

JINTS number of ﬁne meshes within corresponding coarse

meshes in the ydirection for rectangular geometry or

in the z direction for cylindrical geometry

10 in each coarse mesh

KMESH locations of the coarse meshes in the z direction for

rectangular geometry or in the θ direction for

cylindrical geometry

None

KINTS number of ﬁne meshes within corresponding coarse

meshes in the z direction for rectangular geometry or

in the θ direction for cylindrical geometry

10 in each coarse mesh

3-46 April 10, 2000

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DATA CARDS

generator; this reference mesh is analogous to the reference cell used by the cell-based weight

window generator.

For a cylindrical mesh, the AXS and VEC vectors need not be orthogonal but they must not be

parallel; the one half-plane that contains them and the ORIGIN point will deﬁne θ= 0. The AXS

vector will remain ﬁxed. The length of the AXS or VEC vectors must not be zero. The θ coarse

mesh locations are given in revolutions and the last one must be 1.

At least two coarse meshes per coordinate direction must be speciﬁed using IMESH, JMESH, and

KMESH keywords, but the code uses a default value of 10 ﬁne meshes per coarse mesh if IINTS,

JINTS, or KINTS keywords are omitted . If IINTS, JINTS, or KINTS keywords are present, the

number of entries must match the number of entries on the IMESH, JMESH, and KMESH

keywords, respectively. Entries on the IINTS, JINTS, and KINTS keywords must be greater than

zero. A reference point must be speciﬁed using the REF keyword.

A second method of providing a superimposed mesh is to use one that already exists, either written

on the WWOUT ﬁle or on the WWONE ﬁle. To implement this method, use the WWG card with

Ic=0 in conjunction with the MESH card where the only keyword is REF. The reference point must

be within the superimposed mesh and must be provided because there is no reference point in either

WWOUT or WWONE. If the mesh-based weight window generator is invoked by this method,

MCNP expects to read a ﬁle called WWINP. WWOUT or WWONE can be renamed in the local

ﬁlespace or the ﬁles can be equivalenced on the execution line using "WWINP=ﬁlename."

It is not necessary to use mesh-based weight windows from the WWINP ﬁle in order to use the

mesh from that ﬁle. Furthermore, previously generated mesh-based weight windows can be used

(WWP card with SWITCHN < 0 and WWINP ﬁle in mesh format) while the mesh-based weight

window generator is simultaneously generating weight windows for a different mesh (input on the

MESH card). However, it is not possible to read mesh-based weight windows from one ﬁle but a

weight-window generation mesh from a different ﬁle.

The superimposed mesh should fully cover the problem geometry; i.e., the outer boundaries of the

mesh should lie outside the outer boundaries of the geometry, rather than being coincident with

them. This requirement guarantees that particles remain within the weight window mesh. A line

or surface source should not be made coincident with a mesh surface. A point source should never

be coincident with the intersection of mesh surfaces. In particular, a line or point source should

never lie on the axis of a cylindrical mesh. These guidelines also apply to the WWG reference

point speciﬁed using the REF keyword.

If a particle does escape the weight-window generation mesh, the code prints a warning message

giving the coordinate direction and surface number (in that direction) from which the particle

escaped; for example, “warning. particle escaped wwg mesh in z direction” (the mesh index

number appears with NPS on the next line). The code prints the total number of particles escaping

April 10, 2000 3-47

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DATA CARDS

the mesh (if any) after the tally ﬂuctuation charts in the standard output ﬁle. Similarly, if a track

starts outside the mesh, the code prints a warning message giving the coordinate direction that was

missed and which side of the mesh the particle started on; for example, “warning. track started

outside wwg mesh: x too great.” The code prints the total number of particles starting outside the

mesh (if any) after the tally ﬂuctuation charts in the standard output ﬁle.

Ic= 0 on the WWG card with no MESH card is a fatal error. If AXS or VEC keywords are present

and the mesh is rectangular, a warning message is printed and the keyword is ignored. If there are

fatal errors and the FATAL option is on, weight-window generation is disabled.

Example: GEOM=cyl REF=1e–6 1e–7 0 ORIGIN=1 2 3

IMESH 2.55 66.34

IINTS 2 15 $ 2 ﬁne bins from 0 to 2.55, 15 from 2.55 to 66.34

JMESH 33.1 42.1 53.4 139.7

JINTS 6 3 4 13

KMESH .5 1

KINTS 5 5

Example: GEOM=rec REF=1e–6 1e–7 0 ORIGIN=–66.34 –38.11 –60

IMESH –16.5 3.8 53.66

IINTS 10 3 8 $ 10 ﬁne bins from –66.34 to –16.5, etc

13. PDn Detector Contribution Card

Form: PDn P1P2... Pi... PI

n= tally number

Pi= probability of contribution to detector n from cell i

I= number of cells in the problem.

Default: Pi = 1.

Use: Optional. Consider also using the DD card, page 3–102.

The PDn card reduces the number of contributions to detector tallies from selected cells that are

relatively unimportant to a given detector, thus saving computing time. At each collision in cell i,

the detector tallies are made with probability Pi. The tally is then increased by the

factor 1/Pito obtain unbiased results for all cells except those where Pi = 0. This enables you to

increase the running speed by setting Pi<1 for cells many mean free paths from the detectors. It

also selectively eliminates detector contributions from cells by setting the Pi’s to zero.

A default set of probabilities can be established for all tallies by use of a PD0 (zero) card. These

default values will be overridden for a speciﬁc tally n by values entered on a PDn card.

0Pi1≤≤()

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14. DXC DXTRAN Contribution Card

Form: DXCm:n P1P2... Pi... PI

m= which DXTRAN sphere the DXC card applies to. If 0 or absent,

the DXC card applies to all the DXTRAN spheres in the problem.

n=N for neutrons, P for photons, not available for electrons.

Pi= probability of contribution to DXTRAN spheres from cell i

I= number of cells in the problem

Default: m= 0, Pi=1.

Use: Optional. Consider also using the DD card, page 3–102.

This card is analogous to the above PDn card but is used for contributions to DXTRAN spheres.

15. BBREM Bremsstrahlung Biasing Card

Form: BBREM b1b2b3... b49 m1m2... mn

b1= any positive value (currently unused).

b2... b49 = bias factors for the bremsstrahlung energy spectrum.

m1... mn= list of materials for which the biasing is invoked.

Default: None.

Use: Optional.

The bremsstrahlung process generates many low-energy photons, but the higher-energy photons

are often of more interest. One way to generate more high-energy photon tracks is to bias each

sampling of a bremsstrahlung photon toward a larger fraction of the available electron energy. For

example, a bias such as

BBREM 1. 1. 46I 10. 888 999

would create a gradually increasing enhancement (from the lowest to the highest fraction of the

electron energy available to a given event) of the probability that the sampled bremsstrahlung

photon will carry a particular fraction of the electron energy. This biasing would apply to each

instance of the sampling of a bremsstrahlung photon in materials 888 and 999. The sampling in

other materials would remain unbiased. The bias factors are normalized by the code in a manner

that depends both on material and on electron energy, so that although the ratios of the photon

weight adjustments among the different groups are known, the actual number of photons produced

in any group is not easily predictable. For the el03 treatment, there are more than 49 relative photon

energy ratios so the lower energy bins have a linear interpolation between b1and b2for their values.

April 10, 2000 3-49

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In most problems the above prescription will increase the total number of bremsstrahlung photons

produced because there will be more photon tracks generated at higher energies. The secondary

electrons created by these photons will tend to have higher energies as well, and will therefore be

able to create more bremsstrahlung tracks than they would at lower energies. This increase in the

population of the electron-photon cascade will make the problem run more slowly. The beneﬁts of

better sampling of the high-energy domain must be balanced against this increase in run time.

For a more detailed discussion of the bremsstrahlung energy biasing scheme, see Chapter 2.

D. Source Speciﬁcation

Every MCNP problem has one of four sources: general source (SDEF card), surface source (SSR

card), criticality source (KCODE card), or user-supplied source (default if SDEF, SSR, and

KCODE are all missing). All can use source distribution functions, speciﬁed on SIn, SPn, SBn, and

DSn cards.

The following cards are used to specify the source.

The MODE card also serves as part of the source speciﬁcation in some cases by implying the type

of particle to be started from the source.

The source has to deﬁne the values of the following MCNP variables for each particle it produces:

Mnemonic Card Type Page

SDEF General source 3–50

SIn Source information 3–57

SPn Source probability 3–58

SBn Source bias 3–58

DSn Dependent source 3–62

SCn Source comment 3–63

SSW Surface source write 3–65

SSR Surface source read 3–66

KCODE Criticality source 3–71

KSRC Source points 3–71

ACODE Alpha eigenvalue source 3–71

ERG the energy of the particle (MeV). See ∗ below

TME the time when the particle started (shakes)

UUU, VVV, WWW the direction of the ﬂight of the particle

XXX, YYY, ZZZ the position of the particle

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DATA CARDS

Additional variables may have to be deﬁned if there are point detectors or DXTRAN spheres in the

problem.

∗ERG has a different meaning in a special case. If there is a negative IGM on the MGOPT card,

which indicates a special electron–photon multigroup problem, ERG on the SDEF card is

interpreted as an energy group number, an integer.

1. SDEF General Source Card

Form: SDEF source variable = speciﬁcation ...

Use: Required for problems using the general source. Optional for problems using

the criticality source.

The equal signs are optional. The source variables are not quite the same as MCNP variables that

the source must set. Many are intermediate quantities that control the sampling of the ﬁnal

variables. All have default values. The speciﬁcation of a source variable has one of these three

forms:

1. explicit value,

2. a distribution number prefixed by a D, or

3. the name of another variable prefixed by an F, followed by a distribution number

prefixed by a D. Var = Dn means that the value of source variable var is sampled from

distribution n. Var Fvar′Dn means that var is sampled from distribution n that depends

on the variable var′. Only one level of dependence is allowed. Each distribution may be

used for only one source variable.

The above scheme translates into three levels of source description. The ﬁrst level exists when a

source variable has an explicit or default value (for example, a single energy) or a default

distribution (for example, an isotropic angular distribution). The second level occurs when a source

variable is given by a probability distribution. This level requires the SI and/or SP cards. The third

level occurs when a variable depends on another variable. This level requires the DS card.

MCNP samples the source variables in an order set up according to the needs of the particular

problem. Each dependent variable must be sampled after the variable it depends on has been

sampled. If the value of one variable inﬂuences the default value of another variable or the way it

IPT the type of the particle

WGT the statistical weight of the particle

ICL the cell where the particle started

JSU the surface where the particle started, or zero if

the starting point is not on any surface

April 10, 2000 3-51

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DATA CARDS

is sampled, as SUR inﬂuences DIR, they may have to be sampled in the right order. The scheme

used in MCNP to set up the order of sampling is complicated and may not always work. If it fails,

a message will be printed. The ﬁx in such instances may be to use explicit values or distributions

instead of depending on defaults.

Table 3.3 summarizes the source variables and lists their defaults.

TABLE 3.3: Source Variables

Variable Meaning Default

CEL Cell Determined from XXX,YYY,ZZZ

and possibly UUU,VVV,WWW

SUR Surface Zero (means cell source)

ERG Energy (MeV) 14 MeV

TME Time (shakes) 0

DIR µ, the cosine of the angle between

VEC and UUU,VVV,WWW

(Azimuthal angle is always sampled

uniformly in 0oto 360o)

Volume case: µ is sampled

uniformly in −1 to 1 (isotropic)

Surface case: p(µ) = 2µ in 0 to 1

(cosine distribution)

VEC Reference vector for DIR Volume case: required unless

isotropic

Surface case: vector normal to the

surface with sign determined by

NRM

NRM Sign of the surface normal +1

POS Reference point for position sampling 0,0,0

RAD Radial distance of the position from

POS or AXS 0

EXT Cell case: distance from POS along

AXS

Surface case: Cosine of angle from

AXS

AXS Reference vector for EXT and RAD No direction

X x-coordinate of position No X

Y y-coordinate of position No Y

Z z-coordinate of position No Z

CCC Cookie-cutter cell No cookie-cutter cell

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The speciﬁcation of WGT, EFF and PAR must be only an explicit value. A distribution is not

allowed. The allowed value for PAR is 1 for neutron, 2 for photon, or 3 for electron. The default

is the lowest of these three that corresponds to an actual or default entry on the MODE card. Only

one kind of particle is allowed in an SDEF source.

Most of the source variables are scalars. VEC, POS, and AXS are vectors. Where a value of a

source variable is required, as on SDEF, SI, or DS cards, usually a single number is appropriate,

but with VEC, POS, and AXS, the value must actually be a triplet of numbers, the x,y, and z

components of the vector.

The source variables SUR, POS, RAD, EXT, AXS, X, Y, Z, and CCC are used in various

combinations to determine the coordinates (x,y,z) of the starting positions of the source particles.

With them you can specify three different kinds of volume distributions and three different kinds

of distributions on surfaces. Degenerate versions of those distributions provide line and point

sources. More elaborate distributions can be approximated by combining several simple

distributions, using the S option of the SIn and DSn cards.

The three volume distributions are cartesian, spherical, and cylindrical. The value of the variable

SUR is zero for a volume distribution. A volume distribution can be used in combination with the

CEL variable to sample uniformly throughout the interior of a cell. A cartesian, spherical, or

cylindrical region that completely contains a cell is speciﬁed and is sampled uniformly in volume.

If the sampled point is found to be inside the cell, it is accepted. Otherwise it is rejected and another

point is sampled. If you use this technique, you must make sure that the sampling region really does

contain every part of the cell because MCNP has no way of checking for this. Cookie-cutter

rejection, described below, can be used instead of or in combination with CEL rejection.

A cartesian volume distribution is speciﬁed with the variables X,Y, and Z. A degenerate case of the

cartesian distribution, in which the three variables are constants, deﬁnes a point source. A single

point source can be speciﬁed by giving values to the three variables right on the SDEF card. If there

are several source points in the problem, it would usually be easier to use a degenerate spherical

ARA Area of surface (required only for

direct contributions to point detectors

from plane surface source.)

None

WGT Particle weight 1

EFF Rejection efﬁciency criterion for

position sampling .01

PAR Particle type source will emit 1=neutron if MODE N or

N P or N P E

2=photon if MODE P or P E

3=electron if MODE E

April 10, 2000 3-53

CHAPTER 3

DATA CARDS

distribution for each point. Other degenerate cases of the cartesian distribution are a line source and

a rectangular plane source. A cartesian distribution is an efﬁcient shape for the CEL rejection

technique when the cell is approximately rectangular. It is much better than a cylindrical

distribution when the cell is a long thin slab. It is, however, limited in that its faces can only be

perpendicular to the coordinate axes.

A spherical volume distribution is speciﬁed with the variables POS and RAD. X,Y,Z, and AXS

must not be speciﬁed or it will be taken to be a cartesian or cylindrical distribution. The sampled

value of the vector POS deﬁnes the center of the sphere. The sampled value of RAD deﬁnes the

distance from the center of the sphere to the position of the particle. The position is then sampled

uniformly on the surface of the sphere of radius RAD. Uniform sampling in volume is obtained if

the distribution of RAD is a power law with a=2, which is the default in this case. A common use

of the spherical volume distribution is to sample uniformly in the volume between two spherical

surfaces. The two radii are speciﬁed on the SIn card for RAD and the effect of a SPn −21 2 card is

obtained by default (see page 3–58). If RAD is not speciﬁed, the default is zero. This is useful

because it speciﬁes a point source at the position POS. A distribution for POS, with an L on the SIn

card, is the easiest way to specify a set of point sources in a problem.

A cylindrical volume distribution is speciﬁed with the variables POS, AXS, RAD, and EXT. The

axis of the cylinder passes through the point POS in the direction AXS. The position of the particles

is sampled uniformly on a circle whose radius is the sampled value of RAD, centered on the axis

of the cylinder. The circle lies in a plane perpendicular to AXS at a distance from POS which is the

sampled value of EXT. A common use of the cylindrical distribution is to sample uniformly in

volume within a cylindrical shell. The distances of the ends of the cylinder from POS are entered

on the SIn card for EXT and the inner and outer radii are entered on the SIn card for RAD. Uniform

sampling between the two values of EXT and power law sampling between the two values of RAD,

with a=1which gives sampling uniform in volume, are provided by default. A useful degenerate

case is EXT=0, which provides a source with circular symmetry on a plane.

Warning: Never position any kind of degenerate volume distribution in such a way that it lies on

one of the deﬁned surfaces of the problem geometry. Even a bounding surface that extends into the

interior of a cell can cause trouble. If possible, use one of the surface distributions instead.

Otherwise, move to a position just a little way off of the surface. It will not make any detectable

difference in the answers, and it will prevent particles from getting lost.

The value of the variable SUR is nonzero for a distribution on a surface. If X,Y, and Zare speciﬁed,

their sampled values determine the position. You must in this case make sure that the point really

is on the surface because MCNP does not check. If X,Y, and Z are not speciﬁed, the position is

sampled on the surface SUR. The shape of the surface, which can be either a spheroid, sphere, or

plane, determines the way the position is sampled. Sampling with CEL rejection is not available,

but cookie-cutter rejection can be used to do anything that CEL rejection would do. Cylindrical

surface sources must be speciﬁed as degenerate volume sources.

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If the value of SUR is the name of a spheroidal surface, the position of the particle is sampled

uniformly in area on the surface. A spheroid is an ellipse revolved around one of its axes. A

spheroid for this purpose must have its axis parallel to one of the coordinate axes. There is presently

no provision for easy nonuniform or biased sampling on a spheroidal surface. A distribution of

cookie-cutter cells could be used to produce a crude nonuniform distribution of position.

If the value of SUR is the name of a spherical surface, the position of the particle is sampled on that

surface. If the vector AXS is not speciﬁed, the position is sampled uniformly in area on the surface.

If AXS is speciﬁed, the sampled value of EXT is used for the cosine of the angle between the

direction AXS and the vector from the center of the sphere to the position point. The azimuthal

angle is sampled uniformly in the range from to 360o. A nonuniform distribution of position,

in polar angle only, is available through a nonuniform distribution of EXT. A biased distribution of

EXT can be used to start more particles from the side of the sphere nearest the tallying regions of

the geometry. The exponential distribution function (−31; see page 3–61) is usually most

appropriate for this.

If the value of SUR is the name of a plane surface, the position is sampled on that plane. The

sampled value of POS must be a point on the plane. You must make sure that POS really is on the

plane because MCNP, for the sake of speed, does not check it. The sampled position of the particle

is at a distance from POS equal to the sampled value of RAD. The position is sampled uniformly

on the circle of radius RAD centered on POS. Uniform sampling in area is obtained if the

distribution of RAD is a power law with a= 1, which is the default in this case.

Cookie-cutter rejection is available for both cell and surface sources. If CCC is present, the position

sampled by the above procedures is accepted if it is within cell CCC and is resampled if it is not,

exactly like CEL rejection in the cell source case. You must be careful not to specify a cookie-cutter

cell such that MCNP mistakes it for a real cell. There should be no trouble if the cookie-cutter cells

are bounded by surfaces used for no other purpose in the problem and if the cookie-cutter cell cards

are at the end of the list of cell cards. Don’t make a cookie-cutter cell more complicated than it has

to be. For a surface source, the only thing that matters is the intersection of the cookie-cutter cell

with the source surface. An inﬁnitely long cell of uniform cross section, bounded by planes and

cylinders, is usually adequate for a plane surface source.

Warning: The combination of either CEL or CCC rejection with biased sampling of the position is

nearly always an unfair game. If you use this combination, you must make sure that it really is a

fair game because MCNP is not able to detect the error.

The source variables SUR, VEC, NRM, and DIR are used to determine the initial direction of ﬂight

of the source particles. The direction of ﬂight is sampled with respect to the reference vector VEC,

which, of course, can itself be sampled from a distribution. The polar angle is the sampled value of

the variable DIR. The azimuthal angle is sampled uniformly in the range from 0 to 360o.IfVEC

and DIR are not speciﬁed for a volume distribution of position (SUR=0), an isotropic distribution

0°

April 10, 2000 3-55

CHAPTER 3

DATA CARDS

of direction is produced by default. If VEC is not speciﬁed for a distribution on a surface (SUR 0),

the vector normal to the surface, with the sign determined by the sign of NRM, is used by default.

If DIR is not speciﬁed for a distribution on a surface, the cosine distribution p(DIR)=2∗DIR,

0<DIR<1, is used by default. A biased distribution of DIR can be used to make more source

particles start in a direction towards the tallying regions of the geometry. The exponential

distribution function (−31; see page 3–61) is usually most appropriate for this.

Discrete values of DIR are allowed. DIR=1 gives a monodirectional source in the direction of VEC.

This is sometimes useful as an approximation to an actual source that is at a large distance from

the geometry of the problem. In most cases discrete values of DIR will prevent direct contributions

to point detectors from being scored. The direct contribution will be scored only if the source is on

a plane surface, is sampled uniformly in area within a circle (using RAD sampled from SP −21 1),

VEC is perpendicular to the surface (the default), and DIR=1. A cookie-cutter cell is allowed and

a value of ARA is necessary. Discrete values of DIR with DXTRAN are generally wrong because

p(µ)=.5 is assumed. See page 2–152.

The efﬁciency criterion EFF applies to both CCC and CEL rejection. If in any source cell or cookie-

cutter cell the acceptance rate is too low, the problem is terminated for inefﬁciency. The criterion

for termination is MAX(number of successes, 10) < EFF ∗ number of tries. The default value of

EFF, 0.01, lets a problem get by at rather low efﬁciency, but for the rare problem in which low

source efﬁciency is unavoidable, you may need to specify a lower value for EFF.

This section discusses a source in a geometry that we will call a repeated structure and is one that

includes FILL, U, or LAT cards. We strongly emphasize that print table 110 should be studied

carefully to ensure that the proper source path and position are being sampled when repeated

structures are used in a source description.

Warning: Deﬁning a source on a lattice cell bounding plane or on a “window” cell plane coincident

with a lattice surface generally does not work and is not recommended.

The only part of the MCNP source speciﬁcation that is different when the source is in a repeated

structure part of the geometry is the use of the CEL parameter on the SDEF card. CEL must have

a value that is a path from level 0 to level n, which is not necessarily the bottom:

c0 : c1 : ... : cn

ci is a cell in the universe that ﬁlls cell ci-1, or is zero, or is Dm for a distribution of cells in the

repeated structure case. Dm is not valid for a lattice. cican have a minus sign and is discussed more

below. Dm cannot have a minus sign. If ci=0, the cell at that level is searched for. Recall that level

n is not necessarily the bottom level in the problem. If ci is one speciﬁc element in a lattice, it is

indicated as: ... : ci (j1 j2 j3) : ...

≠

3-56 April 10, 2000

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DATA CARDS

The coordinate system for position and direction sampling (pds) is the coordinate system of the

ﬁrst negative or zero ciin the source path. Each entry in the source path represents a geometry level,

where level zero is the ﬁrst source path entry, level one the second, etc., and level zero is above

level one, level two is below level one. The pds level is the level associated with the pds cell or pds

coordinate system. All levels above the pds level must be included in the source path. Levels below

the pds level need not be speciﬁed, and when given, may include one or more zero entries. The

default pds level is the last entry in the source path when the path has no negative or zero entry.

Position rejection is done in cells at all levels where ci 0, but if any cihas a negative universe

number on its cell card and is at or above the pds level, higher level cells are not checked.

The following chart illustrates the idea of the pds level.

Lattice cell elements that are deﬁned using the expanded FILL card (see page 3–29) can be

uniformly sampled automatically. This feature is applied to lattice cell entries in the source path

that lack an explicit lattice index AND that are at or above the pds level. Lattice cells not deﬁned

by the expanded FILL card must include an explicit lattice index when at or above the pds level.

Rejection of automatically sampled lattice elements depends on the entry after the lattice cell

number in the source path.

Assume the following cell cards:

7 0 surfaces lat=1 u=1 ﬁll=0:2 0:0 0:0 1 2 3

cells 8 and 9 belong to universe 2

cells 10 and 11 belong to universe 3

Cell 7 is a lattice with three existing elements: (0 0 0) is ﬁlled by itself [u=1], (1 0 0) is ﬁlled by

cells 8 and 9 [u=2], and (2 0 0) is ﬁlled by cells 10 and 11 [u=3]. The following combinations show

what elements are accepted and what are rejected.

CEL Source Path Cell of pds Level pds Level

8:7:6:5 5 3

8:−7:6 7 1

8:7:−6:0:4:0 6 2

8:−(1 0 0):6(0 0 0):0 7 1

8:7(1 0 0):6(0 0 0):0 Will be determined 3

CEL Source Path Accepted Rejected

7 All elements None

7:0 All elements None

7:8 (1 0 0) (0 0 0), (2 0 0)

7:10 (2 0 0) (0 0 0), (1 0 0)

≠

April 10, 2000 3-57

CHAPTER 3

DATA CARDS

The sampling efﬁciency for cell 7 in the OUTP ﬁle will reﬂect the element rejections. Lattice cell

entries that lack an explicit lattice index AND are below the pds level are not sampled. Instead, the

appropriate lattice element is determined by the input source position.

Lattice element sampling is independent from position sampling. First a lattice element is chosen,

then a position is chosen. If the sampled position is not in the sampled lattice element, the position

is resampled until it is in the speciﬁed source path and in the lattice element chosen or until an

efﬁciency error occurs. The lattice elements will not be resampled to accommodate the sampled

position. Lattice element rejection is done only as described above.

Using the previous description of lattice cell 7, add that cell 6 is ﬁlled by cell 7. The source path

becomes 6:7:0. Three elements of the lattice exist [ﬁll=0:2 0:0 0:0] but element (0 0 0) now is cut

off by cell 6. Lattice element (0 0 0) still will be sampled 1/3 of the time. The ﬁrst time element

(0 0 0) is sampled a fatal error will occur because the sampled position, no matter what it is, will

be rejected because element (0 0 0) does not exist. CAUTION: Implement automatic lattice

sampling carefully and ensure that all of the lattice elements speciﬁed on the expanded FILL card

really do exist.

See Chapter 4 for an example of specifying a source in a lattice geometry.

Example: SDEF (no entries)

This card speciﬁes a 14-MeV isotropic point source at position 0,0,0 at time 0 with weight 1 (all

defaults).

2. SIn Source Information Card

3. SPn Source Probabiy Card

4. SBn Source Bias Card

Form: SIn option I1... Ik

n= distribution number (n= 1,999)

option = how the Ii’s are to be interpreted. Allowed values are:

omitted or H—bin boundaries for a histogram distribution,

for scalar variables only. This is the default.

L—discrete source variable values

A—points where a probability density distribution is deﬁned

S—distribution numbers

I1... Ik=source variable values or distribution numbers

Default: SIn H Ii... Ik

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DATA CARDS

Form: SPn option P1... Pk

or: SPn fab

n= distribution number (n= 1,999)

option = how the Pi are to be interpreted. Allowed values are:

omitted—same as D for an H or L distribution. Probability

density for an A distribution on SI card.

D—bin probabilities for an H or L distribution on SI card.

This is the default.

C—cumulative bin probabilities for an H or L distribution

on SI card.

V—for cell distributions only. Probability is proportional

to cell volume (times Pi if the Pi are present).

Pi... Pk= source variable probabilities

f= designator (negative number) for a built-in function

a,b = parameters for the built-in function (see Table 3.4)

Default: SPn D P1...Pk

Form: SBn option B1... Bk

or: SBn fa b

n, option, f,a, and b are the same as for the SPn card, except that the

only values allowed for f are −21 and −31

Bi... Bk = source variable biased probabilities

Default: SBn D B1... Bk

The ﬁrst form of the SP card, where the ﬁrst entry is positive or nonnumeric, indicates that it and

its SI card deﬁne a probability distribution function. The entries on the SI card are either values of

the source variable or, when the S option is used, distribution numbers. The entries on the SP card

are probabilities that correspond to the entries on the SI card.

When the H option is used, the numerical entries on the SI card are bin boundaries and must be

monotonically increasing. The ﬁrst numerical entry on the SP card must be zero and the following

entries are bin probabilities or cumulative bin probabilities, depending on whether the D or C

option is used. The probabilities need not be normalized. The variable is sampled by ﬁrst sampling

a bin according to the bin probabilities and then sampling uniformly within the chosen bin.

When the A option is used, the entries on the SI card are values of the source variable at which the

probability density is deﬁned. The entries must be monotonically increasing, and the lowest and

highest values deﬁne the range of the variable. The numerical entries on the SP card are values of

the probability density corresponding to the values of the variable on the SI card. They need not be

normalized. In the sampling process, the probability density is linearly interpolated between the

April 10, 2000 3-59

CHAPTER 3

DATA CARDS

speciﬁed values. The ﬁrst and last entries on the SP card will typically be zero, but nonzero values

are also allowed.

When the L option is used, the numerical entries on the SI card are discrete values of the source

variable, such as cell numbers or the energies of photon spectrum lines. The entries on the SP card

are either probabilities of those discrete values or cumulative probabilities, depending on whether

the D or C option is used. The entries on the SI card need not be monotonically increasing.

The S option allows sampling among distributions, one of which is chosen for further sampling.

This feature makes it unnecessary to fold distributions together and is essential if some of the

distributions are discrete and others are linearly interpolated. The distributions listed on an SI card

with the S option can themselves also have the S option. MCNP can handle this structure to a depth

of about 20, which should be far more than necessary for any practical problem. Each distribution

number on the SI card can be preﬁxed with a D, or the D can be omitted. If a distribution number

is zero, the default value for the variable is used. A distribution can appear in more than one place

with an S option, but a distribution cannot be used for more than one source variable.

The V option on the SP card is a special case used only when the source variable is CEL. This

option is useful when the cell volume is a factor in the probability of particle emission. If MCNP

cannot calculate the volume of such a cell and the volume is not given on a VOL card, you have a

fatal error.

The SB card is used to provide a probability distribution for sampling that is different from the true

probability distribution on the SP card. Its purpose is to bias the sampling of its source variable to

improve the convergence rate of the problem. The weight of each source particle is adjusted to

compensate for the bias. All rules that apply to the ﬁrst form of the SP card apply to the SB card.

The second form of the SP card, where the ﬁrst entry is negative, indicates that a built-in analytic

function is to be used to generate a continuous probability density function for the source variable.

Built-in functions can be used only for scalar variables. See Table 3.3 for a description of these

functions.

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CHAPTER 3

DATA CARDS

TABLE 3.4: Built-In Functions for Source Probability and Bias Speciﬁcation

f= −2Maxwell ﬁssion energy spectrum: p(E) = C E1/2 exp(−E/a), where a is a temperature in

MeV.

Default: a= 1.2895 MeV

f= −3Watt ﬁssion energy spectrum: p(E) = C exp(−E/a) sinh(bE)1/2.

Defaults: a= 0.965 MeV, b= 2.29 MeV−1. See Appendix H for additional parameters

appropriate to neutron-induced ﬁssion in various materials and for spontaneous ﬁssion.

f= −4Gaussian fusion energy spectrum: p(E) =Cexp[−((E-b)/a)2], where ais the width in MeV

and bis the average energy in MeV. Width here is deﬁned as the ∆Eabove bwhere the value of the

exponential is equal to e−1. If a< 0, it is interpreted as a temperature in MeV and b must also be

negative. If b= −1, the D–T fusion energy is calculated and used for b. If

b=−2, the D–D fusion energy is calculated and used for b. Note that a is not the “full-width-at-

half-maximum,” but is related to it by FWHM =a (ln 2)1/2.

Defaults: a= −0.01 MeV, b= −1 (DT fusion at 10 keV).

f= −5 Evaporation energy spectrum: p(E) = C E exp(−E/a).

Default: a= 1.2895 MeV.

f= −6 Muir velocity Gaussian fusion energy spectrum: p(E) = C exp − ((E1/2 − b1/2)/a)2,

where a is the width in MeV1/2, and b is the energy in MeV corresponding to the average speed.

Width here is deﬁned as the change in velocity above the average velocity b1/2, where the value of

the exponential is equal to e−1. To get a spectrum somewhat comparable to f=−4, the width can be

determined by a= (b+a4)1/2 −b1/2, where a4 is the width used with the Gaussian fusion energy

spectrum. If a< 0, it is interpreted as a temperature in MeV. If b= −1, the D–T fusion energy is

calculated and used for b. If b=−2, the D–D fusion energy is calculated and used for b.

Defaults: a= −0.01 MeV, b= −1 (DT fusion at 10 keV).

SourceVariable Function No. and Input

Parameters Description

ERG −2aMaxwell ﬁssion spectrum

ERG −3ab Watt ﬁssion spectrum

ERG −4ab Gaussion fusion spectrum

ERG −5aEvaporation spectrum

ERG −6ab Muir velocity Gaussian fusion spectrum

ERG −7ab Spare

DIR, RAD, or EXT −21 aPower law p(x) = c|x|a

DIR or EXT −31 aExponential: p(µ)= ceaµ

TME −41 a b Gaussian distribution of time

April 10, 2000 3-61

CHAPTER 3

DATA CARDS

f= −7 Spare energy spectrum. The basic framework for another energy spectrum is in place to

make it easier for a user to add a spectrum of his own. The subroutines to change are SPROB,

SPEC, SMPSRC, and possibly CALCPS.

f= −21 Power law: p(x) = c|x|a.

The default depends on the variable. For DIR, a=1. For RAD, a=2, unless AXS is deﬁned or JSU

0, in which case a= 1. For EXT, a= 0.

f= −31 Exponential: p(µ)=cea\µ.

Default: a= 0.

f= −41 Gaussian distribution of time: p(t) =cexp[−(1.6651092(t−b)/a)2], where ais the width at

half maximum in shakes and b is the mean in shakes.

Defaults: a=no default, b= 0.

The built-in functions can be used only for the variables shown in Table 3.3. Any of the built-in

functions can be used on SP cards, but only −21 and −31 can be used on SB cards. If a function is

used on an SB card, only that same function can be used on the corresponding SP card. The

combination of a regular table on the SI and SP cards with a function on the SB card is not allowed.

A built-in function on an SP card can be biased or truncated or both by a table on SI and SB cards.

The biasing affects only the probabilities of the bins, not the shape of the function within each bin.

If it is biased, the function is approximated within each bin by nequally probable groups such that

the product of nand the number of bins is as large as possible but not over 300. Unless the function

is −21 or −31, the weight of the source particle is adjusted to compensate for truncation of the

function by the entries on the SI card.

Special defaults are available for distributions that use built-in functions.

1. If SB f is present and SP f is not, an SP f with default input parameters is, in effect,

provided by MCNP.

2. If only an SI card is present for RAD or EXT, an SP −21 with default input parameters

is, in effect, provided.

3. If only SP −21 or SP −31 is present for DIR or EXT, an SI 0 1, for −21,orSI−1 1, for −

31, is, in effect, provided.

4. If SI x and SP −21 are present for RAD, the SI is treated as if it were SI 0 x.

5. If SI x and SP −21 or SP −31 are present for EXT, the SI is treated as if it were SI −x x.

≠

3-62 April 10, 2000

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DATA CARDS

5. DSn Dependent Source Distribution Card

Form: DSn option J1... Jk

or: DSn T I1J1... IkJk

or: DSn Q V1S1... VkSk

n= distribution number (n= 1,999)

option = how the Ji are to be interpreted. Allowed values are:

blank or H—source variable values in a continuous distribution, for scalar

variables only

L—discrete source variable values

S—distribution numbers

T= values of the dependent variable follow values of the independent variable,

which must be a discrete scalar variable

Ii= values of the independent variable

Ji= values of the dependent variable

Q= distribution numbers follow values of the independent variable, which must be

a scalar variable

Vi= monotonically increasing set of values of the independent variable

Si= distribution numbers for the dependent variable

Default: DSn H J1... Jk

The DS card is used instead of the SI card for a variable that depends on another source variable,

as indicated on the SDEF card. No SP or SB card is used. MCNP ﬁrst determines the value of the

independent variable as usual by sampling the probability function of the independent variable.

Then the value of the dependent variable is determined according to the form of the DS card.

The ﬁrst form of the DS card has several possibilities. If the SI card of the independent variable has

a histogram distribution of nbins (n+1entries) and the DS card has the blank or H option, the DS

card must have n+1entries to specify nbins. The ﬁrst entry need not be zero. If the sampled value

of the independent variable is Ii+[f(Ii+1 −Ii)], then the value of the dependent variable is Ji+[f(Ji+1

−Ji)], where the terms in f are used only for continuous distributions. The interpolation factor f

always exists whether or not it is needed for the independent distribution.

If the L or S option is used on the DS card, nentries are required to specify ndiscrete values. It is

not necessary for the distributions of the independent and dependent variables to be both discrete

or both continuous. All combinations work correctly.

When the T option is used on a DS card, the sampled value of the independent variable is sought

among the Ii, and if a match is found, the independent variable gets the value Ji. If no match is

found, the dependent variable gets its default value. The purpose of the T option is to shorten the

input when a dependent variable should usually get the default value.

April 10, 2000 3-63

CHAPTER 3

DATA CARDS

When the Q option is used on a DS card, the Vi deﬁne a set of bins for the independent variable.

The sampled value of the independent variable is compared with the Vi, starting with V1, and if the

sampled value is less than or equal to Vi, the distribution Si is sampled for the value of the

dependent variable. The value of Vk must be greater than or equal to any possible value of the

independent variable. If a distribution number Siis zero, the default value for the variable is used.

The Q option is the only form of the DS card that can be used when the distribution of the

independent variable is a built-in function.

6. SCn Source Comment Card

Form: SCn comment

n = distribution number (n=1,999)

The comment is printed as part of the header of distribution nin the source distribution table

and in the source distribution frequency table. The & continuation symbol is considered as

part of the comment, not as a continuation command.

Default: No comment.

Examples of the General Source

Example 1: SDEF ERG=D1 POS=x y z WGT=w

SI1 H E1E2... Ek

SP1 D 0 P2... Pk

SB1 D 0 B2... Bk

This is a point isotropic source at x,y,z with a biased continuous-energy distribution and average

source particle weight w. The starting cell is not speciﬁed. MCNP will determine it from the values

of x,y, and z.

Example 2: SDEF SUR=mAXS=ijk EXT=D6

SB6 −31 1.5

This is a source on surface m. The presence of AXS and EXT implies that surface m is a sphere

because AXS and EXT are not otherwise used together for sources on a surface. By default, the

particles are emitted in a cosine distribution. They are emitted outward if the positive normal to the

sphere is outward, which it is for all the spherical surface types but might not be if the sphere is

speciﬁed as type SQ. The position on the surface is biased toward the direction i,j,k by an

exponential bias (speciﬁed by −31) such that the maximum and minimum source particle weights

are e1.5 = 4.48 and e−1.5 = 0.223. By default, MCNP provides the effect of two cards: SI6 −1 1 and

SP6 −31 0.

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DATA CARDS

Example 3: SDEF SUR=mNRM=−1 DIR=D1 WGT=w

SB1 −21 2

This is an inward-directed source on spherical surface m, assuming the positive normal of the

surface is directed outward. If w=πr2, where ris the radius of sphere m, this source in conjunction

with a VOID card, a VOL card, and type 2 and type 4 tallies, is suitable for estimating the areas of

surfaces and the volumes of cells. See page 2–183. By default, MCNP provides the effect of two

cards: SI1 0 1 and SP1 −21 1. The directional bias by the SB1 card causes higher track density

toward the center of the sphere, where presumably the cells of greatest interest lie, than it would be

if the unbiased cosine distribution were used. This bias, incidentally, provides a zero-variance

estimate of the (known) volume of the sphere m.

Example 4: SDEF ERG=D1 POS=x y z CEL=m RAD=D2

EXT=D3 AXS=i j k

SP1 −3

SI2 r1r2

SI3 l

This source is distributed uniformly in volume throughout cell m, which presumably approximates

a cylinder. The cell is enclosed by a sampling volume centered at x,y,z. The axis of the sampling

volume is the line through x,y,z in the direction i,j,k. The inner and outer radii of the sampling

volume are r1and r2, and it extends along i,j,k for a distance from x,y,z. The user has to make

sure that the sampling volume totally encloses cell m. The energies of the source particles are

sampled from the Watt ﬁssion spectrum using the default values of the two parameters, making it

a Cranberg spectrum. By default, MCNP interprets SI3 l as if it were actually SI3 −l+l and

provides the effect of two cards: SP2 −21 1 and SP3 −21 0.

Example 5: SDEF SUR=mPOS=x y z RAD=D1 DIR=1 CCC=n

SI1 r

This is a monodirectional source emitted from surface min the direction of the positive normal to

the surface. The presence of POS and RAD implies that surface mis a plane because POS and RAD

are not otherwise used together for sources on a surface. The position is sampled uniformly in area

on the surface within radius r of point x,y,z. The user must make sure that point x,y,z actually lies

on surface m. The sampled position is rejected and resampled if it is not within cookie-cutter cell

n. The starting cell is found from the position and the direction of the particle. By default, MCNP

interprets SI1 r as if it were actually SI1 0 rand provides the effect of card SP1 −21 1.

Example 6: SDEF POS D1 ERG FPOS D2

SI1 L 5 3.3 6 75 3.3 6

SP1 .3 .7

DS2 S 3 4

1±

April 10, 2000 3-65

CHAPTER 3

DATA CARDS

SI3 H21014

SP3 D012

SI4 −3 ab

This is a point isotropic source in two locations, shown by two x,y,z’s on the SI1 card. The code

will determine the starting cell. With probability .3 the ﬁrst location will be picked and with

probability .7 the second location will be chosen. Each location has a different energy spectrum,

pointed to by the DS2 card. All other needed source variables will use their default values.

7. SSW Surface Source Write Card

Form: SSW S1S2 (C1Ck)S

3 Snkeyword=values

The = signs are optional.

Si= problem surface number, with the appropriate sense of inward or outward

particle direction, for which particle–crossing information is to be written to the

surface source ﬁle WSSA. Macrobody surfaces are not allowed.

Ci= problem cell number. A positive entry denotes an other–side cell. A negative

entry speciﬁes a just–left cell.

SYM m symmetry option ﬂag

m=0, no symmetry assumed.

m=1, spherical symmetry assumed. The list of problem surface numbers must

contain only one surface and it must be a sphere.

m=2, write particles to a surface bidirectionally. Otherwise, only particles going out

of a positive surface and into a negative surface are recorded.

PTY n1n2tracks to record

absent=record all tracks. This is the default.

ni=N, record neutron tracks

ni=P, record photon tracks

ni=E, record electron tracks

CEL C1C2Cn

list of names of all the cells from which KCODE ﬁssion source neutrons are to be

written, active cycles only.

Default: SYM=0 PTY absent=record all particle types

Use: Optional, as needed.

This card is used to write a surface source ﬁle or KCODE ﬁssion volume source ﬁle for use in a

subsequent MCNP calculation. Care must be taken to include enough geometry beyond the

speciﬁed surfaces to account for albedo effects. The card allows a list, in parentheses, of one or

more cell names, positive or negative, after any of the surface names. If the list of cells is absent,

any track that crosses the surface in the correct direction will be recorded. If the list is present, a

⋅⋅⋅

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DATA CARDS

track will be recorded if it crosses the surface in the correct direction and is either entering a cell

in the list whose entry is positive or leaving a cell in the list whose entry is negative.

If the SYM=1 option is used, the geometry inside the surface must be spherically symmetric and

the materials must be symmetric. In very few cases will the SYM=1 option apply. The user must

determine whether SYM=1 is appropriate for the problem. If the SYM=1 option is used, fewer

words per particle need to be written to the surface source ﬁle and certain biasing options become

available when reading the surface source ﬁle. In a KCODE calculation, particles are written only

for active cycles.

Example: SSW 4 −7 19 (45 −46)16−83 (49)

A track that crosses surface 19 in the correct direction will be recorded only if it is either entering

cell 45 or leaving cell 46. A track that crosses surface 83 in the correct direction will be recorded

only if it is entering cell 49. A track that crosses surface 4 or 7 or 16 in the correct direction will be

recorded regardless of what cells it happens to be leaving or entering.

Fission volume sources from a KCODE calculation can be written from active cycles only. The

ﬁssion neutrons and prompt photons can then be transported in a subsequent calculation using the

SSR surface source read ﬁxed–source capability. In a KCODE criticality calculation the ﬁssion

neutron sources and prompt photons produced from ﬁssion during each cycle are written to the

WSSA surface source ﬁle if the SSW card has the CEL keyword followed by the names of all the

cells from which ﬁssion source neutrons are to be written. Particles crossing speciﬁed surfaces can

also be written by specifying Si. The SYM=1 option (spherically symmetric surface source) cannot

be used if CEL is speciﬁed.

Example: SSW 1 2 (3 4) CEL 8 9

A track that crosses surface 2 in the correct direction will be recorded only if it enters cell 3 or 4.

A track crossing surface 1 in the correct direction always will be recorded. And particles created

from ﬁssion events in cells 8 and 9 will be recorded.

During execution, surface source information is written to the scratch ﬁle WXXA. Upon normal

completion, WXXA becomes WSSA. If the run terminates abnormally, the WXXA ﬁle will appear

instead of WSSA and must be saved along with the RUNTPE ﬁle. The job must be continued for

at least one more history. At the subsequent normal termination, WXXA disappears and the correct

surface source ﬁle WSSA is properly written.

8. SSR Surface Source Read Card

Form: SSR keyword=values keyword=values

The = signs are optional.

⋅⋅⋅

April 10, 2000 3-67

CHAPTER 3

DATA CARDS

OLD S1S2... Sn

list of problem surface numbers, a subset of the surfaces on the SSW card that

created the ﬁle WSSA, now called RSSA. Macrobody surfaces are not allowed.

CEL C1C2... Cn

like OLD but for cells in which KCODE ﬁssion neutrons or photons were

written

NEW Sa1 Sa2 ... San Sb1 Sb2 ... Sbn

problem surface numbers upon which the surface source is to start particles in

this run. The n entries may be repeated to start the surface source in a,b,...

transformed locations.

COL m collision option ﬂag

m= −1 start from the surface source ﬁle only those particles that came directly

from the source without a collision

m= 1 start from the surface source ﬁle only those particles that had collisions

before crossing the recording surface

m= 0 start particles without regard to collisions

WGT x Each particle weight is multiplied by the constant x as it is accepted for

transport.

TR n transformation number. Track positions and velocities are transformed from

the auxiliary coordinate system (the coordinate system of the problem that wrote

the surface source ﬁle) into the coordinate system of the current problem, using

the transformation on the TRn card, which must be present in the INP ﬁle of the

current problem.

TR Dn Distribution number for a set of SIn, SPn, and SBn cards. If the surface

source is transformed into several locations, the SIn card lists the transformation

numbers and the SPn and SBn cards give the probabilities and bias of each

transformation.

PSC c a nonnegative constant that is used in an approximation to the PSC evaluation

for the probability of the surface source emitting a particle into a speciﬁed angle

relative to the surface normal.

The following four keywords are used only with spherically symmetric surface sources, that is,

sources generated with SYM=1 on the SSW card.

AXS uvw Direction cosines that deﬁne an axis through the center of the surface

sphere in the auxiliary (original) coordinate system. This is the reference vector

for EXT.

EXT Dn n is the number of a distribution (SIn, SPn, and SBn cards) that will bias the

sampling of the cosine of the angle between the direction AXS and the vector

from the center of the sphere to the starting point on the sphere surface.

POA c Particles with a polar angle cosine relative to the source surface normal that

falls between 1 and c will be accepted for transport. All others are disregarded

3-68 April 10, 2000

CHAPTER 3

DATA CARDS

and no weight adjustment is made.

BCW rzbze 0 < zb <ze All particles with acceptable polar angles relative to the

surface normal are started so that they will pass through a cylindrical window

of radius r, starting at zb from the center of the source sphere, and ending at ze

from the center. The axis of the cylinder is parallel to the z-axis of the auxiliary

(original) coordinate system and contains the center of the source sphere. The

weight of each source particle is adjusted to compensate for this biasing of

position and direction.

Defaults: OLD accept all surfaces in the original run.

CEL accept all cells in the original run.

NEW the surfaces in the OLD list.

COL m= 0

WGT 1

TR no transformation

AXS no axis

EXT no position bias

POA c= 0

BCW no cylindrical window

PSC no default value

Use: Required for surface source problems.

The particle type is determined primarily by the type of the particle on the RSSA ﬁle, but particles

incompatible with the problem mode are rejected without weight adjustment.

An exact treatment of point detectors or DXTRAN spheres with a surface source is not possible

because the values required for the source contribution are not readily available. (See the

description of detector tallies in Chapter 2.) To use detectors or DXTRAN with a surface source,

an approximate must be speciﬁed on the SSR card. The most common azimuthally

symmetric approximation for an angular emission probability density function is given by

The PSC=value entered is n, the power to which is raised. Cnis a normalization constant

calculated in MCNP and is the angle between the direction vector to the point detector and the

surface normal at the point where the particle is to be started. Because surface crossings are

recorded in only one direction speciﬁed on the SSW card, the limits on µare always between 1 and

0. A PSC entry of zero speciﬁes an isotropic angular distribution on the surface. An entry of 1

speciﬁes a cosine angular distribution that produces an isotropic angular ﬂux on the surface. In the

case of a 1D spherical surface source of radius R, a cosine distribution is adequate if the point

detector or DXTRAN sphere is more than 4R away from the source.

pθcos()

pθcos()Cnθcos()

nn0≥=

pθcos()

April 10, 2000 3-69

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DATA CARDS

WARNING: Remember that the value entered is only an approximation. If the point detector or

DXTRAN sphere is close to the source sphere and the approximation is poor, the answers will be

WRONG.

Fission neutrons and photons written to the surface source ﬁle in a KCODE calculation can be used

as a volume–distributed source in a subsequent calculation. A NONU card should be used so that

ﬁssion neutrons and photons are not counted twice. Generally a TOTNU card also should be used.

Total is the default for KCODE sources but prompt is the default for non–KCODE sources.

Delayed gammas are ignored in MCNP. The keyword CEL speciﬁes which ﬁssion cells to accept

of those from the KCODE calculation that wrote the RSSA ﬁle.

Example 1: Original run: SSW 1 2 3

Current run: SSR OLD 3 2 NEW 6 7 12 13 TR D5 COL 1

SI5 L 4 5

SP5 .4 .6

SB5 .3 .7

Particles starting on surface 1 in the original run will not be started in the current run because 1 is

absent from the list of OLD surface numbers. Particles recorded on surface 2 in the original run

will be started on surfaces 7 and 13 and particles recorded on surface 3 in the original run will be

started on surfaces 6 and 12, as prescribed by the mapping from the OLD to the NEW surface

numbers. The COL keyword causes only particles that crossed surfaces 2 and 3 in the original

problem after having undergone collisions to be started in the current problem. The TR entry

indicates that distribution function 5 describes the required surface transformations. According to

the SI5 card, surfaces 6 and 7 are related to surfaces 3 and 2, respectively, by transformation TR4;

surfaces 12 and 13 are related to 3 and 2 by TR5. The physical probability of starting on surfaces

6 and 7 is 40% according to the SP5 card, and the physical probability of starting on surfaces 12

and 13 is 60%. The SB5 card causes the particles from surfaces 3 and 2 to be started on surfaces 6

and 7 30% of the time with weight multiplier 4/3 and to be started on surfaces 12 and 13 70% of

the time with weight multiplier 6/7.

Example 2: Original run: SSW 3 SYM 1

Current run: SSR AXS001 EXTD99

SI99 −1 .5 1

SP99 C .75 1

SB99 .5 .5

All particles written to surface 3 in the original problem will be started on surface 3 in the new

problem, which must be exactly the same because no OLD, NEW, COL, or TR keywords are

present. Because this is a spherically symmetric problem, indicated by the SYM 1 ﬂag in the

original run, the position on the sphere can be biased. It is biased in the z-direction with a cone

bias described by distribution 99.

υυ

3-70 April 10, 2000

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9. KCODE Criticality Source Card

Form: KCODE NSRCK RKK IKZ KCT MSRK KNRM MRKP KC8 ALPHA

NSRCK = number of source histories per cycle

RKK = initial guess for keff

IKZ = number of cycles to be skipped before beginning tally accumulation

KCT = number of cycles to be done

MSRK = number of source points to allocate storage for

KNRM = normalize tallies by 0=weight / 1=histories

MRKP = maximum number of cycle values on MCTAL or RUNTPE

KC8 = summary and tally information averaged over:

0 = all cycles

1 = active cycles only.

ALPHA = imposed value of alpha.

Defaults: NSRCK=1000; RKK=1.0; IKZ=30; KCT=IKZ+100; MSRK=4500 or 2*

NSRCK; KNRM=0; MRKP=6500;KC8=1; ALPHA=none.

Use: This card is required for criticality calculations.

The KCODE card speciﬁes the MCNP criticality source that is used for determining keff. The

criticality source uses total ﬁssion nubar values unless overridden by a TOTNU NO card and

applies only to neutron problems. In a MODE N,P problem, secondary photon production from

neutrons is turned off during inactive cycles. SSW particles are not written during inactive cycles.

If alpha is positive, use default implicit capture. See Chapter 1 for further information.

The NSRCK entry is the nominal source size for each cycle. The IKZ entry is the number of cycles

to skip before beginning tally accumulation (this is important if the initial source guess is poor).

The KCT entry speciﬁes the number of cycles to be done before the problem ends. A zero entry

means never terminate on the number of cycles but terminate on time. The MSRK is the maximum

number of source points for which storage will be allocated. If an SRCTP ﬁle with a larger value

of MSRK is read for the initial source, the larger value is used.

Fission sites for each cycle are those points generated by the previous cycle. For the initial cycle,

ﬁssion sites can come from an SRCTP ﬁle from a similar geometry, from a KSRC card, or from a

volume distribution speciﬁed by an SDEF card.

If in the ﬁrst cycle the source being generated overruns the current source, the initial guess (RKK)

is probably too low. The code then proceeds to print a comment, continues without writing a new

source, calculates , reads the initial source back in, and begins the problem using instead

of RKK. If the generated source again overruns the current source after the ﬁrst cycle, the job

terminates and either a better initial guess (RKK) or more source space (MSRK) should be

speciﬁed on the next try.

k′

eff k′

eff

April 10, 2000 3-71

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DATA CARDS

KC8=0 causes tallies and summary table information to be for both active and inactive cycles and

should not be used. KC8=0 also results in a strange MCTAL ﬁle normalization.

10. KSRC Source Points for KCODE Calculation

Form: KSRC x1y1z1x2y2z2...

xi,yi,zi = location of initial source points

Default: None. If this card is absent, an SRCTP source ﬁle or SDEF card must be supplied

to provide initial source points for a criticality calculation.

Use: Optional card for use with criticality calculations.

This card contains up to NSRCK (x,y,z) triplets that are locations of initial source points for a

KCODE calculation. At least one point must be in a cell containing ﬁssile material and points must

be away from cell boundaries. It is not necessary to input all NSRCK coordinate points. MCNP

will start approximately (NSRCK/number of points) particles at each point. Usually one point in

each ﬁssile region is adequate, because MCNP will quickly calculate and use the new ﬁssion source

distribution. The energy of each particle in the initial source is sampled from a Watt ﬁssion

spectrum hardwired into MCNP, with a= 0.965 MeV and b= 2.29 MeV−1.

An SRCTP ﬁle from a previous criticality calculation can be used instead of a KSRC card. If the

current problem has a lot in common with the previous problem, using the SRCTP ﬁle may save

some computer time. Even if the problems are quite different, the SRCTP ﬁle may still be usable

if some of the points in the SRCTP ﬁle are in cells containing ﬁssile material in the current

problem. Points in void or zero importance cells will be deleted. The number of particles actually

started at each point will be such as to produce approximately NSRCK initial source particles.

An SDEF card also can be used to sample initial source points in ﬁssile material regions. The SDEF

card parameters applicable to volume sampling can be used: CEL, POS, RAD, EXT, AXS, X, Y,

Z; and CCC, ERG, and EFF. If a uniform volume distribution is chosen, the early values of keff will

likely be low because too many particles are put near where they can escape, just the opposite of

the usual situation with the KSRC card. Do not change the default value of WGT for a KCODE

calculation.

11. ACODE Alpha Eigenvalue Source Card

Form: ACODE NSRCK RKK IKZ KCT MSRK KNRM KALPHA KALSAV

KALREG MRKP ALMIN

NSRCK = number of source histories per cycle

RKK = initial guess for keff

IKZ = number of cycles to be skipped before beginning tally accumulation

3-72 April 10, 2000

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DATA CARDS

KCT = number of cycles to be done

MSRK = number of source points to allocate storage for

KNRM = normalize tallies by 0=weight / 1=histories

KALPHA = alpha estimator: 1/2/3 = col/abs/trk

4 = differential operator perturbation estimator

KALSAV = cycle to start averaging alphas

KALREG = cycle to start ln-ln regression and reduce number of internal

settle cycles

MRKP = maximum number of cycle values on MCTAL or RUNTPE

ALPHA = initial guess of alpha

ALMIN = minimum ﬂoor value of alpha.

Defaults: NSRCK=1000; RKK=1.0; IKZ=30; KCT=IKZ+100; MSRK=4500 or 2*

NSRCK; KNRM=0; KALPHA=1; KALSAV=automatic;

KALREG=KALSAV+2; MRKP=6500; ALPHA=0; ALMIN=0

The alpha eigenvalue is , where Nis the neutron population at some time tthat builds

up from some initial population No.Alpha is of interest for comparison to benchmark calculations

and comparison to deterministic transport codes such as DANTSYS. The alpha capability allows

positive and negative alpha searches and a ﬁxed positive or negative alpha value to be used in a keff

eigenvalue calculation. Negative values of alpha can result in a time creation (n,2n) delta scattering

reaction. Positive alpha is treated as time absorption. We recommend the use of default implicit

capture for positive alpha problems.

12. Subroutines SOURCE and SRCDX

If SDEF, SSR, or KCODE cards are not present in the INP ﬁle, a user supplied source is assumed

and is implemented by calling subroutine SOURCE, which the user must provide. Chapter 4 has

examples of a SOURCE subroutine and discusses the SRCDX subroutine. The parameters that

must be speciﬁed within the subroutine are listed and deﬁned on page 3–49. Prior to calling

subroutine SOURCE, isotropic direction cosines u,v,w (UUU,VVV,WWW) are calculated and

need not be speciﬁed if you want an isotropic distribution.

The SIn, SPn, and SBn cards also can be used with the SOURCE subroutine, although

modiﬁcations to other parts of MCNP may be required for proper initialization and to set up

storage. A random number generator RANG( ) is available for use by SOURCE for generating

random numbers between 0 and 1. Up to 50 numerical entries can be entered on each of the IDUM

and RDUM cards for use by SOURCE. The IDUM entries must be integers and the RDUM entries

ﬂoating point numbers.

If you are using a detector or DXTRAN and your source has an anisotropic angular distribution,

you will need to supply an SRCDX subroutine to specify PSCs for each detector or DXTRAN

sphere (see Chapters 2 and 4).

oeαt

April 10, 2000 3-73

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DATA CARDS

There are unused variables stored in the particle bank that are reserved for the user called

SPARE(M), M=1,MSPARE, where MSPARE=3. Depending on the application, you may need to

reset them to 0 in SOURCE for each history; MCNP does not reset them.

E. Tally Speciﬁcation

The tally cards are used to specify what type of information the user wants to gain from the Monte

Carlo calculation; that is, current across a surface, ﬂux at a point, heating in a region, etc. This

information is requested by the user by using a combination of the following cards. To obtain tally

results, only the Fn card is required; the other tally cards provide various optional features.

The nis a user-chosen tally number < 999; choices of nare given in the following section. When a

choice of nis made for a particular tally type, any other input card used with that tally (such as En

for energy bins) is given the same value of n by the user.

Much of the information on these cards is used to describe tally “bins,” subdivisions of the tally

space into discrete and contiguous increments such as cosine, energy, or time. Usually when the

user subdivides a tally into bins, MCNP can also provide the total tally summed over appropriate

bins (such as over energy bins). Absence of any bin speciﬁcation card results in one unbounded bin

Mnemonic Card Type Page

Fna Tally type 3–74

FCn Tally comment 3–83

En Tally energies 3–83

Tn Tally times 3–84

Cn Tally cosines 3–85

FQn Tally print hierarchy 3–86

FMn Tally multiplier 3–87

DEn/DFn Dose energy/Dose function 3–91

EMn Energy multiplier 3–92

TMn Time multiplier 3–93

CMn Cosine multiplier 3–93

CFn Cell ﬂagging 3–94

SFn Surface ﬂagging 3–95

FSn Tally segment 3–95

SDn Segmented volume/area 3–97

FUn TALLYX input 3–98

TFn Tally ﬂuctuation print 3–100

DD Detector and DXTRAN diagnostics 3–102

DXT DXTRAN 3–103

FTn Special treatments 3–105

3-74 April 10, 2000

CHAPTER 3

DATA CARDS

rather than one bin with a default bound. No information is printed about the limits on the

unbounded bin.

If there are reﬂecting surfaces or periodic boundaries in the problem, the user may have to

normalize the tallies in some special way (this can be done by setting the weight of the source

particles or by using the FMn card).

Printed with each tally bin is the relative error of the tally corresponding to one standard deviation.

These errors cannot be believed reliable (hence neither can the tally itself) unless the error is fairly

low. Results with errors greater than 50% are useless, results between 20% and 50% can be

believed to within a factor of a few, results between 10% and 20% are questionable, results less

than 10% are generally (but not always) reliable except for detectors, and detector results are

generally reliable below 5%. One bin of every tally is designated for the tally ﬂuctuation charts at

the end of the output ﬁle. This bin is also used for the weight window generator. It also is subject

to ten statistical checks for tally convergence, including calculation of the variance of the variance

(VOV). The VOV can be printed for all bins in a tally by using the DBCN card.

1. Fna Tally Cards

Seven basic neutron tally types, six basic photon tally types, and four basic electron tally types are

available in MCNP as standard tallies. All are normalized to be per source particle unless changed

by the user with a TALLYX subroutine or normed by weight in a criticality (KCODE) calculation.

The tallies are identiﬁed by tally type and particle type as follows. Tallies are given the numbers 1,

2, 4, 5, 6, 7, 8, or increments of 10 thereof, and are given the particle designator :N, :P, or :E (or

Mnemonic Tally Description Fn units ∗Fn units

F1:N or F1:P or F1:E Current integrated over a surface particles MeV

F2:N or F2:P or F2:E Flux averaged over a surface particles/cm2MeV/cm2

F4:N or F4:P or F4:E Flux averaged over a cell particles/cm2MeV/cm2

F5a:N or F5a:P Flux at a point or ring detector particles/cm2MeV/cm2

F6:N or F6:N,P or F6:P Energy deposition averaged over

a cell MeV/g jerks/g

F7:N Fission energy deposition

averaged over a cell MeV/g jerks/g

F8:P or F8:E or F8:P,E

+F8:E

Energy distribution of pulses

created in a detector

Charge deposition

pulses

charge

MeV

N/A

April 10, 2000 3-75

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DATA CARDS

:N,P only in the case of tally type 6 or :P,E only in the case of tally type 8). Thus you may have as

many of any basic tally as you need, each with different energy bins or ﬂagging or anything else.

F4:N, F14:N, F104:N, and F234:N are all legitimate neutron cell ﬂux tallies; they could all be for

the same cell(s) but with different energy or multiplier bins, for example. Similarly F5:P, F15:P,

and *F305:P are all photon point detector tallies. Having both an F1:N card and an F1:P card in the

same INP ﬁle is not allowed. The tally number may not exceed three digits.

Tally types 1, 2, 4, and 5 are normally weight tallies (particles in the above table); however, if the

Fn card is ﬂagged with an asterisk (for example, ∗F1:N), energy times weight will be tallied. The

asterisk ﬂagging can also be used on tally types 6 and 7 to change the units from MeV/g to jerks/g

(1 jerk = 1 GJ = 109J). The asterisk on a tally type 8 converts from a pulse height tally to an energy

deposition tally. All of the units are shown in the above table.

Tally type 8 can also be ﬂagged with a plus (+) to convert it from an energy deposition tally (ﬂagged

with an asterisk) to a charge deposition tally. The tally is the negative particle weight for electrons

and the positive weight for positrons. The +F8 tally can be checked against an F1:E type surface

tally. See page 3–79 for an example.

Only the F2 surface ﬂux tally requires the surface area. The area calculated is the total area of the

surface that may bound several cells, not a portion of the surface that bounds only a particular cell.

If you need only the segment of a surface, you might segment the full surface with the FSn card

(see page 3–95) and use the SDn card (see page 3–97) to enter the appropriate values. You can also

redeﬁne the geometry as another solution to the problem. The detector total is restricted to 20. The

tally total is limited to 100. Note that a single type 5 tally may create more than one detector.

1. Surface and Cell Tallies (tally types 1, 2, 4, 6, and 7)

Simple Form: Fn:pl S1... Sk

General Form: Fn:pl S1 (S2... S3) (S4... S5)S6S7...

n = tally number.

pl = N or P or N,P or E

Si= problem number of surface or cell for tallying, or T.

Only surfaces bounding cells and listed in the cell card description can be used on F1 and F2 tallies.

Tally 6 does not allow E. Tally 7 allows N only.

In the simple form above, MCNP creates k surface or cell bins for the requested tally, listing the

results separately for each surface or cell. In the more general form, a bin is created for each surface

or cell listed separately and for each collection of surfaces or cells enclosed within a set of

parentheses. Entries within parentheses also can appear separately or in other combinations.

Parentheses indicate that the tally is for the union of the items within the parentheses. For

unnormalized tallies (tally type 1), the union of tallies is a sum, but for normalized tallies (types 2,

3-76 April 10, 2000

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DATA CARDS

4, 6, and 7), the union results in an average. See page 3–79 for an explanation of the repeated

structure and lattice tally format.

The symbol T entered on surface or cell Fn cards is shorthand for a region that is the union of all

of the other entries on the card. A tally is made for the individual entries on the Fn card plus the

union of all the entries.

If a tally label of the surfaces or cells in the output requires more than eleven characters, including

spaces, MCNP deﬁnes an alphabetical or numerical designator for printing purposes. The

designator [for example, G is (1 2 3 4 5 6)] is printed with the tally output. This labeling scheme is

usually required for tallies over the union of a long list of surfaces or cells.

Example 1: F2:N 1 3 6 T

This card speciﬁes four neutron ﬂux tallies, one across each of the surfaces 1, 3, and 6 and one

which is the average of the ﬂux across all three of the surfaces.

Example 2: F1:P (1 2) (3 4 5) 6

This card provides three photon current tallies, one for the sum over surfaces 1 and 2; one for the

sum over surfaces 3, 4, and 5; and one for surface 6 alone.

Example 3: F371:N (1 2 3) (1 4) T

This card provides three neutron current tallies, one for the sum over surfaces 1, 2, and 3; one for

the sum over surfaces 1 and 4; and one for the sum over surfaces 1, 2, 3, and 4. The point of this

example is that the T bin is not confused by the repetition of surface 1.

Another case for study is in the DEMO example in Chapter 5.

2. Detector Tallies (tally type 5)

Form for point detectors: Fn:pl X Y Z

n= tally number.

pl = N for neutrons or P for photons,

X Y Z = location of the detector point.

= radius of the sphere of exclusion:

in centimeters, if Ro is entered as positive,

in mean free paths, if entered as negative. (Negative entry illegal in a void.)

April 10, 2000 3-77

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DATA CARDS

Form for ring detectors: Fna:pl aor

n= tally number.

a= the letter X, Y, or Z.

pl = N for neutrons or P for photons.

ao= distance along axis “a” where the ring plane intersects the axis.

r= radius of the ring in centimeters.

= same meaning as for point detectors, but describes a sphere about the point

selected on the ring.

Default: None.

Use: You are encouraged to read about detectors in Chapter 2 before using them because

they are very susceptible to unreliable results if used improperly. Remember that

contributions to a detector are not made through a region of zero importance. Ring

(rather than point) detectors should be used in all problems with axial symmetry. A

detector located right on a surface probably will cause trouble. Detectors and

DXTRAN can be used in problems with the S(α,β) thermal treatment, but the S(α,β)

contributions are approximate (see page 2–53. Detectors used with reﬂecting, white,

or periodic surfaces give wrong answers (see page 2–92). Consider using the PDn and

DDn cards.

For more than one detector with the same n or na designation, sets of the above input parameters

(quadruplets for Fn or triplets for Fna) are simply continued on the same Fn or Fna card. If more

than one detector of the same type (an F5:N and an F15:N, for example) are at the same location,

the time-consuming contribution calculation upon collision is made only once and not

independently for each detector, according to the rules in Chapter 2. Thus it is inexpensive to add

more than one detector (each with a different response function, for example) at the same location

as another.

The printout for detectors is normally in two parts: (1) the total of all contributions to the detector

(as a function of any deﬁned bins such as energy) and (2) the direct (or uncollided) contribution to

the detector from the source. The direct contribution is always included in the total of all

contributions. Adding the symbol ND at the end of a type 5 detector tally card inhibits the separate

printing of the direct contribution for that tally. In coupled neutron/photon problems, the direct

contribution in photon tallies is from photons created at neutron collisions.

Rules of Thumb for Ro:Ro should be about 1/8 to 1/2 mean free path for particles of average

energy at the sphere and zero in a void. Supplying Roin terms of mean free path will increase the

variance and is not recommended unless you have no idea how to specify it in centimeters. Romust

not encompass more than one material. MCNP cannot check this and the consequences may be

wrong answers.

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DATA CARDS

3. Pulse Height Tally (tally type 8)

Simple Form: Fn:pl S1... Sk

General Form: Fn:pl S1 (S2... S3) (S4... S5)S6S7...

n= tally number.

pl = P, E or P,E

Si= problem number of cell for tallying, or T.

The F8 tally provides the energy distribution of pulses created in a detector by radiation and is

called a pulse height tally. See page 3–79 for an explanation of the repeated structure and lattice

tally format. The F8 card is used to list the cell bins, just like an F4 tally. The union of tallies

produces a tally sum, not an average. Cell, user, and energy bin cards are allowed. Flagging,

segment, multiplier, time, and cosine bins are not allowed. The energy bins accumulate the energy

deposited in a cell by all the tracks of a history rather than the energy of the scoring tracks. Both

photons and electrons will be tallied if present, even if only E or only P is on the F8 card. In other

words, the F8:P, F8:E, and F8:P,E are all equivalent tallies. An asterisk on the F8 card converts the

tally from a pulse height tally to an energy deposition tally. A plus on the F8 card converts the tally

from a pulse height tally to a charge deposition tally in units of electron charge. Energy binning is

not recommended.

If the F8 card has an asterisk or plus and there is no energy binning (E8 card), variance reduction

of all kinds is allowed. In this special case, the total energy deposition or charge is correct even

though the tallies in energy bins would not be correct. However, there still can be problems as

described on page 2–83.

Care must be taken when selecting energy bins for a pulse height tally. It is recommended that a

zero bin and an epsilon bin be included such as

E8 0 1E-5 1E-3 1E-1 ...

The zero bin will catch nonanalog knock–on electron negative scores. The epsilon (1E-5) bin will

catch scores from particles that travel through the cell without depositing energy. See page 2–83.

The pulse height tally is a radical departure from other MCNP tallies. All other tallies are estimates

of macroscopic variables, such as ﬂux, whose values are determined by very large numbers of

microscopic events. The pulse height tally records the energy or charge deposited in a cell by each

source particle and its secondary particles. For other tallies it is not necessary to model microscopic

events realistically as long as the expectation values of macroscopic variables were correct. For the

pulse height tally, microscopic events must be modeled much more realistically.

The departures from microscopic realism in MCNP are everywhere. The number, energies, and

directions of the secondary neutrons and photons from a neutron collision are sampled without any

April 10, 2000 3-79

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DATA CARDS

correlation between the particles and with no regard for the conservation of energy. Modeling the

ﬂuctuations in the number of ﬁssion neutrons is limited to choosing between the integer next larger

and the integer next smaller than the average number of ﬁssion neutrons. The ﬂuctuations in the

energy loss rate of an electron are not correlated with the production of knock-ons and x rays. The

variance reduction schemes in MCNP create unrealistic histories that nevertheless give correct

results for macroscopic tallies.

Problems that give correct pulse height tallies are severely limited. The only possible variance

reduction scheme is biasing of the source itself. CAUTION: The pulse height tally does not work

well with neutrons (and is not allowed) because of the nonanalog nature of neutron transport that

departs from microscopic realism at every turn. One can have a neutron source in a MODE N,P or

N,P,E problem, but only the photons and electrons can be tallied on the F8 card. The F8 tally can

be used effectively in photon problems. Electron problems may give correct results as long as the

tally cells are thick enough for the errors in the energy loss rate to average out. MCNP tries to

detect conditions in a problem that would invalidate pulse height tallies, but it is not able to catch

all of them. The user must ascertain that his problem does not violate the necessary conditions for

obtaining correct answers.

Scoring the pulse height tally is done at the end of each history. A temporary account of energy is

kept for each pulse height tally cell bin during the history. At the source, the source cell is credited

with the energy times the weight of the source particle. When a particle crosses a surface, the

energy times the weight of the particle is subtracted from the account of the cell that it is leaving

and is added to the account of the cell that it is entering. The energy here is the kinetic energy of

the particle plus 1.022016 MeV if it is a positron. At the end of the history, the account in each tally

cell is divided by the source weight. The resulting energy determines which energy bin the score

is put in. The value of the score is the source weight, or, if there is an asterisk on the F8 card, the

source weight times the energy in the account. If there is a plus on the F8 card, the tally is the

negative particle weight for electrons and the positive weight for positrons. The +F8 charge

deposition tally can be checked against an electron F1:E surface tally with the FT ELC option if

the volume of the +F8 is exactly enclosed by the surfaces on the F1:E card. For example, if cell 1

is enclosed by spherical surface 2, then the following tallies give the same result provided the two

F1 current tally bins (in – out) are properly subtracted.

+F8:E and F1:E 2

FT1 ELC 1

C101

Note that the meaning of the energy bins of a pulse height tally is entirely different from the

meaning of the energy bins of the other tallies in MCNP. The normal meaning of energy bins is the

energy of a scoring track. The meaning of the energy bins of a pulse height tally is the energy

deposited in a cell bin by all the tracks of a history.

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4. Surface, Cell, and Pulse Height Tallies for Repeated Structures and Lattices

(tally types 1, 2, 4, 6, 7, and 8)

Simple Form: Fn:pl S1... Sk

General Form: Fn:pl S1 (S2... S3) ((S4S5)< (C1C2[I1... I2]) < (C3C4C5)) ...

n= tally number.

pl = N or P or N,P or E.

Si= problem number of a surface or cell for tallying, U=#, or T.

Ci= problem number of a cell ﬁlled with a universe, or U=#.

# = problem number of a universe used on a ﬁll card.

Ii= index data for a lattice cell element, with three possible formats:

I1Indicating the I1th lattice element of cell C2,

as deﬁned by the FILL array.

I1:I2I3:I4I5:I6Range of one or more lattice elements.

Use the same format as on the FILL card.

I1I2I3,I4I5I6Indicating lattice element (I1,I2,I3), (I4,I5,I6), etc

See LAT and FILL cards for indices explanation.

In the simple form, MCNP creates k surface or cell bins for the requested tally, listing the results

separately for each surface or cell. In the more general form, a bin is created for each surface or

cell listed separately and for each collection of surfaces or cells enclosed within a set of

parentheses. A tally bin can involve a single tally level or multiple tally levels. Tallies involving

repeated structure and lattice geometries can use either form.

If a tally label of the surface or cells in a given bin exceeds eleven characters, including spaces, an

alphabetical or numerical designator is deﬁned for printing purposes. The designator

[G is (123456)], for example, would be printed with the tally output. This labeling scheme is

usually required for tallies over the union of a long list of surfaces or cells or with repeated structure

tallies.

Some operators and nomenclature need to be introduced before the explanation of repeated

structures and lattice tallies. The left arrow or less than symbol <is used to identify surfaces or cells

within levels of repeated structures. See page 3–26 for explanation of geometry levels. A tally bin

that includes one or more left arrows implies multiple levels, called a chain. Multiple entries

enclosed by parentheses at any level of a tally chain indicates the union of the items. Brackets [ ]

immediately following a ﬁlled lattice cell identify one or more elements of that lattice. The input

tally bin chain involving multiple levels MUST be enclosed by an outer set of parentheses.

F4:N (5 < 4 < 2 [1 0 0])

This example could specify an F4 tally in cell 5 when it is in cell 4, when cell 4 is in cell 2, which

is a lattice, and only in lattice element [1,0,0]. While any cell (lattice, ﬁlled, or simple) can be

April 10, 2000 3-81

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entered as a tally cell (e.g., S1 through S5), only cells ﬁlled with a universe can be used in higher

levels (e.g., C1 through C5.) See General Form, page 3–79.

Multiple bin format: In addition to multiple levels, multiple entries can be used in each level of the

tally chain resulting in multiple output bins. Within the parentheses required around the tally bin

chain, other sets of parentheses can be used to indicate the union of cells as in a simple tally

description, resulting in fewer output tally bins.

((S4S5)< (C1C2 [I1... I2]) < (C3C4C5))

This example results in one output tally bin and will be the union of the tally in S4plus S5that ﬁll

C1or C2[elements I1... I2] and when C1or C2ﬁlls cells C3,C4,orC5. Removing the ﬁrst and third

inner parentheses: (S4S5< (C1C2 [I1... I2]) <C3C4C5)

results in the creation of 2*1*3=6 bins as follows:

(S4< (C1C2 [I1... I2]) <C3), (S5< (C1C2 [I1... I2]) <C3),

(S4< (C1C2 [I1... I2]) <C4), (S5< (C1C2 [I1... I2]) <C4),

(S4<(C1C2[I1 ... I2]) <C5), (S5< (C1C2 [I1... I2]) <C5),

The repeated structure/lattice input tally bin format with levels that have multiple entries

automatically creates multiple output tally bins. The total number of bins generated is the product

of the number of entries at each level. If parentheses enclose all entries at a level, the number of

entries at that level is one and results in the union of all those entries. See Chapter 4, page 4–46,

for a caution when tallying a union of overlapping regions. For unnormalized tallies (type 1, 8), the

union is a sum. For normalized tallies (type 2, 4, 6, 7), the union is an average. A symbol Ton the

tally line creates an additional tally bin that is the union or total of all the other tally bins.

Brackets: Brackets [ ] enclose index data for lattice cell elements. Brackets make it possible to tally

on a cell or surface only when it is within the speciﬁed lattice elements. The brackets must

immediately follow a ﬁlled lattice cell. Listing a lattice cell without brackets will produce a tally

when the tally cell or surface is in any element of the lattice, provided the tally cell or surface ﬁlls

an entry at all other levels in the chain. The use of brackets is limited to levels after the ﬁrst <in

the tally speciﬁcation.

To tally within lattice elements of a real world (level zero) lattice cell, use the special syntax that

follows. Cell 3 contains material 1 and is bounded by four surfaces. The F4 card speciﬁes a tally

only in lattice element (0,0,0). This syntax is required because brackets can only follow a <.

31−1.0 −1234lat=1

F4:N (3 <3 [0 0 0])

⋅

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Universe format: The universe format, U=#, is a shorthand method of including all cells and lattice

elements ﬁlled by universe #. This format can be used in any level of the tally chain. The following

example illustrates valid shorthand U=# descriptions in the left column. The right column shows

the tally after the shorthand has been expanded. Cells 4 and 5 are ﬁlled with universe 1.

shorthand expanded

F4:N u=1 4 5

(u=1) (4 5)

(u=1 < 2 < 3) (4 5 < 2 < 3)

((u = 1) <2<3) ((4 5)< 2 < 3)

(1 <u = 1 < 2 < 3)(1 < 4 5 < 2 < 3)

(1 < (u = 1)< 2 < 3)(1 < (4 5) < 2 < 3)

In complex geometries, the U=# format should be used sparingly, especially with the multiple bin

format. If 100 cells are ﬁlled by universe 1 and 10 cells are ﬁlled by universe 2, then the tally

F4:N (u = 1 < u= 2)

will create 1000 output tally bins. However,

F4:N ((u = 1) <(u = 2))

will create only one output tally bin.

Use of SDn card: When making tallies in repeated structure and lattice geometries, often a volume

or area is required and MCNP will be unable to calculate it. Possibly the geometry causes the

calculation to fail. A universe can be repeated a different number of times in different cells and the

code has no way to determine this. There are two distinct options for entries on the SDn card

relating to repeated structures and they cannot be mixed within a single tally.

The ﬁrst option is to enter a value for each ﬁrst level entry on the related F card. If the entry on the

F card is the union of cells, the SD card value will be the volume of the union of the cells. The

following examples illustrate Fn card tally descriptions in the left column. The right column shows

the SDn card entries.

F4:N (1 < 4 5 6 < 7 8) SD4 V1

(1 2 3 < 4 5 6 < 7 8)V

1 V2 V3

(1 2 3 < (4 5 6) < (7 8)) V1 V2 V3

((1 2 3) < 4 5 6 < 7 8)V

123L

Vi=volume of cell iand V123 =volume of the union of cells 1, 2, and 3. Even though the ﬁrst line

creates six tally bins, only one SD value is entered. This divisor is applied to all bins generated by

the input tally bin. You do not need to know the number of bins generated by each input tally bin

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in order to use the SD card. The last line is the union of cells 1, 2, and 3 and only one divisor is

entered on the SD card.

The second option is to enter a value for each bin generated by the Fn card.

F4:N (1 < 4 5 6 < 7 8) SD4

(1 2 3 <4 5 6 <7 8) ...

(1 2 3 <(4 5 6) <(7 8))

((1 2 3) <4 5 6 <7 8)

=volume of cell ifor bin jand =volume of the union of cells 1, 2, and 3 for bin j. If cell

iis repeated the same number of times in all six bins generated by the ﬁrst line above, then all six

SD values for this input bin will be the same ( ). However, if cell 1 is repeated a

different number of times in each bin, then different SD values should be entered. The volume is

multiplied by the number of times it is repeated. In these cases, the total cell 1 volume for each

generated bin will not be calculated. The bin generation order is explained previously in the Fn card

section. For the ﬁrst line above, the bin order is (1<4<7), (1<5<7), (1<6<7), (1<4<8), (1<5<8), and

(1<6<8). The second line above generated 18 tally bins, and 18 SD values are required in the proper

sequence. This option requires the knowledge of both the number and sequence of bins generated

by each input tally bin.

2. FCn Tally Comment Card

Form: FCn any desired information

Default: No comment.

Use: Recommended for modiﬁed tally.

Anything entered after FCn will appear as the title heading of tally Fn. This card is particularly

useful when tallies are modiﬁed in some way, so later readers of the output will be warned of

modiﬁed or nonstandard tallies. The FCn card can be continued only by blanks in columns 1–5 on

suceeding lines. The & continuation symbol is considered as part of the comment, not as a

continuation command.

3. En Tally Energy Card

Form: En E1... Ek

n=tally number.

Ei= upper bound (in MeV) of the ith energy bin for tally n.

1V1

2V1

3V1

4V1

5V1

1V2

2V3

3V1

4V2

5V3

6V1

7V1

16 V2

17 V3

V1V2V3

V123

1V123

2V123

3V123

4V123

5V123

jV123

1V1

2V1

3…==

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Default: If the En card is absent, there will be one bin over all energies unless this

default has been changed by an E0 card.

Use: Required if EMn card is used.

The entries on the En card must be entered in the order of increasing magnitude. If a particle has

an energy greater than the last entry, it will not be tallied, but you will be warned that this has

happened. If the last entry is greater than the upper energy limit Emax speciﬁed on the PHYS card,

the last bin will be lowered to Emax. If there are several bins above Emax, the extra bins are

eliminated.

An E0 (zero) card can be used to set up a default energy bin structure for all tallies. A speciﬁc En

card will override the default structure for tally n.

MCNP automatically provides the total over all speciﬁed energy bins, but the total can be inhibited

for a given tally by putting the symbol NT at the end of the En card for that tally. The symbol C at

the end of the line causes the bin values to be cumulative and the last energy bin is also the total

over all energy.

Example: E11 .1 1 20

This will separate an F11 current tally into four energy bins: (1) from the energy cutoff to 0.1 MeV,

(2) from 0.1 to 1.0 MeV, (3) from 1.0 to 20.0 MeV, and (4) a total over all energy.

4. Tn Tally Time Card

Form: Tn T1... Tk

n=tally number.

Ti=upper bound (in shakes) of the ith time bin for tally n.

Default: If the Tn card is absent, there will be one bin over all times unless this default has

been changed by a T0 card.

Use: Required if TMn card is used. Consider FQn card.

The times on the Tn card must be entered in order of increasing magnitude. If a particle has a time

greater than the last entry on the Tn card, it will not be tallied, but you will be warned that this has

happened. The last time bin entry should always be less than or equal to the time cutoff (see CUT

card) except for point detectors. If time bins greater than the time cutoff are entered for tallies other

than point detectors, the ﬁrst bin limit over the cutoff will be lowered to the cutoff. All higher bins

will be eliminated. For point detector tallies, time bins can exceed the time cutoff so that particles

will score at detectors remote from the main body of the system. Setting the time cutoff lower than

April 10, 2000 3-85

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DATA CARDS

the last time bin will inhibit unproductive transport of slow neutrons in the system and will increase

the efﬁciency of the problem.

A T0 (zero) card can be used to set up a default time bin structure for all tallies. A speciﬁc Tn card

will override the default structure for tally n.

MCNP automatically provides the total over all speciﬁed time bins, but the total can be inhibited

for a given tally by putting the symbol NT at the end of the Tn card for that tally. The symbol C at

the end of the line causes the bin values to be cumulative and the last time bin is also the total over

all time.

Example: T2 −1 1 1.0+37 NT

This will separate an F2 ﬂux surface tally into three time bins: (1) from to −1.0 shake, (2) from

−1.0 shake to 1.0 shake, and (3) from 1.0 shake to 1.0e37 shakes, effectively inﬁnity. No total bin

will be printed in this example.

5. Cn Cosine Card (tally type 1 only)

Form: Cn C1... Ck

n=tally number.

Ci=upper cosine limit of the ith angular bin for surface current tally n.

C1 > −1.Ck = 1.

Default: If the Cn card is absent, there will be one bin over all angles unless this default

has been changed by a C0 card.

Use: Tally type 1. Required if CMn card is used. Consider FQn card.

The angular limits described by the Cn card are deﬁned with respect to the positive normal to the

surface at the particle point of entry. An FTn card with an FRVUVWoption can be used to make

the cosine bins relative to the vector u,v,w. The positive normal to the surface is always in the

direction of a cell that has positive sense with respect to that surface. The cosines must be entered

in increasing order, beginning with the cosine of the largest angle less than 180 to the normal and

ending with the normal (cos=1). A lower bound of −1 is set in the code and should not be entered

on the card. The last entry must always be 1.

A C0 (zero) card can be used to set up a default angular bin structure for all tallies. A speciﬁc Cn

card will override the default structure for tally n. Note that the selection of a single cosine bin for

an F1 tally gives the total and not the net current crossing a surface.

∞–

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MCNP does not automatically provide the total over all speciﬁed cosine bins, but the total can be

generated for a given tally by putting the symbol T at the end of the Cn card for that tally. The

symbol C at the end of the line causes the bin values to be cumulative and the last cosine bin is also

the total over all cosine bins.

Example: C1 −.866 −.5 0 .5 .866 1

This will tally currents within the angular limits (1) 180οto 150ο, (2) 150οto 120ο, (3) 120οto 90ο,

(4) 90ο to 60ο, (5) 60ο to 30ο, and (6) 30ο to 0ο with respect to the positive normal. No total will

be provided.

As an example of the relation between a surface normal and sense for the C1 card, consider a source

at the origin of a coordinate system and a plane (PY) intersecting the +yaxis. An entry of 0 and 1

on the C1 card will tally all source particles transmitted through the plane in the 0 to 1 cosine bin

(0οto 90ο) and all particles scattered back across the plane in the −1 to 0cosine bin (90οto 180ο).

A plane (PY) intersecting the −yaxis will result in a tally of all source particles transmitted through

the second plane in the −1 to 0 bin (90οto 180ο) and all particles scattered back across the plane in

the 0 to 1 bin (0ο to 90ο). Note that the positive normal direction for both planes is the same, the

+y axis.

6. FQn Print Hierarchy Card

Form: FQn a1a2... a8

n= tally number

ai= F—cell, surface, or detector

D—direct or ﬂagged

U—user

S—segment

M—multiplier

C—cosine

E—energy

T—time

Default: Order as given above.

Use: Recommended where appropriate.

The ai’s are the letters representing all eight possible types of tally bins. This card can be used to

change the order in which the output is printed for the tallies. For a given tally, the default order is

changed by entering a different ordering of the letters, space delimited. An example of this card is

in the DEMO example in Chapter 5.

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A subset of the letters can be used, in which case MCNP places them at the end of the FQn card

and precedes them with the unspeciﬁed letters in the default order. The ﬁrst letter is for the

outermost loop of the nest in the tally printout coding. The last two sets of bins make a table: the

next to last set goes vertically; the last set of bins goes horizontally in the table.

Note that the default order is a table in E and T; any other bins in a tally will be listed vertically

down the output page. Thus if you have a tally with only cell, user, and energy bins, the print for

that tally will be a hard-to-read vertical list. Specifying U E as the only entries or last two entries

on the FQn card will produce the same output, but in an easy-to-read table.

An FQ0 (zero) card can be used to change the default order for all tallies. A speciﬁc FQn card will

then override that order for tally number n.

An example of this card is in the DEMO example in Chapter 5.

7. FMn Tally Multiplier Card

Form: FMn (bin set 1) (bin set 2) ... T

n= tally number

(bin set i) = ((multiplier set 1) (multiplier set 2) ... (attenuator set))

T= absent for no total over bins

= present for total over all bins

C= cumulative tally bins

attenuator set = C −1 m1px1m2px2...

multiplier set i= C m (reaction list 1) (reaction list 2) ...

special multiplier set i=C−k

C= multiplicative constant

−1 = ﬂag indicating attenuator rather than multiplier set

m= material number identiﬁed on an Mm card

px = density times thickness of attenuating material;

atom density if positive, mass density if negative

k= special multiplier option;

(reaction list i) = sums and products of ENDF or

special reaction numbers, described below.

Parentheses:

1. If a given multiplier set contains only one reaction list, the parentheses surrounding the

reaction list can be omitted. Parentheses within a reaction list are forbidden.

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DATA CARDS

2. If a given bin set consists of more than a single multiplier or attenuator set, each

multiplier or attenuator set must be surrounded by parentheses, and the combination

must also be surrounded by parentheses.

3. If the FMn card consists only of a single bin set, and that bin set consists only of a single

multiplier or attenuator bin, surrounding parentheses can be omitted.

Default: If the C entry is negative (for type 4 tally only), C is replaced by |C| times

the atom density of the cell where the tally is made.

Use: Optional. Use the attenuators only when they are thin. Use only the multiplicative

constant for tally types 6 and 7. Disallowed for tally type 8.

The FMn card is used to calculate any quantity of the form

where is the energy-dependent ﬂuence (particles/cm2) and R(E) is an operator of additive

and/or multiplicative response functions from the MCNP cross-section libraries or specially

designated quantities. Note that some MCNP cross–section library reaction numbers are different

from ENDF/B reaction numbers. See below. The constant Cis any arbitrary scalar quantity that can

be used for normalization. The material number m must appear on an Mm card, but need not be

used in a geometrical cell of the problem.

A reaction list consists of one or more reaction numbers delimited by spaces and/or colons. A space

between reaction numbers means multiply the reactions. A colon means add the reactions. The

hierarchy of operation is multiply ﬁrst and then add. One bin is created for each reaction list. Thus,

if R1,R2, and R3 are three reaction numbers, the form R1R2 : R3 represents one reaction list (one

bin) calling for reaction R3 to be added to the product of reactions R1 and R2. No parentheses are

allowed within the reaction list. The product of R1with the sum of R2and R3would be represented

by the form R1R2:R1R3rather than by the form R1(R2:R3). The latter form would produce two

bins with quite a different meaning (see Examples 1 and 2 below).

The reaction cross sections are microscopic (with units of barns) and not macroscopic. Thus, if the

constant Cis the atomic density (in atoms per barn cm), the results will include the normalization

“per cm3.” The examples in Chapter 4 illustrate the normalization.

Any number of ENDF/B or special reactions can be used in a multiplier set as long as they are

present in the MCNP cross-section libraries, or in special libraries of dosimetry data. If neither a

material nor any reactions are given, the tally is multiplied by the constant C.

A multiplier set that has only two entries, C−k, has special meaning. If k= −1, the tally is

multiplied by 1/weight and the tally is the number of tracks (or collisions for the F5 tally.) If

k= −2, the tally is multiplied by 1/velocity and the tally is the neutron population integrated over

CϕE()RmE()Ed

∫

ϕE()

⋅

April 10, 2000 3-89

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DATA CARDS

time, or the prompt removal lifetime. See Chapter 2 page 2–169, Chapter 4 example 3

page 4–39 and the KCODE problem in Chapter 5.

In addition to most of the approximately one hundred standard ENDF reaction numbers available

(for example, R = 1, 2, 16, representing σtot,σel,σn,2n), the following nonstandard special R

numbers may be used:

Neutrons: −1 total cross section without thermal

−2 absorption cross section

−3 elastic cross section without thermal

−4 average heating number (MeV/collision)

−5 gamma-ray production cross section, barns

−6 total ﬁssion cross section

−7 ﬁssion ν

−8 ﬁssion Q (MeV/ﬁssion)

Photons: −1 incoherent scattering cross section

−2 coherent scattering cross section

−3 photoelectric cross section

−4 pair production cross section

−5 total cross section

−6 photon heating number

Multigroup: −1 total cross section

−2 ﬁssion cross section

−3 nubar data

−4 ﬁssion chi data

−5 absorption cross section

−6 stopping powers

−7 momentum transfers

A list of many of the ENDF reaction numbers can be found in Appendix G. The total and elastic

cross sections, R= 1 and R= 2, are adjusted for temperature dependence. All other reactions are

interpolated directly from the library data tables. Note that for tritium production, the R number

differs from one nuclide to another. Note also that tally types 6 and 7 already include reactions, so

the FMn card makes little sense for n=6or 7. Only the constant-multiplier feature should be used

for these tally types, generally. Photon production reactions can be speciﬁed according to the

MTRP prescription in Table F.6 in Appendix F.

An attenuator set of the form C1mpx, where m is the material number and px is the product of

density and thickness, allows the tally to be modiﬁed by the factor representing an

exponential line-of-sight attenuator. This capability makes it possible to have attenuators without

eσtot

–px

3-90 April 10, 2000

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DATA CARDS

actually modeling them in the problem geometry. Caution: The assumption is made that the

attenuator is thin, so that simple exponential attenuation without buildup from scattering is valid.

The attenuator set can include more than one layer:

in which case the factor is . The attenuator set can also be part of a bin set, for

example,

((C1m1R1) (C2m2R2) (C3−1 m3px3))

in which case the attenuation factor is applied to every bin created by the multiplier sets. Note that

both the inner and the outer parentheses are required for this application.

Tallies are posted in all multiplier bins for each score. MCNP does not automatically provide the

total over all speciﬁed multiplier bins for a particular tally. The total is available for a tally,

however, by putting the symbol T at the end of the FMn card for that tally.

In perturbed problems (see PERT card, page 3–141), the perturbation keyword RXN can affect the

cross sections used with the FM card tally multipliers. If a tally in a cell is dependent on a cross

section that is perturbed, then and a correction is made to the R1j′= 0 case (see page 2-

XII.A.??.) For this required R1j′correction to be made, the user must ensure that the R reactions on

the FM card are the same as the RXN reactions on the PERT card AND that the FM card

multiplicative constant C is negative, indicating multiplication by the atom density to get

macroscopic cross sections. For example, if R = –6 for ﬁssion on the FM card, you should not use

RXN=18 for ﬁssion on the PERT card. If C > 0, the cross sections are not macroscopic, it is

assumed that there is no tally dependence on a perturbed cross section, R1j′=0, and no correction

is made. The same correction is automatically made for the F6 tally and the KCODE keff

calculation, and for an F7 tally if the perturbation reaction is ﬁssion because these three tallies all

have implicit associated FM cards

Example 1: FMn CmR

1R2 : R1R3

Example 2: FMn CmR

1 (R2 : R3)

These two examples reiterate that parentheses cannot be used for algebraic hierarchy within a

reaction list. The ﬁrst example produces a single bin with the product of reaction R1with the sum

of reactions R2 and R3. The second case creates two bins, the ﬁrst of which is reaction R1 alone;

the second is the sum of R2 and R3, without reference to R1.

C1m1px1m2px2

–

eσ1px1

–σ2px2

–

Rij′0≠

April 10, 2000 3-91

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Example 3: F2:N 1234

FM2 (C1)(C2)(C3)(C4)T

Example 4: F12:N 1234

FM12 C1

Example 5: F22:N (1 2 3) 4 T

FM22 (C1) (C2)(C3)(C4)

These three examples illustrate the syntax when only the constant-multiplier feature is used. All

parentheses are required in these examples. Tally 2 creates 20 bins: the ﬂux across each of surfaces

1, 2, 3, and 4 with each multiplied by each constant C1,C2,C3,C4, and the sum of the four

constants. Tally 12 creates 4 bins: the ﬂux across each of surfaces 1, 2, 3, and 4 with each multiplied

by the constant C1. Tally 22 creates 12 bins: the ﬂux across surface 1 plus surface 2 plus surface 3,

the ﬂux across surface 4, and the ﬂux across all four surfaces with each multiplied by each constant

C1,C2,C3, and C4. An FQn card with an entry ofFMorMFwould print these bins of the tallies

in an easy-to-read table rather than strung out vertically down the output page.

Several more examples of the FMn card are in Chapter 4. The DEMO example in Chapter 5 also

illustrates the general form of the card.

Using MCNP tallies, there are two ways to obtain the energy deposited in a material in terms of

rads (1 rad = 100 ergs/g). When the actual material of interest is present in the MCNP model, the

simplest way is to use the heating tally with units MeV/g in conjunction with C=1.602E−08 on the

companion FMn card, where C=(1.602E−06 ergs/MeV)/(100 ergs/g). When the material is not

present in the model, rads can be obtained from type 1, 2, 4, and 5 tallies by using an FMn card

where C is equal to the factor above times , where No is Avogadro’s number and

ηand Aare the number of atoms/molecule and the atomic weight, respectively, of the material of

interest. This value of C equals as discussed on page 2–82. The implicit assumption when

the material is not present is that it does not affect the radiation transport signiﬁcantly. In the

reaction list on the FM card, you must enter −4 1 for neutron heating and

−5 −6 for photon heating. See page 2–82 and 4–38 for examples. For both F4 and F6, if a heating

number from the data library is negative, it is set to zero by the code.

8. DEn Dose Energy Card

DFn Dose Function Card

Form: DEn A E1... Ek

DFn B F1... Fk

n= tally number.

Ei= an energy (in MeV).

Fi= the corresponding value of the dose function.

Noη10 24–

×A⁄

ρaρg

⁄

3-92 April 10, 2000

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DATA CARDS

A = LOG or LIN interpolation method for energy table.

B = LOG or LIN interpolation method for dose function table.

Defaults: If A or B is missing, LOG is chosen for that table.

Use: Tally comment recommended.

This feature allows you to enter a pointwise response function (such as ﬂux-to-dose conversion

factors) as a function of energy to modify a regular tally. Both cards must have the same number

of numerical entries and they must be monotonically increasing in energy. Particle energies outside

the energy range deﬁned on these cards use either the highest or lowest value.

By default MCNP uses log-log interpolation between the points rather than a histogram function

as is done for the EMn card. The energy points speciﬁed on the DEn card do not have to equal the

tally energy bins speciﬁed with the En card for the Fn tally. Unlike EMn card use, there can be

many points on the DEn and DFn cards, but the response can be tallied in only a few energy bins

such as one unbounded energy bin.

If nis zero on these two cards, the function will be applied to all tallies that do not have DEn and

DFn cards speciﬁcally associated with them.

LIN or LOG can be chosen independently for either table. Thus any combination of interpolation

(log-log, linear-linear, linear-log, or log-linear) is possible. The default log-log interpolation is

appropriate for the ANSI/ANS ﬂux-to-dose rate conversion factors (they are listed in Appendix H);

kermas for air, water, and tissue; and energy absorption coefﬁcients.

Example: DE5 E1E2E3E4... Ek

DF5 LIN F1F2F3F4... Fk

This example will cause a point detector tally to be modiﬁed according to the dose function F(E)

using logarithmic interpolation on the energy table and linear interpolation on the dose function

table.

9. EMn Energy Multiplier Card

Form: EMn M1... Mk

n= tally number.

Mi= multiplier to be applied to the ith energy bin.

Default: None.

Use: Requires En card. Tally comment recommended.

April 10, 2000 3-93

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This card can be used with any tally (speciﬁed by n) to scale the usual current, ﬂux, etc. by a

response function. There should be one entry for each energy entry on the corresponding En card.

When a tally is being recorded within a certain energy bin, the regular contribution is multiplied

by the entry on the EMn card corresponding to that bin. For example, a dose rate can be tallied with

the appropriate response function entries. Tallies can also be changed to be per unit energy if the

entries are 1/∆ E for each bin. Note that this card modiﬁes the tally by an energy-dependent

function that has the form of a histogram and not a continuous function. It also requires the tally to

have as many energy bins as there are histograms on the EMn card. If either of these two effects is

not desired, see the DEn and DFn cards.

A set of energy multipliers can be speciﬁed on an EM0 (zero) card that will be used for all tallies

for which there is not a speciﬁc EMn card.

10. TMn Time Multiplier Card

Form: TMn M1... Mk

n= tally number.

Mi= multiplier to be applied to the ith time bin.

Default: None.

Use: Requires Tn card. Tally comment recommended.

This card is just like the EMn card except that the entries multiply time bins rather than energy bins.

The Tn and TMn cards must have the same number of entries. Note that this card modiﬁes the tally

by a time-dependent function that has the form of a histogram and not a continuous function.

A set of time multipliers can be speciﬁed on a TM0 (zero) card that will be used for all tallies for

which there is not a speciﬁc TMn card.

For example, if the entries are 1/∆ T, where ∆Tis the width of the corresponding time bin, the tally

will be changed to be per unit time with the units of 1/∆ T.

11. CMn Cosine Multiplier Card (tally type 1 only)

Form: CMn M1... Mk

n= tally number.

Mi= multiplier to be applied to the ith cosine bin.

Default: None.

Use: Tally type 1. Requires Cn card. Tally comment recommended.

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This card is just like the EMn and TMn cards except that the entries multiply cosine bins. The

number of entries on the CMn card must be the same as on the Cn card. Note that this card modiﬁes

the tally by an angular-dependent function that has the form of a histogram and not a continuous

function.

A set of cosine multipliers can be speciﬁed on a CM0 (zero) card that will be used for all type 1

tallies for which there is not a speciﬁc CMn card.

For example, if you want the directionally dependent F1 tally results to be per steradian, the ith

entry on the CM1 card is

where is 180ο.

12. CFn Cell-Flagging Card (tally types 1, 2, 4, 6, 7)

Form: CFn C1... Ck

n= tally number.

Ci= problem cell numbers whose tally contributions are to be ﬂagged.

Default: None.

Use: Not with detectors or pulse height tallies. Consider FQn card.

Particle tracks can be “ﬂagged” when they leave designated cells and the contributions of these

ﬂagged tracks to a tally are listed separately in addition to the normal total tally. This method can

determine the tally contribution from tracks that have passed through an area of interest.

Cell ﬂagging cannot be used for detector tallies. The same purpose can be accomplished with an

FTn card with the ICD option.

The cell ﬂag is turned on only upon leaving a cell. A source particle born in a ﬂagged cell does not

turn the ﬂag on until it leaves the cell.

In MODE N P the ﬂagged neutron tallies are those caused by neutrons leaving the ﬂagged cell, but

the ﬂagged photon tallies can be caused by either a photon leaving a ﬂagged cell or a neutron

leaving a ﬂagged cell and then leading to a photon which is tallied.

2πθ

iθi1–

cos–cos()

---------------------------------------------------

θo

April 10, 2000 3-95

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Example: F4:N 6 10 13

CF4 3 4

In this example the ﬂag is turned on when a neutron leaves cell 3 or 4. The print of Tally 4 is

doubled. The ﬁrst print is the total track length tally in cells 6, 10, and 13. The second print is the

tally in these cells for only those neutrons that have left cell 3 or 4 at some time before making their

contribution to the cell 6, 10, or 13 tally.

13. SFn Surface-Flagging Card (tally types 1, 2, 4, 6, 7)

Form: SFn S1... Sk

n= tally number.

Si= problem surface numbers whose tally contributions are to be ﬂagged.

Default: None.

Use: Not with detectors. Consider FQn card.

This feature is identical to cell ﬂagging except that particles turn the ﬂag on when they cross the

speciﬁed surfaces. Thus a second tally print is given for only those particles that have crossed one

or more of the surfaces speciﬁed on the SFn card.

Surface ﬂagging cannot be used for detector tallies but an FTn card with the ICD option will do the

same thing.

The situation for photon tallies in MODENPislikethat for the CFn card: a photon can be ﬂagged

either because it has crossed a ﬂagged surface or because it was created by a neutron that crossed

a ﬂagged surface.

Both a CFn and an SFn card can be used for the same tally. The tally is ﬂagged if the track leaves

one or more of the speciﬁed cells or crosses one or more of the surfaces. Only one ﬂagged output

for a tally is produced from the combined CFn and SFn card use.

14. FSn Tally Segment Card (tally types 1, 2, 4, 6, 7)

Form: FSn S1... Sk

n= tally number.

Si= signed problem number of a segmenting surface.

Default: No segmenting.

Use: Not with detectors. May require SDn card. Consider FQn card.

3-96 April 10, 2000

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This card allows you to subdivide a cell or a surface into segments for tallying purposes, the

advantage being that it is then not necessary to specify the problem geometry with extra cells just

for tallying. The segmenting surfaces speciﬁed on the FSn card are listed with the regular problem

surfaces, but they need not be part of the actual geometry and hence do not complicate the cell/

surface relationships.

If ksurfaces are entered on the FSn card, k+1surface or volume segments are created. Tally nis

subdivided into k + 1 segment bins according to the order and sense of the segmenting surfaces

listed on the FSn card. If the symbol Tis on the FSn card, there will be an additional total bin. The

symbol C at the end of the FS card causes the bin values to be cumulative. Segmenting is done

according to the following scheme:

Fn:N S(or C)

FSn S1... SkT (optional)

Tally n over surface S (or in cell C) will be subdivided into the following bins:

1. the portion with the same sense with respect to surface S1 as the sign given to S1,

2. the portion with the same sense with respect to surface S2 as the sign given to S2 but

excluding that already scored in a previously listed segment,

k the portion with the same sense with respect to surface Sk as the sign given to Sk but

excluding that already scored in a previously listed segment,

k+1 everything else,

k+2 entire surface or cell if Tis present on FSn card.

If the symbol T is absent from the FSn card, the (k+2)th bin is missing and MCNP calculates the

tally only for each segment (including the “everything else” segment). If multiple entries are on the

Fn card, each cell or surface in the tally is segmented according to the above rules. For tally types

1 or 2, the segmenting surfaces divide a problem surface into segments for the current or ﬂux tallies.

The segmenting surfaces divide a problem cell into segments for tally types 4, 6, or 7. For

normalized tallies, the segment areas (for type 2), volumes (for type 4), or masses (for types 6 and

7) may have to be provided. See the discussion under the SDn card.

Example 1: F2:N 1

FS2 −3 −4

This example subdivides surface 1 into three sections and calculates the neutron ﬂux across each

of them. There are three prints for the F2 tally: (1) the ﬂux across that part of surface 1 that has

negative sense with respect to surface 3, (2) the ﬂux across that part of surface 1 that has negative

sense with respect to surface 4 but that has not already been scored (and so must have positive sense

with respect to surface 3), (3) everything else (that is, the ﬂux across surface 1 with positive sense

with respect to both surfaces 3 and 4).

April 10, 2000 3-97

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It is possible to get a zero score in some tally segments if the segmenting surfaces and their senses

are not properly speciﬁed. In Example 1 above, if all tallies that are positive with respect to surface

3 are also all positive with respect to surface 4, the second segment bin will have no scores.

Example 2: F2:N 1

FS2 −3 4

The order and sense of the surfaces on the FS2 card are important. This example produces the same

numbers as does Example 1 but changes the order of the printed ﬂux. Bins two and three are

interchanged.

Example 3: F1:N 1 2 T

FS1 −3 T

This example produces three current tallies: (1) across surface 1, (2) across surface 2, and (3) the

sum across surfaces 1 and 2. Each tally will be subdivided into three parts: (1) that with a negative

sense with respect to surface 3, (2) that with a positive sense with respect to surface 3, and (3) a

total independent of surface 3.

Several additional examples of the FSn card are in Chapter 4.

15. SDn Segment Divisor Card (tally types 1, 2, 4, 6, 7)

Form: SDn (D11 82... D1m)(D21 D22 ... D2m)... (Dk1 Dk2 ... Dkm)

n= tally number. n cannot be zero.

k= number of cells or surfaces on Fn card, including T if present.

m= number of segmenting bins on the FSn card, including the

remainder segment, and the total segment if FSn has a T.

Dij = area, volume, or mass of jth segment of the ith surface or cell bin

for tally n.

The parentheses are optional.

Hierarchy for obtaining volume, area, or mass:

1. For cell or surface without segmenting (tally types 2, 4, 6, and 7):

a. nonzero entry on SDn card,

b. nonzero entry on VOL or AREA card,

c. volume, area or mass calculated by MCNP,

d. fatal error

2. For cell or surface with segmenting (tally types 2, 4, 6, and 7):

a. nonzero entry on SDn card,

b. volume, area or mass calculated by MCNP

c. fatal error

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3. For surface in a type 1 tally:

a. nonzero entry on SDn card,

b. no divisor.

Use: Not with detectors. May be required with FSn card. Can be used without FSn card.

For segmented cell volumes or surface areas deﬁned by the FSn card that are not automatically

calculated by MCNP, the user can provide volumes, areas, or masses on this segment divisor card

to be used by tally n. This card is similar to the VOL and AREA cards but is used for speciﬁc tallies,

whereas the other two are used for the entire problem geometry. For tally type 2 the entry is area,

for tally type 4 the entry is volume, and for tally types 6 and 7 the entries are masses. Tally type 1

(the current tally) is not normally divided by anything, but with the SD1 card the user can introduce

any desired divisor, for example, area to tally surface current density.

Example F4:N 1 2 3 T

SD4 1 1 1 1

Note that the SDn card can be used to deﬁne tally divisors even if the tally is not segmented. In this

example the tally calculates the ﬂux in the three cells plus the union of the three cells. The VOL

card can be used to set the volume divisor of the three cells (to unity, for example), but it cannot do

anything about the divisor for the union. Its divisor is the sum of the volumes (whether MCNP-

calculated or user-entered) of the three cells. But the divisors for all four of the cell bins can be set

to unity by means of the SDn card. These entries override entries on the VOL and AREA cards.

See page 3–82 for use with repeated structure tallies.

16. FUn TALLYX Input Card

Form: FUn X1X2... Xk

or: FUn blank

n= tally number.

Xi= input parameter establishing user bin i.

Default: If the FU card is absent, subroutine TALLYX is not called.

Use: Used with a user-supplied TALLYX subroutine or FTn card.

This card is used with a user-supplied tally modiﬁcation subroutine TALLYX and some cases of

the FTn card. If the FUn card has no input parameters, TALLYX will be called but no user bins will

be created. The kentries on the FUn card serve three purposes: (1) each entry establishes a separate

user tally bin for tally n, (2) each entry can be used as an input parameter for TALLYX to deﬁne

the user bin it establishes, and (3) the entries appear in the output as labels for the user bins.

IPTAL(LIPT+3,1,ITAL) is the pointer to the location in the TDS array of the word preceding the

location of the data entries from the FUn card. Thus if the FUn card has the form shown above,

April 10, 2000 3-99

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TDS(L+1) = X1

TDS(L+2) = X2

TDS(L+k) = Xk

where L = IPTAL(LIPT+3,1,ITAL)

k= IPTAL(LIPT+3,4,ITAL) − 1

= IPTAL(LIPT+3,3,ITAL) − 1

n= JPTAL(LJPT+1,ITAL)

ITAL = program number of the tally

MCNP automatically provides the total over all speciﬁed user bins. The total can be inhibited for

a tally by putting the symbol NT at the end of the FUn card for that tally as follows:

FUn X1X2... XkNT

and there is one change in the preceding list of variables:

k= IPTAL(LIPT+3,4,ITAL) − 1

= IPTAL(LIPT+3,3,ITAL)

The symbol C at the end of the FU card causes the bin values to be cumulative in which case

IPTAL(LIPT+3,3,ITAL) = IPTAL(LIPT+3,4,ITAL)

IPTAL(LIPT+3,6,ITAL) = 1.

The discussion of the IPTAL and JPTAL arrays in Appendix E and the following description of

TALLYX may be useful.

SUBROUTINE TALLYX User-supplied Subroutine

Use: Called for tally n only if an FUn card is in the INP ﬁle.

TALLYX is called whenever a tally with an associated FUn card but no FTn card is scored. The

locations of the calls to TALLYX are such that TALLYX is the very last thing to modify a score

before it is posted in the tally. TALLYX calls can be initiated by more than one FUn card for

different values of n; a branch must be constructed inside the subroutine based on which tally Fn

is calling TALLYX, where n= JPTAL(LJPT+1,ITAL). TALLYX has the following form:

SUBROUTINE TALLYX(T,IB)

∗CALL CM

User-supplied FORTRAN statements

3-100 April 10, 2000

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DATA CARDS

RETURN

END

The quantity T (ﬁrst argument of TALLYX) that is scored in a standard tally can be multiplied or

replaced by anything. The modiﬁed score T is then put into one of the k user bins established by

the FUn card. In TALLYX(T,IB) the second argument IB is deﬁned to allow for more than one pass

through TALLYX per tally score. By default, IB=0, which means make one pass through the

MCNP coding where user bin tally scores are posted. If the user sets IB<0 in TALLYX, no score

will be made. If the user sets IB>0, passes through the user bin loop including TALLYX will be

made until IB is reset to zero. This scheme allows for tally modiﬁcation and posting in more than

one user bin. The variable IBU is the variable designating the particular user bin established by the

FUn card. Its value is 1 before the ﬁrst pass through the user bin loop. The indices of the current

user, segment, cosine, energy, and time bins (IBU, IBS, IBC, IBE, and IBT, respectively) and the

ﬂag JBD that indicates ﬂagged- or direct-versus-not are in Common for optional modiﬁcation by

TALLYX. Note that the index of the multiplier bin is not available and cannot be modiﬁed. NTX

is a variable in blank Common. It is set equal to NX just before the CALL TALLYX in TALLYD

and TALLY. The variable NX is set to unity just before the start of the user bins loop and is

incremented after the CALL TALLYX, so NTX contains the number of the TALLYX call. An

example of using NTX to tally in every user bin before leaving the user bin loop follows:

SUBROUTINE TALLYX(T,IB)

∗CALL CM

T = whatever

IBU = NTX

IB = 1

IF(NTX.GE.IPTAL(LIPT+3,4,ITAL)-1) IB = 0

RETURN

END

If IBU is out of range, no score is made and a count of out-of-range scores is incremented. If

excessive loops through TALLYX are made, MCNP assumes IB has been incorrectly set and

terminates the job with a BAD TROUBLE error (excessive is greater than the product of the

numbers of bins of all kinds in the tally). Several examples of the FUn card and TALLYX are in

Chapter 4. The procedure for implementing a TALLYX subroutine is the same as for the user-

provided SOURCE subroutine.

17. TFn Tally Fluctuation Card

Form: TFn I1... I8

n= tally number. n cannot be zero.

Ii= bin number for bin type i.

last = IPTAL(LIPT+i,3,ITAL) 1Iilast≤≤

April 10, 2000 3-101

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DATA CARDS

= total number of bins in one of the eight bin types.

Default: 1 1 last last 1las last last

1. first cell, surface, or detector on Fn card

2. total rather than flagged or uncollided flux

3. last user bin

4. last segment bin

5. first multiplier bin on FMn card

6. last cosine bin

7. last energy bin

8. last time bin.

Use: Whenever one or more tally bins are more important than the default bin. Particularly

useful in conjunction with the weight window generator.

At the end of the output, one chart for each tally is printed to give an indication of tally ﬂuctuations;

that is, how well the tally has converged. The tally mean, relative error, variance of the variance,

Pareto slope (see page 2–118), and ﬁgure of merit (FOM = 1/(σ2t), where σ is the relative error

printed with the tally and tis computer time in minutes) are printed as functions of the number of

histories run. The FOM should be roughly constant. The TF card determines for which bin in tally

nthe ﬂuctuations are printed. It also determines which tally bin is optimized by the weight window

generator (WWE and WWG cards).

The TFn card allows you to change the default bin for a given tally and specify for which tally bin

the chart and all the statistical analysis output will be printed. The eight entries on the card

correspond (in order) to the list of bin indices for the eight dimensions of the tally bins array. The

order is ﬁxed and not affected by an FQn card.

The mean printed in a chart will correspond to some number in the regular tally print. If you have

more than one surface listed on an F2 card, for example, the chart will be for the ﬁrst surface only;

charts can be obtained for all surfaces by having a separate tally for each surface.

You may ﬁnd the Jfeature useful to jump over last entries. Remember that totals are calculated for

energy, time, and user bins (unless inhibited by using NT), so that last for eight energy bins is 9. If

one segmenting surface divides a cell or surface into two segments, last in that case is 2, unless T

is used on the FS card, in which case last is 3. If there are no user bins or cosine bins, for example,

last is 1 for each; last is never less than 1.

Example: Suppose an F2 tally has four surface entries, is segmented into two segments (the

segment plus everything else) by one segmenting surface, and has eight energy bins. By default

one chart will be produced for the ﬁrst surface listed, for the part outside the segment, and totaled

over energy. If we wish a chart for the ﬁfth energy bin of the third surface in the ﬁrst segment, we

would use TF2 3 2J 1 2J 5.

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18. DDn Detector Diagnostics Card

Form: DDn k1m1k2m2...

n= 1 for neutron DXTRAN spheres

= 2 for photon DXTRAN spheres

= tally number for speciﬁc detector tally

ki= criterion for playing Russian roulette for detector i

mi= criterion for printing large contributions

Defaults: If ki is not speciﬁed on a DDn card, ki on the DD card is used. If that is not

speciﬁed, k1on the DD card is used. If that is not speciﬁed, ki = 0.1 is used.

A similar sequence of defaults deﬁnes mi, with a ﬁnal default of mi = 1000.

Use: Optional. Remember that Russian roulette will be played for detectors and

DXTRAN unless speciﬁcally turned off by use of the DD card. Consider

also using the PDn or DXC cards.

This card (1) using a Russian roulette game, can speed up calculations signiﬁcantly by limiting

small contributions that are less than some fraction kof the average contribution per history to

detectors or DXTRAN spheres, and (2) can provide more information about the origin of large

contributions or the lack of a sufﬁcient number of collisions close to the detector or DXTRAN

sphere. The information provided about large contributions can be useful for setting cell

importances or source-biasing parameters.

For a given detector or DXTRAN sphere, the Russian roulette criterion works as follows:

1. If ki is positive, all contributions to the detector or sphere are made for the first 200

histories. Then the average contribution per history is computed (and will be updated

from time to time throughout the problem). Thereafter, any contribution to the detector

or sphere larger than ki times this average contribution will always be made, but any

contribution smaller than ki times the average will be subject to the Russian roulette

game. (ki is not allowed to be greater than 1.)

2. If ki is negative, contributions larger than |ki|will always be made, and contributions

smaller than |ki|will be subject to Russian roulette. This rule applies to all histories from

the beginning of the problem, and the 200th history has no significance.

3. If ki is zero, no Russian roulette game will be played for the detector or sphere.

Probably, k= 0.5 is suitable for most problems; the nonzero default value 0.1 means that the game

is always played unless explicitly turned off by the user.

The second entry, mi, determines the condition for printing diagnostics for large contributions. If

the entry is zero, there is no diagnostic print. If the entry is positive, two possibilities exist.

April 10, 2000 3-103

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1. If the corresponding kiis positive or zero, no diagnostic prints will be made for the first

200 histories. Thereafter, the first 100 contributions larger than mitimes the average tally

per history will be printed.

2. If the corresponding kiis negative, the first 100 contributions larger than mitimes |ki|will

be printed.

Remember that when kiis positive the Russian roulette game is played on the basis of the estimated

average contribution per history. Because the estimate improves from time to time, the game is

based on different values for different histories. This can make debugging a problem more

complicated, and the variance estimate does not quite obey the Central Limit Theorem. A

procedure worth considering is to determine the average contribution per history in a preliminary

run and then to use some fraction of the negative of this value in subsequent longer runs. The

Russian roulette game is played without regard to particle time or energy; thus time and energy bins

for which the ultimate tally is small may lose a disproportionate share of scores by the roulette

game.

The DD card eliminates tracks with DXTRAN but only contributions with detectors.

Example: DXT:N x1y1z1RI1RO1

x2y2z2RI2RO2

x3y3z3RI3RO3

DXT:P x4y4z4RI4RO4

F15X:P a1r1R1

a2r2R2

DD .2 100 .15 2000

DD1 −1.1E25 3000 J J J 3000

DD15 .4 10

Detector/sphere k m

sphere 1 −1.1E25 3000

sphere 2 .15 2000

sphere 3 .2 3000

sphere 4 .2 100

detector 1 .4 10

detector 2 .15 2000

Another example of the DD card and a description of its output is in Chapter 5. For a more detailed

discussion of the Russian roulette game, see page 2–95 in Chapter 2.

19. DXT DXTRAN Card

Form: DXT:n x1y1z1RI1RO1x2y2z2RI2RO2... DWC1DWC2DPWT

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DATA CARDS

n=N for neutrons, P for photons, not available for electrons.

xiyizi= coordinates of the point at the center of the ith pair of spheres

RIi= radius of the ith inner sphere in cm

ROi= radius of the ith outer sphere in cm

DWC1= upper weight cutoff in the spheres

DWC2= lower weight cutoff in the spheres

DPWT = minimum photon weight. Entered on DXT:N card only.

Defaults: Zero for DWC1,DWC2, and DPWT.

Use: Optional. Consider using the DXC:N, DXC:P, or DD cards when using

DXTRAN.

DXTRAN is used to improve the particle sample in the vicinity of a tally (see page 2–152). It

should not be misconstrued as a tally itself, such as a detector; it is used in conjunction with tallies

as a variance reduction technique. DXTRAN spheres must not overlap. The inner sphere should

normally cover the tally region if possible. Specifying a tally cell or surface partly inside and partly

outside a DXTRAN sphere usually will make the mean of the tally erratic and the variance huge.

The technique is most effective when the geometry inside the spheres is very simple and can be

costly if the inside geometry is complicated, involving several surfaces. The inner sphere is

intended to surround the region of interest. The outer sphere should surround neighboring regions

that may scatter into the region of interest. In MCNP, the relative importance of the two regions is

ﬁve. That is, the probability density for scattering toward the inner sphere region is ﬁve times as

high as the probability density for scattering between the inner and outer spheres. The weight factor

is 1/5 for particles scattered toward the inner sphere.

Rule of Thumb for RI and RO: The inner radius RI should be at least as large as the tally region,

and RO–RI should be about one mean free path for particles of average energy at the spheres.

DXTRAN can be used around detectors, but the combination may be very sensitive to reliable

sampling.

There can be up to ﬁve sets of X Y Z RI RO on each DXT card. There is only one set of DWC1and

DWC2entries for each particle type. This pair is entered after conclusion of the other data and (with

DXT:N) before the one value of DPWT. The weight cutoffs apply to DXTRAN particle tracks

inside the outer radii and have default values of zero. The DXTRAN photon weight cutoffs have

no effect unless the simple physics is used, with one exception: upon leaving the sphere, track

weights (regardless of what physics is used) are checked against the cutoffs of the CUT:P card. The

DXTRAN weight cutoffs DWC and DWC2 are ignored when mesh-based weight windows are

used.

The minimum photon weight limit DPWT on the DXT:N card parallels almost exactly the

minimum photon weight entries on the PWT card. One slight difference is that in Russian roulette

April 10, 2000 3-105

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DATA CARDS

during photon production inside DXTRAN spheres, the factor for relating current cell importance

to source cell importance is not applied. Thus, the user must have some knowledge of the weight

distribution of the DXTRAN particles (from a short run with the DD card, for example) inside the

DXTRAN sphere, so the lower weight limit for photon production may be intelligently speciﬁed.

As in the case of the PWT entries, a negative entry will make the minimum photon weight relative

to the source particle starting weight. The default value is zero, which means photon production

will occur at each neutron DXTRAN particle collision in a material with nonzero photon

production cross section inside the DXTRAN sphere.

DXTRAN can be used in a problem with the S(α,β) thermal treatment, but contributions to the

DXTRAN spheres are approximate. DXTRAN should not be used with reﬂecting surfaces, white

boundaries, or periodic boundaries (see page 2–92). DXTRAN is incompatible with a

monodirectional source because direct contributions from the source are ignored.

If more than one set of DXTRAN spheres is used in the same problem, they can “talk” to each other

in the sense that collisions of DXTRAN particles in one set of spheres cause contributions to

another set of spheres. The contributions to the second set have, in general, extremely low weights

but can be numerous with an associated large increase in computer time. In this case the DXTRAN

weight cutoffs probably will be required to kill the very-low-weight particles. The DD card can

give you an indication of the weight distribution of DXTRAN particles.

20. FTn Special Treatments for Tallies

Form: FTn ID1P1,1 P1,2 P1,3 ... ID2P2,1 P2,2 P2,3 ...

n= tally number.

IDi= the alphabetic keyword identiﬁer for a special treatment.

FRV ﬁxed arbitrary reference direction for tally 1 cosine binning.

GEB Gaussian energy broadening.

TMC time convolution.

INC identify the number of collisions.

ICD identify the cell from which each detector score is made.

SCX identify the sampled index of a speciﬁed source distribution.

SCD identify which of the speciﬁed source distributions was used.

PTT put different multigroup particle types in different user bins.

ELC electron current tally.

Pi,j = parameters for that special treatment, either a number, a parenthesis

or a colon.

Default: If the FT card is absent, there is no special treatment for tally n.

Use: Optional; as needed.

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The syntax and meaning of the Pi,j is different for each IDi. A special treatment may cause a set of

user bins or possibly a set of some other kind of bins to be created. The information in the Pi,j

allows the number and kind of those bins to be inferred easily. More than one special treatment can

be speciﬁed by a given tally except for combinations of INC, ICD, SCX,SCD, PTT and ELC. Only

one of these special treatments can be used by a tally at one time because all require user bins,

making them mutually exclusive.

A description of the special treatments available follows with an explanation of the allowed

parameters for each.

FRV V1V2V3

The Vi are the xyz components of vector V, not necessarily normalized. If the FRV special

treatment is in effect for a type 1 tally, the direction V is used in place of the vector normal to the

surface as the reference direction for getting the cosine for binning.

GEB a b c

The parameters specify the full width at half maximum of the observed energy broadening in a

physical radiation detector: fwhm , where E is the energy of the particle. The

units of a, b, and c are MeV, MeV1/2, and none, respectively. The energy actually scored is sampled

from the Gaussian with that fwhm. See Chapter 2.

TMC a b

All particles should be started at time zero. The tally scores are made as if the source was actually

a square pulse starting at time a and ending at time b.

INC

No parameters follow the keyword but an FUn card is required. Its bin boundaries are the number

of collisions that have occurred in the track since the creation of the current type of particle,

whether at the source or at a collision where some other type of particle created it. If the INC

special treatment is in effect, the call to TALLYX that the presence of the FUn card would normally

trigger does not occur. Instead IBU is set by calling JBIN with the number of collisions as the

argument.

ICD

No parameters follow the keyword but an FUn card is required. Its bins are the names of some or

all of the cells in the problem. If the cell from which a detector score is about to be made is not in

the list on the FUn card, the score is not made. TALLYX is not called. The selection of the user bin

is done in TALLYD.

SCX k

abEcE

++=

April 10, 2000 3-107

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The parameter k is the name of one of the source distributions and is the k that appears on the SIk

card. One user bin is created for each bin of source distribution k plus a total bin. The scores for

tally n are then binned according to which bin of source distribution k the source particle came

from. The score of the total bin is the score you would see for tally n without the special treatment,

if source distribution k is not a dependent distribution.

CAUTION: For a dependent distribution, the score in the total bin is the subtotal portion of the

score from dependent distribution k.

SCD

No parameters follow the keyword but an FUn card is required. Its bins are a list of source

distribution numbers from SIk cards. The scores for tally n are then binned according to which

distribution listed on the FUn card was sampled. This feature might be used to identify which of

several source nuclides emitted the source particle. In this case, the source distributions listed on

the FUn card would presumably be energy distributions. Each energy distribution is the correct

energy distribution for some nuclide known to the user and the probability of that distribution being

sampled from is proportional to the activity of that nuclide in the source. The user might want to

include an FCn card that tells to what nuclide each energy distribution number corresponds.

CAUTION: If more than one of the source distributions listed on the FU card is used for a given

history, only the ﬁrst one used will score.

PTT

No parameters follow the keyword but an FUn card is required. Its bins are a list of atomic weights

in units of MeV of particles masquerading as neutrons in a multigroup data library. The scores for

tally n are then binned according to the particle type as differentiated from the masses in the

multigroup data library. For example, .511 0 would be for electrons and photons masquerading

as neutrons.

ELC c

The single parameter c of ELC speciﬁes how the charge on an electron is to affect the scoring of

an F1 tally. Normally, an electron F1 tally gives particle current without regard for the charges of

the particles. There are 3 possible values for c:

c=1 to cause negative electrons to make negative scores

c=2 to put positrons and negative electrons into separate user bins

c=3 for the effect of both c=1 and c=2

If c=2 or 3, three user bins, positrons, electrons and total are created.

F. Material Speciﬁcation Cards

The cards in this section specify the isotopic composition of the materials in the cells and which

cross-section evaluations are to be used.

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1. Mm Material Card

Form: Mm ZAID1fraction1ZAID2fraction2... keyword=value ...

m corresponds to the material number on the cell cards

ZAIDi= either a full ZZZAAA.nnX or partial ZZZAAA element or

nuclide identiﬁer for constituent i, where ZZZ is the atomic

number, AAA is the atomic mass, nn is the library identiﬁer,

and X is the class of data

fractioni= atomic fraction (or weight fraction if entered as a negative

number) of constituent i in the material.

keyword = value, where = sign is optional. Keywords are:

GAS = mﬂag for density–effect correction to electron stopping power.

m = 0 calculation appropriate for material in the condensed

(solid or liquid) state used.

m = 1 calculation appropriate for material in the gaseous state

used.

ESTEP = ncauses the number of electron substeps per energy step to

be increased to n for the material. If n is smaller than the

built–in default found for this material, the entry is ignored.

Both the default value and the ESTEP value actually used

are printed in Table 85.

NLIB = id changes the default neutron table identiﬁer to the string id.

The neutron default is a blank string, which selects the

ﬁrst matching entry in XSDIR.

PLIB = id changes the default photon table identiﬁer to id.

ELIB = id changes the default electron table identiﬁer to id.

COND = id sets conduction state of a material only for el03 evaluation.

<0 nonconductor

=0 (default) nonconductor if at least one nonconducting

component; otherwise a conductor

>0 conductor if at least one conducting component.

Mnemonic Card Type Page

Mm Material 3–108

DRXS Discrete reaction 3–109

TOTNU Total ﬁssion 3–110

NONU Fission turnoff 3–111

AWTAB Atomic weight 3–112

XSn Cross-section ﬁles 3–112

VOID Negates materials 3–112

PIKMT Photon–production bias 3–113

MGOPT Multigroup card 3–114

April 10, 2000 3-109

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Default: None for ZAID fraction; GAS=0; ESTEP internally set; NLIB, PLIB, and

ELIB=ﬁrst match in XSDIR; COND=0.

Use: Optional, but required if you want materials in cells.

Neutrons. For naturally occurring elements, AAA = 000. Thus, ZAID = 74182.55 represents the

isotope 182W and ZAID = 74000.55 represents the element tungsten. Natural elements not

available from among those listed in Appendix G must be constructed on an Mm card by adding

together the individual isotopes if they are available. If the density for cells with AAA = 000 is

input in g/cm3, MCNP will assume the atomic weight for the natural element. The ZZZ and AAA

quantities are determined for neutrons by looking at the list of cross sections in Appendix G and

ﬁnding the appropriate ZAID associated with an evaluation that you want.

Photons and electrons. If neutrons are not being run, the AAA can be set to 000. Cross sections are

speciﬁed exactly like the neutron cross sections, but ZZZAAA.nnX equals ZZZ000. There is no

distinction between isotope and element for photons and electrons. However, if the isotopic

distribution for the element differs from the natural element, the atom density should be entered on

the cell cards to ensure the correct atom density for these cells.

Nuclide Fraction. The nuclide fractions can be normalized to 1.0 or left unnormalized. For

instance, if the material is H2O the atom fractions for H and O can be entered as 0.667 and 0.333

or as 2 and 1, respectively. If the fractions are entered with negative signs they are assumed to be

weight fractions. Weight fractions and atom fractions cannot be mixed on the same Mm card.

There is no limit to the number of “nuclide fraction” entries or the total number of different cross-

section tables allowed.

Default Library Hierarchy. When NLIB=id is included on an Mm card, the default neutron table

identiﬁer for that material is changed to id. Fully specifying a ZAID on that Mm card,

ZZZAAA.nnX, overrides the NLIB=id default.

Example: M1 NLIB=50D 1001 2 8016.50C 1 6012 1

This material consists of three isotopes. Hydrogen (1001) and carbon (6012) are not fully speciﬁed

and will use the default neutron table that has been deﬁned by the NLIB entry to be 50D, the

discrete reaction library. Oxygen (8016.50C) is fully speciﬁed and will use the continuous energy

library. The same default override hierarchy applies to photon and electron speciﬁcations.

2. DRXS Discrete Reaction Cross-Section Card

Form: DRXS ZAID1ZAID2... ZAIDi...

or blank

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ZAIDi= Identifying number of the form ZZAAA.nn, where ZZ is the

atomic number, AAA the mass number, and nn the neutron library

identiﬁer.

Default: Continuous-energy cross-section treatment if DRXS is absent.

Use: Optional. Applies only to neutron cross sections.

Nuclides listed on the optional DRXS card are given a discrete energy treatment instead of the

regular fully continuous-energy cross-section treatment if the necessary discrete data are available.

Check the list in Appendix G for availability. If the DRXS card is present but has no entries after

the mnemonic, discrete cross sections will be used for every nuclide, if available.

All discrete reaction libraries are based on a 262 energy group structure. Groups below 1 eV make

the discrete treatment appropriate for thermal neutron problems near room temperature. All

discrete reaction libraries have photon production data given in expanded format.

It is not recommended that this card be used unless you are transporting neutrons in an energy

region where resonances and hence self-shielding are of little importance. However, if the problem

under consideration meets this criterion, using the DRXS card can reduce computer storage

requirements and enhance timesharing.

Use of these discrete cross sections will not result in the calculation being what is commonly

referred to as a multigroup Monte Carlo calculation because the only change is that the cross

sections are represented in a histogram form rather than a continuous-energy form. The angular

treatment used for scattering, energy sampling after scattering, etc., is performed using identical

procedures and data as in the continuous-energy treatment. The user wanting to make a truly

multigroup Monte Carlo calculation should use the MGOPT card multigroup capability.

3. TOTNU Total Fission Card

Form: TOTNU NO

or blank

Default: If the TOTNU card is absent, prompt is used for non-KCODE calculations

and total is used for KCODE calculations.

Use: All steady-state problems should use this card.

In a non-KCODE problem, prompt is used for all ﬁssionable nuclides for which prompt values

are available if the TOTNU card is absent. If a TOTNU card is present but has no entry after it, total

sampling both prompt and delayed will be used for those ﬁssionable nuclides for which

υ υ

υ,υ,

April 10, 2000 3-111

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DATA CARDS

prompt and delayed values are available. A TOTNU card with NO as the entry is the same as if the

card were absent, that is, prompt is used.

In a KCODE calculation, total including both prompt and delayed as available, is used for

all ﬁssionable nuclides if the TOTNU card is absent. If a TOTNU card is present but has no entry

after it, total using both prompt and delayed is again used. A TOTNU card with NO as the

entry causes prompt to be used for all ﬁssionable nuclides for which prompt values are available.

The nuclide list of Appendix G indicates data available for each ﬁssionable nuclide. The MCNP

neutron cross-section summary print from XACT will show whether prompt or total was used.

4. NONU Fission Turnoff Card

Form: NONU a1a2... ai... amxa

or blank

ai= 0 ﬁssion in cell i treated as capture; gammas produced

= 1 ﬁssion in cell i treated as real; gammas produced

= 2 ﬁssion in cell i treated as capture; gammas not produced

mxa = number of cells in the problem

Default: If the NONU card is absent, ﬁssion is treated as real ﬁssion.

Use: Optional, as needed.

This card turns off ﬁssion in a cell. The ﬁssion is then treated as simple capture and is accounted

for on the loss side of the problem summary as the “Loss to ﬁssion” entry. If the NONU card is not

used, all cells are given their regular treatment of real ﬁssion, that is, the same as if all entries were

one. If the NONU card is present but blank, all ai’s are assumed to be zero and ﬁssion in all cells

is treated like capture. The NONU card cannot be added to a continue-run.

A value of 2 treats ﬁssion as capture and, in addition, no ﬁssion gamma rays are produced. This

option should be used with KCODE ﬁssion source problems written to surface source ﬁles.

Suppressing the creation of new ﬁssion neutrons and photons is important because they are already

accounted for in the source.

Sometimes it is desirable to run a problem with a ﬁxed source in a multiplying medium. For

example, an operating reactor power distribution could be speciﬁed as a function of position in the

core either by an SDEF source description or by writing the ﬁssion source from a KCODE

calculation to a WSSA ﬁle with a CEL option on an SSW card. The non-KCODE calculation

would be impossible to run because of the criticality of the system and because ﬁssion neutrons

have already been accounted for. Using the NONU card in the non-KCODE mode allows this

problem to run correctly by treating ﬁssion as simple capture.

υ,υ

υ,υ,

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5. AWTAB Atomic Weight Card

Form: AWTAB ZAID1AW1ZAID2AW2...

ZAIDi=ZAID used on the Mm material card excluding the X for

class of data speciﬁcation.

AWi= atomic weight ratios.

Default: If the AWTAB card is absent, the atomic weight ratios from the cross–section

directory ﬁle XSDIR and cross–section tables are used.

Use: Optional, as needed.

Entries on this card override the existing atomic weight ratios as contained in both the cross–

section directory ﬁle XSDIR and the cross–section tables. The AWTAB card is needed when

atomic weights are not available in an XSDIR ﬁle. Also, for ﬁssion products, ZAID=50120.35, the

atomic weight of tin ( ) will be used, so the following AWTAB card is needed:

AWTAB 50120.35 116.490609

WARNING: Using atomic weight ratios different from the ones in the cross–section tables in a

neutron problem can lead to negative neutron energies that will cause the problem to terminate

prematurely.

6. XSn Cross-Section File Card

n= 1 to 999

Use: Optional, as an alternative to the directory part of the XSDIR ﬁle.

The XSn card can be used to load cross–section evaluations not listed in the XSDIR ﬁle directory.

You can use XSn cards in addition to the XSDIR ﬁle. Each XSn card describes one cross section

table. The entries for the XSn card are identical to those in XSDIR except that

the +is not used for continuation. A detailed description of the required entries is provided in

Appendix F.

7. VOID Material Void Card

Form: VOID no entries

or: VOID C1C2... Ci

Ci= cell number

Default: None.

Use: Debugging geometry and calculating volumes.

120Sn

April 10, 2000 3-113

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DATA CARDS

The ﬁrst form is used when calculating volumes stochastically (see page 2–183) and when

checking for geometry errors (see page 3–8). When the VOID card is blank, the material number

and density is set to zero for all cells, FM cards are turned off, heating tallies are turned into ﬂux

tallies, and, if there is no NPS card, the effect of an NPS 100000 card is created. If there is a

TALLYX subroutine, it may need to be changed, too.

The second form is used to selectively void cells instead of setting the material number and density

to zero by hand on cell cards. It is a convenience if you want to check whether the presence of some

object in your geometry makes any signiﬁcant difference in the answers.

8. PIKMT Photon–Production Bias Card

Form: PIKMT Z1IPIK1MT1,1 PMT1,1 ...

ZnIPIKnMTn,1 PMTn,1 ...

Zi= the ZAID of the ith entry. Full or partial ZAIDs can be speciﬁed;

that is, 29000 is equivalent to 29000.50.

IPIKi= the parameter that controls the biasing for ZAIDi.

0 = no biasing for ZAIDi; photons from ZAIDi are produced with the

normal sampling technique.

−1 = no photons are produced from ZAIDi.

> 0 = there is biasing for ZAIDi. The value of IPIKiis the number

of partial photon–production reactions to be sampled.

MTi,j and PMTi,j are only required for ZAIDs with IPIKi>0, where IPIKi

pairs of entries of MTs and PMTs are necessary. The MTs are the

identiﬁers for the partial photon–production reactions to be sampled.

The PMTs control, to a certain extent, the frequency with which the

speciﬁed MTs are sampled. The entries need not be normalized. For

a ZAID with a positive value of IPIK, any reaction that is not

identiﬁed with its MT on the PIKMT card will not be sampled.

Default: If the PIKMT card is absent, there is no biasing of neutron–induced photons.

If PIKMT is present, any ZAID not listed has a default value of IPIKi=−1.

Use: Optional; see caveats below.

For several classes of coupled neutron–photon calculations, the desired result is the intensity of a

small subset of the entire photon energy spectrum. Two examples are discrete–energy (line)

photons and the high–energy tail of a continuum spectrum. In such cases, it may be proﬁtable to

bias the spectrum of neutron–induced photons to produce only those that are of interest.

1. WARNING: Use of the PIKMT card can cause nonzero probability events to be

completely excluded and the biasing game may be not necessarily a fair one. While

MT1IPIK1

,PMT1IPIK1

MTnIPIK

,PMTn IPIKn

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DATA CARDS

neutron tallies will be unaffected (within statistics), the only reliable photon tallies will

be those with energy bins immediately around the energies of the discrete photons

produced.

2. Users need information about the MT identifiers of the reactions that produce discrete--

energy photons. This information is available on the web.

3. The feature is also useful for biasing the neutron–induced photon spectrum to produce

very high energy photons (for example, ). Without biasing, these high–

energy photons are produced very infrequently; therefore, it is difficult to extract reliable

statistical information about them. An energy cutoff can be used to terminate a track

when it falls below the energy range of interest. Los Alamos users interested in using the

PIKMT card for this application should see X–5 regarding an internal code (NIPE) that

is useful for optimizing such problems.

Example: PIKMT 26000.55 1 102001 1 7014 0

29000 2 3001 2 3002 1

8016 −1

This example results in normal sampling of all photon–production reactions for 14N. All photons

from neutron collisions with Fe are from the reaction with MT identiﬁer 102001. Two photon–

production reactions with Cu are allowed. Because of the PMT parameters the reaction with MT

identiﬁer 3001 is sampled twice as frequently relative to the reaction with MT identiﬁer 3002 than

otherwise would be the case. No photons are produced from 16O or from any other isotopes in the

problem that are not listed on the PIKMT card.

9. MGOPT Multigroup Adjoint Transport Option

Form: MGOPT MCAL IGM IPLT ISB ICW FNW RIM

MCAL = F for forward problem

A for adjoint problem

IGM = the total number of energy groups for all kinds of particles in the

problem. A negative total indicates a special electron–photon

problem.

IPLT = indicator of how weight windows are to be used.

= 0 means that IMP values set cell importances. Weight windows, if

any, are ignored for cell importance splitting and Russian roulette.

= 1 means that weight windows must be provided and are

transformed into energy–dependent cell importances. A zero

weight–window lower bound produces an importance equal to

the lowest nonzero importance for that energy group.

= 2 means that weight–windows do what they normally do.

ISB = Controls adjoint biasing for adjoint problems only (MCAL=A).

Eγ10 MeV≥

April 10, 2000 3-115

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DATA CARDS

= 0 means collisions are biased by inﬁnite–medium ﬂuxes.

= 1 means collisions are biased by functions derived from

weight–windows, which must be supplied.

= 2 means collisions are not biased.

ICW = name of the reference cell for generated weight windows.

= 0 means weight windows are not generated.

0 requires volumes be supplied or calculated for all cells of

nonzero importance.

FNW = normalization value for generated weight windows. The value of

the weight–window lower bound in the most important energy

group in cell ICW is set to FNW.

RIM = compression limit for generated weight windows. Before

generated weight windows are printed out, the weight

windows in each group separately are checked to see that

the ratio of the highest to the lowest is less than RIM.

If not, they are compressed.

Default: IPLT=0, ISB=0, ICW=0, FNW=1, RIM=1000. MCAL and IGM must be

speciﬁed.

Use: Required for multigroup calculation.

MCAL and IGM are required parameters. The others are optional. “J” is not an acceptable value

for any of the parameters.

At this time, the standard MCNP multigroup neutron cross sections are given in 30 groups and

photons are given in 12 groups. Thus, an existing continuous–energy input ﬁle can be converted to

a multigroup input ﬁle simply by adding one of the following cards:

MGOPT F 30 $MODE N

MGOPT F 42 $MODE N P

MGOPT F 12 $MODE P

A negative IGM value allows a single cross–section table to include data for more than one sort of

particle. This feature applies currently to electron/photon multigroup calculations only. A problem

with 50 electron groups followed by 30 photon groups in one table would have

IGM=−80. Also all tables must have the same group structure. A negative IGM value will use the

energy variable on the source or tally card as a group index unless it is associated with a

distribution. For an energy distribution on the source card, there should be IGM increasing integer

entries for each group on the SI card. On a tally energy card, if there are less than IGM entries, they

will be taken as energies in MeV; otherwise, the bins will be according to group index. The

particles can be separated in tallies by using the PTT option on the FTn tally card.

≠

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An input ﬁle for an adjoint problem can have both an IMP card and weight window cards

(IPLT=0 ISB=1). The entries on the weight window cards are not weight windows in the normal

sense but biasing functions. If IPLT=1 the values on a weight window card become energy–

dependent cell importances. Until now, importances have been energy independent.

See Appendix G for a more complete discussion of multigroup libraries.

G. Energy and Thermal Treatment Speciﬁcation

The following cards control energy and other physics aspects of MCNP. All energies are in units

of MeV and all times are in shakes

1. PHYS Energy Physics Cutoff Card

a) Neutrons

Form: PHYS:N EMAX EMCNF IUNR DNB

EMAX = upper limit for neutron energy, MeV.

EMCNF = energy boundary above which neutrons are treated with implicit

capture and below which they are treated with analog capture.

IUNR = 0/1 = on/off unresolved resonance range probability tables.

DNB = number of delayed neutrons produced from ﬁssion

–1/0/>0 = natural sampling/no delayed neutrons produced/DNB

delayed neutrons per ﬁssion.

DNB > 0 not allowed in KCODE calculation.

Default: EMAX = very large; EMCNF = 0.0 MeV; IUNR = 0; DNB = –1

Use: Optional.

EMAX is the upper limit for neutron energy. All neutron cross-section data above EMAX are

expunged. If EMAX is not speciﬁed, there is no upper energy expunging of cross-section data to

save computer storage space. The physics of MCNP is such that if a neutron energy is greater than

the maximum energy in a table (typically 20 MeV), the cross section for the maximum energy is

Mnemonic Card Type Page

PHYS Energy physics cutoff 3–103

TMP Free-gas thermal temperature 3–108

THTME Thermal times 3–108

MTm S(α,β) material 3–109

April 10, 2000 3-117

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DATA CARDS

used with no extrapolation. If a particle is born above EMAX, either by source or collision, it is

rejected and the particle energy is resampled.

EMCNF controls the type of capture. Any neutron with energy greater than EMCNF will receive

the implicit capture treatment; below EMCNF, it will receive analog capture. This parameter is

analogous to EMCPF on the PHYS:P card and is useful in eliminating low-energy histories when

using a thermal treatment. Substantial computer time may be saved in a region of low absorption

(especially if the region is heterogeneous and bounded by a reﬂecting surface) simply by reducing

the number of tracks. EMCNF should be set to operate when a neutron enters a thermal regime,

typically a few kT. However, analog capture may undesirably kill important particles before they

are tallied or before they participate in physics important to the problem.

If EMCNF = EMAX, analog capture is used regardless of the value of WC1 on the CUT card. If

WC1 = 0, analog capture is used regardless of the value of EMCNF.

IUNR controls the treatment of cross sections in the unresolved energy range. The probability table

treatment (IUNR=0) should be left on for better physics but can be turned off (IUNR=1) to measure

the effect of the probability table treatment or to speed calculations when unresolved resonances

are unimportant.

DNB controls the number of delayed neutrons produced from ﬁssion and can be used only when

TOTNU is speciﬁed for ﬁssionable nuclides for which delayed and prompt νvalues are available.

If DNB is not speciﬁed, the number of delayed neutrons produced per ﬁssion is determined from

the ratio of delayed ν to total ν. The nuclide list of Appendix G indicates data available for each

ﬁssionable nuclide.

b) Photons

Form: PHYS:P EMCPF IDES NOCOH

EMCPF = upper energy limit for detailed photon physics treatment, MeV.

IDES = 0 photons will produce electrons in MODE E problems or

bremsstrahlung photons with the thick target bremsstrahlung

model.

= 1 photons will not produce electrons as above.

NOCOH = 0 coherent scattering occurs.

= 1 coherent scattering will not occur.

Default: EMCPF = 100 MeV; IDES = 0; NOCOH = 0.

Use: Optional.

Photons with energy greater than EMCPF will be tracked using the simple physics treatment. If

WC1 = 0 on the CUT:P card, analog capture is used in the energy region above EMCPF. Otherwise

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capture is simulated by weight reduction with Russian roulette on weight cutoff. Photons with

energy less than EMCPF will be treated with the more detailed physics that always includes analog

capture. For a detailed discussion of the simple and detailed photon physics treatments, see Chapter

The simple physics treatment, intended primarily for higher energy photons, considers the

following physical processes: photoelectric effect without ﬂuorescence, Compton scattering from

free electrons without the use of form factors, and pair production. The highly forward peaked

coherent Thomson scattering is ignored.

In the detailed physics treatment, photoelectric absorption can result in ﬂuorescent emission, the

Thomson and Klein-Nishina differential cross sections are modiﬁed by appropriate form factors

taking electron binding effects into account, and coherent scattering is included.

To turn off the production of secondary electrons generated by photons, the switch IDES can be

set, either on the PHYS:P or on the PHYS:E card. If either of these cards sets IDES = 1, photons

will NOT produce electrons, even if IDES = 0 is set on the other. In a photon-only problem, turning

off secondary electrons causes the thick-target bremsstrahlung model to be bypassed. This option

should be exercised only with great care because it alters the physics of the electron-photon cascade

and will give erroneously low photon results when bremsstrahlung and electron transport are

signiﬁcant.

NOCOH is a switch to allow coherent scattering to be turned off for photons with energies below

EMCPF. Thus, coherent scattering can be suppressed within the detailed physics treatment without

losing the other advantages of the detailed model. When NOCOH = 1, the cross section for

coherent scattering will be set to zero. This approximation can be useful in problems with bad

point detector variances.

c) Electrons

Form: PHYS:E EMAX IDES IPHOT IBAD ISTRG BNUM XNUM RNOK ENUM

NUMB

EMAX = upper limit for electron energy in MeV.

IDES = 0/1 = photons will/will not produce electrons.

IPHOT = 0/1 = electrons will/will not produce photons.

IBAD = 0 full bremsstrahlung tabular angular distribution.

= 1 simple bremsstrahlung angular distribution approximation.

ISTRG = 0 sampled straggling for electron energy loss.

= 1 expected-value straggling for electron energy loss.

BNUM <0 only applicable for el03 evaluation. See below for details.

= 0 bremsstrahlung photons will not be produced

>0 produce BNUM times the analog number of bremsstrahlung

April 10, 2000 3-119

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photons. Radiative energy loss uses the bremsstrahlung energy

of the ﬁrst sampled photon.

XNUM >0 produce XNUM times the analog number of electron-induced

x–rays.

= 0 x-ray photons will not be produced by electrons.

RNOK >0 produce RNOK times the analog number of knock-on electrons.

= 0 knock-on electrons will not be produced.

ENUM >0 produce ENUM times the analog number of photon-induced

secondary electrons.

= 0 photon-induced secondary electrons will not be produced.

NUMB > 0 produce bremsstrahlung on each substep

= 0 nominal bremsstrahlung production

Defaults: EMAX = 100 MeV; IDES, IPHOT, IBAD, ISTRG = 0;

BNUM, XNUM, RNOK, ENUM = 1., NUMB = 0

Use: Optional.

EMAX is the upper electron energy limit in MeV. Electron cross sections and related data are

generated on a logarithmic energy grid from EMAX down to an energy at least as low as the global

energy cutoff for electrons. Setting the value of EMAX too high results in longer processing times

and larger storage requirements for electron data. EMAX should be set to the highest electron

energy encountered in your problem.

IDES is a switch to turn off electron production by photons. The default (IDES = 0) is for photons

to create electrons in all photon-electron problems and for photons to produce bremsstrahlung

photons using the thick-target bremsstrahlung approximation in photon problems run without

electrons. In either case the electron default cross section library will be read, which requires

considerable processing time. Electron transport is also very slow. However, the neglect of electron

transport and bremsstrahlung production will cause erroneously low photon results when these

effects are important. IDES = 1 turns off electron production, but it does not turn off the pair

production--produced annihilation photons. See ENUM.

IPHOT is a switch to turn off photon production by electrons. Because photon transport is fast

relative to electron transport and is usually required for an accurate physical model, the default

(IPHOT = 0, which leaves photon production on) is recommended.

IBAD is a switch to turn on the simple approximate bremsstrahlung angular distribution treatment

and turn off the full, more detailed model. The electron transport random walk can be done with

either the simple or full treatment, but photon contributions to detectors and DXTRAN can use

only the simple treatment. The full detailed physics model is more accurate and just as fast as the

simple approximate treatment for the electron transport random walk, and is therefore the default

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DATA CARDS

(IBAD = 0) even though it is inconsistent with the way bremsstrahlung photons contribute to

detectors and DXTRAN spheres. Setting IBAD = 1 causes the simple treatment to be used for

detectors and DXTRAN and the electron random walk, which is self-consistent.

ISTRG is a switch to control the electron continuous-energy slowing down treatment. If

ISTRG = 1, the expected value for each collision is used; if ISTRG = 0 (default), the more realistic

sampled value is used. The option of using the expected value is useful for some comparisons to

deterministic electron transport calculations.

BNUM, XNUM, RNOK, and ENUM are biasing parameters for speciﬁc classes of electron or

photon production processes. For each parameter the default is 1.0, which invokes an analog

treatment for the associated process. Other values allow biasing of the sampling of the processes.

The processes associated with the four parameters follow.

BNUM is used to control the sampling of bremsstrahlung photons produced along electron

substeps. The default value (BNUM = 1) results in the analog number of bremsstrahlung tracks

being sampled. If BNUM >0, the number of bremsstrahlung photons produced is BNUM times

the number that would be produced in the analog case. If the number of tracks is increased, an

appropriate weight reduction is made; if the biasing reduces the number of tracks, the weight is

increased. If BNUM = 0, the production of bremsstrahlung photons is turned off. In the el1

treatment, BNUM > 0 produces BNUM times the number of analog identical photons with

appropriately modiﬁed weights. In the el03 treatment, BNUM > 0 produces BNUM times the

number of analog photons, each sampled independently for energy and angle with appropriately

modiﬁed weights. Such a scheme is similar to the one used in ITS3.0 and recommended by

Bielajew, et. al. (A. F. Bielajew, R. Mohan, and C. S. Chui, “Improved Bremsstrahlung Photon

Angular Aampling in the EGS4 Code System,” Nov. 1989, PIR-0203.) In either case radiative

energy loss uses the bremsstrahlung energy of the ﬁrst sampled photon. BNUM < 0 (only for el03)

produces BNUMtimes the number of analog photons, each sampled independently for energy

and angle with appropriately modiﬁed weights. However, the radiative energy loss uses the average

energy of all the bremsstrahlung photons sampled. Such a scheme conserves energy more closely

but becomes more like a continuous slowing down approximation energy loss model.

XNUM is used to control the sampling of x-ray photons produced along electron substeps. The

default value (XNUM = 1) results in the analog number of tracks being sampled. If

XNUM >0, the number of photons produced is XNUM times the number that would be produced

in the analog case, and an appropriate weight adjustment is made. If XNUM = 0, the production

of x-ray photons by electrons is turned off.

RNOK is used to control the number of knock-on electrons produced in electron interactions. The

default value (RNOK = 1) results in the analog number of tracks being sampled. If RNOK >0, the

number of knock-on electrons produced is RNOK times the analog number, and an appropriate

weight adjustment is made. If RNOK = 0, the production of knock-on electrons is turned off.

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ENUM is used to control the generation of photon-induced secondary electrons. The default value

(ENUM = 1) results in an analog treatment. If ENUM > 0, ENUM times the analog number of

secondaries will be produced, and an appropriate weight adjustment is made. If ENUM = 0, the

generation of secondary electrons by photons will be turned off. ENUM = 0 differs from IDES =

1. If ENUM = 0, pair production is totally turned off. If IDES = 1, the pair production–produced

annihilation photons are still produced.

NUMB generates bremsstrahlung on each electron substep. Only a real event, one that has been

sampled to have a bremsstrahlung interaction, causes energy loss. The weights of the

bremsstrahlung photons are multiplied by the probability of interaction in a substep. If two or more

photons are produced in a real event, the weight of the second or more photons is the unadjusted

value because there is no Poisson sampling, except for real events.

In any of these biasing schemes, increasing the population of photons also increases the population

of electrons because the additional photon tracks create photoelectrons, Compton recoil electrons,

pair production electrons, etc. Similarly, increasing the number of electrons will propagate an

increase in the population of subsequent generations of the cascade. Because electron transport is

slow, a judicious use of ENUM < 1 may often be appropriate. When BNUM is set by the user,

ENUM=1/BNUM in the el03 treatment unless the user sets ENUM. When NUMB>0, ENUM=1%

by default.

The use of the switches, or of zero values for the biasing parameters, to turn off various processes

goes beyond biasing, and actually changes the physics of the simulation. Therefore such actions

should be taken with extreme care. These options are provided primarily for purposes of

debugging, code development, and special-purpose studies of the cascade transport process.

2. TMP Free-Gas Thermal Temperature Card

Form: TMPn T1n T2n ... Tin ... TIn

n= index of time on the THTME card.

Tin = temperature of ith cell at time n, in MeV.

I= number of cells in the problem.

Default: 2.53 x 10−8 MeV, room temperature.

Use: Optional. Required when THTME card is used. Needed for low-energy

neutron transport at other than room temperature. A fatal error occurs if a

zero temperature is speciﬁed for a nonvoid cell.

The TMP cards provide MCNP the time-dependent thermal cell temperatures that are necessary

for the free-gas thermal treatment of low-energy neutron transport described on page 2–28. This

treatment becomes important when the neutron energy is less than about 4 times the temperature

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DATA CARDS

of heavy nuclei or less than about 400 times the temperature of light nuclei. Thus the TMP cards

should be used when parts of the problem are not at room temperature and neutrons are transported

with energies within a factor of 400 from the thermal temperature.

Thermal temperatures are entered as a function of time with a maximum of 99 time entries allowed.

These times are entered on a thermal time (THTME) card. The thermal temperatures at time t1n are

listed, cell by cell, on the TMP1 card; the cell thermal temperatures at time t2n are listed on the

TMP2 card, etc. A linear interpolation is used to determine the cell thermal temperatures at times

between two entries. Time values before t1n or after tIn use the thermal temperatures at the nearest

time entry.

We use kT to denote the thermal temperature of a cell and use units of MeV. The following formulas

can be used to provide the values of kT for temperatures in degrees Kelvin, Celsius, Rankine, and

Fahrenheit.

kT(MeV) = 8.617 10−11T where T is in degrees K

=8.617 10−11(T+ 273.15) where Tis in degrees C

=4.787 10−11Twhere Tis in degrees R

=4.787 10−11(T+ 459.67) where Tis in degrees F

3. THTME Thermal Times Card

Form: THTME t1t2... tn... tN

tn= time in shakes at which thermal temperatures are speciﬁed on

the TMP card.

N= total number of thermal times speciﬁed.

Default: Zero; temperature is not time dependent.

Use: Optional. Use with TMP card.

The THTME card speciﬁes the times at which the thermal temperatures on the TMPn cards are

provided. The temperatures on the TMP1 card are at time t1on the THTME card, the temperatures

on the TMP2 card are at time t2on the THTME card, etc. The times must be monotonically

increasing: tn<tn+1. For each entry on the THTME card there must be a TMPn card.

4. MTm S(α,β) Material Card

Form: MTm X1X2...

Xi=S(α,β) identiﬁer corresponding to a particular component on the

Mm card.

Default: None.

Use: Optional, as needed.

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For any material deﬁned on an Mn card, a particular component of that material (represented by a

ZAID number) can be associated through an MTm card with an S(α,β) data set if that data set

exists. The S(α,β) data for that ZAID are used in every cell in which that material is speciﬁed. For

a particular ZAID in a material, the free-gas treatment can be used down to the energy where S(α,β)

data are available. At that point, the S(α,β) treatment automatically overrides the free-gas treatment

(that is, there is no mixing of the two treatments for the same ZAID in the same material at a given

energy). Typically the free-gas model is used for a particular ZAID of a material down to 4 eV and

then the S(α,β) treatment will take over. In general, S(α,β) effects are most signiﬁcant below 2 eV.

The S(α,β) treatment is invoked by identiﬁers on MTm cards. The mrefers to the material m

deﬁned on a regular Mm card. The appearance of an MTm card will cause the loading of the

corresponding S(α,β) data from the thermal data ﬁle. The currently available S(α,β) identiﬁers for

the MTm card are listed in Table G.1 of Appendix G. S(α,β) contributions to detectors or

DXTRAN spheres are approximate.

Examples: M1 1001 2 8016 1 $ light water

MT1 LWTR.07

M14 1001 2 6012 1 $ polyethylene

MT14 POLY.03

M8 6012 1 $ graphite

MT8 GRPH.01

H. Problem Cutoff Cards

The following cards can be used in an initiate-run or a continue-run input ﬁle to specify parameters

for some of the ways to terminate tracks in MCNP.

1. CUT Cutoffs Card

Form: CUT:n T E WC1 WC2 SWTM

n=N for neutrons, P for photons, E for electrons.

T= time cutoff in shakes, 1 shake=10−8 sec.

E= lower energy cutoff in MeV.

WC1 and WC2 = weight cutoffs.

Mnemonic Card Type Page

CUT Cutoffs 3–123

ELPT Cell–by–cell energy cutoff 3–125

NPS History cutoff 3–125

CTME Computer time cutoff 3–126

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SWTM = minimum source weight.

Use: Optional, as needed.

Neutron default: T=very large, E=0.0 MeV, WC1 = −0.50,WC2 = −0.25,

SWTM=minimum source weight if the general source is used.

If a neutron’s time becomes greater than T, its transport is stopped and it is killed. Even though

MCNP is time dependent, neutron decay is not considered. Any neutron with energy lower than E

is killed.

If a neutron’s weight WGT falls below WC2 times the ratio Rof the source cell importance to the

current cell importance, then with probability WGT/(WC1 ∗R), the neutron survives and is

assigned WGT =WC1 ∗R. If negative values are entered for the weight cutoffs, the values

∗Ws and |WC2| ∗ Ws

will be used for WC1 and WC2, respectively, where Wsis the minimum weight assigned to a source

neutron from an MCNP general source. These negative entries are recommended for most

problems. If only WC1 is speciﬁed, then WC2 = 0.5 ∗ WC1. See page 2–139 for a discussion of

weight cutoffs.

In a coupled neutron/photon problem, photons are generated before the neutron weight cutoff game

is played.

If WC1 is set to zero, capture is treated explicitly by analog rather than implicitly by reducing the

neutrons’s weight according to the capture probability. If EMCNF =Emax on the PHYS card,

analog capture is used regardless of the value of WC1 except for neutrons leaving a DXTRAN

sphere.

SWTM (source weight minimum) can be used to make the weight cutoffs relative to the minimum

starting weight of a source particle for user source as is done automatically for the general source.

The entry will in general be the minimum starting weight of all source particles, including the

effects of energy and direction biasing. The entry is also effective for the general source as well.

Then SWTM is multiplied by the W entry on the SDEF card but is unaffected by any directional or

energy biasing. This entry is ignored for a KCODE calculation.

Photon default: T=neutron cutoff, E=0.001 MeV, WC1 = −0.50,WC2 = −0.25,

SWTM=minimum source weight if the general source is used.

If there are pulse height tallies, WC1 =WC2 = 0.

The CUT:P weight cutoffs are analogous to the CUT:N card except that they are used only for

energies above the EMCPF entry on the PHYS:P card (see page 3–117). If WC1=0, analog capture

is speciﬁed for photons of energy greater than EMCPF, just as it is for neutrons. For energies below

EMCPF, analog capture is the only choice with one exception: photons leaving a DXTRAN

WC1

April 10, 2000 3-125

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DATA CARDS

sphere. Their weight is always checked against the CUT:P weight cutoff upon exiting. If only WC1

is speciﬁed, then WC2 = 0.5 ∗ WC1.

In a coupled neutron/photon problem, the photon weight cutoffs are the same as the neutron weight

cutoffs unless overridden by a CUT:P card. Again, the photon weight cutoffs have no effect at

energies below EMPCF (except with DXTRAN as noted above).

MCNP allows only analog capture below 0.001 MeV. Because the photoelectric cross section is

virtually 100% of the total cross section below that energy for all isotopes, tracks will be quickly

captured and terminated.

Electron default: T=neutron cutoff, E=0.001 MeV, WC1 = 0,WC2 = 0,

SWTM=minimum source weight if the general source is used.

The CUT:E weight cutoff entries have the same meaning as the neutron entries have.

2. ELPT Cell–by–cell Energy Cutoff

Form: ELPT:n x1x2... xi... xI

n=N for neutrons, P for photons, E for electrons.

xi= lower energy cutoff of cell i

I= number of cells in the problem.

A separate lower energy cutoff can be speciﬁed for each cell in the problem. The higher of either

the value on the ELPT:n card or the global value E on the CUT:n card applies.

3. NPS History Cutoff Card

Form: NPS N

N= number of particle histories.

Default: None.

Use: As needed to terminate the calculation. In a criticality calculation, the NPS

card has no meaning and a warning error message is issued if it is used.

The single entry N on this card is used to terminate the Monte Carlo calculation after Nhistories

have been transported—unless the calculation is terminated earlier for some other reason such as

computer time cutoff.

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In a continue-run, NPS is the total number of particles including runs before the continue-run; it is

cumulative. However, a negative NPS entry means to print an output ﬁle at the time of the last

history run and then stop.

In a surface source problem, either more or less than all of the particle histories on the RSSA

surface source ﬁle will be run, depending on the value N entered on the NPS card. If N<NP1,

where NP1 is the number of original histories, Russian roulette with weight adjustment will be

played with each history in the ﬁle, using a survival probability of N/NP1.IfN>NP1, the histories

will be split N/NP1 to 1, and the fractional part is taken care of by sampling. This can be done

equally well for nonspherical sources by cell importance splitting. With a spherical source, each

multiple occurrence of the history is sampled for a different starting location on the source sphere,

possibly improving the spatial statistics of the results. In either case, the use of the NPS card will

not provide additional information about the original source distributions or the transport to the

recording surface crossing.

4. CTME Computer Time Cutoff Card

Form: CTME x

x= maximum amount of computer time (in minutes) to be spent in the

Monte Carlo calculation.

Default: None.

Use: As needed.

For a continue-run job the time on the CTME card is the time relative to the start of the continue-

run; it is not cumulative.

Five normal ways to terminate an MCNP calculation are the NPS card, the CTME card, the job

time limit, the end of a surface source ﬁle, and the number of cycles on a KCODE card. If more

than one is in effect, the one encountered ﬁrst will control. MCNP checks the computer time

remaining in a running problem and will terminate the job itself, leaving enough time to wrap up

and terminate gracefully.

I. User Data Arrays

Two arrays, IDUM and RDUM, are in MCNP variable COMMON and are available to the user.

They are included in the dumps on the RUNTPE ﬁle and can therefore be used for any purpose,

including accumulating information over the entire course of a problem through several continue-

runs. Each array is dimensioned 50, and they can be ﬁlled by cards in the input ﬁle. IDUM is an

integer array and RDUM is a ﬂoating point array.

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1. IDUM Integer Array Card

Form: IDUM I1... In,

Default: All array values zero.

Use: Useful only in user-modiﬁed versions of MCNP.

Entries (up to 50) ﬁll the IDUM array with integer numbers. If ﬂoating point numbers are entered,

they will be truncated and converted to integers.

2. RDUM Floating Point Array Card

Form: RDUM R1... Rn,

Default: All array values zero.

Use: Useful only in user-modiﬁed versions of MCNP.

Entries (up to 50) ﬁll the RDUM array with ﬂoating point numbers.

J. Peripheral Cards

The following cards offer a variety of conveniences:

1. PRDMP Print and Dump Cycle Card

Form: PRDMP NDP NDM MCT NDMP DMMP

NDP = increment for printing tallies

NDM = increment for dumping to RUNTPE ﬁle

MCT = ﬂag to write MCTAL ﬁle and for OUTP comparisons

Mnemonic Card Type Page

PRDMP Print and dump cycle 3–127

LOST Lost particle 3–129

DBCN Debug information 3–129

FILES Create user ﬁles 3–133

PRINT Printing control 3–134

MPLOT Plot tally while problem is running 3–136

PTRAC Particle track output card 3–137

PERT Perturbation Card 3–141

1n50≤≤

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NDMP = maximum number of dumps on RUNTPE ﬁle

DMMP Sequential MCNP Multiprocessing MCNP

TFC entries every TFC entries and

rendezvous every

< 0 1000 particles 1000 particles

= 0 1000 particles 10 during the run

(see discussion below)

> 0 DMMP particles DMMP particles

Default: Print only after the calculation has successfully ended. Dump every

15 minutes and at the end of the problem. Do not write a MCTAL ﬁle.

Write all dumps to the RUNTPE ﬁle. DMMP=0 (see table above).

Use: Recommended, especially for complex problems.

The PRDMP card allows the user to control the interval at which tallies are printed to the OUTP

ﬁle and information is dumped to the RUNTPE ﬁle. Positive entries mean that after every NDP

histories the summary and tallies are printed to the output ﬁle, and after every NDM histories a

dump is written to the run ﬁle. A negative entry changes the unit from histories to minutes of

computer time.In a criticality calculation, positive entries for NDP and NDM on the PRDMP card

are interpreted as the number of cycles rather than the number of particles started. Printing and

dumping are done only at the ends of cycles.

If the third entry MCT on the PRDMP card is nonzero, a MCTAL ﬁle is written at the problem end.

The MCTAL ﬁle is an ASCII ﬁle of tallies that can be subsequently plotted with the MCNP

MCPLOT option (see description elsewhere). The MCTAL ﬁle is also a convenient way to store

tally information in a format that is stable for use in the user’s own auxiliary programs. For

example, if the user is on a system that cannot use the MCNP MCPLOT option, the MCTAL ﬁle

can be manipulated into whatever format is required by the user’s own local plotting algorithms. If

MCT=−1, references to code name, version number, problem ID, ﬁgure of merit, and anything else

having to do with running time are omitted from MCTAL and OUTP so that tracking runs (identical

random walks) yield identical MCTAL and OUTP ﬁles. MCT=−2 turns off additional prints in

OUTP to assist in comparing multitasking output.

The PRDMP card also allows the user to control the size of the RUNTPE ﬁle by specifying the

maximum number of dumps, NDMP, to be written. The RUNTPE ﬁle will contain the last NDMPs

that were written. For example, if NDMP = 4, after dump 20 is written only dumps 17, 18, 19, and

20 will be on the RUNTPE ﬁle. In all cases, the ﬁxed data and cross section data at the front of the

RUNTPE ﬁle are preserved.

The ﬁfth entry DMMP has several possible meanings. For sequential MCNP, a value of

DMMP~ results in TFC entries every 1000 particles initially. This value doubles to 2000 after0≤

April 10, 2000 3-129

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DATA CARDS

20 TFC entries. A positive value of DMMP produces TFC entries every DMMP particles initially.

For distributed memory multiprocessing, DMMP < 0 produces TFC entries and task rendezvous

every 1000 particles initially, the same as does the sequential version. DMMP=0, the default value,

produces ten TFC entries and task rendezvous, rounded to the nearest 1000 particles, based on

other cutoffs such as NPS, CTME, etc. This selection optimizes speedup in conjunction with TFC

entries. If detectors/DXTRAN are used with default Russian roulette criteria (DD card default), the

DMMP=0 entry is changed by MCNP to < 0, ensuring tracking with the sequential version (i.e.,

TFC entries and rendezvous every 1000 particles). As with the sequential version, DMMP > 0

produces TFC entries and task rendezvous every DMMP particles, even with detectors/DXTRAN

with default Russian roulette criteria. Setting DMMP to a large positive number minimizes

communication time and maximizes speedup. However, the TFC may not have many entries,

possibly only one, if DMMP=NPS.

2. LOST Lost Particle Card

Form: LOST LOST(1) LOST(2)

LOST(1) = number of particles which can be lost before the job

terminates with BAD TROUBLE

LOST(2) = maximum number of debug prints that will be made for

lost particles

Defaults: 10 lost particles and 10 debug prints.

Use: Discouraged. Losing more than 10 particles is rarely justiﬁable.

The word “lost” means that a particle gets to an ill-deﬁned section of the geometry and does not

know where to go next. This card should be used cautiously: you should know why the particles

are being lost, and the number lost should be statistically insigniﬁcant out of the total sample. Even

if only one of many particles gets lost, there could be something seriously wrong with the geometry

speciﬁcation. Geometry plots in the area where the particles are being lost can be extremely useful

in isolating the reason that particles are being lost. See page 3–8.

3. DBCN Debug Information Card

Form: DBCN X1X2X3... X20

X1= the starting pseudorandom number. Default =(519)152917;

X2= debug print interval;

X3 and X4= history number limits for event log printing;

X5= maximum number of events in the event log to print per history.

Default = 600;

X6= unused.

X7= 1 produces a detailed print from the volume and surface area

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calculations;

X8= number of the history whose starting pseudorandom number

is to be used to start the ﬁrst history of this problem;

X9= closeness of coincident repeated structures surfaces.

Default = 1.E-4;

X10 = seconds between time interrupts. Default = 100 seconds;

X11 = 1 causes collision lines to print in lost particle event log;

X12 = expected number of random numbers;

X13 = random number stride. Default = 152917;

X14 = random number multiplier. Default = 519;

X15 = 1 prints the shifted conﬁdence interval and the variance of

the variance for all tally bins;

X16 = scale the score grid for the accumulation of the empirical

f(x) in print tables 161 and 162;

X17 = 0 default angular treatment for partial substeps to generation

sites of secondary particles;

>0 alternate angular treatment for secondary generation;

<0 MCNP4A treatment of electron angles at secondary

generation sites;

X18 = 0 default “MCNP–style” energy indexing algorithm;

1 “ITS–style” energy indexing algorithm;

X20 = track previous version.

Use: Optional.

The entries on this card are used primarily for debugging problems and the code itself. The ﬁrst 12

can be changed in a continue run which is useful for diagnosing troubles that occur late in a long-

running problem.

1. X1 is the random number used for starting the transport of the first particle history in a

run. See also entry X8, which for repeating particle histories, is the preferred method of

changing the pseudorandom number sequence. See the caution after the last DBCN item

listed below.

2. X2 is used to print out information about every X2th particle. The information consists

of: (a) the particle history number, (b) the total number of neutron, photon, and electron

collisions, (c) the total number of random numbers generated, and (d) the random

number at the beginning of the history. This information is printed at the beginning of

the history and is preceded by the letters DBCN in the output to aid in a pattern search.

3. and 4. Event log printing is done for histories X3 through X4, inclusively. The

information includes a step-by-step account of each history, such as where and how a

particle is born, which surface it crosses and which cell it enters, what happens to it in a

cell, etc. See X11.

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5. X5is the maximum number of events the event log will print per history. The default is

600.

6. Unused.

7. X7 = 1 will cause a detailed print from the volume and surface area calculations and is

useful only to MCNP code developers.

8. The X8th entry causes the starting random number of the problem to be the random

number that would normally be picked for the X8th history. If a surface source is used,

the X8th surface source history will be taken from the RSSA file at the problem start. The

purpose of this entry is to let the X8th history be the first history of a problem for

debugging purposes or to select a random number sequence different from that in an

identical problem to compare statistical convergence. See the caution after the last

DBCN item listed below.

9. X9defines the distance allowed between coincident repeated structures surfaces for them

still to be considered coincident. The default is 1.E−4. A value of 1.E−30 reproduces the

earlier treatment where coincident repeated structures surfaces was not allowed. X9

should not have to be changed unless geometries have dimensions greater than 1.E5 or

unless surfaces at different levels are intended to be closer than 2.E−4.

10. X10 is the seconds between time interrupts for checking if a history has run too long or

is in an infinite loop. The default is 100 seconds. If in two consecutive time interrupts the

random walk is in the same history, MCNP assumes that something is wrong and stops

the job. If histories should legitimately take longer than X10 seconds the job can be

continued with a larger value for X10 specified on the DBCN card in the continue-run

INP file. This entry also affects the time increment MCNP reserves for itself to terminate

a job before the job time limit is reached. The increment for interactive jobs is 2X10 or

1% of the time limit, whichever is greater.

11. X11 = 1 causes collision lines to print in the lost particle event log.

12. X12 is the expected number of random numbers for this calculation. Entering X12 will

cause the last line of the output file to print X12 and the actual number of random numbers

used so that a quick comparison can be made to see if two problems tracked each other.

13. X13 is the random number stride, S. The default is S = X13 = 152917. Each source history

starts with a random number S numbers up the pseudorandom number sequence from the

random number of the previous history. If any history requires more than S random

numbers, the number of times S was exceeded is printed in the problem summary of the

OUTP file. The maximum number of random numbers required for a history is always

printed in the problem summary. Exceeding the random number stride will cause a

correlation between histories and should be avoided because variances may be

underestimated. However, if the stride is too large, the period of the random number

sequence, E13, will be exceeded.246 7.04≈

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S should be chosen so that NPS∗S< 246. Exceeding the period will underestimate

variances, particularly if S is a power of 2.

14. X14 is the random number multiplier. The default is X14 =5

19 =19073486328125, which

is adequate for all known problems. If a new entry is such that the sum of its left and right

24-bit halves is not less than 224 then the input value is rejected. If X14 is even it is

rejected because the random number sequence rapidly converges to zero.

15. A nonzero X15th entry causes the shifted confidence interval and the variance of the

variance (VOV) to be calculated and printed for all tally bins. An extra line of tally

output is created for each tally that contains nonzero information. The shifted confidence

interval center is followed by the estimated VOV. If the tally mean and relative error

(RE) are all zeros, the VOV line is not printed because it is all zero also. Changing X15

from nonzero to zero in a CONTINUE run will cause the VOV information not to be

printed. X15 cannot be changed from zero to nonzero in a CONTINUE run.

16. MCNP uses a logarithmically spaced history score grid in print table 161 for f(x),

producing a straight line for f(x) on a log–log plot for 1/xnbehavior, covering 60 decades

of unnormalized tally magnitudes from 1E−30 to 1E30. This range can be multiplied by

the X16th entry when the range is not sufficient. A negative entry means that negative

history scores will be accrued in the score grid f(−x) and the absolute value of X16 will

be used as the score grid multiplier. Positive history scores will then be lumped into the

lowest bin with this option. This scaling can be done only in the original problem, not

in a CONTINUE run.

17. If 0, the default angular treatment for partial substeps to generation sites of secondary

particles is invoked. This treatment accounts for the probability of the delta function first,

then interpolates in the cosine of the deflection angle. It does not preserve the plane in

which the deflection angle will lie at the end of the full substep.

If > 0, an angular treatment for secondary generation is invoked as follows. The cosine

of the electron angle is interpolated and the end–of–substep plane is preserved, but the

changing probability of the delta function along the substep is ignored. This option is

preserved for further testing of angular algorithms because results have been known to

be sensitive to these details.

If < 0, the MCNP4A treatment of electron angles at secondary generation sites is

invoked. Used with dbcn(20)=0, comparisons to the earlier treatment can be made.

18. If 0, the default “MCNP–style” energy indexing algorithm is used, also called the “bin–

centered” treatment.

If , the “ITS–style” energy algorithm is used, also called the “nearest group

boundary” treatment. Allows us to match ITS results as closely as possible.

19. A nonzero X20th entry causes MCNP to track the previous version of the code, except in

the few cases when bugs are too hard to duplicate with this option. Because bug

corrections, new features, and enhanced physics must be undone, should be

used only for debugging purposes.

0≠

X20 0≠

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CAUTION: When trying to duplicate a particle history by setting the starting random number with

either X1 or X8, the random number sequence may be altered by a default Russian Roulette game

on contributions to detectors or DXTRAN spheres. If a problem has detectors or DXTRAN, the

only ways to reproduce histories with X1 or X8are: (a) turn off the Russian Roulette game on the

DD card by setting k = 0; (b) play the roulette game with a ﬁxed criterion by setting k < 0 on the

DD card; or (c) reproduce a history with NPS < 200.

4. FILES File Creation Card

Form: FILES unit no. ﬁlename access form record length

unit no. = 1 to 99

ﬁlename = name of the ﬁle

access = sequential or direct

form = formatted or unformatted

record length = record length in a direct access ﬁle

Default: None; none; sequential; formatted if sequential, unformatted if direct; not

required if sequential, no default if direct.

Use: When a user-modiﬁed version of MCNP needs ﬁles whose characteristics

may vary from run to run. Not legal in a continue-run.

If this card is present, the ﬁrst two entries are required and must not conﬂict with existing MCNP

units and ﬁles. The words “sequential,” “direct,” “formatted,” and “unformatted” can be

abbreviated. If more than one ﬁle is on the FILES card, the defaults are not much help but the

abbreviations will keep it brief. The maximum number of ﬁles allowed is six, unless the dimension

of the KUFIL array in Fixed Common is increased.

Example: FILES 21 ANDYSF022 MIKE D U 512

If the ﬁlename is DUMN1 or DUMN2, the user can optionally use the execution line message to

designate a ﬁle whose name might be different from run to run, for instance in a continue-run.

Example: FILES 17 DUMN1

MCNP INP=TEST3 DUMN1=POST3

Caution: The names of any user ﬁles in a continue-run will be the same as in the initial run. The

names are not automatically sequenced if a ﬁle of the same name already exists; therefore, a second

output ﬁle from a continue-run will clobber an existing ﬁle of the same name. If you are using the

FILES card for an input ﬁle and do a continue-run, you will have to provide the coding for keeping

track of the record number and then positioning the correct starting location on the ﬁle when you

continue or MCNP will start reading the ﬁle at the beginning.

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5. PRINT Output Print Tables

Form: PRINT x

x= no entry gives the full output print

x=x1x2... prints basic output plus the tables speciﬁed by the table

numbers x1,x2,...

x=−x1−x2... prints full output except the tables speciﬁed by x1,x2,...

Default: No PRINT card in the INP ﬁle or no PRINT option on the execution line will

result in a reduced output print.

Use: Optional.

The following output will be printed automatically, as applicable:

• a listing of the input ﬁle,

• the problem summary of particle creation and loss,

• KCODE cycle summaries,

• tallies,

• tally ﬂuctuation charts, and

• the tables listed below marked basic and default.

You will always get the information indicated by the ﬁrst ﬁve bullets above and the tables labelled

“basic” below. They cannot be turned off. Tables marked “default” will be printed automatically

but they can be turned off with the PRINT card.

To get all optional print tables applicable to your problem, indicated in the table below as blank

type, use the PRINT card in the INP ﬁle or the PRINT option on the execute line. The execute line

takes precedence over the input card. Absence of a PRINT card or a PRINT option produces only

the tables marked “basic,” “default,” and “shorten.” Entries are allowed only on the PRINT card,

not following the PRINT option. Entries on the PRINT card can be in any order.

The PRINT card entries are table numbers of optional and default tables, and control turning the

table off or on. If all the entries are positive, you will get the “basic” tables plus the tables requested

on the PRINT card. If any entry is negative, you will get all tables applicable to your problem

except those turned off by the negative entries.

The table number appears in the upper righthand corner of the table, providing a convenient pattern

when scanning the output ﬁle with an editor. The pattern is PRINT TABLE n, where nis preceded

always by one space and is a two- or three-digit number. The table numbers and titles and type are

summarized in the table below. Tables that can not be controlled by the PRINT card are marked as

type “basic.” Tables that are automatically printed but can be turned off are marked as type

“default.” Tables with no type (blank) can be turned off and on with the PRINT card or option.

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Tables 160, 161, and 162 are different from the other tables. If you turn off table 160, tables 161

and 162 will not appear either. If table 160 is printed, they will all be printed. They are all

automatically printed if there is no PRINT card or if there is a blank PRINT card. If a PRINT card

has a positive entry, tables 160, 161, and 162 will not appear, unless table 160 is explicitly

requested. If the entry is negative, they will appear, unless table 160 is explicitly turned off.

Table 175 can not be turned off completely, but the output can be greatly shortened to every 100

cycles plus the last ﬁve cycles. PRINT −175 and PRINT 110 both will produce the short version of

Table 175.

Table 128, the repeated structure universe map, is special. If table 128 is not turned on in an initial

run, it CANNOT be turned on in a subsequent continue–run because the (often large) storage arrays

have not been set up. Table 128 is the only print table that affects storage. The information in the

other tables is always stored, whether or not it is printed. A warning will be printed in a repeated

structures problem if you do not request the universe map/lattice activity table in the original run.

The PRINT control can be used in a continue–run to recover all or any applicable print tables, even

if they were not requested in the original run. A continue ﬁle with NPS −1 and PRINT will create

the output ﬁle for the initial run starting with the Problem Summary (located after table 110). Table

128 can never be printed if it was not requested in the original run.

Table Number Type Table Description

10 Source coefﬁcients and distribution

20 Weight window information

30 Tally description

35 Coincident detectors

40 Material composition

50 Cell volumes and masses, surface areas

60 basic Cell importances

62 basic Forced collision and exponential transform

70 Surface coefﬁcients

72 basic Cell temperatures

85 Electron range and straggling tables

multigroup: ﬂux values for biasing adjoint calcs

86 Electron bremsstrahlung and secondary production

90 KCODE source data

98 Physical constants and compile options

100 basic Cross section tables

102 Assignment of S(α,β) data to nuclides

110 First 50 starting histories

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Example: PRINT 110 40 150

The output ﬁle will contain the “basic” tables plus tables 40, 110, and 150, not 160, 161, 162 (the

“default” tables), and the shortened version of 175.

Example: PRINT 170 −70 −110

The output ﬁle will contain all the “basic” tables, all the “default” tables, the long version of table

175, and all the optional tables except tables 70, 110, and 170 applicable to your problem.

6. MPLOT Plot tally while problem is running

Form: MPLOT MCPLOT keyword=parameter

Default: None.

Use: Optional.

This card speciﬁes a plot of intermediate tally results that is to be produced periodically during the

run. The entries are MCPLOT commands for one picture. The = sign is optional. During the run,

as determined by the FREQ n entry, MCRUN will call MCPLOT to display the current status of

one or more of the tallies in the problem. If a FREQ n command is not included on the MPLOT

card, n will be set to 5000. The following commands can not appear on the MPLOT card:

RMCTAL, RUNTPE, DUMP, and END. All of the commands on the MPLOT card are executed

for each displayed picture, so coplots of more than one bin or tally are possible. No output is sent

120 Analysis of the quality of your importance function

126 basic Particle activity in each cell

128 Universe map

130 Neutron/photon/electron weight balance

140 Neutron/photon nuclide activity

150 DXTRAN diagnostics

160 default TFC bin tally analysis

161 default f(x) tally density plot

162 default Cumulative f(x) and tally density plot

170 Source distribution frequency tables, surface source

175 shorten Estimated keff results by cycle

178 Estimated keff results by batch size

180 Weight window generator bookkeeping summary

controlled by WWG(7), not print card

190 basic Weight window generator summary

198 Weight windows from multigroup ﬂuxes

200 basic Weight window generated windows

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to COMOUT. MCPLOT will not take plot requests from the terminal and returns to MCRUN after

each plot is displayed. See Appendix B for a complete list of MCPLOT commands available.

Another way to plot intermediate tally results is to use the TTY interrupt <ctrl–c>IMCPLOT or

<ctrl–c>IM that allows interactive plotting during the run. At the end of the history that is running

when the interrupt occurs, MCRUN will call MCPLOT, which will take plot requests from the

terminal. No output is sent to the COMOUT ﬁle. The following commands can not be used:

RMCTAL, RUNTPE, DUMP and END.

7. PTRAC Particle Track Output Card

Form: PTRAC keyword=parameter(s) keyword=parameter(s)

Default: See Table 3.5.

Use: Optional.

This card generates an output ﬁle, default name PTRAC, of user–ﬁltered particle events. The name

PTRAC can be changed on the execution line or within the message block. Using this card without

any keywords causes all particle events to be written to the PTRAC ﬁle. CAUTION: an extremely

large ﬁle likely will be created unless NPS is small. Use of one or more keywords listed in Table

3.5 will reduce signiﬁcantly the PTRAC ﬁle size. In Table 3.5 the keywords are arranged into three

categories: output control keywords, event ﬁlter keywords, and history ﬁlter keywords. The output

control keywords provide user control of the PTRAC ﬁle and I/O. The event ﬁlter keywords ﬁlter

particle events on an event–by–event basis. That is, if the history meets the ﬁlter criteria, all ﬁltered

events for that history are written to ﬁle PTRAC. The PTRAC card keywords can be entered in any

order and, in most cases, the corresponding parameter values can appear in any order (exceptions

noted below.) The PTRAC card is not legal in a continue–run input ﬁle because a change in the

PTRAC input would require a readjustment in dynamically allocated storage.

When multiple keywords are entered on the PTRAC card, the ﬁlter criteria for each keyword must

be satisﬁed to obtain an output event. For example:

PTRAC FILTER=8,9,erg EVENT=sur NPS=1,50 TYPE=e CELL=3,4

will write only surface crossing events for 8–9 MeV electrons generated by histories 1–50 that have

entered cells 3 or 4.

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BUFFER Determines the amount of storage available for ﬁltered events. A small value

results in increased I/O and a decrease in required memory, whereas a large

value minimizes I/O and increases memory requirements.

FILE Controls ﬁle type. One of the following values can be entered:

asc—generates an ASCII output ﬁle.

bin—generates a binary output ﬁle. This is the default.

MAX Sets the maximum number of events to write to the PTRAC ﬁle. A negative

value terminates MCNP when this value is reached.

MEPH Determines the maximum number of events per history to write to the PTRAC

Default: write all events.

WRITE Controls what particle parameters are written to the PTRAC ﬁle.

pos—only x, y, z location with related cell and material numbers.

all—additionally, u, v, w direction cosines, energy, weight, and time.

If the size of the PTRAC ﬁle is a concern and the additional parameters are not

needed, the default value of “pos” is recommended.

EVENT Speciﬁes the type of events written to PTRAC. One or more of the

following parameter values can be entered:

src—initial source events

bnk—bank events

TABLE 3.5: PTRAC Keywords, Parameter Values, and Defaults

Keyword Parameter Values Default Entries

OUTPUT CONTROL KEYWORDS

BUFFER Integer > 0 100 1

FILE asc, bin bin 1

MAX Integer 10000 1

MEPH Integer >0 ∗ 1

WRITE pos, all pos 1

EVENT FILTER KEYWORDS

EVENT src, bnk, sur, col, ter ∗1–5

FILTER Real, Integer, Mnemonic ∗2–72

TYPE n, p, e ∗1–3

HISTORY FILTER KEYWORDS

NPS Integer > 0 ∗ 1–2

CELL Integer > 0 ∗ Unlimited

SURFACE Integer > 0 * Unlimited

TALLY Integer ∗Unlimited

VALUE Real, Integer ∗Unlimited

0≠

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sur—surface events

col—collision events

ter—termination events

The bank events include secondary sources, e.g., photons produced by

neutrons, as well as particles created by variance reduction techniques, e.g.,

DXTRAN and energy splitting. See page I-5 for a complete list.

FILTER Speciﬁes additional MCNP variables for ﬁltering. The parameter

values consist of one or two numerical entries and a variable mnemonic that

corresponds to a variable in the PBLCOM common block. See Table 3.6 for

available mnemonics. A single numerical entry requires an exact value.

EXAMPLE: FILTER=2,icl writes only those events that occur in cell 2.

Two numerical entries represent a range.

EXAMPLE: FILTER=0,10,x writes only those events in which

the particle’s x–coordinate is between 0 and 10 cm. When a range is

speciﬁed, the ﬁrst entry must be less than or equal to the second. Multiple

sets of numerical entries and mnemonics are also allowed.

EXAMPLE: FILTER=0.0,10.0,x 0,1,u 1.0,2,erg writes only those

events in which the particle’s x–coordinate is between 0 and 10 cm

and the particle’s x–axis cosine in between 0 and 1 and the particle’s

energy is between 1 and 2 MeV.

Default: No additional ﬁltering.

TYPE Filters events based on particle type. One or more of the following parameter

values can be entered:

n—neutron events; p—photon events; e—electron events

EXAMPLE: TYPE=p,e writes only photon and electron events.

Default: Events for all particle types are written.

NPS Sets the range of particle histories for which events will be output. A single

value produces ﬁltered events only for the speciﬁed history.

EXAMPLE: NPS=10 writes events only for particle number 10.

Two entries indicate a range and will produce ﬁltered events for all histories

within that range. The ﬁrst entry must be less than or equal to the second.

EXAMPLE: NPS=10,20 writes events for particles 10 through 20.

Default: Events for all histories.

CELL, SURFACE, TALLY The cell, surface, or tally numbers entered after these

keywords are used for history ﬁltering. If any track of the history enters listed

cells or crosses listed surfaces or contributes to the TFC bin of listed tallies,

all ﬁltered events for the history are written to the PTRAC ﬁle. See page 3–100

for speciﬁcation of the TFC bin.

EXAMPLE: CELL=1,2 writes all ﬁltered events for those histories that

enter cell 1 or 2.

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EXAMPLE: TALLY=4 writes all ﬁltered events for those histories that

contribute to tally 4 (see VALUE keyword for ﬁlter criteria.) The number of

entries following CELL, SURFACE, and TALLY is unlimited. A negative

TALLY entry indicates that the corresponding VALUE entry is a multiplier

rather than an absolute value.

Default: No history ﬁltering.

VALUE Speciﬁes the tally cutoff above which history events will be written.

The number of entries must match those of the TALLY keyword.

EXAMPLE: Tally=4 VALUE=2.0 writes all ﬁltered events of any history

that contributes 2.0 or more to the TFC bin of tally 4. A negative TALLY

value indicates that the corresponding VALUE entry is a multiplier.

EXAMPLE: TALLY=–4 VALUE=2.0 writes all ﬁltered events of any

history that contributes more than 2.0∗Ta to tally 4, where Ta is the

average tally of the TFC bin. The values for Ta are updated every DMMP

histories. Typically, DMMP=1000. See the PRDMP card, page 3–127.

Filtering based on the Ta values will occur only when they become nonzero.

Thus, when using a multiplier, PTRAC events may not be written for several

thousand particles, or at all, if scores are seldom or never made to the TFC bin

of the speciﬁed tally. In such cases, it is best to enter an absolute value.

EXAMPLE: TALLY=4 VALUE=0.0 writes all ﬁltered events of every

history that scores to tally 4.

Default: A multiplier of 10.0 for each tally associated with the TALLY

keyword .

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8. PERTn Perturbation Card

Form: PERTn:pl keyword=parameter(s) keyword=parameter(s)

n= unique, arbitrary perturbation number.

pl = N, P, or N,P. Not available for electrons.

keyword = See Table 3.7.

Default: Some keywords are required. See Table 3.7.

Use: Optional.

This card allows perturbations in cell material density, composition, or reaction cross-section data.

The perturbation analysis uses the ﬁrst and second order differential operator technique described

TABLE 3.6: Mnemonic Values for the FILTER Keyword

Mnemonic MCNP Variable Description

X XXX X–coordinate of particle position (cm)

Y YYY Y–coordinate of particle position (cm)

Z ZZZ Z–coordinate of particle position (cm)

U UUU Particle X–axis direction cosine

V VVV Particle Y–axis direction cosine

W WWW Particle Z–axis direction cosine

ERG ERG Particle energy (MeV)

WGT WGT Particle weight

TME TME Time at the particle position (shakes)

VEL VEL Speed of the particle (cm/shake)

IMP1 FIML(1) Neutron cell importance

IMP2 FIML(2) Photon cell importance

IMP3 FIML(3) Electron cell importance

SPARE1 SPARE(1) Spare banked variable

SPARE2 SPARE(2) Spare banked variable

SPARE3 SPARE(3) Spare banked variable

ICL ICL Problem number of current cell

JSU JSU Problem number of current surface

IDX IDX Number of current DXTRAN sphere

NCP NCP Count of collisions for current branch

LEV LEV Geometry level of particle location

III III 1st lattice index of particle location

JJJ JJJ 2nd lattice index of particle location

KKK KKK 3rd lattice index of particle location

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in Chapter 2, page 2–191. Using this technique, the perturbation estimates are made without

actually changing the input material speciﬁcations. Multiple perturbations can be applied in the

same run, each speciﬁed by a separate PERT card. There is no limit to the number of perturbations

because dynamic memory is used for perturbation storage. The entire tally output is repeated for

each perturbation, giving the estimated differential change in the tally or this change can be added

to the unperturbed tally (see the METHOD keyword). For this reason, the number of tallies and

perturbations should be kept to a minimum. A track length estimate of perturbations to keff is

automatically estimated and printed for KCODE problems. The CELL keyword that identiﬁes one

or more perturbed problem cells is required. Also, either the MAT or RHO keyword must be

speciﬁed.

CELL Indicates which cells are perturbed. At least one entry is required, and there is

no limit to the number of entries. A comma or space delimiter is required

between entries:

CELL=1,2,3,4

CELL=1 10i 12

MAT Speciﬁes the perturbation material number, which must have a corresponding

M card. Composition changes can only be made through the use of this

keyword. If the RHO keyword is omitted, the MAT keyword is required. Note

in the CAUTIONS below that certain composition changes are prohibited.

RHO Speciﬁes the perturbed density of the cells listed after the CELL keyword. A

positive entry indicates units of atoms/cm3 and a negative entry indicates units

of g/cm3. If the MAT keyword is omitted, the RHO keyword is required.

METHOD Speciﬁes the number of terms to include in the perturbation estimate.

1 — include ﬁrst and second order (default)

2 — include only ﬁrst order

TABLE 3.7: PERT Keywords, Parameter Values, and Defaults

Keyword Parameter Values Default Entries

BASIC KEYWORDS

CELL Integer > 0 Required Unlimited

MAT Integer > 0 ∗ 1

RHO Real, integer ∗1

ADVANCED KEYWORDS

METHOD 1 1

ERG Real, Integer > 0 All Energies 2

RXN Integer 1 Unlimited

123,,±

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3 — include only second order

A positive entry produces perturbation tallies that give the estimated

differential change in the unperturbed tally (default). A negative entry adds

this change to the unperturbed tally. The ability to produce ﬁrst and second

order terms separately enables the user to determine the signiﬁcance of

including the second-order estimator for subsequent runs. If the second-order

results are a signiﬁcant fraction (20-30%) of the total, then higher order terms

are necessary to accurately predict the change in the unperturbed tally. In such

cases, the magnitude of the perturbation should be reduced to satisfy this

condition. Typically, this technique is accurate to within a few percent for

up to 30% changes in the unperturbed tally. It is strongly recommended

that the magnitude of the second order term be determined before the user

continues with this capability.

ERG The two entries specify an energy range in which the perturbation is applied.

The default range includes all energies. This keyword is usually used with

the RXN keyword to perturb a speciﬁc cross-section over a particular energy

range.

RXN Entries must be ENDF/B reaction types that identify one or more speciﬁc

reaction cross-sections to perturb. A list of available ENDF/B reaction types

is given in Table I, Appendix G. This keyword allows the user to perturb a

single reaction cross-section of a single nuclide in a material, all reaction

types of a single nuclide, a single reaction for all nuclides in a material, and

a set of cross-sections for all nuclides in a material. The default reaction is the

total cross section (RXN=1 for neutrons and multigroup, RXN=-5 for

photons.) Relevant nonstandard special R numbers on page 3–88 can be used.

Those that cannot be used are −4,−5,−7, and −8 for neutrons; −6 for photons;

and −3,−4,−6, and −7 for multigroup problems. If these irrevelant R numbers

are used, the following fatal error will be printed: “fatal error. reaction # illegal

in perturbation #.”

RXN=2 elastic cross section

RXN=−2 absorption cross section

RXN reaction numbers must be consistent with FM card reaction numbers (see page 3–88) if the

perturbation affects the tally cross section. RXN=−6 is most efﬁcient for ﬁssion, although MT=18,

MT=19, or MT=−2 (multigroup) also work for keff and F7 tallies.

3-144 April 10, 2000

CHAPTER 3

DATA CARDS

CAUTIONS

1. There is no limit to the number of perturbations, but they should be kept to a minimum as each

perturbation can degrade performance by 10-20%.

2. It is not possible to take a region originally speciﬁed as void and put in a material with the

perturbation technique. However, you can specify a region as containing a material and use

the PERT card to make it void by setting RHO=0.

3. It is not possible to introduce a new nuclide into a material composition. However, you can

set up the problem with a mixture of all nuclides of interest and use PERT cards to remove

one or more (see the examples below).

4. The track length estimate of keff in KCODE criticality calculations assumes the fundamental

eigenvector (ﬁssion distribution) is unchanged in the perturbed conﬁguration.

5. Use caution in selecting the multiplicative constant and reaction number on FM cards used

with F4 tallies in perturbation problems. The track length correction term R1j′is made only if

the multiplicative constant on the FM card is negative (indicating macroscopic cross sections

with multiplication by the atom density of the cell). If the multiplicative constant on the FM

card is positive, it is assumed that any FM card cross sections are independent of the perturbed

cross sections. If there is a reaction (RXN) speciﬁed on the PERT card, the track length

correction term R1j is set only if the exact same reaction is speciﬁed on the FM card. For

example, an entry of RXN=2 on the PERT card is not equivalent to the special elastic reaction

−3 on the FM card (should either enter 2 and 2 or −3 and −3).

6. DXTRAN, F5 point detector tallies, and F8 pulse height tallies are not compatible with the

PERT card. DXTRAN will give a fatal error; F5 and F8 will give zero perturbations.

7. Large perturbations require higher than second order terms to avoid inaccurate tallies. Refer

to the METHOD keyword for a more complete discussion.

Examples of the PERT Card

Example 1: PERT1:n,p CELL=1 RHO=0.03

This perturbation speciﬁes a density change to 0.03 atoms/cm3in cell 1. This change is applied to

both neutron and photon interactions.

Example 2: 3 1 −1 −1 2−34−5 6 $ mat 1 at 1 g/cm3

12 1 −1 −7 8−9 10 −11 12 $ mat 1 at 1 g/cm3

…

April 10, 2000 3-145

CHAPTER 3

DATA CARDS

C M1 material is semiheavy water

M1 1001 .334 1002 .333 8016 .333

C M8 material is heavy water

M8 1002 .667 8016 .333

PERT2:n CELL=3,12 MAT=8 RHO=−1.2

This perturbation changes the material composition of cells 3 and 12 from material 1 to material 8.

The MAT keyword on the PERT card speciﬁes the perturbation material. The material density was

also changed from 1.0 to 1.2 g/cm3 to change from water to heavy water.

Example 3: PERT3:n,p CELL=1 10i 12 RHO=0 METHOD=−1

This perturbation makes cells 1 through 12 void for both neutrons and photons. The estimated

changes will be added to the unperturbed tallies.

Example 4: 60 13 −2.34 105 −106 −74 73 $ mat 13 at 2.34 g/cm3

M13 1001 −.2 8016 −.2 13027 −.2 26000 −.2 29000 −.2

M15 1001 −.2 8016 −.2 13027 −.2 26000 −.2 29000 −.4

PERT1:p CELL=60 MAT=15 RHO=−2.808 RXN=51 9i 61,91

ERG=1,20

PERT2:p CELL=60 RHO=−4.68 RXN=2

This example illustrates sensitivity analysis. The ﬁrst PERT card generates estimated changes in

tallies caused by a 100% increase in the Cu (n,n’) cross section (ENDF/B reaction types 51–61 and

91) above 1 MeV. To effect a 100% increase, double the composition fraction (−.2 to −.4) and

multiply the ratio of this increase by the original cell density

(RHO=[1.2/1.0] ∗ −2.34 =−2.808 g/cm3, where the composition fraction for material 13 is 1.0 and

that for material 15 is 1.2.) A change must be made to RHO to maintain the other nuclides in their

original amounts. Otherwise, after MCNP normalizes the M15 card, it would be as follows, which

is different from the composition of the original material M13:

M15 1001 −.167 8016 −.167 13027 −.167 26000 −.167 29000 −.333

The second PERT card (PERT2:p) gives the estimated tally change for a 100% increase in the

elastic (RXN=2) cross section of material 13. RHO=−2.34 ∗ 2 = −4.68 g/cm3

Example 5: M4 6000.60C .5 6000.50C .5

M6 6000.60C 1

M8 6000.50C 1

PERT1:n CELL=3 MAT=6 METHOD=−1

PERT2:n CELL=3 MAT=8 METHOD=−1

…

3-146 April 10, 2000

CHAPTER 3

SUMMARY OF MCNP INPUT FILE

The perturbation capability can be used to determine the difference between one cross–section

evaluation and another. The difference between these perturbation tallies will give an estimate of

the effect of using different cross section evaluations.

Example 6: 1 1 0.05 −12−3 $ mat 1 at 0.05 x 1024 atoms/cm3

M1 1001 .1 8016 .2 92235 .7

M9 1001 .1 8016 .22 92235 .7

F14:n 1

FM14 (−11−6 −7 $ keff estimator for cell 1

PERT1:n CELL=1 MAT=9 RHO=0.051 METHOD=1

PERT2:n CELL=1 MAT=9 RHO=0.051 METHOD=−1

These perturbations involve a 10% increase in the oxygen atom fraction of material 1 (RHO=0.05

x [1.02/1.0] = 0.051). The effect of this perturbation on tally 14, which is a track length estimate

of keff, will be provided as a differential change (PERT1) as well as with this change added to the

unperturbed estimate of keff (PERT2). Note: if the RHO keyword is omitted from the PERT cards,

the 235U composition will be perturbed, which can produce invalid results (see Caution #4.)

Example 7: 1 1 −1.5 −12−34−5 6 $ mat 1 at 1.5 g/cm3

M1 1001 −.4333 6000 −.2000 8016 −.3667 $ half water

$ half plastic

M2 1001 −.6666 8016 −.3334 $ water

M3 1001 −.2000 6000 −.4000 8016 −.4000 $ plastic

PERT1:n CELL=1 MAT=2 RHO=−1.0 METHOD=−1

PERT2:n CELL=1 MAT=3 RHO=−2.0 METHOD=−1

This example demonstrates how to make signiﬁcant composition changes (e.g., changing a region

from water to plastic.) The unperturbed material is made from a combination of the two desired

materials, typically half of each. PERT1 gives the predicted tally as if cell 1 were ﬁlled with water

and PERT2 gives the predicted tally as if cell 1 were ﬁlled with plastic. The difference between

these perturbation tallies is an estimate of the effect of changing cell 1 from water to plastic.

V. SUMMARY OF MCNP INPUT FILE

A. Input Cards

The following table lists the various input cards and when they are required. Two kinds of defaults

are involved in the following table: (1) if a particular entry on a given card has a default value, that

…

April 10, 2000 3-147

CHAPTER 3

SUMMARY OF MCNP INPUT FILE

value is listed in the appropriate location on the card, and (2) the omission of a card from the input

ﬁle sometimes has a default meaning, and if so, the default description is preceded by an asterisk.

TABLE 3.8: Summary of MCNP Input Cards

Use Card and Defaults

General Categories Page

optional Message block plus blank terminator 3–1

required Problem title card 3–2

required Cell cards plus blank terminator 3–10

required Surface cards plus blank terminator 3–12

required Data cards plus blank terminator 3–22

optional C Comment card 3–4

Problem type card page 3–23

(a) MODE N

(a) Required for all but MODE N

Geometry cards page 3–23

optional VOL 0

optional AREA 0

optional U 0

optional TRCL 0

optional LAT 0

optional FILL 0

optional TRn none

Variance reduction cards page 3–32

required IMP required unless weight windows used

optional ESPLT *no energy splitting or roulette

optional PWT −1 MODE N P or N P E only

optional EXT 0

optional VECT none

optional FCL 0

optional WWE none

required WWN required unless importances used

optional WWP 5 3 5 0 0 0

optional WWG none

optional WWGE single energy or time interval

3-148 April 10, 2000

CHAPTER 3

SUMMARY OF MCNP INPUT FILE

optional MESH none

optional PDn 1

optional DXC 1

optional BBREM none electron photon transport only

Source speciﬁcation cards page 3–49

optional SDEF ERG=14 TME=0 POS=0,0,0 WGT=1

optional SIn H Ii ... Ik

optional SPn D Pi... Pk

optional SBn D Bi... Bk

optional DSn H Ji... Jk

optional SCn none

optional SSW SYM 0

optional SSR OLD NEW COL m=0

(b) KCODE 1000 1 30 130 MAX(4500,2∗NSRCK) 0

6500 1 none

(b) ACODE 1000 1 30 130 MAX(4500,2∗NSRCK) 0

1 automatic KALSAV+2 6500 0 0

(b) neutron criticality problems only

Tally speciﬁcation cards page 3–73

optional FnaR

o = 0 for n = 5

optional FCn none

optional En very large

optional Tn very large

optional Cn 1

optional FQn F D U S M C E T

optional FMn 1

optional DEn/DFn none

optional EMn 1

optional TMn 1

optional CMn 1

optional CFn none

optional SFn none

optional FSn none

optional SDn 0

TABLE 3.8: Summary of MCNP Input Cards

April 10, 2000 3-149

CHAPTER 3

SUMMARY OF MCNP INPUT FILE

optional FUn (Requires SUBROUTINE TALLYX)

optional TFn 1 1 last last 1 last last last

optional DD 0.1 1000

optional DXT – – – – – 0 0 0

optional FTn none

Material speciﬁcation cards page 3–107

optional Mm no ZAID default; 0; set internally; ﬁrst match in

XSDIR; .01p; .01e

(d) DRXS ∗fully continuous

(d) TOTNU *prompt for non-KCODE; total for KCODE

(d) NONU *ﬁssion treated as real ﬁssion

optional AWTAB *atomic weights from cross-section tables

optional XSn none

optional VOID none

optional PIKMT *no photon–production biasing

optional MGOPT *fully continuous

(d) neutron problems only

Energy and Thermal cards page 3–116

optional PHYS:N *very large 0 0

optional PHYS:P *100 0 0

optional PHYS:E *10000001111

(e) TMP 2.53 x 10−8

(e) THTME 0

(e) MTm none

(e) neutron problems only

Problem cutoffs page 3–123

optional CUT:N very large 0 −0.5 −0.25 SWTM

optional CUT:P very large .001 −0.5 −0.25 SWTM

optional CUT:E very large .001 0 0 SWTM

optional EPLT cut card energy cutoff

optional NPS none

optional CTME none

TABLE 3.8: Summary of MCNP Input Cards

ν ν

3-150 April 10, 2000

CHAPTER 3

SUMMARY OF MCNP INPUT FILE

B. Storage Limitations

Table 3.9 summarizes some of the more important limitations that have to be considered when

setting up a problem. It may be necessary to modify MCNP to change one or more of these

restrictions for a particular problem.

User arrays page 3–126

optional IDUM 0

optional RDUM 0

Peripheral cards page 3–127

optional PRDMP end −15 0 all 10 rendezvous points

optional LOST 10 10

optional DBCN (1519)152917 0006000001.E−410000

152917 519 000000

optional FILES none none sequential formatted –

optional PRINT *short output

optional MPLOT none

optional PTRAC none

optional PERT none

*This describes the effect of not using this particular card.

TABLE 3.9: Storage Limitations

Entries in the description of a cell *1000 after processing

Total number of tallies NTALMX = 100

Detectors MXDT = 20

Neutron DXTRAN spheres MXDX = 5

Photon DXTRAN spheres MXDX = 5

NSPLT or PSPLT card entries *10

Entries on IDUM card *50

Entries on RDUM card *50

*Set as a dimension in an array

TABLE 3.8: Summary of MCNP Input Cards

18 December 2000 4-1

CHAPTER 4

GEOMETRY SPECIFICATION

CHAPTER 4

EXAMPLES

In this chapter, cookbook examples of several topics provide instructive, real examples that you can

follow and learn from. They should be studied in conjunction with the theory and instructions of

Chapters 1, 2, and 3. You must understand the geometry discussions in Chapters 1 and 2 before

studying the following examples. The concept of combining regions of space bounded by surfaces

to make a cell must be fully appreciated; the following examples should help solidify this concept.

The use of macrobodies will simplify many geometry deﬁnition situations.

Following the geometry speciﬁcation examples are examples of coordinate transformation,

repeated structure and lattice geometries, tally options, source speciﬁcations, a SOURCE

subroutine, and SRCDX subroutines for point detectors and/or DXTRAN spheres. The tally

examples include the FMn, FSn, and FTn cards and the TALLYX subroutine for user-deﬁned

tallies using the FUn card.

I. GEOMETRY SPECIFICATION

Several more examples of the union and complement operators are given to help you understand

these features. In all examples, the cell numbers will be circled; the surface numbers will not be

circled but will appear next to the surface they represent. All cells are voids.

All examples in this chapter are available at Los Alamos from CFS under the /x6code/manual/

examples/chap4 node. The input ﬁle for the ﬁrst example is called exp1, etc. You are encouraged

to experiment with these ﬁles by plotting and modifying them.

The next several examples become progressively more difﬁcult and usually take advantage of what

you learned in the preceding ones. Remember that unless altered by parentheses, the hierarchy of

operations is that intersections are performed ﬁrst and then unions.

Example 1: In Figure 4.1a, surfaces 2 and 4 are cylinders and the others are planes with their

positive sides to the right. Cells 1 and 2 are easy to specify:

101−2−3

203−4−5

Cell 3 is harder, and you need to have in mind Figure 1.5 and its explanation. Remember that a

union adds regions and an intersection gives you only the areas that overlap or are common to both

regions. Regions can be added together more than once–or duplicated–with the union operator.

4-2 18 December 2000

CHAPTER 4

GEOMETRY SPECIFICATION

Let us start the deﬁnition of cell 3 at surface 2 (this is not a requirement). The expression 2 −3

deﬁnes the following region: everything in the world outside surface 2 intersected with everything

to the left of surface 3. This region is hatched in Figure 4.1b. Let us examine in detail how

Figure 4.1b was derived. First look at each region separately. The area with a positive sense with

respect to surface 2 is shown in Figure 4.1c. It includes everything outside surface 2 extending to

inﬁnity in all directions. The area with negative sense with respect to surface 2 is undeﬁned so far.

The area with negative sense with respect to surface 3 is shown in Figure 4.1d. It includes

everything to the left of surface 3 extending to inﬁnity, or half the universe. Recall that an

intersection of two regions gives only the area common to both regions or the areas that overlap.

Superimposing Figures 4.1c and 4.1d results in Figure 4.1e. The cross-hatched regions show the

space common to both regions. This is the same area hatched in Figure 4.1b.

Figure 4-1a. Figure 4.1b

Figure 4.1c Figure 4.1d

Let us now deal with surface 1. To the quantity 2 −3 we will add everything with a negative sense

with respect to surface 1 as indicated by the expression 2 −3: −1, or (2 −3): −1 if you prefer. Recall

(1) that in the hierarchy of operations, intersections are performed ﬁrst and then unions (so the

parentheses are unnecessary in the previous expression), and (2) that a union of two regions results

18 December 2000 4-3

CHAPTER 4

GEOMETRY SPECIFICATION

in a space containing everything in the ﬁrst region plus everything in the second region; this

includes everything common to both regions. Superimposing the region shown in Figure 4.1b and

the region to the left of surface 1 results in Figure 4.1f. Our geometry now includes everything

hatched plus everything crosshatched and has added part of the tunnel which is interior to surface 2.

By the same method we will deal with surface 4. To the quantity 2 −3: −1 we will add everything

with a positive sense with respect to surface 4, written as 2 −3: −1: 4. Figure 4.1g shows our new

geometry. It includes everything in Figure 4.1f plus everything outside surface 4.

Our ﬁnal step is to block off the large tunnel extending to inﬁnity to the right by adding the region

with a positive sense with respect to surface 5 to the region shown in Figure 4.1g. The ﬁnal

expression that deﬁnes cell 3 of Figure 4.1a is 2 −3: −1: 4: 5.

Figure 4.1e Figure 4.1f

Figure 4-1g.

There is more than one way to deﬁne cell 3. Starting with surface 1, we can add the region to the

left of 1 to the region outside surface 2 or −1: 2, which is illustrated in Figure 4.1h. We wish to

intersect this space with the space having a negative sense with respect to surface 3. Superimposing

4-4 18 December 2000

CHAPTER 4

GEOMETRY SPECIFICATION

Figure 4.1h and the region to the left of surface 3 results in Figure 4.1i. The cross-hatched area

indicates the area common to both regions and is the result of the intersection. Note that the cross-

hatched area of Figure 4.1i is identical to the entire hatched plus crosshatched area of Figure 4.1f.

Therefore, we have deﬁned the same geometry in both ﬁgures but have used two different

approaches to the problem. To ensure that the intersection of −3 is with the quantity −1: 2 as we

have illustrated, we must use parentheses giving the expression (−1: 2) −3. Remember the order in

which the operations are performed. Intersections are done before unions unless parentheses alter

the order. The ﬁnal expression is (−1: 2) −3: 4: 5.

Figure 4.1h Figure 4.1i

Another tactic uses a somewhat different approach. Rather than deﬁning a small region of the

geometry as a starting point and adding other regions until we get the ﬁnal product, we shall start

by deﬁning a block of space and adding to or subtracting from that block as necessary. We

arbitrarily choose our initial block to be represented by 4: −1: 5, illustrated in Figure 4.1j.

Figure 4-1j.

To this block we need to add the space in the upper and lower left corners. The expression 2 −3

isolates the space we need to add. Adding 2 −3 to our original block, we have 4: −1: 5: (2 −3). The

parentheses are not required for correctness in this case but help to illustrate the path our reasoning

has followed. Figure 4.1k depicts the union of 2 −3 with the block of space we originally chose.

18 December 2000 4-5

CHAPTER 4

GEOMETRY SPECIFICATION

Figure 4-1k.

Let us arbitrarily choose a different initial block, 4: 5: −3, all the world except cell 2. From this

region we need to subtract cell 1. If we intersect the region (2: −1) with (4: 5: −3), as shown in

Figure 4.1l, we will have introduced an undeﬁned tunnel to the right of surface 5. To correct this

error, deﬁne an area (2: −1: 3) or (2: −1: 5) and intersect this region with the initial block.

Figure 4.1l.

Another approach is to intersect the two regions −1: 2 and −3: 4, then add that to the region to the

right of surface 5 by (−1: 2)(−3: 4): 5. In the above paragraph the expression

(4:5:−3)(2 : −1: 5) can have the common quantity: 5 factored out, also resulting in

(−1: 2)(−3 : 4): 5.

Finally, another approach is to forget about the reality of the geometry and for cell 3 take the inverse

(or complement) of all the cells bounding cell 3, which is cells 1 and 2. This says that cell 3 is all

of the world excluding that which has already been deﬁned to be in cells 1 and 2. The advantage

of this is that cells 1 and 2 are easy to specify and you don’t get bogged down in details for cell 3.

Cell 3 thus becomes (−1:2:3)(−3:4:5).Note that the speciﬁcations for cells 1 and 2 are reversed.

Intersections become unions. Positive senses become negative. Then each piece is intersected with

the other. There is a complement operator in MCNP that is a shorthand notation for the above

4-6 18 December 2000

CHAPTER 4

GEOMETRY SPECIFICATION

expression; it is the symbol #, which can be thought of as meaning not in. Therefore, cell 3 is

speciﬁed by #1 #2, translated as everything in the world that is not in cell 1 and not in cell 2.

Example 2:

Figure 4-2.

Cell 1 is everything interior to the surfaces 1 and 2:

10−1: −2

201 2

Example 3:

Figure 4-3.

In this geometry of four cells deﬁned by three spheres, cell 3 is disconnected. Cell 3 is the region

inside surface 3 but outside surfaces 1 and 2 plus the region enclosed between surfaces 1 and 2:

10−12

20−21

18 December 2000 4-7

CHAPTER 4

GEOMETRY SPECIFICATION

30−312:−2−1

403

Cell 3 could also be written as

30(−312):(−2−1) The parentheses are not required.

Example 4:

Figure 4-4.

In this example all vertical lines are planes with their positive sides to the right and all horizontal

lines are cylinders. Cells 1, 2, and 3 are simple; they are deﬁned by 1 -2 -3, 3 −4−5, and 5 −6−7,

respectively. Cell 4 is simple if the complement operator is used; it is #1 #2 #3 #5 or #1 #2 #3 −8.

Cell 5 is also simple; it is no more than 8 (or verbally, everything in the world with a positive sense

with respect to surface 8).

If cell 5 were deﬁned as just #4, it would be incorrect. That says cell 5 is everything in the universe

not in cell 4, which includes cells 1, 2, and 3. The speciﬁcation #4 #1 #2 #3 is correct but should

not be used because it tells MCNP that cell 5 is bounded by surfaces 1 through 7 in addition to

surface 8. This will cause MCNP to run signiﬁcantly more slowly than it should because anytime

a particle enters cell 5 or has a collision in it, the intersection of the particle’s trajectory with each

bounding surface has to be calculated.

Specifying cell 4 exclusively with the complement operator is very convenient and

computationally efﬁcient in this case. However, it will be instructive to set up cell 4 explicitly

without complements. There are many different ways to specify cell 4; the following approach

should not be considered to be the way.

First consider cell 4 to be everything outside the big cylinder of surface 4 that is bounded on each

end by surfaces 1 and 7. This is speciﬁed by (−1:4:7). The parentheses are not necessary but may

1 2 3 4 5

4-8 18 December 2000

CHAPTER 4

GEOMETRY SPECIFICATION

add clarity. Now all that remains is to add the corners outside cylinders 2 and 6. The corner outside

cylinder 2 is (2 −3), whereas it is (5 6) outside cylinder 6. Again the parentheses are optional. These

corners are then added to what we already have outside cylinder 4 to get

(−1:4:7):(2 −3):(5 6)

The region described so far does not include cells 1, 2, or 3 but extends to inﬁnity in all directions.

This region needs to be terminated at the spherical surface 8. In other words, cell 4 is everything

we have deﬁned so far that is also common with everything inside surface 8 (that is, everything so

far intersected with −8). So as a ﬁnal result,

((−1:4:7):(2 −3):(5 6)) −8

The inner parentheses can be removed, but the outer ones are necessary (remember the hierarchy

of operations) to give us

(−1:4:7:2 −3:5 6) −8

If the outer parentheses are removed, the intersection of −8 will occur only with 5 and 6, an event

that is clearly incorrect.

Example 5:

Figure 4-5.

This example is similar to the previous one except that a vertical cylinder (surface 4) is added to

one side of the horizontal cylinder (surface 3).

Cell 1 is (1 −3−2), cell 3 is #1 #2 #4, and cell 4 is just 6.

3 4

18 December 2000 4-9

CHAPTER 4

GEOMETRY SPECIFICATION

Cell 2 is more than might initially meet the eye. It might appear to be simply (−5−4 3), but this

causes a mirror image of the cell 2 we want to show up on the bottom half of cell 1, as represented

by the dashed lines in Figure 4.5. We need to add an ambiguity surface to keep cell 2 above the y-

axis. Let surface 7 be an ambiguity surface that is a plane at z= 0. This surface appears in the

MCNP input ﬁle as any other surface. Then cell 2 becomes (−5−4 3 7) for the ﬁnal result. You

should convince yourself that the region above surface 7 intersected with the region deﬁned by −5

−4 3 is cell 2 (don’t even think of surface 7 as an ambiguity surface but just another surface deﬁning

some region in space). The mirror problem can also be avoided by deﬁning cells 1 and 2 as a right

circular cylinder (rcc) macrobodies. The necessary cards for deﬁning the macrobodies would be

1 rcc 0 -2 0 0 4 0 4

2 rcc 0 0 0 0 0 7 1

In this case cells 1,2 and 3 would simply be (-1), (-2 1), and (1 2 -6) respectively. Notice that to get

the interface between the cylinders correct macrobody 2 extends into cell 1 and is then truncated

by the deﬁnition of cell 1.

Example 6:

Figure 4-6.

This is three concentric spheres with a box cut out of cell 3. Surface 8 is the front of the box and 9

is the back of the box. The cell cards are

10−1

20−21

30−32 (−4:5:−6:7:8:−9) $ These parentheses are required.

123

4-10 18 December 2000

CHAPTER 4

GEOMETRY SPECIFICATION

403

504−5 6 −7−8 9

Cell 3 is everything inside surface 3 intersected with everything outside surface 2 but not in cell 5.

Therefore, cell 3 could be written as

30−3 2 #(4 −5 6 −7−8 9)

or 3 0 −3 2#5

or 3 0 −3 2(−4:5:−6:7:8:−9)

Cell 5 could also be speciﬁed using a RPP macrobody. The correct cell and surface cards for this

would be

5 0 -4 $

4 rrp 2 4 7.5 8.5 -2 2

Example 7:

Figure 4-7.

This is three concentric boxes, a geometry very challenging to set up using only intersections,

easier with unions, and almost trivial with the BOX macrobody. Surfaces 5, 11, and 17 are the back

sides of the boxes (smaller to larger, respectively); 6, 12, and 18 are the fronts:

10−2 −3 41 5 −6

20−7 −8 910 11−12

(2 : 3 : −4 :−1 :−5 :6)

30−13 −14 15 16 17 −18

(7 : 8 : −9 :−10 :−11 : 12)

4013:14:−15:−16 :−17 :18

15 1 2 3 4

18 December 2000 4-11

CHAPTER 4

GEOMETRY SPECIFICATION

Example 8:

Figure 4-8.

This is two concentric spheres with a torus attached to cell 2 and cut out of cell 1:

10−1 4

20−2 (1 : −4)

302

If the torus were attached to cell 1 and cut out of cell 2, this bug-eyed geometry would be:

10−1 :−4

20−2 14

302

Example 9:

123

1722

4-12 18 December 2000

CHAPTER 4

GEOMETRY SPECIFICATION

Figure 4-9.

Cell 9 is a box cut out of the left part of spherical cell 17; surface 9 is the front of the box and 8 is

the rear. Cell 17 is disconnected; the right part is the space interior to the spheres 6 and 7. An F4

tally in cell 17 would be the average ﬂux in all parts of cell 17. An F2 surface tally on surface 7

would be the ﬂux across only the solid portion of surface 7 in the ﬁgure. The cell speciﬁcations are:

90−3 −2 4 1 8 −9

170−5 (3 : −4 : −1 : 2 : 9 : −8) : −6 : −7

22 0 5 6 7

A variation on this problem is for the right portion of cell 17 to be the intersection of the interiors

of surfaces 6 and 7 (the region bounded by the dashed lines in the above ﬁgure):

90−3 −2 4 1 8 −9

170−5 (3 : −4 : −1 : 2 : 9 : −8) : −6 −7

22 0 5 (6 : 7)

Example 10:

Figure 4-10.

This is a box with a cone sitting on top of it. Surface 6 is the front of the box and 7 is the rear. You

should understand this example before going on to the next one.

1012−3 (-4 : −5)−6 7

20−1 : −2 : 3 : 4 5 : 6 : −7

This problem could be simpliﬁed by replacing surfaces 1-6 with a BOX macrobody. The resulting

cell and surface cards would be

c cell cards

1 0 -8:(-5 8.5)

2 0 #1 $ or -8.4:-8.6:8.3:(8.5 5):8.1:-8.2

18 December 2000 4-13

CHAPTER 4

GEOMETRY SPECIFICATION

c surface cards

5 kz 8 0.25 -1

8 box -2.5 -2.5 0 5 0 0 0 5 0 0 0 5

Example 11: Surfaces 15 and 16 are cones, surface 17 is a sphere, and cell 2 is disconnected.

10−1 23 (−4 :−16) 5 −6 (12 : 13 : −14)

(10 : −9 :−11 :−7 : 8) 15

20−10 9117−8 −1 : 2 −12 14 −6 −13 3

30−17 (1 : −2 :−5 : 6 : −3 :−15 : 16 4)

4017

Figure 4-11.

Example 12:

Figure 4-12.

910

11 16

4-14 18 December 2000

CHAPTER 4

GEOMETRY SPECIFICATION

Cell 1 consists of two cylinders joined at a 45° angle. Cell 2 is a disk consisting of a cylinder

(surface 8) bounded by two planes. Surface 5 is a diagonal plane representing the intersection of

the two cylinders. The problem is to specify the disk (cell 2) in one cell formed by the two cylinders

(cell 1). A conﬂict arises in specifying cell 1 since, from the outside cell 3, corner abetween

surfaces 1 and 3 is convex, but on the other side of the cell the same two surfaces form a concave

corner at b. The dilemma is solved by composing cell 1 of two disconnected cells, each bounded

by surface 5 between the corners aand b. Surface 5 must be included in the list of surface cards in

the MCNP input ﬁle. When the two parts are joined to make cell 1, surface 5 does not appear.

Convince yourself by plotting it using an origin of 0 0 24 and basis vectors 0 1 1 0 −1 1. See

Appendix B for an explanation of plotting commands.

10(2−1 −5 (7:8:−6)):(4 −3 5(−6:8:7))

20−8 6−7

30(−2:1:5)(−4:3:−5)

A more efﬁcient expression for cell 1 is

10(2−1 −5:4 −3 5)(−6:8:7)

Example 13:

This example has the most complicated geometry so far, but it can be described very simply. The

input ﬁle is called antares and is available from the /x6code/manual/examples/chap4 node.

You can see that Example 13 is similar to Example 1. There is just a lot more of it.It is possible to

set this geometry up by any of the ways mentioned in Example 1. However, going around the outer

surfaces of the cells inside cell 10 is tedious. There is a problem of visualization and also the

problem of coming up with undeﬁned tunnels going off to inﬁnity as in Example 1.

18 December 2000 4-15

CHAPTER 4

GEOMETRY SPECIFICATION

Figure 4-13.

The way to handle this geometry is by the last method in Example 1. Set up the cell/surface

relations for each interior cell, then just take the complement for cell 10. For the interior cells,

101−2 −23

20−3 25 −24 2

303−5 12 −15 16 −11

405−6 12 −17 18 −11

506 −8 12 −13 −19 20

608 −9 −26

70−12 4 −7 −27

80−12 7−10 14 −21 22

902−3 −25

Cell 10 is surrounded by the spherical surface 28. Considering cell 10 to be everything outside cells

1 through 9 but inside surface 28, one can reverse the senses and replace all intersections with

unions to produce

35 6 8

15 17 19

16 18 20

123 4 5 10

35 6 8

13 26

107

3 4 5

4-16 18 December 2000

CHAPTER 4

GEOMETRY SPECIFICATION

100(−1:2:23)(3:−25:24:−2)

(−3:5:−12:15:−16:11)

(−5:6:−12:17:−18:11)

(−6:8:−12:13:19:−20)

(−8:9:26)(12:−4:7:27)

(12:−7:10:−14:21:−22)

(−2:3:25) −28

Note how easy cell 10 becomes when the complement operator is used:

10 0 #1 #2 #3 #4 #5 #6 #7 #8 #9 −28

Once again this example can be greatly simpliﬁed by replacing all but cell 7 with macrobodies.

However the deﬁnition of cell 7 must then be changed to use the facets of the surrounding

macrobodies instead of surfaces 12 and 7. The facets of macrobodies can be visualized using the

MBODY OFF option of the geometry plotter.

Example 14:

Figure 4-14.

This example illustrates some necessary conditions for volume and area calculations. The

geometry has three cells, an outer cube, an inner cube, and a sphere at the center. If cell 3 is

described as

308−9 −10 11 −12 13 #2 #1

(and #1 must be included to be correct), the volume of cell 3 cannot be calculated. As described, it

is not bounded by all planes so it is not a polyhedron, nor is it rotationally symmetric. If cell 3 is

described by listing all 12 bounding surfaces explicitly, the volume can be calculated.

18 December 2000 4-17

CHAPTER 4

COORDINATE TRANSFORMATIONS

II. COORDINATE TRANSFORMATIONS

In most problems, the surface transformation feature of the TRn card will be used with the default

value of 1 for M. When M= 1 applies, most of the geometry can be set up easily in an (x,y,z)

coordinate system and only a small part of the total geometry will be difﬁcult to specify. For

example, a box with sides parallel to the (x,y,z) coordinate system is simple to describe, but inside

might be a tilted object consisting of a cylinder bounded by two planes. Since the axis of the

cylinder is neither parallel to nor on the x,y,orzaxis, a general quadratic must be used to describe

the surface of the cylinder. The GQ surface card has 10 entries that are usually difﬁcult to

determine. On the other hand, it is simple to specify the entries for the surface card for a cylinder

centered on the y-axis. Therefore, we deﬁne an auxiliary coordinate system (x′,y′,z′) so the axis of

the cylinder is one of the primed axes, y′ for example. Now we will use the TRn card to describe

the relationship between one coordinate system and the other. M=1requires that the coordinates

of a vector from the (x,y,z) origin to the (x′,y′,z′) origin be given in terms of (x,y,z).

Only in rare instances will M=−1be needed. Some unusual circumstances may require that a small

item of the geometry must be described in a certain system which we will call (x,y,z), and the

remainder of the surfaces would be easily described in an auxiliary system (x′,y′,z′). The Oientries

on the TRn card are then the coordinates of a vector from the (x′,y′,z′) origin to the (x,y,z) origin

given in terms of the primed system.

Example 1: The following example consists of a can whose axis is in the yz plane and is tilted 30°

from the y-axis and whose center is at (0,10,15) in the (x,y,z) coordinate system. The can is bounded

by two planes and a cylinder, as shown in Figure 4.15.

The surface cards that describe the can in the simple (x′,y′,z′) system are:

11CY 4

21PY−7

31PY 7

Figure 4-15.

Y’

Z’

(X,Y,Z)=(0,10,15)

4-18 18 December 2000

CHAPTER 4

COORDINATE TRANSFORMATIONS

The 1 before the surface mnemonics on the cards is the n that identiﬁes which TRn card is to be

associated with these surface cards. The TRn card indicates the relationship of the primed

coordinate system to the basic coordinate system.

We will specify the origin vector as the location of the origin of the (x′,y′,z′) coordinate system with

respect to the (x,y,z) system; therefore, M= 1. Since we wanted the center of the cylinder at

(0,10,15), the Oientries are simply 0 10 15. If, however, we had wanted surface 2 to be located at

(x,y,z) = (0,10,15), a different set of surface cards would accomplish it. If surface 2 were at y′=0

and surface 3 at y′=14 the Oi entries would remain the same. The signiﬁcant fact to remember

about the origin vector entries is that they describe one origin with respect to the other origin. The

user must locate the surfaces about the auxiliary origin so that they will be properly located in the

main coordinate system.

The Bi entries on the TRn card are the cosines of the angles between the axes as listed on page 3–

26 in Chapter 3. In this example, the x-axis is parallel to the x′-axis. Therefore, the cosine of the

angle between them is 1. The angle between yand x′is 90° with a cosine of 0. The angle between

zand x′and also between xand y′is 90° with a cosine of 0. The angle between yand y′is 30° with

a cosine of 0.866. The angle between z and y′is 60° with 0.5 being the cosine. Similarly, 90° is

between xand z′;120° is between yand z′; and 30° is between zand z′. The complete TRn card is

TR1 0 10 15 1 0 0 0 .866 .5 0 −.5 .866

An asterisk preceding TRn indicates that the Bi entries are the angles in degrees between the

appropriate axes. The entries using the ∗TRn mnemonic are

∗TR1 0 10 15 0 90 90 90 30 60 90 120 30

The default value of 1 for M, the thirteenth entry, has been used and is not explicitly speciﬁed.

The user need not enter values for all of the Bi. As shown on page 3–26, Bimay be speciﬁed in any

of ﬁve patterns. Pattern #1 was used above, but the simplest form for this example is pattern #4

since all the skew surfaces are surfaces of revolution about some axis. The complete input card then

becomes

∗TR1 0 10 15 3J 90 30 60

Example 2: The following example illustrates another use of the TRn card. The ﬁrst part of the

example uses the TR1 card and an M=1transformation; the second part with the TR2 card uses

an M=−1 transformation. Both parts and transformations are used in the following input ﬁle.

EXAMPLE OF SURFACE TRANSFORMATIONS

20 −4 3−5

60 −14 −13 :−15 41 −42

31 PX−14

18 December 2000 4-19

CHAPTER 4

COORDINATE TRANSFORMATIONS

41 X −14 10 0 12 14 10

5 1 PX 14

13 2 SX −15 70

14 2 CX 30

15 2 Y 75 0 30 16

41 2 PY 0

42 2 PY 75

TR1 20 31 37 .223954 .358401 .906308

TR2 −250 −100 −65 .675849 .669131 .309017

J J .918650 J J −.246152 −1

A. TR1 and M = 1 Case:

Cell 2 is bounded by the plane surfaces 3 and 5 and the spheroid surface 4, which is a surface of

revolution about the skew axis x’ in Figure 4.16.

Figure 4-16.

To get the coefﬁcients of surfaces 3, 4, and 5, deﬁne the x′axis as shown in the drawings (since the

surfaces are surfaces of revolution about the x′ axis, the orientation of the y′ and z′ axes does not

matter), then set up cell 2 and its surfaces, with coefﬁcients deﬁned in the x′y′z′coordinate system.

On the TR1 card, the origin vector is the location of the origin of the x′y′z′coordinate system with

respect to the main xyz system of the problem. The pattern #4 on page 3–26 in Chapter 3 is

Z(up)

x’

center is at

(X,Y,Z)=(20,31,37)

tilted 25 from vertical

x’

28 24

x’

vertical plane containing x’

axis is 32 from YZ plane

4-20 18 December 2000

CHAPTER 4

COORDINATE TRANSFORMATIONS

appropriate since the surfaces are all surfaces of revolution about the x′ axis. The components of

one vector of the transformation matrix are the cosines of the angles between x′and the x,y, and z

axes. They are obtained from spherical trigonometry:

B. TR2 and M = −1 Case:

Cell 6 is the union of a can bounded by spherical surface 13 and cylindrical surface 14 and a conical

piece bounded by conical surface 15 and ambiguity surfaces 41 and 42, which are planes. (Surface

42 is required because when surface 15 is deﬁned in x′y′z′it is as a type Y surface, which becomes

a cone of one sheet; when it is transformed into the xyz system it becomes a type GQ surface, which

in this case is a cone of two sheets. Weird, but that’s the way it has to be.) Surfaces 13 and 14 are

surfaces of revolution about one axis, and surfaces 15, 41, and 42 are surfaces of revolution about

an axis perpendicular to the ﬁrst axis. Both axes are skewed with respect to the xyz coordinate

system of the rest of the geometry.

Deﬁne the auxiliary x′y′z′coordinate system as shown in Figure 4.17. Set up cell 6 with its surfaces

speciﬁed in the x′y′z′ coordinate system as part of the input ﬁle and add a second transformation

card, TR2.

Because the location of the origin of the xyz coordinate system is known relative to the x′y′z′system

(rather than the other way around, as in the ﬁrst part of the example), it is necessary to use the

reverse mapping. This is indicated by setting M= −1. In this reverse mapping the origin vector (−

250,−100,−65) is the location of the origin of the xyz system with respect to the x′y′z′system. For

the components of the transformation matrix, pattern #3 out of the four possible choices from

Chapter 3 is most convenient here. The xyz components of z′and the x′y′z′components of zare easy

to get. The components of xand of yare not. The whole transformation matrix is shown here with

X’

90F

G=25

cos E = cos 58˚ x sin 25˚ = .223954

cos F = cos 32˚ x sin 25˚ = .358401

cos G = cos 25˚ = .906308

18 December 2000 4-21

CHAPTER 4

REPEATED STRUCTURE AND LATTICE EXAMPLES

Figure 4-17.

the components that are obtained from Figure 4.17 written in:

The zz′component is −SQRT(1. −.309107∗∗2−.918650∗∗2) = −.246152, and the xx′component

is SQRT(1. − .669131∗∗2− .309017∗∗2) = .675849, with the signs determined by inspection of the

ﬁgure.

III. REPEATED STRUCTURE AND LATTICE EXAMPLES

Example 1: This example illustrates the use of transformations with simple repeated structures.

The geometry consists of a sphere enclosing two boxes that each contain a cylindrical can. Cell 2

is ﬁlled by universe 1. Two cells are in universe 1—the cylindrical can, cell 3, and the space outside

the can, cell 4. Cell 2 is deﬁned and the LIKE m BUT card duplicates the structure at another

location. The TRCL entry identiﬁes a TRn card that deﬁnes the displacement and rotational axis

transformation for cell 5. To plot type:

b100 010 ex11 or3.53.50

xy z

x′.675849 cos 48° = .669131 cos 72° = .309017

y′cos 15° × cos 18° = .918650

z′ −.246152

Z axis is 18 from

the Y’Z’ plane

YY axis is 42 from

the Y’Z’ plane

250

41 X’

Y’

60 6

13 13

85 55

Z’

Y’

projection

of the Z axis

on the Y’Z’

plane is 15

from the X’Y’plane

100

4-22 18 December 2000

CHAPTER 4

REPEATED STRUCTURE AND LATTICE EXAMPLES

simple repeated structures

1 0 -27 #2 #5 imp:n=1

2 0 1 -2 -3 4 -5 6 ﬁll=1 imp:n=1

3 0 -10 -11 12 u=1 imp:n=1

4 0 #3 u=1 imp:n=1

5 like 2 but trcl=3

7 0 27 imp:n=0

1px−3

2px 3

3py 3

4py−3

5 pz 4.7

6 pz –4.7

10 cz 1

11 pz 4.5

12 pz –4.5

27 s 3.5 3.5 0 11

sdef pos 3.5 3.5 0

f2:n 1

tr3∗7 7 0 40 130 90 50 40 90 90 90 0

nps 10000

18 December 2000 4-23

CHAPTER 4

REPEATED STRUCTURE AND LATTICE EXAMPLES

Example 2: This example illustrates the LIKE m BUT construct, the FILL card, the U card, two

forms of the TRCL card, and a multiple source cell deﬁnition. The following line will plot the view

shown on the left:

b100 010 ex21 la0

In this example ﬁve cells, numbers 2 through 6, are identical except for their locations. Cell 2 is

described fully and the other four are declared to be like cell 2 but in different locations. Cell 2 is

deﬁned in an auxiliary coordinate system that is centered in the cell for convenience. That

coordinate system is related to the main coordinate system of the problem by transformation

number 2, as declared by the TRCL = 2 entry and the TR2 card. Cells 2 through 6 are all ﬁlled with

universe number 1. Because no transformation is indicated for that ﬁlling, universe 1 inherits the

transformation of each cell that it ﬁlls, thereby establishing its origin in the center of each of those

ﬁve cells. Universe 1 contains three inﬁnitely long tubes of square cross section embedded in cell

11, which is unbounded. All four of these inﬁnitely large cells are truncated by the bounding

surfaces of each cell that is ﬁlled by universe 1, thus making them effectively ﬁnite. The

transformations that deﬁne the locations of cells 8, 9 and 10 are entered directly on the cell cards

after the TRCL symbol rather than indirectly through TR cards as was done for cells 2 through 6

to illustrate the two possible ways of doing this. Cells 8, 9 and 10 are each ﬁlled with universe 2,

which consists of ﬁve inﬁnite cells that are truncated by the boundaries of higher level cells. The

simplicity and lack of repetition in this example were achieved by careful choice of the auxiliary

cell 8 cell 9

cell 10

cell 2

cell 3

cell 4cell 5

cell 6

4-24 18 December 2000

CHAPTER 4

REPEATED STRUCTURE AND LATTICE EXAMPLES

coordinate systems at all levels. All of the location information is contained in just a few TRCL

entries, some direct and some pointing to a few TR cards.

The source deﬁnition is given on the SDEF, SIn and SPn cards. The source desired is a cylindrical

volume distribution, equally probable in all the cylindrical rods. The energies are given by

distribution 1. The entry for CEL shows that level 0 cells are given by distribution 2 and level 1

cells by distribution 3. The zero means that cells are searched for at level 2 and also that the

sampled position and direction will apply to the universe indicated by the entry just preceding the

ﬁrst entry that is ≤0. In this case the position and direction will be deﬁned in the coordinate system

of the cell sampled by distribution 3 at level 1. The SI2 card lists all the cells at level 0 that will

contain the source. SP2 indicates equal probability. SI3 lists the cells in level 1 and the positions

on the SI7 card are given in the coordinates of this level. A cylindrical volume distribution is

speciﬁed by RAD, EXT, AXS, and POS. The radius on the SI5 card is from 0 to .1. The ends of the

cylinder are at -2 and 2 (SI6) and the four sets of entries on the SI7 card are the origins of the four

cylinders of cells 12–15. These parameters describe exactly the four cells 12–15.

chapter 4 example 2

11−.5 −7 #2 #3 #4 #5 #6 imp:n=1

2 0 1 -2 -3 4 5 -6 imp:n=2 trcl=2 fill=1

3 like 2 but trcl=3

4 like 2 but trcl=4

5 like 2 but trcl=5 imp:n=1

6 like 2 but trcl=6

7 0 7 imp:n=0

8 0 8 -9 -10 11 imp:n=1 trcl=(−.9 .9 0) fill=2 u=1

9 like 8 but trcl=(.9 .9 0)

10 like 8 but trcl=(.1 -.9 0)

11 2 −18 #8 #9 #10 imp:n=1 u=1

12 2 -18 -12 imp:n=1 trcl=(-.3 .3 0) u=2

13 like 12 but trcl=(.3 .3 0)

14 like 12 but trcl=(.3 -.3 0)

15 like 12 but trcl=(-.3 -.3 0)

16 1 -.5 #12 #13 #14 #15 u=2 imp:n=1

1 px -2

2 py 2

3 px 2

4py−2

5pz−2

6 pz 2

7 so 15

8px−.7

18 December 2000 4-25

CHAPTER 4

REPEATED STRUCTURE AND LATTICE EXAMPLES

9py.7

10 px .7

11 py −.7

12 cz .1

sdef erg=d1 cel=d2:d3:0 rad=d5 ext=d6 axs=0 0 1 pos=d7

# si1 sp1 sb1

100

3 .22 .05

4 .08 .05

5 .25 .1

6 .18 .1

7 .07 .2

8.1.2

9 .05 .1

11 .05 .2

si2 l 2 3 4 5 6

sp2 1 1 1 1

si3 l 8 9 10

sp3 1 1 1

si5 0 .1

sp5 -21 1

si6 -2 2

sp6 0 1

si7 l .3 .3 0 .3 -.3 0 -.3 .3 0 -.3 -.3 0

sp7 1 1 1 1

m1 6000 1

m2 92235 1

drxs

tr2 -6 7 1.2

tr3 7 6 1.1

tr4 8 -5 1.4

tr5∗-1 -4 1 40 130 90 50 40 90 90 90 0

tr6 -9 -2 1.3

f4:n 2 3 4 5 6 12 13 14 15

e4 1 3 5 7 9 11 13

sd4 5j 1.8849555921 3r

fq f e

cut:n 1e20 .1

nps 100000

4-26 18 December 2000

CHAPTER 4

REPEATED STRUCTURE AND LATTICE EXAMPLES

Cell 2 could be replaced with a RPP macrobody that can then be replicated and translated

identically to cell 2 above.

Example 3:

This is a simple example illustrating the use of the FILL, U, and LAT cards to create an object

within several cells of a lattice. A hexahedral lattice is contained within a cylinder of radius 45 cm.

Cell 1 is the interior of the cylinder, and cell 5 is everything outside (all surfaces are inﬁnite in the

z-direction). Cell 1 is ﬁlled by universe 1. Cell 2 is deﬁned to be in universe 1. Surfaces 301-304

deﬁne the dimensions of the square lattice.

When ﬁlling the cells of a lattice, all cells visible, even partially, must be speciﬁed by the FILL

card. In this case, the “window” created by the cylinder reveals portions of 25 cells (5x5 array). A

FILL card with indices of –2 to 2 in the x- and y-directions will place the [0,0,0] element at the

center of the array. Universe 2, described by cells 3 and 4, is the interior and exterior, respectively,

of an inﬁnite cylinder of radius 8 cm. The cells in universe 1 not ﬁlled by universe 2 are ﬁlled by

universe 1, in effect they are ﬁlled by themselves. The following ﬁle describes a cylinder that

contains a square lattice, with the inner 3x3 array of cells containing a small cylinder in each cell.

simple lattice

1 0 –1 fill=1 imp:n=1

2 0 –301 302 –303 304 lat=1 u=1 imp:n=1 fill=-2:2 –2:2 0:0

1 1 1 1 1 1 2 2 2 1 1 2 2 2 1 1 2 2 2 1 1 1 1 1 1

18 December 2000 4-27

CHAPTER 4

REPEATED STRUCTURE AND LATTICE EXAMPLES

3 0 –10 u=2 imp:n=1

4 0 #3 imp:n=1 u=2

5 0 1 imp:n=0

1 cz 45

10 cz 8

301 px 10

302 px –10

303 py 10

304 py –10

Example 4:

This example illustrates a lattice geometry and uses the FILL entries followed by transformations,

the U card, and the LAT card. Cell 2 is the bottom half of the large sphere outside the small sphere

(cell 1), is ﬁlled by universe 1, and the transformation between the ﬁlled cell and the ﬁlling universe

immediately follows in parentheses.

Cell 6 describes a hexahedral lattice cell (LAT=1) and by the order of speciﬁcation of its surfaces,

also describes the order of the lattice elements. The (0,0,0) element has its center at

(–6 –6.5 0) according to the transformation information on the card for cell 2. Element (1,0,0) is

0,0,0

cell 9

cell 8

cell 7 cell 7

4-28 18 December 2000

CHAPTER 4

REPEATED STRUCTURE AND LATTICE EXAMPLES

beyond surface 5, element (–1,0,0) beyond surface 6, (0,1,0) beyond surface 7, etc. Cell 6 is ﬁlled

by universe 3, which consists of two cells: cell 8 inside the ellipsoid and cell 9 outside the ellipsoid.

When a lattice cell is deﬁned with a macrobody, the lattice element indexing is somewhat

predetermined. The ﬁrst, third and ﬁfth facets are used to deﬁne the direction of increasing indices.

For the RPP, the second index increases in the positive y direction and the third index increases in

the positive z direction. For the BOX, the order of deﬁning the three vectors will determine the axis

each index will increase in a positive direction.

Cell 3 is the top left-hand quarter of the sphere; cell 4 is the top right-hand quarter. Both are ﬁlled

by universe 2. Both FILL entries are followed by a transformation. The interorigin vector portion

of the transformation is between the origin of the ﬁlled cell and the origin of the ﬁlling universe,

with the universe considered to be in the auxiliary coordinate system. The (0,0,0) lattice element is

located around the auxiliary origin and the lattice elements are identiﬁed by the ordering of the

surfaces describing cell 7. The skewed appearance is caused by the rotation part of the

transformation.

The source is centered at (0,–5,0) (at the center of cell 1). It is a volumetric source ﬁlling cell 1,

and the probability of a particle being emitted at a given radius is given by the power law function.

For RAD the exponent defaults to 2, so the probability increases as the square of the radius,

resulting in a uniform volumetric distribution.

example 4

–

1 imp:n=1

20 1

–

4 fill=1 (

–

–6

.5 0) imp:n=1

30 2

−

∗

fill=2 (

−

7 5 0 30 60 90 120 30 90) imp:n=1

4 0 2 3

−

∗

fill=2 (4 8 0 15 105 90 75 15 90) imp:n=1

5 0 4 imp:n=1

−

5 6

−

$7 8

−

9 10 fill=3 u=1 lat=1 imp:n=1

−

2.7

−

11 12

−

13 14

−

15 16 u=2 lat=1 imp:n=1

−

17 u=3

9 0 17 u=3

1sy

−

5 3

2 py 0

3 px 0

4 so 15

5 px 1.5

6px

−

1.5

7 py 1

8py

−

9 pz 3

18 December 2000 4-29

CHAPTER 4

REPEATED STRUCTURE AND LATTICE EXAMPLES

−

11 p 1

−

.5 0 1.3

12 p 1

−

.5 0

−

1.3

13 py .5

14 py

−

15 pz 3

16 pz

−

17 sq 1 2 0 0 0 0

−

1 .2 0 0

sdef pos 0

−

5 0 erg d1 rad d2

si1 0 10

sp1 0 1

si2 3

sp2

−

e0 1 2 3 4 5 6 7 8 9 10 11 12

f2:n 3

sd2 1

f4:n 8 9

sd4 1 1

m1 4009 1

m2 6000 1

m3 13027 1

nps 100000

dbcn 0 0 1 4

4-30 18 December 2000

CHAPTER 4

REPEATED STRUCTURE AND LATTICE EXAMPLES

Example 5:

This example illustrates a more complicated lattice geometry and uses the FILL card followed by

the array speciﬁcation. It builds on the expertise from example 4. There are three “main” cells: cell

1 is inside surface 5, cell 3 is the outside world, and cell 2 is the large square (excluding cell 1) that

is ﬁlled with a lattice, some of whose elements are ﬁlled with 3 different universes.

Universe 1 is a hexahedral lattice cell inﬁnite in the z direction. Looking at the FILL parameters,

we see that the lattice has ﬁve elements in the ﬁrst direction numbered from -2 to 2, nine elements

in the second direction numbered from -4 to 4 and one element in the third direction. The remaining

entries on the card are the array that identiﬁes what universe is in each element, starting in the lower

left hand corner with (-2,-4,0), (-1,-4,0), (0,-4,0), etc. An array entry, in this case 1, the same as the

number of the universe of the lattice means that element is ﬁlled by the material speciﬁed for the

lattice cell itself. Element (1,-3,0) is ﬁlled by universe 2, which is located within the element in

accordance with the transformation deﬁned on the TR3 card. Element (-1,-2,0) is ﬁlled by universe

3. Cell 7, part of universe 3, is ﬁlled by universe 5, which is also a lattice. Note the use of the X

card to describe surface 13. The quadratic surface, which is symmetric about the x-axis, is deﬁned

by specifying three coordinate pairs on the surface.

The source is a volumetric source of radius 3.6 which is centered in and completely surrounds cell

1. CEL rejection is used to uniformly sample throughout the cell. That is, the source is sampled

uniformly in volume and any points outside cell 1 are rejected. The same effect could have been

18 December 2000 4-31

CHAPTER 4

REPEATED STRUCTURE AND LATTICE EXAMPLES

achieved using cookie-cutter rejection. The PRINT card results in a full output print, and the VOL

card sets the volumes of all the cells to unity.

example 5

1 1 -.6 -5 imp:n=1

2 0 -1 2 -3 4 5 -22 23 imp:n=1 fill=1

3 0 1:-2:3:-4:22:-23 imp:n=0

4 2 -.8 -6 7 -8 9 imp:n=1 lat=1 u=1

fill=-2:2 -4:4 0:0 1 1111 1112(3) 1 1 3111

1 2 3 2 1 1 1 1 1 1 1 4(2) 2 1 1 1 1 3 4(1) 1

1 2 3 1 1 1 1 1 1 1

5 3 -.5 -11 10 12 imp:n=1 u=2

6 4 -.4 11:-10:-12 imp:n=1 u=2

7 0 -13 imp:n=1 u=3 fill=5

8 3 -.5 13 imp:n=1 u=3

9 4 -.4 -14 15 -16 17 imp:n=1 lat=1 u=5

10 3 -.5 -18 19 -20 21 imp:n=1 u=4

11 4 -.4 18:-19:20:-21 imp:n=1 u=4

1 px 15

2 px -15

3 py 15

4 py -15

5 s 7 2.1 0 3.5

6 px 4

7 px -5

8 py 2

9 py -2

10 p .7 −.7 0 −2.5

11 p .6 .8 0 .5

12 py −1

13 x −4.5 0 −.5 1.7 3.5 0

14 px 1.6

15 px −1.4

16 py 1

17 py −1.2

18 px 3

19 px −3

20 py .5

21 py −.6

22 pz 6

23 pz −7

4-32 18 December 2000

CHAPTER 4

REPEATED STRUCTURE AND LATTICE EXAMPLES

sdef erg d1 pos 7 2 0 cel=1 rad d2

si2 3.6

si1 0 10

sp1 0 1

f4:n 10

e4 1 3 5 7 9 11

m1 4009 1

m2 6000 1

m3 13027 1

m4 1001 2 8016 1

nps 100000

dbcn 0 0 1 4

*tr 0 0 0 10 80 90 100 10 90

*tr2 1 0 0 2 88 90 92 2 90

tr3 3 0 0

vol 1 10r

Example 6: This example primarily illustrates a fairly complex source description in a lattice

geometry. The geometry consists of two “main” cells, each ﬁlled with a different lattice. Cell 2, the

left half, is ﬁlled with a hexahedral lattice, which is in turn ﬁlled with a universe consisting of a cell

of rectangular cross section and a surrounding cell. The relation of the origin of the ﬁlling universe,

1, to the ﬁlled cell, 2, is given by the transformation in parentheses following FILL=1. Cell 3, the

right half, is ﬁlled with a different hexahedral lattice, in turn ﬁlled by universes 4 and 5. Lattice

cells must be completely speciﬁed by an expanded FILL card if the lattice contains a source (cell

5) or by selecting a coordinate system of a higher level universe (SI7 L –2:4:8). Check print table

110 to see the lattice elements that are being sampled. Become familiar with the geometry before

proceeding to the source description.

18 December 2000 4-33

CHAPTER 4

REPEATED STRUCTURE AND LATTICE EXAMPLES

In brief, a volume distributed source located in each of the ten boxes and eight circles (in two

dimensions) is desired. The cells involved are given by distribution 6. The S on the SI6 card

indicates distribution numbers will follow. The four distributions will describe the cells further. The

probabilities for choosing each distribution of cells is given by the SP6 card.

The SI7 card shows the entire path from level 0 to level n for the nine boxes on the left. The

expanded FILL notation is used on the cell 4 card to describe which elements of the lattice exist

and what universe each one is ﬁlled with. All nine are ﬁlled by universe 3. SI12 says x is sampled

from –4 to 4 and SI14 says y is sampled from –3 to 3. Used together with the expanded FILL,

MCNP will sample source points from all nine lattice elements. Without the expanded FILL, only

the [0,0,0] element would have source points. Another method would be to use the following input

cards:

40−11 12 –14 13 imp:n=1 lat=1 u=1 fill=3

si7 l –2:4:8

si12 –46 –4

si14 –17 17

The minus sign by the 2 means the sampled position and direction will be in the coordinate system

of the level preceding the entry ≤0. There is no preceding entry so they will be in the coordinate

system of cell 2. If a point is chosen that is not is cell 8, it is rejected and the variable is resampled.

SI8 describes a path from cell 3 to element (0,0,0) of cell 5 to cell 11, from cell 3 to element (1,0,0)

to cell 11, etc. Element (1,2,0) is skipped over and will be treated differently. SI9 is the path to cell

13, the circle in element (1,2,0) and SI10 is the path to cell 15, the box in element (1,2,0). All the

other source variables are given as a function of cell and follow explanations given in the manual.

cell 2 cell 3

[0,0,0]

4-34 18 December 2000

CHAPTER 4

REPEATED STRUCTURE AND LATTICE EXAMPLES

example 6

1 0 1:$-$3:–4:5:6:$-$7 imp:n=0

2 0 –2 3 4 –5 –6 7 imp:n=1 fill=1 (–25 0 0)

3 0 –1 2 4 –5 –6 7 imp:n=1 fill=2 (0 –20 0)

4 0 –11 12 –14 13 imp:n=1 lat=1 u=1 fill=-1:1 -1:1 0:0 3 8r

5 0 –15 2 –18 17 imp:n=1 lat=1 u=2

fill=0:1 0:3 0:0 4 4 4(5 0 0) 4 4 5 4 4

6 1 –.9 21:–22:–23:24 imp:n=1 u=3

7 1 –.9 19 imp:n=1 u=4

8 2 –18 –21 22 23 –24 imp:n=1 u=3

9 1 –.9 20(31:–32:–33:34) imp:n=1 u=5

11 2 –18 –19 imp:n=1 u=4

13 2 –18 –20 imp:n=1 u=5

15 2 –18 –31 32 33 –34 imp:n=1 u=5

1 px 50

2 px 0

3 px –50

4 py –20

5 py 20

6 pz 60

7 pz –60

11 px 8.334

12 px –8.334

13 py –6.67

14 py 6.67

15 px 25

17 py 0

18 py 10

19 c/z 10 5 3

20 c/z 10 5 3

21 px 4

22 px –4

23 py –3

24 py 3

31 px 20

32 px 16

33 py 3

34 py 6

m1 6000 .4 8016 .2 11023 .2 29000 .2

m2 92238 .98 92235 .02

sdef erg fcel d1 cel d6 x fcel d11 y fcel d13 z fcel d15

18 December 2000 4-35

CHAPTER 4

REPEATED STRUCTURE AND LATTICE EXAMPLES

rad fcel d17 ext fcel d19 pos fcel d21 axs fcel d23

ds1 s d2 d3 d4 d5

sp2 –2 1.2

sp3 –2 1.3

sp4 –2 1.4

sp5 –2 1.42

si6 s d7 d8 d9 d10

sp6 .65 .2 .1 .05

si7 l 2:4:8

sp7 1

si8 l 3:5(0 0 0):11 3:5(1 0 0):11 3:5(0 1 0):11 3:5(1 1 0):11

3:5(0 2 0):11 3:5(0 3 0):11 3:5(1 3 0):11

sp8 1 1 1 1 1 1 1

si9 l 3:5(1 2 0):13

sp9 1

si10 l 3:5(1 2 0):15

sp10 1

ds11 s d12 0 0 d25

si12 –4 4

sp12 0 1

ds13 s d14 0 0 d26

si14 –3 3

sp14 0 1

ds15 s d16 0 0 d16

si16 –60 60

sp16 0 1

ds17 s 0 d18 d18 0

si18 0 3

sp18 –21 1

ds19 s 0 d20 d20 0

si20 –60 60

sp20 0 1

ds21 s 0 d22 d22 0

si22 l 10 5 0

sp22 1

ds23 s 0 d24 d24 0

si24 l 0 0 1

sp24 1

si25 16 20

sp25 0 1

si26 3 6

sp26 0 1

4-36 18 December 2000

CHAPTER 4

REPEATED STRUCTURE AND LATTICE EXAMPLES

f2:n 1

e2 .1 1 20

f6:n 2 4 6 8 3 5 7 9 11 13 15

sd6 1 1 1 1 1 1 1 1 1 1 1

nps 5000

Example 7:

This example illustrates a hexagonal prism lattice and shows how the order of speciﬁcation of the

surfaces on a cell card identiﬁes the lattice elements beyond each surface. The (0,0,0) element is

the space described by the surfaces on the cell card, perhaps inﬂuenced by a TRCL entry. The user

chooses where the (0,0,0) element will be. The user chooses the location of the (1,0,0) element—

it is beyond the ﬁrst surface entered on the cell card. The (–1,0,0) element MUST be in the opposite

direction from (1,0,0) and MUST be beyond the second surface listed. The user then chooses where

the (0,1,0) element will be—it must be adjacent to the (1,0,0) element—and that surface is listed

next. The (0,–1,0) element MUST be diagonally opposite from (0,1,0) and is listed fourth. The ﬁfth

and sixth elements are deﬁned based on the other four and must be listed in the correct order:

(–1,1,0) and (1,–1,0). Pairs can be picked in any order but the pattern must be adhered to once set.

Illustrated is one pattern that could be selected and shows how the numbering of elements in this

example progresses out from the center.

hexagonal prism lattice

1 0 –1 –19 29 fill=1 imp:n=1

2 0 –301 302 –303 305 –304 306 lat=2 u=1 imp:n=1

302

305 303

301

306304

[0,0,0]

[-1,0,0][-2,0,0] [1,0,0] [2,0,0]

[-2,1,0] [-1,1,0] [0,1,0] [1,1,0] [2,1,0]

[-2,2,0] [-1,2,0] [0,2,0]

[-1,-1,0] [0,-1,0] [1,-1,0] [2,-1,0]

[-1,-2,0] [0,-2,0] [1,-2,0] [2,-2,0] [3,-2,0]

‘

18 December 2000 4-37

CHAPTER 4

REPEATED STRUCTURE AND LATTICE EXAMPLES

3 0 1:19:-29 imp:n=0

1 cz 20

19 pz 31.75

29 pz –31.75

301 px 1

302 px -1

303 p 1 1.7320508076 0 2

304 p -1 1.7320508076 0 2

305 p 1 1.7320508076 0 -2

306 p –1 1.7320508076 0 -2

sdef

f1:n 1

nps 2000

One of the most powerful uses of macrobodies is for the speciﬁcation of hexagonal prisms. The

example above can be simpliﬁed by using the RHP (also caled HEX) macrobody as follows:

hexagonal prism lattice

C Cell Cards

1 0 -2 fill=1 imp:n=1

2 0 -1 lat=2 u=1 imp:n=1

3 0 2 imp:n=0

C Surface Cards

1 rhp 0 0 -31.75 0 0 63.5 2 0 0

2 rcc 0 0 -31.75 0 0 63 20

4-38 18 December 2000

CHAPTER 4

REPEATED STRUCTURE AND LATTICE EXAMPLES

Example 8:

This example shows how the LIKE m BUT and TRCL cards can be used to create an array of

nonidentical objects within each cell of a lattice. Only one lattice element is shown in the plot

above. A lattice of hexahedral subassemblies, each holding an array of 25 cylindrical rods, is

contained within a cylindrical cell. Cell 1 is the space inside the large cylinder and is ﬁlled with

universe 1. Cell 2 is the only cell in universe 1 and is the hexahedral lattice that ﬁlls cell 1. The

lattice is a 7x7x1 array, indicated by the array indices on the FILL card, and is ﬁlled either by

universe 2 or by itself, universe 1. Cell 3, a fuel rod, is in universe 2 and is the space inside the

cylindrical rod. The other fuel cells, 5–24, are like cell 3 but at different x,y locations. The material

in these 21 fuel cells is slightly enriched uranium. Cells 25–28 are control rods. Cell 25 is like 3

but the material is changed to cadmium, and the density and the x,y location are different. Cells

26–28 are like cell 25 but at different x,y locations. Cell 4 is also in universe 2 and is the space

outside all 25 rods. To describe cell 4, each cell number is complimented. Notice in the plot that all

the surfaces except for the center one have a new predictable surface number—1000 * cell no +

surface no. These numbers could be used in the description of cell 4 if you wanted.

The KCODE and KSRC cards specify the criticality source used in calculating keff. There are 1000

particles per cycle, the initial guess for keff is 1, 5 cycles are skipped before the tally accumulation

begins, and a total of 10 cycles is run.

example of pwrlat

1 0 -1 -19 29 fill=1 imp:n=1

2 2 -1 -301 302 -303 304 lat=1 u=1 imp:n=1 fill=-3:3 -3:3 0:0

18 December 2000 4-39

CHAPTER 4

REPEATED STRUCTURE AND LATTICE EXAMPLES

1111111 1122211 1222221 1222221

1 2 2 2 2 2 1 1 1 2 2 2 1 1 1 1 1 1 1 1 1

3 1 -18 -10 u=2 imp:n=1

4 2 -1 #3 #5 #6 #7 #8 #9 #10 #11 #12 #13 #14 #15 #16 #17 #18

#19 #20 #21 #22 #23 #24 #25 #26 #27 #28 imp:n=1 u=2

5 like 3 but trcl=(-6 6 0)

6 like 3 but trcl=(-3 6 0)

7 like 3 but trcl=(0 6 0)

8 like 3 but trcl=(3 6 0)

9 like 3 but trcl=(6 6 0)

10 like 3 but trcl=(-6 3 0)

11 like 3 but trcl=(0 3 0)

12 like 3 but trcl=(6 3 0)

13 like 3 but trcl=(-6 0 0)

14 like 3 but trcl=(-3 0 0)

15 like 3 but trcl=(3 0 0)

16 like 3 but trcl=(6 0 0)

17 like 3 but trcl=(-6 -3 0)

18 like 3 but trcl=(0 -3 0)

19 like 3 but trcl=(6 -3 0)

20 like 3 but trcl=(-6 -6 0)

21 like 3 but trcl=(-3 -6 0)

22 like 3 but trcl=(0 -6 0)

23 like 3 but trcl=(3 -6 0)

24 like 3 but trcl=(6 -6 0)

25 like 3 mat=3 rho=-9 trcl=(-3 3 0)

26 like 25 but trcl=(3 3 0)

27 like 25 but trcl=(-3 -3 0)

28 like 25 but trcl=(3 -3 0)

50 0 1:19:-29 imp:n=0

1 cz 60

10 cz 1.4

19 pz 60

29 pz -60

301 px 10

302 px -10

303 py 10

304 py -10

kcode 1000 1 5 10

ksrc 0 0 0

4-40 18 December 2000

CHAPTER 4

TALLY EXAMPLES

m1 92235 .02 92238 .98

m2 1001 2 8016 1

m3 48000 1

IV. TALLY EXAMPLES

This section contains examples of the FMn, FSn, and FTn tally cards, a complicated repeated

structures/lattice example, and the TALLYX subroutine. Refer also to page 3–75 for the FMn card,

to page 3–83 for the FSn card, to page 3–93 for the FTn card, to page 3–69 for the basic repeated

structure/lattice tally, and to page 3–87 for TALLYX before trying to understand these examples.

A. FMn Examples (Simple Form)

Example 1: Consider the following input cards.

F4:N 10

FM4 0.04786 999 102

M999 92238.13 1

The F4 neutron tally is the track length estimate of the average ﬂuence in cell 10. Material 999 is

238U with an atomic fraction of 100%.

C = 0.04786 normalization factor (such as atom/barn ⋅cm)

M= 999 material number for 238U as deﬁned on the material card

(with an atom density of 0.04786 atom/barn⋅cm)

R1= 102 ENDF reaction number for radiative capture

cross section (microscopic)

The average ﬂuence is multiplied by the microscopic (n,γ) cross section of 238U (with an atomic

fraction of 1.0) and then by the constant 0.04786 (atom/barn⋅cm). Thus the tally 4 printout will

indicate the number of 239U atoms/cm3produced as a result of (n,γ) capture with 238U.

Standard F6 and F7 tallies can be duplicated by F4 tallies with appropriate FM4 cards. The FM4

card to duplicate F6 is

FM4 C M 1 –4.

For F7 it is

FM4 C M –6 –8.

C=10−24 x number of atoms per gram

R1= 1 ENDF reaction number for total cross section (barns)

R2=−4 reaction number for average heating number (MeV/collision)

R1=−6 reaction number for total ﬁssion cross section (barns)

R2=−8 reaction number for ﬁssion Q (MeV/ﬁssion)

18 December 2000 4-41

CHAPTER 4

TALLY EXAMPLES

This technique applied to F2 tallies can be used to estimate the average heating over a surface rather

than over a volume. It provides the surface equivalents of F6 and F7 tallies, which are not available

as standard tallies in MCNP.

Example 2: Consider a point detector.

F25:N 0 0 0 0

FM25 0.00253 1001 –6 –8

M1001 92238.60 .9 92235.60 .1

This F25 neutron tally is the ﬁssion heating per unit volume of material 1001 at the origin. Material

1001 does not actually have to be in a cell at the origin. The FM25 card constants are:

C= 0.00253 atoms per barn⋅cm (atomic density) of material 1001

M= 1001 material number for material being heated

R1=−6 reaction number for total ﬁssion cross section (barn)

R2=−8 reaction number for ﬁssion Q (MeV/ﬁssion)

Example 3: Lifetime calculation

F4:N 1

SD4 1

FM4 (-1 1 16:17) $ bin 1 = (n,xn) reaction rate

(-1 1 -2) $ bin 2 = capture (n,0n) reaction rate

(-1 1 -6) $ bin 3 = ﬁssion reaction rate

(-1 -2) $ bin 4 = prompt removal lifetime=ﬂux/velocity

M1 92235 –94.73 92238 –5.27

This F4 neutron ﬂux tally from a Godiva criticality problem is multiplied by four FM bins and will

generate four separate tally quantities. The user can divide bins 1, 2, and 3 by bin 4 to obtain the

(n,xn) lifetime, the (n,0n) lifetime, and the (n,f) lifetime, respectively. The FM4 card entries are:

C=−1 multiply by atomic density of material 1

M= 1 material number on material card

R1= 16:17 reaction number for (n,2n) cross section plus

reaction number for (n,3n) cross section

R2=−2 reaction number for capture cross section

R3= –6 reaction number for total ﬁssion cross section

=1–2prompt removal lifetime = ﬂux/velocity = time integral of population

More examples: (Remember C= –1 =ρfor type 4 tally)

F5:N 0000 Neutron heating per cm3 with an atom density of

FM5 ρM1−4 ρ of material M at a point detector

4-42 18 December 2000

CHAPTER 4

TALLY EXAMPLES

F5Y:P 10 5 0 Photon heating per cm3 of material M

FM5 ρM−5 −6 with an atom density ρ at a ring detector

F1:N 1 2 3 Number of neutron tracks crossing surfaces 1, 2, and 3

FM1 1 0 per neutron started

F35:P 0000 Number of photon collisions per source particle

FM35 1 0 that contribute to point detector

M99 3007 1 7Li tritium production per cm3 in cell 10

F4:N 10

FM4 −1 99 91

F104:N 8 Number of reactions per cm3 of type R in cell 8

FM104 −1 MR of material M of atom density ρ

B. FMn Examples (General Form)

Remember that the hierarchy of operation is multiply ﬁrst and then add and that this hierarchy can

not be superseded by the use of parentheses.

Example 1:

F4:N 1

FM4 (ρ 1 (1 –4)(–2)) (ρ 1 1) where C = ρ= atomic density (atom/barn⋅cm)

M1 6012.10 1

In this example there are three different tallies, namely

(a) ρ11−4

(b) ρ1−2

Thus tally (a) will yield the neutron heating in MeV/cm3 from 12C in cell 1. The advantage in

performing the multiplication 1 −4 in tally (a) is that the correct statistics are determined for the

desired product. This would not be true if tally (a) were to be done as two separate tallies and the

product formed by hand after the calculation.

Example 2: F4:N 1

FM4 (0.04635 1 (105:91))

M1 3006.50 0.0742 3007.50 0.9258

18 December 2000 4-43

CHAPTER 4

TALLY EXAMPLES

In this example we obtain the total tritium production per cm3 from natural lithium (ENDF/B-V

evaluation) in cell 1. The constant C on the FM4 card is the atomic density of natural lithium. A

subtle point is that the R = 105 reaction number contains the reaction data for just the 6Li reaction

and R = 91 contains the reaction data for the 7Li reaction (p.524 Appendix G). However, this

examples uses both sets of reaction data in the FM4 card to calculate the tritium production in a

media composed of both 6Li and 7Li. Thus, four calculations are carried out (two for 6Li using

R = 91,105, and two for 7Li using R = 91,105). Note that two of these calculations (6Li with

R = 91, and 7Li with R = 105) will contribute nothing to the total tritium production.

Example 3: Suppose we have three reactions—R1,R2, and R3—and wish to add R2 and R3and

multiply the result by R1. The following would NOT be valid: FMn (C m R1 (R2:R3)).

The correct card is: FMn (C m (R1R2:R1R3)).

C. FSn Examples

The FSn card allows you to subdivide your tally into geometry segments, avoiding overspecifying

the problem geometry with unnecessary cells.

The entries on the FS card are the names and senses of surfaces that deﬁne how to segment any

surface or cell tally.

Example 1: Consider a 1-MeV point isotropic source at the center of a 2 cm cube of carbon. We

wish to calculate the ﬂux through a 1-cm2window in the center of one face on the cube. The input

ﬁle calculating the ﬂux across one entire face is shown in Figure 4.18.

EXAMPLE 1, SIMPLE CUBE

11−2.22 1 2 −3 −4 −5 6 IMP:N=1

2 0 #1 IMP:N=0

1 PY 0

2PZ−1

3 PY 2

4 PZ 1

5PX1

6 PX –1

SDEF POS = 0 1 0 ERG = 1

M1 6012.60 –1

F2:N 3

The FS card retains the simple cube geometry and four more surface cards are required,

x121 3

Figure 4-18.

4-44 18 December 2000

CHAPTER 4

TALLY EXAMPLES

7PX.5

8PX−.5

9PZ.5

10 PZ −.5

FS2 7 −10 −8 9

The four segmenting surface cards are

listed with the other surface cards, but they

are not part of the actual geometry and

hence do not complicate the cell-surface

relationships.

The F2 tally is subdivided into ﬁve separate

tallies as shown in Figure 4.19: (1) the ﬁrst is the ﬂux of particles crossing surface 3 but with a

positive sense to surface 7; (2) the second is the remaining ﬂux with negative sense to surface 7

crossing surface 3 but with a negative sense to surface 10; (3) the third is the remaining ﬂux

(negative sense to 7 and positive sense to 10) crossing 3 but with a negative sense to 8; (4) the

remaining ﬂux with positive sense to 9; and (5) everything else. In this example, the desired ﬂux in

the window is in the ﬁfth subtally—the “everything else” portion.

The FS segmenting card could have been set up other ways. For example:

FS2 −10 79 −8 and

FS2 −8 9−10 7

Each works, but the order of the subtallies is changed. A way to avoid the ﬁve subtallies and to get

only the window of interest is to use the TALLYX subroutine described later.

Example 2: Consider a source at the center of a 10-cm radius sphere called cell 1. We want to

determine the ﬁssion heating in a segment of the sphere deﬁned by the intersection of the 10-cm

sphere, an 8-cm inner sphere, and a 20°cone whose vertex is at the source and is about the Y-axis.

This is accomplished by using

F7:N 1

FS7 −2 −3

where surface 2 is the 8-cm surface and surface 3 is the cone. This breaks the F7 tally up into three

portions: (1) the heating inside the 8-cm sphere; (2) the heating outside the 8-cm sphere but within

the cone—this is the desired portion; and (3) everything else, which is a 2-cm shell just inside the

10-cm sphere but outside the cone.

5678

III

Figure 4-19.

18 December 2000 4-45

CHAPTER 4

TALLY EXAMPLES

D. FTn Examples

Example 1: Consider the following input cards.

F1:N 2

FT1 FRV V1V2V3

The FTn card is the special treatment for tallies card. Various tally treatments are available for

certain speciﬁc tally requirements. The FTn tally with the FRV card used in conjunction with tally

type 1 will redeﬁne the vector normal to the tally surface. In this case, the current over surface 2

(tally type 1) uses the vector V as its reference vector for getting the cosine for binning.

Example 2:

F5:P 4

FT5 ICD

FU5 1 3

In this example the photon ﬂux at detector 5 is being tallied. However, only the contributions to the

detector tally from cells 1 and 3 are of interest. The ICD keyword allows the user to create a

separate bin for each cell, and only contributions from one of the speciﬁed cells are scored. The

FUn card speciﬁes the cells from which tallies are to be made, but TALLYX is not called.

Example 3: When keeping track of charged particle current across a surface, it is sometimes

desirable to track both positive and negative score contributions, applicable in cases that include

electrons and positrons. Consider a photon source that is enclosed in a spherical shell of lead. If a

surface current tally is taken over the sphere and it is desirable to tally both the positron and

electron current separately, then the special treatment card option is invoked.

11−.001124 −11 imp:e=1 imp:p=1

22−11.0 11 −21 imp:e=1 imp:p=1

3 0 21 imp:e=0 imp:p=0

11 so 30

21 so 32

m1 6012 .000125 7014 .6869 8016 .301248 18040 .011717

m2 82000 1.

mode p e

sdef pos = 0. 0. 0. erg = 2.5

f1:e 21

ft1 elc 2

f2:p 21

4-46 18 December 2000

CHAPTER 4

TALLY EXAMPLES

e2 1e-3 1e-2 0.1 0.5 1.0 1.5 2.0 2.5 C

nps 10000

The input deck shown above models a sphere ﬁlled with dry air surrounded by a spherical shell of

lead. The centrally located source emits 2.5 MeV photons that travel through the air into the lead

shell. The F1 surface current tally has been modiﬁed with the ELC special tally option. The

parameter value of 2 that follows the ELC keyword speciﬁes that positrons and electrons be placed

into separate tally user bins. Once this option has been invoked, the user can inspect the output tally

bins for the respective scoring of either particle.

The F2 tally scores photon ﬂux crossing surface 21, scored into energy bins deﬁned on the E2 card.

The C at the end of the energy bin card indicates that the bins are cumulative. Therefore the bin

with an upper limit of 1 MeV would contain scores from particles that cross surface 21 with energy

less than or equal to 1 MeV.

Example 4: Consider the following two point sources, each with a different energy distribution:

sdef pos=d1 erg=fpos d2

si1 L 536 7536

sp1 .3 .7

ds2 S 3 4

si3 H21014

sp3 D 0 1 2

si4 H.528

sp4 D0 31

f2:n 2

ft2 scd

fu2 3 4

The SCD option causes tallies to be binned according to which source distribution was sampled.

The FUn card is used to list the distribution numbers of interest. Thus, the tallies in this example

are placed in one of two bins, depending on which of the two sources emitted the particle. The two

sources may represent two nuclides with different energy distributions, for instance, with the use

of the SCD option allowing the user to determine each nuclide’s contribution to the ﬁnal tally.

E. Repeated Structure/Lattice Tally Example

An explanation of the basic repeated structure/lattice tally format can be found on page 3–69 in

Chapter 3. The example shown here illustrates more complex uses. Figures 4.20(a–f) indicate the

tally regions for each tally line. The number of bins generated by MCNP is shown at the end of

each tally line following the $.

18 December 2000 4-47

CHAPTER 4

TALLY EXAMPLES

example 1 – repeated structure lattice tally example

1 0 –1 –2 3 13 fill=4

2 0 –1 –2 3 –13 fill=1

3 0 –4 5 –6 7 u=1 lat=1

fill=–2:2 –2:0 0:0 1 1 3 1 1 1 3 2 3 1 3 2 3 2 3

4 0 –8 9 –10 11 u=2 fill=3 lat=1

5 1 –0.1 –12 u=3

6 0 12 u=3

7 0 –14 –2 3 u=4 fill=3 trcl=(-60 40 0)

8 like 7 but trcl=(-30 40 0)

9 like 7 but trcl=(0 40 0)

10 like 7 but trcl=(30 40 0)

11 like 7 but trcl=(60 40 0)

12 0 #7 #8 #9 #10 #11 u=4

13 0 1:2:-3

1 cz 100

2 pz 100

3 pz -100

4 px 20

5 px -20

6 py 20

7 py -20

8 px 10

9 px -10

10 py 10

11 py -10

12 cz 5

13 py 19.9

14 cz 10

f4:n 5 6 (5 6 3) $ 3 bins

(5<3) (5<(3[–2:2 –2:0 0:0])) $ 2 bins

(5<(7 8 9 10 11)) (5<7 8 9 10 11<1) (5<1) $ 7 bins

((5 6)<3[0 –1 0]) ((5 6)<3[0:0 –1:-1 0:0]) ((5 6)<3[8]) $ 3 bins

(5<(4[0 0 0]3[8]))(5<4[0 0 0]<3[8])

(3<(3[1]3[2]3[4]3[5]3[6]3[10])) $ 3 bins

5<u=3) $ 12 bins

4-48 18 December 2000

CHAPTER 4

TALLY EXAMPLES

Figure 4-20.

Fig 4.20a

Fig 4.20b

Fig 4.20c

Fig 4.20d

Fig 4.20e

Fig 4.20f

18 December 2000 4-49

CHAPTER 4

TALLY EXAMPLES

Tally line 1: This ﬁrst line creates three tally output bins: cell 5, cell 6, and the union of cells 5, 6,

and 3, indicated in Figure 4.20a. Notice that cell 3 is ﬁlled entirely by cells 5 and 6. Cell 5 plus cell

6 equals cell 3. If a particle is tallied in cell 5 and tallied in cell 3, it will be tallied twice in the bin

(5 6 3). CAUTION: A true union IS performed when ﬁrst level cells overlap (or ﬁll) another cell.

This probably is NOT a tally you often want. If an average of cell 3 and region (5 6) outside cell 3

is desired, separate bins must be deﬁned and properly combined using correct volume weighting.

Tally line 2: These two input tally bins result in identical output tallies and demonstrate the use of

lattice index brackets that include all existing lattice elements, thereby making the two tallies

equivalent. The simpler format will execute faster. Tally region shown in Fig. 4.20b.

Tally line 3: This line illustrates omission of geometry levels and a single output bin vs. multiple

bins. All three input bins tally cell 5 within cells 7 through 11. The second bin speciﬁes the entire

path explicitly. Because the only place cell 5 exists within cell 1 is in cells 7–11, the 7–11

speciﬁcation can be omitted, as in the third input bin. In the second input bin, the parentheses

around cells 7–11 are omitted, creating multiple output bins. Five tally bins are produced: (5<7<1),

(5<8<1), (5<9<1), (5<10<1), and (5<11<1). The sum of these ﬁve bins should equal the tally in the

ﬁrst and last output bins on this line. Tally region shown in Fig. 4.20c.

Tally line 4: This line illustrates the union of multiple tally cells, (5 6), and various ways of

specifying lattice index data. The three input tally bins create three output tally bins with identical

values because the three different lattice descriptions refer to the same lattice element, the eighth

entry on the FILL array. If the parentheses around (5 6) were removed, two output bins would be

created for each input bin, namely (5<3[0 –1 0]) and (6<3[0 –1 0]), etc. Tally region shown in Fig.

4.20d.

Tally line 5: This line illustrates tallys in overlapping regions in repeated structures in a lattice and

a tally in lattice elements ﬁlled with themselves. Three tally output bins are produced. In the ﬁrst

input bin, a particle is tallied only once when it is in cell 5 and in 4[0 0 0] and in 3[0 –1 0]. Figure

4.20f shows all the cell 5’s included in this tally bin. This tally is probably more useful than the

overlapping regions in tally line 1. Input bin two demonstrates a tally for a nested lattice. A tally is

made when a particle is in cell 5 and in cell 4, element [0 0 0] and in cell 3, element [0 –1 0]. Note

that 3[0 –1 0] is indeed ﬁlled with cell 4 (u=2). If that were not true, a zero tally would result in

this bin. The ﬁnal input tally bin demonstrated a tally in lattice elements that are ﬁlled with their

own universe number. This method is the only way to tally in these elements separate from the rest

of cell 3. Tally region shown in Fig. 4.20e.

Tally line 6: This line illustrates the universe format. The single input bin includes all possible

chains involving cell 5. Because u=3 is not within parentheses, the input is expanded into twelve

output bins: (5<3[3], etc. The format 3[3] indicates the third lattice element of cell 3 as entered on

the cell 3 FILL array. Note that the third element is ﬁlled by universe 3, consisting of cells 5 and 6.

Tally region shown in Fig. 4.20f.

4-50 18 December 2000

CHAPTER 4

TALLY EXAMPLES

F. TALLYX Subroutine Examples

An explanation of the TALLYX subroutine arguments can be found on page 3–87 in Chapter 3.

Only examples illustrating some uses of TALLYX will be found here.

Example 1: In the example of the FSn card to get the ﬂux through a window on the face of a cube,

instead of using the FS2 card, which established ﬁve sub tallies, TALLYX could have been used to

get only the desired window tally. Two input cards are used:

FU2 1

RDUM −.5 .5 −.5 .5

The following subroutine (which is implemented just like a user-provided SOURCE subroutine by

replacing lines TX.2 through TX.18) does the job. Note that IB=0 and IBU=1 upon entry into

TALLYX.

SUBROUTINE TALLYX(T,IB)

∗CALL CM

IF(X.LT.RDUM(1).OR.X.GT.RDUM(2))IB = –1

IF(Z.LT.RDUM(3).OR.Z.GT.RDUM(4))IB = –1

RETURN

END

The subroutine was generalized a bit by using the RDUM input card, although the card could have

been avoided by hard wiring the dimensions of the window into TALLYX.

Example 2: Dump 18 words of the GPBLCM array to a BCD ﬁle called UOUT each time a

neutron crosses surface 15. The input cards are

F2:N 15

FU2 1

FILES 7 UOUT

The user-provided subroutine is

SUBROUTINE TALLYX(T,IB)

∗CALL CM

WRITE(7,20) (GPBLCM(I),I=1,10),NPA,ICL,JSU,IPT,IEX,NODE,IDX,NCP

20 FORMAT(5E14.6/5E14.6/7I10/2I10)

RETURN

END

18 December 2000 4-51

CHAPTER 4

TALLY EXAMPLES

Every time surface 15 is crossed and the F2 tally is scored, TALLYX is called and part of the

GPBLCM array is written to the ﬁle UOUT. If more discrimination is desired, such as dump the

GPBLCM array only for neutrons with energy between 2.5 and 4.5 MeV and crossing surface 15

at less than 30° with respect to the normal (assume surface 15 has been deﬁned by a PY card), add

the following two lines before the WRITE statement:

IF(VVV.LT.0.866)RETURN

IF(ERG.LT.2.5.OR.ERG.GT.4.5)RETURN

To write a binary ﬁle, the FILES card entries are 7 UOUT S U and the WRITE statement in

TALLYX is unformatted:

SUBROUTINE TALLYX(T,IB)

∗CALL CM

WRITE(7) (GPBLCM(I),I=1,10),NPA,ICL,JSU,IPT,IEX,NODE,IDX,NCP

RETURN

END

The advantage of a BCD ﬁle is that it is easy to look at and manipulate, but it requires more I/O

time and a larger ﬁle. A binary ﬁle is more compact than a BCD ﬁle and requires less I/O time to

write; however, it may be more difﬁcult to use.

Example 3: Calculate the number of neutron tracks exiting cell 20 per source neutron. This is also

done in Chapter 5 with the TEST1 example using the FMn card. The input cards are

F4:N 20

FU4 1

SD4 1

and TALLYX becomes

SUBROUTINE TALLYX(T,IB)

∗CALL CM

T=1.0

IF(PMF.LT.DLS) IB = –1

RETURN

END

The quantity T=1.0 is scored every time a track exits cell 20. The variables used in this subroutine,

PMF (the distance to collision) and DLS (distance to the boundary), are available to TALLYX from

the MCNP COMMON.

4-52 18 December 2000

CHAPTER 4

TALLY EXAMPLES

Example 4: Divide the point detector scores into separate tallies (that is, user bins) depending upon

which of the 20 cells in a problem geometry caused the contributions. The input cards are

F5:N 0 0 0 0

FU5 1 18I 20

and TALLYX is

SUBROUTINE TALLYX(T,IB)

∗CALL CM

IBU=ICL9

RETURN

END

The FU5 card establishes 20 separate user bins, one for each cell in the problem. Note the use of

the “nI” input format, described in Chapter 3, which creates 18 linear interpolates between 1 and

20. The variable ICL9 contains, for the current history, the number of the cell which produced the

original particle.

Example 5: Determine the quantity ∫ϕ(E)f(E)dE in cell 14 where f(E)=eαt. The input cards are

F4:N 14

FU4 α

where α is a numerical value and TALLYX is

SUBROUTINE TALLYX(T,IB)

∗CALL CM

T=T*EXP(TDS(IPTAL(LIPT+3,1,ITAL)+1)*TME)

RETURN

END

The FU4 card establishes a single user bin, and the value of αis stored in

TDS(IPTAL(LIPT+3,1,ITAL)+1) and used for the tally label.

Example 6: Tally the number of neutrons passing through cell 16 which have had 0, 1, 2, 3, or 4

collisions. The input cards are

F4:N 16

FU4 0 1 2 3 4

SD4 1

and TALLYX is

18 December 2000 4-53

CHAPTER 4

SOURCE EXAMPLES

SUBROUTINE TALLYX(T,IB)

∗CALL CM

IBU=INT(SPARE1)+1

IF(IBU.GT.5)IB = –1

T=WGT

RETURN

END

The subroutine can be generalized by replacing the 5 in the IF statement with

IPTAL(LIPT+3,3,ITAL), which is the number of entries on the FU4 card.

If no tracks were put into the bank (from ﬁssion, geometry splitting, etc.,), INT(SPARE1) could

be replaced by NCH(1), the number of neutron collisions per history. However, to be general, use

the quantity SPARE1, which goes into the bank with the rest of the banked track parameters.

SPARE1 must be set to 0 when a source particle is started and then incremented at collisions. This

is done with the following patch in the STARTP and HSTORY subroutines:

∗I,SP.30

SPARE1=0.

∗I,HS.184

SPARE1=SPARE1+1.

If the IF statement in this TALLYX is omitted, a count will be made of the cases of ﬁve or more

collisions, and in these cases no score will be tallied but a count will be printed of the times that

the tally was unable to be made because IBU was a value where no bin existed.

In the ﬁve user bins, T is the number of neutrons per source neutron passing through cell 16 that

has undergone 0, 1, 2, 3, or 4 collisions, respectively. Note that the FU4 card has ﬁve entries to

establish the ﬁve user bins and provide labels. Note also that in this example, the neutrons are

calculated so that T=T∗ renormalization factor (which preserves the weight associated with the

tracks), where in Example 3 the neutron tracks are calculated so that T=1. Again the value of

SPARE1 is available from COMMON. Finally, note that if SPARE1 >5 (six or more collisions) no

tally is made because IB is set to be less than zero. If an E4 card were added, the neutrons would

be tallied as a function of energy for each user bin.

V. SOURCE EXAMPLES

Some examples of the general source are given here to illustrate the power and complexity of this

feature. Refer to Chapter 3 for the more complete explanation and other examples.

Example 1: SDEF ERG = D1 DIR FERG D2

SUR=1 CEL=2 POS=X Y Z RAD D5 VEC = U V W

SI1 H 10−7 10−5 …13.5 14 …20 $ Level 1

SP1 D 0 10−4 …10−2 10−1 ….3

4-54 18 December 2000

CHAPTER 4

SOURCE EXAMPLES

DS2 S 3 …34…4

SI3 0 .2 …1 $ Level 2

SP3 D 0 10−4….1

SI4 0 .1 … 1

SP4 D 0 10−2 … .1

SI5 37

SP5 −21 1 $ Optional card

This example of the general source illustrates two levels of dependency. Let us assume a duct

streaming problem where the source at the duct opening has been obtained from a reactor

calculation. Energies above 13.5 MeV have one angular distribution and energies below 13.5 MeV

have a different angular distribution. The source has a uniform spatial distribution on a circular disk

of radius 37 cm centered at x,y,z on planar surface 1 going into cell 2.

This example can be expanded by having the source in two ducts instead of one (with the same

energy and angular distribution as before). The SI1, SP1, DS2, SI3, SP3, SI4, and SP4 cards remain

unchanged. The SDEF card is changed as shown below and the other cards are added.

SDEF ERG = D1 DIR FERG D2 SUR = D6 CEL FSUR D7

POS FSUR D8 RAD FSUR D9 VEC FSUR D10

SI6 L 1 7

SP6 D .6 .4

DS7 L 2 8

DS8 L x1 y1 z1x2 y2 z2

DS9 S 11 12

DS10 L u1 v1 w1u2 v2 w2

SI11 0 37

SP11 −21 1

SI12 0 25

SP12 −21 1

Example 2: This example is a two-source-cell problem where the material in one cell is uranium

and in the other is thorium. The uranium cell has two isotopes, 235U and 238U, and the thorium has

one, 232Th. Each isotope has many photon lines from radioactive decay. The following input cards

describe this source.

SDEF CEL = D1 ERG FCEL D2 …

SC1 source cells $ Level 1

SI1 L 1 2

SP1 D 2 1

SC2 source “spectra”

DS2 S 3 4

SC3 uranium nuclides $ Level 2

18 December 2000 4-55

CHAPTER 4

SOURCE EXAMPLES

SI3 S 5 6

SP3 D 1 3

SC4 thorium nuclide

SI4 S 7

SP4 D 1

SC5 U235 photon lines $ Level 3

SI5 L E1…EI

SP5 D I1…II

SC6 U238 photon lines

SI6 L E1…EJ

SP6 D I1…IJ

SC7 Th232 photon lines

SI7 L E1…EK

SP7 D I1…IK

Example 3: SDEF SUR = D1 CEL FSUR D2 ERG FSUR D6

X FSUR D3 Y FSUR D4 Z FSUR D5

SI1 L 10 0

SP1 .8 .2

DS2 L 0 88

DS6 S 61 62

SP61 −3 .98 2.2

SP62 −3 1.05 2.7

DS3 S 0 31

SI31 20 30

SP31 0 1

DS4 S 0 41

SI41 −17 36

SP41 0 1

DS5 S 0 51

SI51 −10 10

SP51 0 1

Of the particles from this source, 80% start on surface 10, and the rest start in cell 88. When a

particle starts in cell 88, its position is sampled, with rejection, in the rectangular polyhedron

bounded by x=20to 30, y= −17 to 36, and z= −10 to 10. When a particle starts on surface 10, its

cell is found from its position and direction. The energy spectrum of the particles from surface 10

is different from the energy spectrum of the particles from cell 88. A zero after the S option invokes

the default variable value.

Example 4: SDEF ERG=D1 DIR FERG D2 SUR=m

SI1 E1E2…Ek

SP1 0 P2…Pk

4-56 18 December 2000

CHAPTER 4

SOURCE SUBROUTINE

DS2 Q .3 21 .8 22 1.7 23 20. 24

SP21 −21 1

SP22 −21 1.1

SP23 −21 1.3

SP24 −21 1.8

This is an example of using the Q option. The low-energy particles from surface mcome out with

a cosine distribution of direction, but the higher-energy particles have a more nearly radial

distribution. The energy values on the DS2 card need not be the same as any of the Ei on the SI1

card.

VI. SOURCE SUBROUTINE

When possible, you should take advantage of the standard sources provided by the code rather than

write a source subroutine. When you write your own source subroutine, you lose features such as

sampling from multiple distributions, using dependent distributions, and having frequency prints

for each tabular distribution. Also, subroutine SRCDX is needed.

The standard sources, however, cannot handle all problems. If the general source (SDEF card),

surface source (SSR), or criticality source (KCODE card) is unsuitable for a particular application,

MCNP provides a mechanism to furnish your own source-modeling capability. The absence of

SDEF, SSR, or KCODE cards causes MCNP to call subroutine SOURCE, which you must supply.

Subroutine SOURCE speciﬁes the coordinates, direction, weight, energy, and time of source

particles as listed and deﬁned on page 3–40. If the value of IPT (particle type) set by STARTP,

which calls SOURCE, is not satisfactory, SOURCE must also specify IPT. STARTP sets IPT=1

(neutron) for MODE N, N P, and N P E; sets IPT=2 (photon) for MODE P and P E; and sets IPT=3

(electron) for MODE E. MCNP checks the user’s source for consistency of cell, surface, direction,

and position. If the source direction is anisotropic and there are point detectors or DXTRAN

spheres, an SRCDX subroutine is also required (see page 4–52). The SOURCE subroutine can be

put into MCNP with PRPR.

The following example of a subroutine SOURCE uses SIn, SPn, and SBn cards and demonstrates

the use of MCNP subroutines SMPSRC, ROTAS, CHKCEL, and the function NAMCHG. The

geometry is a 5-cm-long cylinder centered about the y-axis, divided into 5 cells by PY planes at 1-

cm intervals. The 1-MeV monoenergetic source is a biased isotropic distribution that is also biased

along the y-axis. The input distribution cards are

SI1 –1 0 1 $ These 3 cards

SP1 0 1 1 $ represent a biased

SB1 0 1 2 $ isotropic distribution.

SI2 012345$ These 3 cards

SP2 042211$ represent a biased

18 December 2000 4-57

CHAPTER 4

SOURCE SUBROUTINE

SB2 011224$ distribution in y.

RDUM 1 $ cylindrical radius

IDUM 246810 $ source cells

This problem can be run with the general source by removing the RDUM and IDUM cards and

adding:

SDEF ERG=1 VEC=0 1 0 AXS=0 1 0 DIR=D1 EXT=D2 RAD=D3

SI3 0 1 $ represents a covering surface of radius 1

SP3 −21 1 $ samples from the power law with k=1

∗IDENT SRCEX

∗D,SO.11

DIMENSION A(3)

WGT=1.

C RDUM(1)--RADIUS OF SOURCE CYLINDER.

C SAMPLE RADIUS UNIFORM IN AREA.

R=RDUM(1)*SQRT(RANG())

C Y COORDINATE POSITION, PROBABILITY AND BIAS ARE

C DEFINED IN DISTRIBUTION 2 BY THE SI2, SP2, SB2 CARDS.

C SAMPLE FOR Y.

C IB RETURNS THE INDEX SAMPLED AND FI THE INTERPOLATED FRACTION.

C NEITHER ARE USED IN THIS EXAMPLE.

CALL SMPSRC(YYY,2,IX]B,FI)

C SAMPLE FOR X AND Z.

TH=2.*PIE*RANG()

XXX=-R*SIN(TH)

ZZZ=R*COS(TH)

C DIRECTION IS ISOTROPIC BUT BIASED IN CONE ALONG Y-AXIS

C DEFINED AS DISTRIBUTION 1 BY THE SI1, SP1, SB1 CARDS.

C SAMPLE FOR CONE OPENING C=COS(NU).

C ROTAS SAMPLES A DIRECTION U,V,W at AN ANGLE ARCCOS(C)

C FROM THE REFERENCE VECTOR UOLD(3)

C AND AT AN AZIMUTHAL ANGLE SAMPLED UNIFORMLY.

CALL SMPSRC(C,1,IB,FI)

UOLD(1)=0.

UOLD(2)=1.

UOLD(3)=0.

CALL ROTAS(C,UOLD,A,LEV,IRT)

UUU=A(1)

VVV=A(2)

WWW=A(3)

C CELL SOURCE - FIND STARTING CELL.

4-58 18 December 2000

CHAPTER 4

SRCDX SUBROUTINE

C IDUM(1)-IDUM(5)--LIST OF SOURCE CELLS (PROGRAM NAME).

JSU=0

DO 10 I=1,5

ICL=NAMCHG(1,IDUM(I))

CALL CHKCEL(ICL,2,J)

IF(J.EQ.0) GO TO 20

10 CONTINUE

CALL EXPIRE(1,’SOURCE’,

1 ’SOURCE IS NOT IN ANY CELLS ON THE IDUM CARD.’)

20 ERG=1.

TME=0.

VII. SRCDX SUBROUTINE

If a user has supplied a subroutine SOURCE that does not emit particles isotropically (uniform

emission in all directions) and is using either a detector tally or DXTRAN in the calculations, then

subroutine SRCDX must also be supplied to MCNP. The structure of this subroutine is the same as

for subroutine SOURCE, except that usually only a single parameter, PSC, needs to be speciﬁed

for each detector or set of DXTRAN spheres. PSC as deﬁned in SRCDX is used to calculate the

direct contribution from the source to a point detector, to the point selected for the ring detector, or

DXTRAN sphere. Other parameters may also be speciﬁed in SRCDX. For example, if a quantity

such as particle energy and/or weight is directionally dependent, its value must be speciﬁed in both

subroutine SOURCE and SRCDX. When using detectors and a subroutine SOURCE with an

anisotropic distribution, check the direct source contribution to the detectors carefully to see if it is

close to the expected result.

In general, it is best to have as few directionally-dependent parameters as possible in subroutine

SOURCE. Directionally dependent parameters must also be dealt with in subroutine SRCDX.

The most general function for emitting a particle from the source in the laboratory system can be

expressed as p(µ,ϕ), where µ is the cosine of the polar angle and ϕ is the azimuthal angle in the

coordinate system of the problem. Most anisotropic sources are azimuthally symmetric and p(µ,ϕ)

=p(µ)/2π. The quantity p(µ) is the probability density function for the µvariable only (that is, ∫

p(µ) dµ = 1,p(µ)≥0). PSC is p(µo), where µois the cosine of the angle between the direction

deﬁning the polar angle for the source and the direction to a detector or DXTRAN sphere point in

the laboratory system. (MCNP includes the 2π in the calculation automatically.) Note that p(µo)

and hence PSC may have a value greater than unity and must be non-negative. It is valuable to point

out that every source must have a cumulative distribution function based on p(µ,ϕ) from which to

sample angular dependence. The probability density function p(µ,ϕ) needs only to be considered

explicitly for those problems with detectors or DXTRAN.

18 December 2000 4-59

CHAPTER 4

SRCDX SUBROUTINE

Table 4.1 gives the equations for PSC for six continuous source probability density functions. More

discussion of probability density functions is given in the detector theory section of Chapter 2 (see

page 2–75). The isotropic case is assumed in MCNP; therefore SRCDX is required only for the

anisotropic case.

As an example of calculating µo, consider a spherical surface cosine source (type 2 in Table 4.1)

with several point detectors in the problem. Assume that a point on the spherical surface has been

selected at which to start a particle. The value of µofor a detector is given by the scalar (or dot)

product of the two directions; that is,

TABLE 4.1:

Continuous Source Distributions and their Associated PSC’s

Source

Description Source

Distribution PSC Range of µo

2. Isotropic

Surface Cosine Uniform

µ0.5

2|µo|−1≤µ

o≤1

0≤µ

o≤ 1

(or −1 ≤ µo≤0)

0 −1≤µ

o< 0

(or 0 < µ ≤ 1)

4. Point Cosine

Point Cosine*

*The quantities aand bmust have values such that PSC is always nonnegative and finite over the range

of µo.

|µ|

a+bµ|µo| −1≤µ

o≤1

0 ≤ µo≤ 1

(−1 ≤ µo≤ 0)

0 −1 ≤ µo< 0

(or 0 < µo≤ 1)

5. Point Cosine* a+bµ,a≠ 0 −1≤µ

o≤ 1

6. Point Cosine* a + b|µ| −1 ≤ µo≤ 1

2abµo

+()

2ab+

--------------------------

2abµo

+()

2ab–

--------------------------





abµo

------------------

abµo

2ab+

---------------------

4-60 18 December 2000

CHAPTER 4

SRCDX SUBROUTINE

(4.1)

where u,v, and w are the direction cosines of the line from the source point to the point detector

location and u’,v’, and w’are the direction cosines for either the outward normal if the surface

source is outward or the inward normal if the source is inward.

If u=u’,v=v’, and w=w’, then µo=1, indicating that the point detector lies on the normal line.

The value of PSC for the detector point is

where the parenthetical values of µo are for the inward-directed cosine distribution.

For |µo|less than 0.25 in case 2 of Table 4.1, PSC is less than 0.5, which is the value for an isotropic

source. This means that source emissions for these values of |µo|are less probable than the isotropic

case for this source distribution. The converse is also true. Note that if |µo|is greater than 0.5, PSC

is greater than one, which is perfectly valid.

An example of a subroutine SRCDX with appropriate PRPR lines for a surface outward cosine

distribution is shown in Figure 4.21. This is basically the technique that is used in MCNP to

calculate PSC for a spherical surface source in a cosine distribution; the only difference is that

MCNP uses the cosines of the direction from the center of the sphere used to select the source point

because this is the normal to the spherical surface. The primed direction cosines were calculated in

Figure 4.21 to aid in illustrating this example. The direction cosines u,v, and was deﬁned in

Equation (4.1) have already been calculated in subroutine DDDET when SRCDX is called and are

available through COMMON.

∗I,SX.5

C CALCULATE PSC FOR A SURFACE (SPHERE) OUTWARD COSINE DIST

C FIND THE DIRECTION COSINES FOR THIS EXAMPLE BASED

C ON THE SOURCE POINT ON THE SPHERE (X,Y,Z).

UP = (XXX - RDUM(1))/RDUM(4)

VP = (YYY - RDUM(2))/RDUM(4)

WP = (ZZZ - RDUM(3))/RDUM(4)

C (RDUM(1),RDUM(2),RDUM(3)) ARE THE COORDINATES OF THE CENTER

C OF THE SPHERE FROM THE RDUM CARD. RDUM(4) IS THE RADIUS.

C U, V, AND W HAVE BEEN CALCULATED FOR THE CURRENT

C POINT DETECTOR IN SUBROUTINE DDDET

PSC=2.*MAX(ZERO,UUU*UP+VVV*VP+WWW*WP)

Figure 4-21.

µouu′vv′ww′,++=

PSC 2µo,µo0µo0<()>=

0, µo0µo0≥(),≤=

18 December 2000 4-61

CHAPTER 4

SRCDX SUBROUTINE

The PRPR cards in Figures 4.21 and 4.22 are the recommended procedure for replacing the

existing dummy SRCDX subroutine.

For many sources, a discrete probability density function will be used. In this situation, a

cumulative distribution function P(µ) is available and is deﬁned as

where pjis an average value of the probability density function in the interval ∆µj. Thus, the

probability density function is a constant pjin the interval ∆µj. For this case, there are Nvalues of

Pi with Each value of Pihas an associated value of µi.

Because PSC is the derivative of P(µo), then

(4.2)

This is an average PSC between µi-1 and µi and is also an average value of p(µ) in the speciﬁed

range of µ.

Frequently, the cumulative distribution function is divided into Nequally probable intervals. For

this case,

This is precisely the form used in MCNP for calculating contributions to the point detector for

elastic scattering with N= 32.

An example of a subroutine SRCDX for a discrete probability density function is shown in

Figure 4.22. This subroutine would work with the subroutine SOURCE example on page 4–51, and

would calculate PSC = 1/2 for the isotropic distribution.

A biased anisotropic distribution can also be represented by

SIn µoµ1…µ

SPn 0 p1…pn

SBn 0 q1…qn

A reference vector u’,v’,w’ for this distribution is also needed.

Pµ() pµ′()µ′andP

i1+

1–

∫pj∆µj,

j1i,=

∑

P10PN1+

,1.0 and Pi1– Pi.<==

PSC PiPi1–

–

µiµi1–

–

-----------------------µi1– µoµi.<≤,=

PSC 1

----1

µiµi1–

–

---------------------- .=

4-62 18 December 2000

CHAPTER 4

SRCDX SUBROUTINE

The subroutine SOURCE input cards can be modiﬁed for this case by changing the SI1, SP1, SB1,

and RDUM cards as follows:

SI1 −1 0 1 $ These 3 cards

SP1 0 2 1 $ represent a biased

SB1 0 1 2 $ anisotropic distribution.

RDUM 1 0 1 0 $ cylindrical radius and reference vector

SOURCE would sample this anisotropic distribution and SRCDX would calculate the appropriate

PSC.

∗I,SX.5

C THE VARIABLY DIMENSIONED BLOCK SPF HOLDS THE SI, SP, SB ARRAYS.

C THE KSD ARRAY IS A POINTER BLOCK TO THE SPF ARRAY.

C THE FOLLOWING STATEMENT FUNCTION IS DEFINED.

K(I,J)=KSD(LKSD+I,J)

C RDUM(2),RDUM(3),RDUM(4)--DIRECTION COSINES

C FOR THE SOURCE REFERENCE DIRECTION.

AM=UUU*RDUM(2)+VVV*RDUM(3)+WWW*RDUM(4)

C K(4,1) IS THE LENGTH OF THE DISTRIBUTION.

C K(13,1) IS THE OFFSET INTO THE SPF BLOCK.

DO 10 I=1,K(4,1)-1

10 IF(SPF(K(13,1)+1,I).LE.AM.AND.SPF(K(13,1)+1,I+1).GE.AM)

1 GO TO 20

GO TO 30

20 PSC=(SPF(K(13,1)+2,I+1)-SPF(K(13,1)+2,I))/

1 (SPF(K(13,1)+1,I+1)-SPF(K(13,1)+1,I))

PSC=PSC*SPF(K(13,1)+3,I+1)

RETURN

30 PSC=0.

Figure 4-22.

It is extremely important to note that the above case applies only when the source is anisotropic

with azimuthal symmetry. For the general case,

The 2π factor must be applied by the user because MCNP assumes azimuthal symmetry and, in

effect, divides the user-deﬁned PSC by 2π.

For a continuous p(µ,ϕ) function, PSC is calculated as above. In the case of a discrete probability

density function,

PSC 2πpµoϕo

,().=

18 December 2000 4-63

CHAPTER 4

SRCDX SUBROUTINE

where is an average probability density function in

the speciﬁed values of µoand µoand Pi−Pi-1 is the probability of selecting µoand µoin these

intervals. For N equally probable bins and n equally spaced ∆ϕ’s, each 2π/n wide,

Another way to view this general case is by considering solid angles on the unit sphere. For an

isotropic source, the probability (Pi−Pi-1) of being emitted into a speciﬁed solid angle is the ratio

of the total solid angle (4π) to the speciﬁed solid angle (∆ϕ∆µ). Then, PSC ≡ 0.5. Thus, for the

general case (normed to PSC ≡ 0.5 for an isotropic source)

Note that PSC is greater than 0.5 if the speciﬁed solid angle ∆µ∆ϕiis less than (Pi−Pi-1)4π. This

is the same as the previous general expression.

CAUTIONS:

You are cautioned to be extremely careful when using your own subroutine SOURCE with either

detectors or DXTRAN. This caution applies to the calculation of the direct contribution from the

source to a point detector, point on a ring, or point on a DXTRAN sphere. Not only is there the

calculation of the correct value of PSC for an anisotropic source, but there may also be problems

with a biased source.

For example, if an isotropic source is biased to start only in a cone of a speciﬁed angle (for example,

ψ), the starting weight of each particle should be WGT∗(1−cos ψ)/2, where WGT is the weight of

the unbiased source (that is, WGT is the expected weight from a total source). The weight in

SRCDX must be changed to the expected weight WGT to calculate the direct contribution to a point

detector correctly if PSC is deﬁned to be 0.5.

This example can be viewed in a different way. The probability density function for the above

biased source is

PSC 2πpµoϕo

,()⋅2πPiPi1–

–()

µiµi1–

–()ϕ

iϕi1–

–()

-------------------------------------------------------

2πPiPi1–

–()

∆µi∆ϕi

----------------------------------

µi1– µoµiϕ, i1– ϕoϕiand pµoϕo

,()<≤<≤

PSC n

----1

∆µi

-------- .=

PSC 0.5()PiPi1–

–()4π

∆µ∆ϕi

----------------------------------------------- 2πPiPi1–

–()

∆µi∆ϕi

---------------------------------- .==

4-64 18 December 2000

CHAPTER 4

SRCDX SUBROUTINE

Thus, PSC is this constant everywhere in the cone and zero elsewhere. Multiplying this PSC and

biased starting weight gives

WGT ∗ (1 − cos ψ) ∗ 0.5/(1 − cos ψ)

or WGT ∗ 0.5, which is the expected result for an isotropic source.

Another source type that requires caution is for a user supplied source that is energy-angle

correlated. For example, assume a source has a Gaussian distribution in energy where the mean of

the Gaussian is correlated in some manner with µ. In subroutine SRCDX, the µoto a point detector

must be calculated and the energy of the starting particle must be sampled from the Gaussian based

on this µo. This must be done for each point detector in the problem, thus guaranteeing that the

direct source contribution to each detector will be from the proper energy spectrum. The original

energy of the starting particle, as well as all of the other starting parameters, selected in subroutine

SOURCE are automatically restored after the direct source contribution to detectors is made. Thus,

the subroutine SOURCE is still sampled correctly.

pµ() 1

1Ψcos–

---------------------- , for Ψµ1≤≤cos=

0 for 1 µΨ.cos<≤–=

18 December 2000 5-1

CHAPTER 5

DEMO PROBLEM AND OUTPUT

CHAPTER 5

OUTPUT

WHAT IS COVERED IN CHAPTER 5

This chapter shows annotated output from four test problems and an event log print:

DEMO illustrates tally ﬂexibility

TEST1 annotated tables produced by PRINT card

CONC output associated with detectors and detector diagnostics

KCODE output from a criticality calculation (GODIVA)

Event log event log and debug prints

Portions of the complete output have been excluded. The line “SKIP nnn LINES IN OUTPUT”

indicates these omissions.

The event log and debug prints help ﬁnd errors if you set up a geometry improperly or modify

the code. The DBCN input card also is useful when ﬁnding errors but is not discussed here.

MCNP prints out warning messages if needed. Do not ignore these warning messages. Look up

the pertinent section in the manual if you need explanation to help you understand what you are

being warned about.

I. DEMO PROBLEM AND OUTPUT

DEMO has a point isotropic neutron source (SDEF) in the center of a tungsten cube (M2), with

energy uniformly distributed from 0.1 to 10 MeV (SI1,SP1). Flux is calculated across each facet

of the cube (F2), across the sum of all facets (F22), and across the sum of some of the facets

(F12). A pulse height tally (F8) is made in the tungsten cell. Selected pages of the output ﬁle

follow.

The FQ card in the DEMO input ﬁle changes the printing order. Depending upon what you are

interested, the tally output can be made more readable. FQ2 causes energy to be printed as a

function of surface. FQ22 causes surface to be printed as a function of energy. FQ62 prints

multiplier bins as a function of energy for the two surfaces desired. The NT and T features also

are illustrated in tallies 62 and 22, respectively. The generalized FM62 card used with the F62:N

tally is a useful feature for normalization, unit conversion, reaction rate, etc., and has three

multipliers instead of one. Finally, the TF62 card causes the tally ﬂuctuation chart for the second

surface, the ﬁrst multiplier bin, and the second energy bin to be printed. By default the

ﬂuctuation chart for tally 62 would contain information for the ﬁrst surface, the ﬁrst multiplier

bin, and the last energy bin.

5-2 18 December 2000

CHAPTER 5

DEMO PROBLEM AND OUTPUT

The F8:E card provides a pulse height tally in cell 1. The F8 tally capability is limited to an

analog problem. The default implicit capture is turned off by the CUT:N card. Analog capture is

the default for photons and electrons, so CUT:P and CUT:E cards are not required. The pulse

height tally tracks the energy deposited in a cell by both photons and electrons, even if only E or

P is on the F8 card. The F8 tally is not available for neutrons and will return an error if

attempted. In the following output is a warning that “f8 tally unreliable since neutron transport

nonanalog”. This message means some nonanalog events such as (n,2n) may have occured to the

neutron from the source to the production of a photon, not that there is an F8:N tally or that there

is some neutron variance reduction in the deﬁnition of the problem.

A tally ﬂuctuation chart bin analysis follows each tally. Only an analysis for tally 8 is shown in

this example. This analysis checks the variance of the variance as well as the general behavior of

the probability density function of each tally and provides an additional set of checks to ensure

the reliability of a tally. Ten different statistical checks are run for the tally and presented in

tabular form. The results of the ten checks are presented in pass?yes/no table format. These

checks do not guarantee the absolute reliability of the tally, but they provide a better method of

identifying problems that have not been sampled well. A more complete description of the

signiﬁcance of each entry in the tally ﬂuctuation section is presented in TEST1.

There are three possible physics treatments for problems involving photons. The ﬁrst is the

explicit p,e treatment where photons generate electrons that are the tracked and generate photons

(ad inﬁnium). This is the most accurate model but is costly in terms of runtime. The second

physics treatment is mode p only that uses the default ‘thick target bremsstrahlung’ (TTB) model

where electrons are generated in the direction of the incident photon and are immediately

annihilated after generating bremsstrahlung photons. The third photon physics treatment is a

mode p only with the thick target bremsstrahlung turned off (IDES=1 on the PHYS card). Then

electrons are completely ignored.

The choice of which physics treatment to use depends on the objective of the problem being

solved. Using a test problem similar to the cube above it was found that F4 photon tallies for the

three treatments agreed reasonably well above 2 MeV. Below 1 MeV the results from the

simplest model (photon mode no bremstrahlung) began to diverge from the full physics model

results. Below the annihilation photon peak, the TTB treatment also begins to diverge from the

mode p,e results. The choice of physics treatment had a drastic impact on the runtime of the

problem. To run 1E6 particles on a SGI 2000 mode p with and without thick target

bremstrahlung took 2.25 and 1.70 minutes respectively, while the full physics mode p,e problem

took just over 17 hours. If it is necessary to model photon generation and transport below 0.5

MeV then the full physics model should be used. However if these low energy photons are not

important or if the calculation is for diagnostic purposes, then the mode p with or without thick

target bremstrahlung model is sufﬁcient.

18 December 2000 5-3

CHAPTER 5

DEMO PROBLEM AND OUTPUT

The small table preceding the summary of statistical checks indicates that some of the tally

scores were not made for some reason. In the case of tally 2, 93547 particles did not score in any

of the bins because their energy was greater than that of the upper limit of the highest energy bin.

Tallies 12, 22, and 62 also had signiﬁcant numbers of particles that had energies above the

highest energy bin. This is concerning since for tally 2 the number of particles not scored is

nearly 90% of the initial source particles. This can be ﬁxed by simple increasing the upper limit

of the last energy bin, or adding more bins to cover the energy range up to the maximum energy

of the source (10 MeV).

CHAPTER 5

DEMO PROBLEM AND OUTPUT

5-4 18 December 2000

1mcnp version 4c ld=01/20/00 07/19/00 14:32:23

************************************************************************* probid = 07/19/00 14:32:23

i=demo name=demo.

1- demo: a box with flux across surfaces in various combinations

2- 1 2 -1.6 -1

3- 2 0 1

5- 1 rpp -1 1 -1 1 -1 1

7- cut:n 10000 0.0 0.0 0.0

8- mode n p e

9- sdef pos=0 0 0 cel=1 wgt=1 erg=d1

10- si1 0.1 10

11- sp1 0 1

12- imp:n,p,e 1 0

13- e0 0.2 0.4 0.6 0.8 1

14- f2:n 1.1 1.2 1.3 1.4 1.5 1.6

15- fq2 e f

16- f12:n (1.3 1.5) (1.4 1.6) (1.2 1.1)

17- f22:n 1.1 1.2 1.3 1.4 1.5 1.6 t

18- fq22 f e

19- m1 6000.50c 1

warning. material 1 is used only for a perturbation or tally.

20- m2 74000.55c 1

21- f62:n 1.3 1.4

22- fm62 (1 2(1 -4)(-2))(1 1 1)

23- e62 0.2 0.4 0.6 0.8 1 nt $

24- fq62 m e

25- tf62 2 5j 2

26- f8:e 1

warning. f8 tally unreliable since neutron transport nonanalog.

27- e8 0.001 10i 20 $

28- nps 104000

warning. tally 8 needs zero energy bin for negative f8 scores.

SKIP 482 LINES IN OUTPUT

1tally 22 nps = 104000

tally type 2 particle flux averaged over a surface. units 1/cm**2

18 December 2000 5-5

CHAPTER 5

DEMO PROBLEM AND OUTPUT

tally for neutrons

areas

surface: 1.1 1.2 1.3 1.4 1.5 1.6 total

4.00000E+00 4.00000E+00 4.00000E+00 4.00000E+00 4.00000E+00 4.00000E+00 2.40000E+01

energy: 2.0000E-01 4.0000E-01 6.0000E-01 8.0000E-01 1.0000E+00

surface

1.1 7.30357E-04 0.0942 1.29205E-03 0.0513 1.18025E-03 0.0653 1.17268E-03 0.0651 1.16323E-03 0.0534

1.2 6.57953E-04 0.0686 1.15327E-03 0.0508 1.05268E-03 0.0545 1.13600E-03 0.0519 1.18122E-03 0.0643

1.3 6.33307E-04 0.0756 1.27882E-03 0.0717 1.13639E-03 0.0528 1.17168E-03 0.0521 1.13825E-03 0.0524

1.4 6.52386E-04 0.0705 1.07859E-03 0.0576 1.21496E-03 0.0528 1.25657E-03 0.0495 1.05426E-03 0.0537

1.5 6.11844E-04 0.0711 1.21984E-03 0.0522 1.25661E-03 0.0487 1.26987E-03 0.0496 1.15210E-03 0.0516

1.6 6.60230E-04 0.0686 1.15995E-03 0.0522 1.19812E-03 0.0515 1.10335E-03 0.0526 1.07351E-03 0.0543

total 6.57680E-04 0.0312 1.19709E-03 0.0231 1.17317E-03 0.0221 1.18503E-03 0.0217 1.12710E-03 0.0224

energy: total

surface

1.1 5.53857E-03 0.0284

1.2 5.18112E-03 0.0258

1.3 5.35846E-03 0.0273

1.4 5.25677E-03 0.0248

1.5 5.51027E-03 0.0237

1.6 5.19516E-03 0.0244

total 5.34006E-03 0.0102

SKIP 227 LINES OF OUTPUT

1tally 8 nps = 104000

tally type 8 pulse height distribution. units number

tally for photons electrons

cell 1

energy

1.0000E-03 1.88365E-02 0.0224

1.8191E+00 1.50962E-02 0.0250

3.6372E+00 2.59615E-04 0.1924

5.4553E+00 7.69231E-05 0.3535

7.2734E+00 4.80769E-05 0.4472

9.0915E+00 1.92308E-05 0.7071

1.0910E+01 0.00000E+00 0.0000

1.2728E+01 0.00000E+00 0.0000

1.4546E+01 0.00000E+00 0.0000

1.6364E+01 0.00000E+00 0.0000

1.8182E+01 0.00000E+00 0.0000

2.0000E+01 0.00000E+00 0.0000

total 3.43365E-02 0.0164

1analysis of the results in the tally fluctuation chart bin (tfc) for tally 8 with nps = 104000 print table 160

normed average tally per history = 3.43365E-02 unnormed average tally per history = 3.43365E-02

CHAPTER 5

DEMO PROBLEM AND OUTPUT

5-6 18 December 2000

estimated tally relative error = 0.0164 estimated variance of the variance = 0.0003

relative error from zero tallies = 0.0164 relative error from nonzero scores = 0.0000

number of nonzero history tallies = 3571 efficiency for the nonzero tallies = 0.0343

history number of largest tally = 13 largest unnormalized history tally = 1.00000E+00

(largest tally)/(average tally) = 2.91235E+01 (largest tally)/(avg nonzero tally)= 1.00000E+00

(confidence interval shift)/mean = 0.0001 shifted confidence interval center = 3.43410E-02

if the largest history score sampled so far were to occur on the next history, the tfc bin quantities would change as follows:

estimated quantities value at nps value at nps+1 value(nps+1)/value(nps)-1.

mean 3.43365E-02 3.43458E-02 0.000270

relative error 1.64444E-02 1.64420E-02 -0.000145

variance of the variance 2.51529E-04 2.51451E-04 -0.000310

shifted center 3.43410E-02 3.43410E-02 0.000000

figure of merit 1.28091E+03 1.28128E+03 0.000290

the 100 largest history tallies appear to have a maximum value of about 1.00000E+00

the large score tail of the empirical history score probability density function appears to have no unsampled regions.

===================================================================================================================================

results of 10 statistical checks for the estimated answer for the tally fluctuation chart (tfc) bin of tally 8

tfc bin --mean-- ---------relative error--------- ----variance of the variance---- --figure of merit-- -pdf-

behavior behavior value decrease decrease rate value decrease decrease rate value behavior slope

desired random <0.10 yes 1/sqrt(nps) <0.10 yes 1/nps constant random >3.00

observed random 0.02 yes yes 0.00 yes yes constant random 10.00

passed? yes yes yes yes yes yes yes yes yes yes

===================================================================================================================================

this tally meets the statistical criteria used to form confidence intervals: check the tally fluctuation chart to verify.

the results in other bins associated with this tally may not meet these statistical criteria.

estimated asymmetric confidence interval(1,2,3 sigma): 3.3776E-02 to 3.4906E-02; 3.3212E-02 to 3.5470E-02; 3.2647E-02 to 3.6035E-02

estimated symmetric confidence interval(1,2,3 sigma): 3.3772E-02 to 3.4901E-02; 3.3207E-02 to 3.5466E-02; 3.2643E-02 to 3.6030E-02

fom = (histories/minute)*(f(x) signal-to-noise ratio)**2 = (3.602E+04)*( 1.886E-01)**2 = (3.602E+04)*(3.556E-02) = 1.281E+03

1 some tally scores were not made for various reasons:

18 December 2000 5-7

CHAPTER 5

DEMO PROBLEM AND OUTPUT

beyond last bin not in

tally angle energy time user

2 0 93547 0 0

12 0 93547 0 0

22 0 93547 0 0

62 0 31134 0 0

1status of the statistical checks used to form confidence intervals for the mean for each tally bin

tally result of statistical checks for the tfc bin (the first check not passed is listed) and error magnitude check for all bins

2 missed 2 of 10 tfc bin checks: the figure of merit has a trend during the last half of the problem

passed all bin error check: 36 tally bins all have relative errors less than 0.10 with no zero bins

12 missed 3 of 10 tfc bin checks: the variance of the variance does not monotonically decrease over the last half of problem

passed all bin error check: 18 tally bins all have relative errors less than 0.10 with no zero bins

22 missed 2 of 10 tfc bin checks: the figure of merit has a trend during the last half of the problem

passed all bin error check: 42 tally bins all have relative errors less than 0.10 with no zero bins

62 missed 1 of 10 tfc bin checks: there is insufficient tfc bin tally information to estimate the large tally slope reliably

passed all bin error check: 30 tally bins all have relative errors less than 0.10 with no zero bins

8 passed the 10 statistical checks for the tally fluctuation chart bin result

missed all bin error check: 13 tally bins had 6 bins with zeros and 4 bins with relative errors exceeding 0.10

the 10 statistical checks are only for the tally fluctuation chart bin and do not apply to other tally bins.

the tally bins with zeros may or may not be correct: compare the source, cutoffs, multipliers, et cetera with the tally bins.

warning. 4 of the 5 tally fluctuation chart bins did not pass all 10 statistical checks.

warning. 1 of the 5 tallies had bins with relative errors greater than recommended.

1tally fluctuation charts

tally 2 tally 12 tally 22

nps mean error vov slope fom mean error vov slope fom mean error vov slope fom

8000 4.6911E-03 0.0945 0.0153 0.0 648 5.6745E-03 0.0580 0.0033 0.0 1719 4.6911E-03 0.0945 0.0153 0.0 648

16000 4.8398E-03 0.0645 0.0059 0.0 605 5.6055E-03 0.0416 0.0020 3.2 1458 4.8398E-03 0.0645 0.0059 0.0 605

24000 5.2302E-03 0.0505 0.0035 0.0 600 5.5189E-03 0.0342 0.0013 3.5 1313 5.2302E-03 0.0505 0.0035 0.0 600

32000 5.2548E-03 0.0433 0.0025 2.0 593 5.5052E-03 0.0299 0.0013 2.2 1247 5.2548E-03 0.0433 0.0025 2.0 593

40000 5.3488E-03 0.0401 0.0074 1.7 561 5.3085E-03 0.0272 0.0011 2.4 1215 5.3488E-03 0.0401 0.0074 1.7 561

48000 5.3088E-03 0.0364 0.0056 1.7 558 5.3638E-03 0.0248 0.0009 2.3 1209 5.3088E-03 0.0364 0.0056 1.7 558

56000 5.5486E-03 0.0425 0.0630 1.5 347 5.3419E-03 0.0231 0.0009 2.1 1175 5.5486E-03 0.0425 0.0630 1.5 347

64000 5.6121E-03 0.0383 0.0534 1.6 378 5.3601E-03 0.0216 0.0008 2.0 1194 5.6121E-03 0.0383 0.0534 1.6 378

72000 5.5574E-03 0.0355 0.0468 1.6 391 5.3959E-03 0.0203 0.0007 2.0 1198 5.5574E-03 0.0355 0.0468 1.6 391

80000 5.5380E-03 0.0332 0.0408 1.6 411 5.3570E-03 0.0193 0.0006 2.1 1223 5.5380E-03 0.0332 0.0408 1.6 411

88000 5.5665E-03 0.0316 0.0340 1.6 415 5.3594E-03 0.0192 0.0064 1.9 1127 5.5665E-03 0.0316 0.0340 1.6 415

96000 5.5762E-03 0.0298 0.0302 1.6 422 5.3934E-03 0.0188 0.0091 1.9 1054 5.5762E-03 0.0298 0.0302 1.6 422

CHAPTER 5

DEMO PROBLEM AND OUTPUT

5-8 18 December 2000

104000 5.5386E-03 0.0284 0.0274 1.7 431 5.4344E-03 0.0179 0.0078 1.9 1079 5.5386E-03 0.0284 0.0274 1.7 431

tally 62 tally 8

nps mean error vov slope fom mean error vov slope fom

8000 2.2306E-05 0.2087 0.0992 0.0 133 3.3500E-02 0.0601 0.0034 0.0 1604

16000 2.0345E-05 0.1462 0.0436 0.0 118 3.2250E-02 0.0433 0.0018 10.0 1344

24000 1.9682E-05 0.1195 0.0259 0.0 107 3.2708E-02 0.0351 0.0012 10.0 1243

32000 2.0456E-05 0.1000 0.0167 0.0 111 3.2594E-02 0.0305 0.0009 10.0 1201

40000 2.1405E-05 0.0960 0.0550 0.0 98 3.2850E-02 0.0271 0.0007 10.0 1225

48000 2.1120E-05 0.0866 0.0429 0.0 99 3.3104E-02 0.0247 0.0006 10.0 1217

56000 2.1137E-05 0.0793 0.0335 0.0 100 3.3304E-02 0.0228 0.0005 10.0 1208

64000 2.1388E-05 0.0732 0.0264 0.0 104 3.3734E-02 0.0212 0.0004 10.0 1239

72000 2.2092E-05 0.0691 0.0232 0.0 103 3.4250E-02 0.0198 0.0004 10.0 1259

80000 2.1131E-05 0.0666 0.0212 0.0 102 3.4000E-02 0.0188 0.0003 10.0 1278

88000 2.0994E-05 0.0634 0.0184 0.0 103 3.4284E-02 0.0179 0.0003 10.0 1292

96000 2.1022E-05 0.0602 0.0160 0.0 103 3.4656E-02 0.0170 0.0003 10.0 1286

104000 2.1130E-05 0.0574 0.0140 0.0 105 3.4337E-02 0.0164 0.0003 10.0 1281

***********************************************************************************************************************

dump no. 2 on file demo.r nps = 104000 coll = 9352864 ctm = 2.89 nrn = 46800292

15 warning messages so far.

run terminated when 104000 particle histories were done.

computer time = 2.94 minutes

mcnp version 4c 01/20/00 07/19/00 14:35:44 probid = 07/19/00 14:32:23

18 December 2000 5-9

CHAPTER 5

TEST1 PROBLEM AND OUTPUT

II. TEST1 PROBLEM AND OUTPUT

TEST 1 deﬁnes a disk of concrete 100 cm thick, with a 75-cm radius. A 14.19 MeV neutron

source is incident at a point in the center of a face of the disk and normal to it. Several neutron

and photon tallies are made on surface 18 and in cell 17. There is no energy cutoff and the simple

physics treatment that includes implicit capture is used for photons with energy greater than

0.001 MeV.

The disk is divided into 16 slabs, each 6.25 cm thick, as seen in Fig. 5.1. The neutron importance

of each slab, or cell, varies from 0 in cell 1 to 32 in cell 17. Photon importances are set equal to

neutron importances. The problem ran 10,000 particles and the tally means, errors, and FOMs

shown in the tally ﬂuctuation charts seem to be stable.

Figure 5.1

The weight window generator was used to generate a better importance function for subsequent

runs. The resulting cards are printed at the end of the TEST1 output ﬁle and can be copied into

an input ﬁle to be run a second time. Generation of weight windows did not affect the results of

TEST1 but did slow down the calculation by 14%. When the importances in TEST1 were

replaced by the generated weight windows (WWP and WWN cards), the problem took 14.27

minutes to run 10000 particles vs 10.01 minutes for TEST1. However, the photon FOMs

increased by a factor of 2 to 3 and the errors decreased by half, while the means appeared to stay

stable. The neutron means, errors, and FOMs stayed approximately the same, indicating that they

were already well chosen to optimize tally 12. The use of the mesh-based weight window

generator instead of the cell-based weight window generator for this problem did not

signiﬁcantly improve the FOM because the cell-based weight windows were quite good.

Following is a partial output from TEST1. The symbol Xappearing left of the table title indicates

that table does not appear unless the PRINT option or card is used. If Nn, where n is an integer,

appears before an item on a page or below a column, that item is explained or discussed in Note

Nn in the text following the output.

2 3 4 16 17 18

2 3 16 17 1

CHAPTER 5

TEST1 PROBLEM AND OUTPUT

5-10 18 December 2000

N11mcnp version 4c ld=01/20/00 06/23/00 11:30:40

************************************************************************* probid = 06/23/00 11:30:40

N2i=test1 name=test1.

N31- test1: 100 cm thick concrete disk with 15 splitting surfaces

2- c

3- 1 0 1 : -2 : 18

4- 2 1 -2.2505 -1 -3 2

5- 3 1 -2.2505 -1 -4 3

6- 4 1 -2.2505 -1 -5 4

7- 5 1 -2.2505 -1 -6 5

8- 6 1 -2.2505 -1 -7 6

9- 7 1 -2.2505 -1 -8 7

10- 8 1 -2.2505 -1 -9 8

11- 9 1 -2.2505 -1 -10 9

12- 10 1 -2.2505 -1 -11 10

13- 11 1 -2.2505 -1 -12 11

14- 12 1 -2.2505 -1 -13 12

15- 13 1 -2.2505 -1 -14 13

16- 14 1 -2.2505 -1 -15 14

17- 15 1 -2.2505 -1 -16 15

18- 16 1 -2.2505 -1 -17 16

19- 17 1 -2.2505 -1 -18 17

20-

21- 1 cy 75

22- 2 py 0

23- 3 py 6.25

24- 4 py 12.50

25- 5 py 18.75

26- 6 py 25.00

27- 7 py 31.25

28- 8 py 37.50

29- 9 py 43.75

30- 10 py 50.00

31- 11 py 56.25

32- 12 py 62.50

33- 13 py 68.75

34- 14 py 75.00

35- 15 py 81.25

36- 16 py 87.50

37- 17 py 93.75

38- 18 py 100.00

39-

40- mode n p

41- c the following is los alamos concrete

42- m1 1001.60c 8.47636e-2

18 December 2000 5-11

CHAPTER 5

TEST1 PROBLEM AND OUTPUT

43- 8016.60c 6.04086e-1

44- 11023.60c 9.47250e-3

45- 12000.60c 2.99826e-3

46- 13027.60c 2.48344e-2

47- 14000.60c 2.41860e-1

48- 19000.60c 6.85513e-3

49- 20000.60c 2.04808e-2

50- 26054.60c 2.74322e-4

51- 26056.60c 4.26455e-3

52- 26057.60c 9.76401e-5

53- 26058.60c 1.30187e-5

N454- sdef pos=0 0 0 cel=2 wgt=1 vec=0 1 0 sur=2 dir=1 erg=14.19

55- imp:n 0 1 5r 2 2 2 4 4 4 8 8 16 32

56- imp:p 0 1 5r 2 2 2 4 4 4 8 8 16 32

N557- pwt 0 -10 -9 -8 -7 -6 -5 -4 -3 -2 -1.7 -1.4 -1.0 -0.7 -0.4 -0.3 -0.2

58- f1:p 18

59- f11:n 18

60- fc12 optimize weight window generator on tally 12

61- f12:p 18

62- e12 20

63- wwg 12 2

64- f6:n,p 17

65- e6 .00001 .0001 .001 .01 .05 .1 .5 1 13i 15 20

66- f16:n 17

67- f26:p 17

68- f34:n 17

69- fm34 -1 1 1 -4

N670- e0 .0001 .001 .01 .05 .1 .5 1 13i 15 20

71- phys:n 15 0

72- phys:p .001

73- nps 10000

74- print

X 1source print table 10

N7values of defaulted or explicitly defined source variables

cel 2.0000E+00

sur 2.0000E+00

erg 1.4190E+01

tme 0.0000E+00

dir 1.0000E+00

pos 0.0000E+00 0.0000E+00 0.0000E+00

x 0.0000E+00

y 0.0000E+00

z 0.0000E+00

rad 0.0000E+00

ext 0.0000E+00

axs 0.0000E+00 0.0000E+00 0.0000E+00

CHAPTER 5

TEST1 PROBLEM AND OUTPUT

5-12 18 December 2000

vec 0.0000E+00 1.0000E+00 0.0000E+00

ccc 0.0000E+00

nrm 1.0000E+00

ara 0.0000E+00

wgt 1.0000E+00

eff 1.0000E-02

par 0.0000E+00

order of sampling source variables.

cel sur pos vec dir erg tme

X 1tally 11 print table 30

N8 tally type 1 number of particles crossing a surface.

tally for neutrons

N9warning. energy bin limits adjusted for tally 11

surfaces 18

N10 energy bins

0.00000E+00 to 1.00000E-04 mev

1.00000E-04 to 1.00000E-03 mev

1.00000E-03 to 1.00000E-02 mev

1.00000E-02 to 5.00000E-02 mev

5.00000E-02 to 1.00000E-01 mev

1.00000E-01 to 5.00000E-01 mev

5.00000E-01 to 1.00000E+00 mev

1.00000E+00 to 2.00000E+00 mev

2.00000E+00 to 3.00000E+00 mev

3.00000E+00 to 4.00000E+00 mev

4.00000E+00 to 5.00000E+00 mev

5.00000E+00 to 6.00000E+00 mev

6.00000E+00 to 7.00000E+00 mev

7.00000E+00 to 8.00000E+00 mev

8.00000E+00 to 9.00000E+00 mev

9.00000E+00 to 1.00000E+01 mev

1.00000E+01 to 1.10000E+01 mev

1.10000E+01 to 1.20000E+01 mev

1.20000E+01 to 1.30000E+01 mev

1.30000E+01 to 1.40000E+01 mev

1.40000E+01 to 1.50000E+01 mev

total bin

SKIP 158 LINES IN OUTPUT

X 1material composition print table 40

material

number component nuclide, atom fraction

1 1001, 8.47636E-02 8016, 6.04086E-01 11023, 9.47250E-03 12000, 2.99826E-03

18 December 2000 5-13

CHAPTER 5

TEST1 PROBLEM AND OUTPUT

13027, 2.48344E-02 14000, 2.41860E-01 19000, 6.85513E-03 20000, 2.04808E-02

26054, 2.74322E-04 26056, 4.26455E-03 26057, 9.76401E-05 26058, 1.30187E-05

material

number component nuclide, mass fraction

1 1001, 4.53200E-03 8016, 5.12597E-01 11023, 1.15530E-02 12000, 3.86599E-03

13027, 3.55480E-02 14000, 3.60364E-01 19000, 1.42190E-02 20000, 4.35460E-02

26054, 7.84990E-04 26056, 1.26547E-02 26057, 2.94921E-04 26058, 4.00121E-05

X 1cell volumes and masses print table 50

N11

cell atom gram input calculated reason volume

density density volume volume mass pieces not calculated

1 1 0.00000E+00 0.00000E+00 0.00000E+00 0.00000E+00 0.00000E+00 0 infinite

2 2 7.18983E-02 2.25050E+00 0.00000E+00 1.10447E+05 2.48560E+05 1

3 3 7.18983E-02 2.25050E+00 0.00000E+00 1.10447E+05 2.48560E+05 1

4 4 7.18983E-02 2.25050E+00 0.00000E+00 1.10447E+05 2.48560E+05 1

5 5 7.18983E-02 2.25050E+00 0.00000E+00 1.10447E+05 2.48560E+05 1

6 6 7.18983E-02 2.25050E+00 0.00000E+00 1.10447E+05 2.48560E+05 1

7 7 7.18983E-02 2.25050E+00 0.00000E+00 1.10447E+05 2.48560E+05 1

8 8 7.18983E-02 2.25050E+00 0.00000E+00 1.10447E+05 2.48560E+05 1

9 9 7.18983E-02 2.25050E+00 0.00000E+00 1.10447E+05 2.48560E+05 1

10 10 7.18983E-02 2.25050E+00 0.00000E+00 1.10447E+05 2.48560E+05 1

11 11 7.18983E-02 2.25050E+00 0.00000E+00 1.10447E+05 2.48560E+05 1

12 12 7.18983E-02 2.25050E+00 0.00000E+00 1.10447E+05 2.48560E+05 1

13 13 7.18983E-02 2.25050E+00 0.00000E+00 1.10447E+05 2.48560E+05 1

14 14 7.18983E-02 2.25050E+00 0.00000E+00 1.10447E+05 2.48560E+05 1

15 15 7.18983E-02 2.25050E+00 0.00000E+00 1.10447E+05 2.48560E+05 1

16 16 7.18983E-02 2.25050E+00 0.00000E+00 1.10447E+05 2.48560E+05 1

17 17 7.18983E-02 2.25050E+00 0.00000E+00 1.10447E+05 2.48560E+05 1

X 1surface areas print table 50

surface input calculated reason area

area area not calculated

1 1 0.00000E+00 4.71239E+04

2 2 0.00000E+00 1.76715E+04

3 3 0.00000E+00 1.76715E+04

4 4 0.00000E+00 1.76715E+04

5 5 0.00000E+00 1.76715E+04

6 6 0.00000E+00 1.76715E+04

7 7 0.00000E+00 1.76715E+04

8 8 0.00000E+00 1.76715E+04

CHAPTER 5

TEST1 PROBLEM AND OUTPUT

5-14 18 December 2000

9 9 0.00000E+00 1.76715E+04

10 10 0.00000E+00 1.76715E+04

11 11 0.00000E+00 1.76715E+04

12 12 0.00000E+00 1.76715E+04

13 13 0.00000E+00 1.76715E+04

14 14 0.00000E+00 1.76715E+04

15 15 0.00000E+00 1.76715E+04

16 16 0.00000E+00 1.76715E+04

17 17 0.00000E+00 1.76715E+04

18 18 0.00000E+00 1.76715E+04

X 1cells print table 60

atom gram neutron photon photon wt

cell mat density density volume mass pieces importance importance generation

1 1 0 0.00000E+00 0.00000E+00 0.00000E+00 0.00000E+00 0 0.0000E+00 0.0000E+00 0.000E+00

2 2 1 7.18983E-02 2.25050E+00 1.10447E+05 2.48560E+05 1 1.0000E+00 1.0000E+00 -1.000E+01

3 3 1 7.18983E-02 2.25050E+00 1.10447E+05 2.48560E+05 1 1.0000E+00 1.0000E+00 -9.000E+00

4 4 1 7.18983E-02 2.25050E+00 1.10447E+05 2.48560E+05 1 1.0000E+00 1.0000E+00 -8.000E+00

5 5 1 7.18983E-02 2.25050E+00 1.10447E+05 2.48560E+05 1 1.0000E+00 1.0000E+00 -7.000E+00

6 6 1 7.18983E-02 2.25050E+00 1.10447E+05 2.48560E+05 1 1.0000E+00 1.0000E+00 -6.000E+00

7 7 1 7.18983E-02 2.25050E+00 1.10447E+05 2.48560E+05 1 1.0000E+00 1.0000E+00 -5.000E+00

8 8 1 7.18983E-02 2.25050E+00 1.10447E+05 2.48560E+05 1 2.0000E+00 2.0000E+00 -4.000E+00

9 9 1 7.18983E-02 2.25050E+00 1.10447E+05 2.48560E+05 1 2.0000E+00 2.0000E+00 -3.000E+00

10 10 1 7.18983E-02 2.25050E+00 1.10447E+05 2.48560E+05 1 2.0000E+00 2.0000E+00 -2.000E+00

11 11 1 7.18983E-02 2.25050E+00 1.10447E+05 2.48560E+05 1 4.0000E+00 4.0000E+00 -1.700E+00

12 12 1 7.18983E-02 2.25050E+00 1.10447E+05 2.48560E+05 1 4.0000E+00 4.0000E+00 -1.400E+00

13 13 1 7.18983E-02 2.25050E+00 1.10447E+05 2.48560E+05 1 4.0000E+00 4.0000E+00 -1.000E+00

14 14 1 7.18983E-02 2.25050E+00 1.10447E+05 2.48560E+05 1 8.0000E+00 8.0000E+00 -7.000E-01

15 15 1 7.18983E-02 2.25050E+00 1.10447E+05 2.48560E+05 1 8.0000E+00 8.0000E+00 -4.000E-01

16 16 1 7.18983E-02 2.25050E+00 1.10447E+05 2.48560E+05 1 1.6000E+01 1.6000E+01 -3.000E-01

17 17 1 7.18983E-02 2.25050E+00 1.10447E+05 2.48560E+05 1 3.2000E+01 3.2000E+01 -2.000E-01

N12total 1.76715E+06 3.97696E+06

X 1surfaces print table 70

N13 surface trans type surface coefficients

1 1 cy 7.5000000E+01

2 2 py 0.0000000E+00

3 3 py 6.2500000E+00

4 4 py 1.2500000E+01

5 5 py 1.8750000E+01

6 6 py 2.5000000E+01

7 7 py 3.1250000E+01

8 8 py 3.7500000E+01

9 9 py 4.3750000E+01

10 10 py 5.0000000E+01

11 11 py 5.6250000E+01

18 December 2000 5-15

CHAPTER 5

TEST1 PROBLEM AND OUTPUT

12 12 py 6.2500000E+01

13 13 py 6.8750000E+01

14 14 py 7.5000000E+01

15 15 py 8.1250000E+01

16 16 py 8.7500000E+01

17 17 py 9.3750000E+01

18 18 py 1.0000000E+02

1 cell temperatures in mev for the free-gas thermal neutron treatment. print table 72

N14 all non-zero importance cells with materials have a temperature for thermal neutrons of 2.5300E-08 mev.

minimum source weight = 1.0000E+00 maximum source weight = 1.0000E+00

3 warning messages so far.

N151physical constants print table 98

name value description

huge 1.0000000000000E+37 infinity

pie 3.1415926535898E+00 pi

euler 5.7721566490153E-01 euler constant

avogad 6.0220434469282E+23 avogadro number (molecules/mole)

aneut 1.0086649670000E+00 neutron mass (amu)

avgdn 5.9703109000000E-01 avogadro number/neutron mass (1.e-24*molecules/mole/amu)

slite 2.9979250000000E+02 speed of light (cm/shake)

planck 4.1357320000000E-13 planck constant (mev shake)

fscon 1.3703930000000E+02 inverse fine structure constant h*c/(2*pi*e**2)

gpt(1) 9.3958000000000E+02 neutron mass (mev)

gpt(3) 5.1100800000000E-01 electron mass (mev)

fission q-values: nuclide q(mev) nuclide q(mev)

90232 171.91 91233 175.57

92233 180.84 92234 179.45

92235 180.88 92236 179.50

92237 180.40 92238 181.31

92239 180.40 92240 180.40

93237 183.67 94238 186.65

94239 189.44 94240 186.36

94241 188.99 94242 185.98

94243 187.48 95241 190.83

95242 190.54 95243 190.25

96242 190.49 96244 190.49

other 180.00

the following compilation options were used:

pointer

CHAPTER 5

TEST1 PROBLEM AND OUTPUT

5-16 18 December 2000

cheap

unix

sun

plot

mcplot

gkssim

xlib

xs64

default datapath: /usr/local/codes/data/mc/type2/unix64

N161cross-section tables print table 100

table length

tables from file endf602

1001.60c 2322 1-h-1 from endf-vi.1 mat 125 11/25/93

8016.60c 50346 8-o-16 from endf/b-vi mat 8 25 11/25/93

11023.60c 48471 11-na-23 from endf/b-vi.1 mat1125 11/25/93

12000.60c 52785 12-mg-nat from endf/b-vi mat1200 11/25/93

13027.60c 49407 13-al-27 from endf/b-vi mat1325 11/25/93

14000.60c 100118 14-si-nat from endf/b-vi mat1400 11/25/93

19000.60c 23390 19-k-nat from endf/b-vi mat1900 11/25/93

20000.60c 70573 20-ca-nat from endf/b-vi mat2000 11/25/93

26054.60c 120443 endf/b-vi.1 fe54a mat2625 11/25/93

26056.60c 172174 endf/b-vi.1 fe56a mat2631 11/25/93

26057.60c 133044 endf/b-vi.1 fe57a mat2634 11/25/93

26058.60c 92535 endf/b-vi.1 fe58a mat2637 11/25/93

tables from file mcplib022

1000.02p 623 01/15/93

8000.02p 623 01/15/93

11000.02p 635 01/15/93

12000.02p 643 01/15/93

13000.02p 643 01/15/93

14000.02p 643 01/15/93

19000.02p 643 01/15/93

20000.02p 651 01/15/93

26000.02p 651 01/15/93

total 921363

N17 any neutrons with energy greater than emax = 1.50000E+01 from the source or from a collision will be resampled.

N18 neutron cross sections outside the range from 0.0000E+00 to 1.5000E+01 mev are expunged.

maximum photon energy set to 100.0 mev (maximum electron energy)

18 December 2000 5-17

CHAPTER 5

TEST1 PROBLEM AND OUTPUT

tables from file el032

1000.03e 2329 6/6/98

8000.03e 2333 6/6/98

11000.03e 2337 6/6/98

12000.03e 2337 6/6/98

13000.03e 2337 6/6/98

14000.03e 2339 6/6/98

19000.03e 2343 6/6/98

20000.03e 2343 6/6/98

26000.03e 2345 6/6/98

X 1range table for material 1 (condensed) print table 85

electron substeps per energy step = 4, default = 4. mean ionization energy = 1.41099E+02 ev.

N19X density effect data

non-conductor

z = 1

occ no, be(ev) pairs

1. 13.600

z = 8

occ no, be(ev) pairs

2. 538.000 2. 28.480 4. 13.620

z = 11

occ no, be(ev) pairs

2. 1075.000 2. 66.000 2. 34.000 4. 34.000 -1. 5.139

z = 12

occ no, be(ev) pairs

2. 1308.000 2. 92.000 2. 54.000 4. 54.000 -2. 7.646

z = 13

occ no, be(ev) pairs

2. 1564.000 2. 121.000 2. 77.000 4. 77.000 -3. 9.075

z = 14

occ no, be(ev) pairs

2. 1844.000 2. 154.000 2. 104.000 4. 104.000 2. 13.460 -2. 8.151

z = 19

occ no, be(ev) pairs

2. 3610.000 2. 381.000 2. 299.000 4. 296.000 2. 37.000 2. 19.000

4. 18.700 -1. 4.341

z = 20

occ no, be(ev) pairs

2. 4041.000 2. 441.000 2. 353.000 4. 349.000 2. 46.000 2. 28.000

4. 28.000 -2. 6.113

z = 26

occ no, be(ev) pairs

2. 7117.000 2. 851.000 2. 726.000 4. 713.000 2. 98.000 2. 61.000

4. 59.000 6. 9.000 -2. 7.870

z = 26

CHAPTER 5

TEST1 PROBLEM AND OUTPUT

5-18 18 December 2000

occ no, be(ev) pairs

2. 7117.000 2. 851.000 2. 726.000 4. 713.000 2. 98.000 2. 61.000

4. 59.000 6. 9.000 -2. 7.870

z = 26

occ no, be(ev) pairs

2. 7117.000 2. 851.000 2. 726.000 4. 713.000 2. 98.000 2. 61.000

4. 59.000 6. 9.000 -2. 7.870

z = 26

occ no, be(ev) pairs

2. 7117.000 2. 851.000 2. 726.000 4. 713.000 2. 98.000 2. 61.000

4. 59.000 6. 9.000 -2. 7.870

plas(ev) wt tmin(mev)

30.57106 2.35209 0.36478

N20 energy stopping power range radiation beta**2 density rad/col drange dyield

n collision radiation total yield corr

mev mev cm2/g mev cm2/g mev cm2/g g/cm2 mev cm2/g g/cm2

133 1.0790E-03 7.975E+01 3.695E-03 7.976E+01 7.021E-06 3.708E-06 4.210E-03 0.000E+00 4.633E-05 1.098E-06 4.000E-09

132 1.1766E-03 7.605E+01 3.798E-03 7.605E+01 8.276E-06 7.395E-06 4.589E-03 0.000E+00 4.994E-05 1.254E-06 4.700E-09

131 1.2831E-03 7.241E+01 3.900E-03 7.242E+01 9.711E-06 1.109E-05 5.003E-03 0.000E+00 5.386E-05 1.435E-06 5.527E-09

130 1.3992E-03 6.886E+01 4.003E-03 6.887E+01 1.136E-05 1.482E-05 5.454E-03 0.000E+00 5.813E-05 1.645E-06 6.502E-09

129 1.5259E-03 6.540E+01 4.105E-03 6.541E+01 1.324E-05 1.860E-05 5.945E-03 0.000E+00 6.276E-05 1.888E-06 7.655E-09

128 1.6640E-03 6.205E+01 4.207E-03 6.205E+01 1.541E-05 2.248E-05 6.481E-03 0.000E+00 6.780E-05 2.169E-06 9.015E-09

127 1.8146E-03 5.880E+01 4.308E-03 5.881E+01 1.791E-05 2.646E-05 7.064E-03 0.000E+00 7.327E-05 2.494E-06 1.062E-08

SKIP 122 LINES IN OUTPUT

4 7.7111E+01 1.906E+00 2.563E+00 4.469E+00 2.757E+01 3.582E-01 1.000E+00 4.644E-01 1.344E+00 1.471E+00 3.600E+00

3 8.4090E+01 1.914E+00 2.814E+00 4.728E+00 2.909E+01 3.769E-01 1.000E+00 4.770E-01 1.471E+00 1.518E+00 4.079E+00

2 9.1700E+01 1.921E+00 3.089E+00 5.010E+00 3.065E+01 3.960E-01 1.000E+00 4.898E-01 1.608E+00 1.564E+00 4.613E+00

1 1.0000E+02 1.928E+00 3.390E+00 5.318E+00 3.226E+01 4.152E-01 1.000E+00 5.025E-01 1.759E+00 1.608E+00 5.206E+00

N211electron secondary production for material 1 print table 86

energy stopping power brems thick tgt k x-ray knock-on

n collision radiation total brems

mev mev barn mev barn mev barn barn barn barn

133 1.0790E-03 2.496E+03 1.157E-01 2.496E+03 1.398E+03 5.031E-05 0.000E+00 0.000E+00

132 1.1766E-03 2.380E+03 1.189E-01 2.381E+03 1.327E+03 1.049E-04 0.000E+00 0.000E+00

131 1.2831E-03 2.267E+03 1.221E-01 2.267E+03 1.259E+03 1.642E-04 0.000E+00 0.000E+00

130 1.3992E-03 2.155E+03 1.253E-01 2.156E+03 1.193E+03 2.286E-04 0.000E+00 0.000E+00

129 1.5259E-03 2.047E+03 1.285E-01 2.047E+03 1.130E+03 2.987E-04 0.000E+00 0.000E+00

128 1.6640E-03 1.942E+03 1.317E-01 1.942E+03 1.070E+03 3.749E-04 0.000E+00 0.000E+00

127 1.8146E-03 1.841E+03 1.349E-01 1.841E+03 1.012E+03 4.578E-04 0.000E+00 0.000E+00

SKIP 122 LINES in OUTPUT

4 7.7111E+01 5.967E+01 8.022E+01 1.399E+02 2.275E+01 1.986E+01 1.596E+00 2.409E+03

3 8.4090E+01 5.990E+01 8.808E+01 1.480E+02 2.275E+01 2.097E+01 1.613E+00 2.409E+03

2 9.1700E+01 6.012E+01 9.670E+01 1.568E+02 2.274E+01 2.210E+01 1.629E+00 2.409E+03

1 1.0000E+02 6.035E+01 1.061E+02 1.665E+02 2.274E+01 2.327E+01 1.645E+00 2.409E+03

18 December 2000 5-19

CHAPTER 5

TEST1 PROBLEM AND OUTPUT

1decimal words of dynamically allocated storage

general 148616

tallies 47160

bank 46724

cross sections 1842726

total 2058178 = 8232712 bytes

***********************************************************************************************************************

N22 dump no. 1 on file test1.r nps = 0 coll = 0 ctm = 0.00 nrn = 0

3 warning messages so far.

X 1 starting mcrun. dynamic storage = 2058182 words, 8232728 bytes. cp0 = 0.12 print table 110

N23 test1: 100 cm thick concrete disk with 15 splitting surfaces

nps x y z cell surf u v w energy weight time

1 0.000E+00 0.000E+00 0.000E+00 2 2 0.000E+00 1.000E+00 0.000E+00 1.419E+01 1.000E+00 0.000E+00

2 0.000E+00 0.000E+00 0.000E+00 2 2 0.000E+00 1.000E+00 0.000E+00 1.419E+01 1.000E+00 0.000E+00

3 0.000E+00 0.000E+00 0.000E+00 2 2 0.000E+00 1.000E+00 0.000E+00 1.419E+01 1.000E+00 0.000E+00

4 0.000E+00 0.000E+00 0.000E+00 2 2 0.000E+00 1.000E+00 0.000E+00 1.419E+01 1.000E+00 0.000E+00

5 0.000E+00 0.000E+00 0.000E+00 2 2 0.000E+00 1.000E+00 0.000E+00 1.419E+01 1.000E+00 0.000E+00

6 0.000E+00 0.000E+00 0.000E+00 2 2 0.000E+00 1.000E+00 0.000E+00 1.419E+01 1.000E+00 0.000E+00

7 0.000E+00 0.000E+00 0.000E+00 2 2 0.000E+00 1.000E+00 0.000E+00 1.419E+01 1.000E+00 0.000E+00

8 0.000E+00 0.000E+00 0.000E+00 2 2 0.000E+00 1.000E+00 0.000E+00 1.419E+01 1.000E+00 0.000E+00

9 0.000E+00 0.000E+00 0.000E+00 2 2 0.000E+00 1.000E+00 0.000E+00 1.419E+01 1.000E+00 0.000E+00

10 0.000E+00 0.000E+00 0.000E+00 2 2 0.000E+00 1.000E+00 0.000E+00 1.419E+01 1.000E+00 0.000E+00

11 0.000E+00 0.000E+00 0.000E+00 2 2 0.000E+00 1.000E+00 0.000E+00 1.419E+01 1.000E+00 0.000E+00

12 0.000E+00 0.000E+00 0.000E+00 2 2 0.000E+00 1.000E+00 0.000E+00 1.419E+01 1.000E+00 0.000E+00

13 0.000E+00 0.000E+00 0.000E+00 2 2 0.000E+00 1.000E+00 0.000E+00 1.419E+01 1.000E+00 0.000E+00

14 0.000E+00 0.000E+00 0.000E+00 2 2 0.000E+00 1.000E+00 0.000E+00 1.419E+01 1.000E+00 0.000E+00

15 0.000E+00 0.000E+00 0.000E+00 2 2 0.000E+00 1.000E+00 0.000E+00 1.419E+01 1.000E+00 0.000E+00

16 0.000E+00 0.000E+00 0.000E+00 2 2 0.000E+00 1.000E+00 0.000E+00 1.419E+01 1.000E+00 0.000E+00

17 0.000E+00 0.000E+00 0.000E+00 2 2 0.000E+00 1.000E+00 0.000E+00 1.419E+01 1.000E+00 0.000E+00

18 0.000E+00 0.000E+00 0.000E+00 2 2 0.000E+00 1.000E+00 0.000E+00 1.419E+01 1.000E+00 0.000E+00

19 0.000E+00 0.000E+00 0.000E+00 2 2 0.000E+00 1.000E+00 0.000E+00 1.419E+01 1.000E+00 0.000E+00

20 0.000E+00 0.000E+00 0.000E+00 2 2 0.000E+00 1.000E+00 0.000E+00 1.419E+01 1.000E+00 0.000E+00

21 0.000E+00 0.000E+00 0.000E+00 2 2 0.000E+00 1.000E+00 0.000E+00 1.419E+01 1.000E+00 0.000E+00

22 0.000E+00 0.000E+00 0.000E+00 2 2 0.000E+00 1.000E+00 0.000E+00 1.419E+01 1.000E+00 0.000E+00

23 0.000E+00 0.000E+00 0.000E+00 2 2 0.000E+00 1.000E+00 0.000E+00 1.419E+01 1.000E+00 0.000E+00

24 0.000E+00 0.000E+00 0.000E+00 2 2 0.000E+00 1.000E+00 0.000E+00 1.419E+01 1.000E+00 0.000E+00

25 0.000E+00 0.000E+00 0.000E+00 2 2 0.000E+00 1.000E+00 0.000E+00 1.419E+01 1.000E+00 0.000E+00

26 0.000E+00 0.000E+00 0.000E+00 2 2 0.000E+00 1.000E+00 0.000E+00 1.419E+01 1.000E+00 0.000E+00

27 0.000E+00 0.000E+00 0.000E+00 2 2 0.000E+00 1.000E+00 0.000E+00 1.419E+01 1.000E+00 0.000E+00

CHAPTER 5

TEST1 PROBLEM AND OUTPUT

5-20 18 December 2000

28 0.000E+00 0.000E+00 0.000E+00 2 2 0.000E+00 1.000E+00 0.000E+00 1.419E+01 1.000E+00 0.000E+00

29 0.000E+00 0.000E+00 0.000E+00 2 2 0.000E+00 1.000E+00 0.000E+00 1.419E+01 1.000E+00 0.000E+00

30 0.000E+00 0.000E+00 0.000E+00 2 2 0.000E+00 1.000E+00 0.000E+00 1.419E+01 1.000E+00 0.000E+00

31 0.000E+00 0.000E+00 0.000E+00 2 2 0.000E+00 1.000E+00 0.000E+00 1.419E+01 1.000E+00 0.000E+00

32 0.000E+00 0.000E+00 0.000E+00 2 2 0.000E+00 1.000E+00 0.000E+00 1.419E+01 1.000E+00 0.000E+00

33 0.000E+00 0.000E+00 0.000E+00 2 2 0.000E+00 1.000E+00 0.000E+00 1.419E+01 1.000E+00 0.000E+00

34 0.000E+00 0.000E+00 0.000E+00 2 2 0.000E+00 1.000E+00 0.000E+00 1.419E+01 1.000E+00 0.000E+00

35 0.000E+00 0.000E+00 0.000E+00 2 2 0.000E+00 1.000E+00 0.000E+00 1.419E+01 1.000E+00 0.000E+00

36 0.000E+00 0.000E+00 0.000E+00 2 2 0.000E+00 1.000E+00 0.000E+00 1.419E+01 1.000E+00 0.000E+00

37 0.000E+00 0.000E+00 0.000E+00 2 2 0.000E+00 1.000E+00 0.000E+00 1.419E+01 1.000E+00 0.000E+00

38 0.000E+00 0.000E+00 0.000E+00 2 2 0.000E+00 1.000E+00 0.000E+00 1.419E+01 1.000E+00 0.000E+00

39 0.000E+00 0.000E+00 0.000E+00 2 2 0.000E+00 1.000E+00 0.000E+00 1.419E+01 1.000E+00 0.000E+00

40 0.000E+00 0.000E+00 0.000E+00 2 2 0.000E+00 1.000E+00 0.000E+00 1.419E+01 1.000E+00 0.000E+00

41 0.000E+00 0.000E+00 0.000E+00 2 2 0.000E+00 1.000E+00 0.000E+00 1.419E+01 1.000E+00 0.000E+00

42 0.000E+00 0.000E+00 0.000E+00 2 2 0.000E+00 1.000E+00 0.000E+00 1.419E+01 1.000E+00 0.000E+00

43 0.000E+00 0.000E+00 0.000E+00 2 2 0.000E+00 1.000E+00 0.000E+00 1.419E+01 1.000E+00 0.000E+00

44 0.000E+00 0.000E+00 0.000E+00 2 2 0.000E+00 1.000E+00 0.000E+00 1.419E+01 1.000E+00 0.000E+00

45 0.000E+00 0.000E+00 0.000E+00 2 2 0.000E+00 1.000E+00 0.000E+00 1.419E+01 1.000E+00 0.000E+00

46 0.000E+00 0.000E+00 0.000E+00 2 2 0.000E+00 1.000E+00 0.000E+00 1.419E+01 1.000E+00 0.000E+00

47 0.000E+00 0.000E+00 0.000E+00 2 2 0.000E+00 1.000E+00 0.000E+00 1.419E+01 1.000E+00 0.000E+00

48 0.000E+00 0.000E+00 0.000E+00 2 2 0.000E+00 1.000E+00 0.000E+00 1.419E+01 1.000E+00 0.000E+00

49 0.000E+00 0.000E+00 0.000E+00 2 2 0.000E+00 1.000E+00 0.000E+00 1.419E+01 1.000E+00 0.000E+00

50 0.000E+00 0.000E+00 0.000E+00 2 2 0.000E+00 1.000E+00 0.000E+00 1.419E+01 1.000E+00 0.000E+00

1problem summary

run terminated when 10000 particle histories were done.

+ 06/23/00 11:34:29

test1: 100 cm thick concrete disk with 15 splitting surfaces probid = 06/23/00 11:30:40

N24neutron creation tracks weight energy neutron loss tracks weight energy

(per source particle) (per source particle)

source 10000 1.0000E+00 1.4190E+01 N27escape 13411 4.0545E-01 9.9156E-01

energy cutoff 0 0. 0.

time cutoff 0 0. 0.

weight window 0 0. 0. weight window 0 0. 0.

N25cell importance 35681 2.5286E-01 9.6306E-02 N28cell importance 23893 2.5427E-01 1.0282E-01

N26weight cutoff 0 8.1018E-02 5.4609E-02 N29weight cutoff 8625 8.0533E-02 5.5477E-02

energy importance 0 0. 0. energy importance 0 0. 0.

dxtran 0 0. 0. dxtran 0 0. 0.

forced collisions 0 0. 0. forced collisions 0 0. 0.

exp. transform 0 0. 0. exp. transform 0 0. 0.

upscattering 0 0. 1.0851E-07 N30downscattering 0 0. 8.2722E+00

delayed fission 0 0. 0. N31capture 0 6.0858E-01 4.7835E+00

(n,xn) 496 2.9921E-02 7.3027E-02 loss to (n,xn) 248 1.4960E-02 2.0833E-01

prompt fission 0 0. 0. loss to fission 0 0. 0.

total 46177 1.3638E+00 1.4414E+01 N32 total 46177 1.3638E+00 1.4414E+01

18 December 2000 5-21

CHAPTER 5

TEST1 PROBLEM AND OUTPUT

number of neutrons banked 35929 N33average time of (shakes) cutoffs

neutron tracks per source particle 4.6177E+00 escape 6.2490E+03 tco 1.0000E+34

neutron collisions per source particle 1.1440E+02 capture 1.6237E+04 eco 0.0000E+00

total neutron collisions 1143962 capture or escape 1.2243E+04 wc1 -5.0000E-01

N34net multiplication 1.0150E+00 0.0010 any termination 1.7300E+04 wc2 -2.5000E-01

photon creation tracks weight energy photon loss tracks weight energy

(per source particle) (per source particle)

source 0 0. 0. escape 7391 5.1898E-01 1.0525E+00

energy cutoff 0 0. 0.

time cutoff 0 0. 0.

weight window 0 0. 0. weight window 0 0. 0.

cell importance 9943 1.3568E-01 1.4838E-01 cell importance 6205 1.4357E-01 1.6084E-01

weight cutoff 0 4.8911E-02 4.7908E-03 weight cutoff 32220 5.0010E-02 4.9142E-03

energy importance 0 0. 0. energy importance 0 0. 0.

dxtran 0 0. 0. dxtran 0 0. 0.

forced collisions 0 0. 0. forced collisions 0 0. 0.

exp. transform 0 0. 0. exp. transform 0 0. 0.

from neutrons 14595 1.9430E+00 6.1915E+00 compton scatter 0 0. 4.4296E+00

bremsstrahlung 20279 9.0128E-01 8.9382E-02 capture 0 2.4540E+00 1.4688E-01

N35p-annihilation 1998 2.7534E-01 1.4070E-01 pair production 999 1.3767E-01 7.8001E-01

electron x-rays 0 0. 0.

1st fluorescence 0 0. 0.

2nd fluorescence 0 0. 0.

total 46815 3.3043E+00 6.5747E+00 total 46815 3.3043E+00 6.5747E+00

number of photons banked 45816 N36 average time of (shakes) cutoffs

photon tracks per source particle 4.6815E+00 escape 7.1788E+03 tco 1.0000E+34

photon collisions per source particle 1.7059E+01 capture 1.1333E+04 eco 1.0000E-03

total photon collisions 170585 capture or escape 1.0608E+04 wc1 -5.0000E-01

any termination 1.0995E+04 wc2 -2.5000E-01

computer time so far in this run 6.70 minutes maximum number ever in bank 30

computer time in mcrun 3.29 minutes bank overflows to backup file 0

source particles per minute 3.0400E+03 dynamic storage 2058182 words, 8232728 bytes.

random numbers generated 20699414 most random numbers used was 49909 in history 7348

range of sampled source weights = 1.0000E+00 to 1.0000E+00

1neutron activity in each cell print table 126

tracks population collisions collisions number flux average average

cell entering * weight weighted weighted track weight track mfp

(per history) energy energy (relative) (cm)

2 2 18335 10493 43171 2.8559E+00 4.3317E-03 6.7719E+00 7.2919E-01 6.4461E+00

3 3 21618 9090 64201 3.8597E+00 1.8065E-03 4.4117E+00 6.4965E-01 5.5031E+00

CHAPTER 5

TEST1 PROBLEM AND OUTPUT

5-22 18 December 2000

4 4 22171 8049 74156 4.1862E+00 1.0093E-03 3.1694E+00 6.0279E-01 4.9011E+00

5 5 21004 7371 76022 4.1000E+00 6.3848E-04 2.4071E+00 5.6861E-01 4.5125E+00

6 6 18634 7005 70710 3.6435E+00 4.2679E-04 1.8468E+00 5.3952E-01 4.1749E+00

7 7 15941 7715 62015 3.0870E+00 3.1639E-04 1.4970E+00 5.1823E-01 3.9409E+00

8 8 25940 13029 103223 2.4867E+00 2.3767E-04 1.2220E+00 4.9832E-01 3.7443E+00

9 9 20450 8738 83591 1.9533E+00 1.9586E-04 1.0554E+00 4.8175E-01 3.6338E+00

10 10 16178 8132 67245 1.5263E+00 1.5952E-04 8.8762E-01 4.6817E-01 3.5113E+00

11 11 24770 12543 104079 1.1670E+00 1.3360E-04 7.5248E-01 4.6093E-01 3.4174E+00

12 12 18448 7887 78368 8.6360E-01 1.1375E-04 6.5578E-01 4.5158E-01 3.3338E+00

13 13 13562 7038 57418 6.1837E-01 9.7234E-05 5.8251E-01 4.4111E-01 3.2583E+00

14 14 19237 10654 81605 4.3278E-01 8.6184E-05 5.2830E-01 4.3273E-01 3.2085E+00

15 15 12669 7187 53795 2.8142E-01 8.2713E-05 5.0187E-01 4.2634E-01 3.1945E+00

16 16 15238 10976 65726 1.7060E-01 8.7605E-05 5.2869E-01 4.2251E-01 3.2366E+00

17 17 13964 11774 58637 7.6169E-02 1.0658E-04 6.3919E-01 4.2288E-01 3.3528E+00

total 298159 147681 1143962 3.1309E+01

N37 N38 N39 N40 N41 N42 N43 N44

1photon activity in each cell print table 126

tracks population collisions collisions number flux average average

cell entering * weight weighted weighted track weight track mfp

(per history) energy energy (relative) (cm)

2 2 685 1412 3165 1.9102E+00 1.7128E+00 1.7128E+00 7.7437E+00 7.5138E+00

3 3 1163 1722 5257 2.8973E+00 1.5538E+00 1.5538E+00 7.1560E+00 7.0346E+00

4 4 1330 1703 5440 2.7612E+00 1.4754E+00 1.4754E+00 6.5959E+00 6.8331E+00

5 5 1321 1700 5157 2.3396E+00 1.4202E+00 1.4202E+00 5.8593E+00 6.7537E+00

6 6 1236 1550 4819 1.9547E+00 1.3651E+00 1.3651E+00 5.1434E+00 6.6591E+00

7 7 1101 1522 4174 1.4558E+00 1.5056E+00 1.5056E+00 4.3974E+00 6.9473E+00

8 8 2064 2831 8015 1.2345E+00 1.3998E+00 1.3998E+00 3.7583E+00 6.6404E+00

9 9 1898 2515 7305 8.8062E-01 1.3206E+00 1.3206E+00 2.9399E+00 6.5022E+00

10 10 1723 2444 6274 6.2921E-01 1.3613E+00 1.3613E+00 2.3435E+00 6.5900E+00

11 11 3428 4744 12135 4.7280E-01 1.3471E+00 1.3471E+00 1.8650E+00 6.5684E+00

12 12 3158 4225 11015 3.7026E-01 1.3603E+00 1.3603E+00 1.5276E+00 6.6002E+00

13 13 2994 4232 10925 2.9309E-01 1.3062E+00 1.3062E+00 1.2124E+00 6.4108E+00

14 14 5483 8048 19249 1.9942E-01 1.3322E+00 1.3322E+00 9.3212E-01 6.5211E+00

15 15 4490 7131 16119 1.3955E-01 1.3791E+00 1.3791E+00 7.4475E-01 6.6435E+00

16 16 6871 11772 23766 8.7332E-02 1.3893E+00 1.3893E+00 6.2711E-01 6.7138E+00

17 17 8744 15809 27770 4.8320E-02 1.4586E+00 1.4586E+00 5.8386E-01 6.9410E+00

total 47689 73360 170585 1.7674E+01 N45

X 1neutron weight balance in each cell -- external events print table 130

N46 cell entering source energy time exiting other total

cutoff cutoff

2 2 5.0025E-01 1.0000E+00 0.0000E+00 0.0000E+00 -1.3773E+00 0.0000E+00 1.2300E-01

3 3 1.5253E+00 0.0000E+00 0.0000E+00 0.0000E+00 -1.4235E+00 0.0000E+00 1.0178E-01

4 4 1.4169E+00 0.0000E+00 0.0000E+00 0.0000E+00 -1.3346E+00 0.0000E+00 8.2296E-02

18 December 2000 5-23

CHAPTER 5

TEST1 PROBLEM AND OUTPUT

5 5 1.2472E+00 0.0000E+00 0.0000E+00 0.0000E+00 -1.1795E+00 0.0000E+00 6.7740E-02

6 6 1.0434E+00 0.0000E+00 0.0000E+00 0.0000E+00 -9.9146E-01 0.0000E+00 5.1938E-02

7 7 8.5224E-01 0.0000E+00 0.0000E+00 0.0000E+00 -8.0834E-01 0.0000E+00 4.3898E-02

8 8 6.6384E-01 0.0000E+00 0.0000E+00 0.0000E+00 -6.3099E-01 0.0000E+00 3.2856E-02

9 9 5.0444E-01 0.0000E+00 0.0000E+00 0.0000E+00 -4.7817E-01 0.0000E+00 2.6266E-02

10 10 3.8652E-01 0.0000E+00 0.0000E+00 0.0000E+00 -3.6796E-01 0.0000E+00 1.8566E-02

11 11 2.9006E-01 0.0000E+00 0.0000E+00 0.0000E+00 -2.7559E-01 0.0000E+00 1.4468E-02

12 12 2.1097E-01 0.0000E+00 0.0000E+00 0.0000E+00 -2.0013E-01 0.0000E+00 1.0842E-02

13 13 1.5138E-01 0.0000E+00 0.0000E+00 0.0000E+00 -1.4351E-01 0.0000E+00 7.8751E-03

14 14 1.0557E-01 0.0000E+00 0.0000E+00 0.0000E+00 -9.9876E-02 0.0000E+00 5.6967E-03

15 15 6.8826E-02 0.0000E+00 0.0000E+00 0.0000E+00 -6.4757E-02 0.0000E+00 4.0695E-03

16 16 4.0784E-02 0.0000E+00 0.0000E+00 0.0000E+00 -3.8564E-02 0.0000E+00 2.2201E-03

17 17 1.8644E-02 0.0000E+00 0.0000E+00 0.0000E+00 -1.7607E-02 0.0000E+00 1.0373E-03

total 9.0263E+00 1.0000E+00 0.0000E+00 0.0000E+00 -9.4317E+00 0.0000E+00 5.9455E-01

X 1neutron weight balance in each cell -- variance reduction events print table 130

cell weight cell weight energy dxtran forced exponential total

window importance cutoff importance collision transform

2 2 0.0000E+00 0.0000E+00 -8.1706E-05 0.0000E+00 0.0000E+00 0.0000E+00 0.0000E+00 -8.1706E-05

3 3 0.0000E+00 0.0000E+00 3.4900E-04 0.0000E+00 0.0000E+00 0.0000E+00 0.0000E+00 3.4900E-04

4 4 0.0000E+00 0.0000E+00 8.6370E-05 0.0000E+00 0.0000E+00 0.0000E+00 0.0000E+00 8.6370E-05

5 5 0.0000E+00 0.0000E+00 4.4060E-04 0.0000E+00 0.0000E+00 0.0000E+00 0.0000E+00 4.4060E-04

6 6 0.0000E+00 0.0000E+00 7.8845E-04 0.0000E+00 0.0000E+00 0.0000E+00 0.0000E+00 7.8845E-04

7 7 0.0000E+00 -1.9599E-03 -4.1174E-04 0.0000E+00 0.0000E+00 0.0000E+00 0.0000E+00 -2.3716E-03

8 8 0.0000E+00 0.0000E+00 -1.1306E-04 0.0000E+00 0.0000E+00 0.0000E+00 0.0000E+00 -1.1306E-04

9 9 0.0000E+00 0.0000E+00 -5.4611E-04 0.0000E+00 0.0000E+00 0.0000E+00 0.0000E+00 -5.4611E-04

10 10 0.0000E+00 8.1516E-04 -3.5028E-05 0.0000E+00 0.0000E+00 0.0000E+00 0.0000E+00 7.8013E-04

11 11 0.0000E+00 0.0000E+00 6.2059E-05 0.0000E+00 0.0000E+00 0.0000E+00 0.0000E+00 6.2059E-05

12 12 0.0000E+00 0.0000E+00 2.2170E-05 0.0000E+00 0.0000E+00 0.0000E+00 0.0000E+00 2.2170E-05

13 13 0.0000E+00 7.2159E-05 -7.6281E-05 0.0000E+00 0.0000E+00 0.0000E+00 0.0000E+00 -4.1214E-06

14 14 0.0000E+00 0.0000E+00 -2.6033E-05 0.0000E+00 0.0000E+00 0.0000E+00 0.0000E+00 -2.6033E-05

15 15 0.0000E+00 -3.4469E-04 -1.1228E-05 0.0000E+00 0.0000E+00 0.0000E+00 0.0000E+00 -3.5592E-04

16 16 0.0000E+00 4.7893E-06 4.8176E-05 0.0000E+00 0.0000E+00 0.0000E+00 0.0000E+00 5.2965E-05

17 17 0.0000E+00 0.0000E+00 -1.1216E-05 0.0000E+00 0.0000E+00 0.0000E+00 0.0000E+00 -1.1216E-05

total 0.0000E+00 -1.4125E-03 4.8443E-04 0.0000E+00 0.0000E+00 0.0000E+00 0.0000E+00 -9.2803E-04

X 1neutron weight balance in each cell -- physical events print table 130

cell (n,xn) fission capture loss to loss to total

(n,xn) fission

2 2 1.2693E-02 0.0000E+00 -1.2926E-01 -6.3465E-03 0.0000E+00 -1.2292E-01

3 3 7.3045E-03 0.0000E+00 -1.0578E-01 -3.6522E-03 0.0000E+00 -1.0213E-01

4 4 4.5440E-03 0.0000E+00 -8.4654E-02 -2.2720E-03 0.0000E+00 -8.2382E-02

5 5 1.5039E-03 0.0000E+00 -6.8933E-02 -7.5196E-04 0.0000E+00 -6.8181E-02

6 6 1.0648E-03 0.0000E+00 -5.3259E-02 -5.3242E-04 0.0000E+00 -5.2727E-02

CHAPTER 5

TEST1 PROBLEM AND OUTPUT

5-24 18 December 2000

7 7 1.2633E-03 0.0000E+00 -4.2158E-02 -6.3164E-04 0.0000E+00 -4.1526E-02

8 8 6.6219E-04 0.0000E+00 -3.3074E-02 -3.3109E-04 0.0000E+00 -3.2743E-02

9 9 4.0116E-04 0.0000E+00 -2.5921E-02 -2.0058E-04 0.0000E+00 -2.5720E-02

10 10 1.6792E-04 0.0000E+00 -1.9430E-02 -8.3959E-05 0.0000E+00 -1.9346E-02

11 11 6.9211E-05 0.0000E+00 -1.4564E-02 -3.4606E-05 0.0000E+00 -1.4530E-02

12 12 1.1235E-04 0.0000E+00 -1.0921E-02 -5.6173E-05 0.0000E+00 -1.0865E-02

13 13 3.3433E-05 0.0000E+00 -7.8877E-03 -1.6716E-05 0.0000E+00 -7.8710E-03

14 14 2.8198E-05 0.0000E+00 -5.6848E-03 -1.4099E-05 0.0000E+00 -5.6707E-03

15 15 4.5747E-05 0.0000E+00 -3.7364E-03 -2.2874E-05 0.0000E+00 -3.7135E-03

16 16 1.1042E-05 0.0000E+00 -2.2786E-03 -5.5208E-06 0.0000E+00 -2.2731E-03

17 17 1.6120E-05 0.0000E+00 -1.0342E-03 -8.0602E-06 0.0000E+00 -1.0261E-03

total 2.9921E-02 0.0000E+00 -6.0858E-01 -1.4960E-02 0.0000E+00 -5.9362E-01

X 1photon weight balance in each cell -- external events print table 130

cell entering source energy time exiting other total

cutoff cutoff

2 2 4.7473E-01 0.0000E+00 0.0000E+00 0.0000E+00 -8.3535E-01 0.0000E+00 -3.6062E-01

3 3 7.9509E-01 0.0000E+00 0.0000E+00 0.0000E+00 -9.7217E-01 0.0000E+00 -1.7708E-01

4 4 8.5367E-01 0.0000E+00 0.0000E+00 0.0000E+00 -9.0146E-01 0.0000E+00 -4.7792E-02

5 5 7.7300E-01 0.0000E+00 0.0000E+00 0.0000E+00 -7.9943E-01 0.0000E+00 -2.6432E-02

6 6 6.5537E-01 0.0000E+00 0.0000E+00 0.0000E+00 -6.3816E-01 0.0000E+00 1.7204E-02

7 7 5.0124E-01 0.0000E+00 0.0000E+00 0.0000E+00 -4.9142E-01 0.0000E+00 9.8151E-03

8 8 3.9429E-01 0.0000E+00 0.0000E+00 0.0000E+00 -3.8071E-01 0.0000E+00 1.3571E-02

9 9 2.9763E-01 0.0000E+00 0.0000E+00 0.0000E+00 -2.8463E-01 0.0000E+00 1.3007E-02

10 10 2.2210E-01 0.0000E+00 0.0000E+00 0.0000E+00 -2.1339E-01 0.0000E+00 8.7039E-03

11 11 1.7411E-01 0.0000E+00 0.0000E+00 0.0000E+00 -1.6764E-01 0.0000E+00 6.4640E-03

12 12 1.3066E-01 0.0000E+00 0.0000E+00 0.0000E+00 -1.2545E-01 0.0000E+00 5.2095E-03

13 13 9.9406E-02 0.0000E+00 0.0000E+00 0.0000E+00 -9.2106E-02 0.0000E+00 7.2993E-03

14 14 7.2720E-02 0.0000E+00 0.0000E+00 0.0000E+00 -6.8691E-02 0.0000E+00 4.0291E-03

15 15 5.0601E-02 0.0000E+00 0.0000E+00 0.0000E+00 -4.6222E-02 0.0000E+00 4.3796E-03

16 16 3.2973E-02 0.0000E+00 0.0000E+00 0.0000E+00 -3.0825E-02 0.0000E+00 2.1479E-03

17 17 1.9081E-02 0.0000E+00 0.0000E+00 0.0000E+00 -1.7969E-02 0.0000E+00 1.1125E-03

total 5.5467E+00 0.0000E+00 0.0000E+00 0.0000E+00 -6.0656E+00 0.0000E+00 -5.1898E-01

X 1photon weight balance in each cell -- variance reduction events print table 130

cell weight cell weight energy dxtran forced exponential total

window importance cutoff importance collision transform

2 2 0.0000E+00 0.0000E+00 1.6185E-04 0.0000E+00 0.0000E+00 0.0000E+00 0.0000E+00 1.6185E-04

3 3 0.0000E+00 0.0000E+00 -5.6439E-04 0.0000E+00 0.0000E+00 0.0000E+00 0.0000E+00 -5.6439E-04

4 4 0.0000E+00 0.0000E+00 -3.1260E-04 0.0000E+00 0.0000E+00 0.0000E+00 0.0000E+00 -3.1260E-04

5 5 0.0000E+00 0.0000E+00 -8.6533E-04 0.0000E+00 0.0000E+00 0.0000E+00 0.0000E+00 -8.6533E-04

6 6 0.0000E+00 0.0000E+00 -4.1528E-04 0.0000E+00 0.0000E+00 0.0000E+00 0.0000E+00 -4.1528E-04

7 7 0.0000E+00 -5.5121E-03 6.4045E-04 0.0000E+00 0.0000E+00 0.0000E+00 0.0000E+00 -4.8716E-03

8 8 0.0000E+00 0.0000E+00 1.0003E-04 0.0000E+00 0.0000E+00 0.0000E+00 0.0000E+00 1.0003E-04

18 December 2000 5-25

CHAPTER 5

TEST1 PROBLEM AND OUTPUT

9 9 0.0000E+00 0.0000E+00 -1.0854E-04 0.0000E+00 0.0000E+00 0.0000E+00 0.0000E+00 -1.0854E-04

10 10 0.0000E+00 -2.5548E-03 1.0642E-05 0.0000E+00 0.0000E+00 0.0000E+00 0.0000E+00 -2.5442E-03

11 11 0.0000E+00 0.0000E+00 -2.3652E-04 0.0000E+00 0.0000E+00 0.0000E+00 0.0000E+00 -2.3652E-04

12 12 0.0000E+00 0.0000E+00 1.6010E-04 0.0000E+00 0.0000E+00 0.0000E+00 0.0000E+00 1.6010E-04

13 13 0.0000E+00 3.7151E-04 3.7336E-04 0.0000E+00 0.0000E+00 0.0000E+00 0.0000E+00 7.4487E-04

14 14 0.0000E+00 0.0000E+00 -7.7834E-05 0.0000E+00 0.0000E+00 0.0000E+00 0.0000E+00 -7.7834E-05

15 15 0.0000E+00 -1.5345E-04 9.8355E-05 0.0000E+00 0.0000E+00 0.0000E+00 0.0000E+00 -5.5098E-05

16 16 0.0000E+00 -4.6109E-05 -2.6627E-05 0.0000E+00 0.0000E+00 0.0000E+00 0.0000E+00 -7.2736E-05

17 17 0.0000E+00 0.0000E+00 -3.6924E-05 0.0000E+00 0.0000E+00 0.0000E+00 0.0000E+00 -3.6924E-05

total 0.0000E+00 -7.8949E-03 -1.0993E-03 0.0000E+00 0.0000E+00 0.0000E+00 0.0000E+00 -8.9942E-03

X 1photon weight balance in each cell -- physical events print table 130

cell from brems- p-annihi- electron fluorescence capture pair total

neutrons strahlung lation x-rays production

2 2 4.6576E-01 1.3117E-01 4.3772E-02 0.0000E+00 0.0000E+00 -2.5835E-01 -2.1886E-02 3.6046E-01

3 3 3.9309E-01 1.6619E-01 4.6183E-02 0.0000E+00 0.0000E+00 -4.0474E-01 -2.3092E-02 1.7764E-01

4 4 2.7820E-01 1.2576E-01 3.7790E-02 0.0000E+00 0.0000E+00 -3.7475E-01 -1.8895E-02 4.8105E-02

5 5 2.2175E-01 1.2742E-01 3.6338E-02 0.0000E+00 0.0000E+00 -3.4004E-01 -1.8169E-02 2.7297E-02

6 6 1.5651E-01 7.8617E-02 2.7130E-02 0.0000E+00 0.0000E+00 -2.6548E-01 -1.3565E-02 -1.6789E-02

7 7 1.1943E-01 6.9034E-02 1.9332E-02 0.0000E+00 0.0000E+00 -2.0307E-01 -9.6661E-03 -4.9435E-03

8 8 8.9937E-02 6.1268E-02 1.7977E-02 0.0000E+00 0.0000E+00 -1.7386E-01 -8.9885E-03 -1.3671E-02

9 9 6.4458E-02 3.9925E-02 1.5964E-02 0.0000E+00 0.0000E+00 -1.2526E-01 -7.9818E-03 -1.2899E-02

10 10 4.6276E-02 2.8892E-02 8.8699E-03 0.0000E+00 0.0000E+00 -8.5763E-02 -4.4349E-03 -6.1597E-03

11 11 3.4747E-02 2.1394E-02 6.0028E-03 0.0000E+00 0.0000E+00 -6.5370E-02 -3.0014E-03 -6.2274E-03

12 12 2.6405E-02 1.6839E-02 4.9917E-03 0.0000E+00 0.0000E+00 -5.1109E-02 -2.4958E-03 -5.3696E-03

13 13 1.7351E-02 1.3353E-02 3.8170E-03 0.0000E+00 0.0000E+00 -4.0656E-02 -1.9085E-03 -8.0441E-03

14 14 1.3261E-02 8.8015E-03 3.4989E-03 0.0000E+00 0.0000E+00 -2.7763E-02 -1.7495E-03 -3.9513E-03

15 15 8.3679E-03 6.1235E-03 1.7251E-03 0.0000E+00 0.0000E+00 -1.9678E-02 -8.6255E-04 -4.3245E-03

16 16 5.1510E-03 3.9539E-03 1.2204E-03 0.0000E+00 0.0000E+00 -1.1790E-02 -6.1019E-04 -2.0752E-03

17 17 2.3495E-03 2.5404E-03 7.2662E-04 0.0000E+00 0.0000E+00 -6.3288E-03 -3.6331E-04 -1.0755E-03

total 1.9430E+00 9.0128E-01 2.7534E-01 0.0000E+00 0.0000E+00 -2.4540E+00 -1.3767E-01 5.2798E-01

X 1neutron activity of each nuclide in each cell, per source particle print table 140

N47 cell nuclides atom total collisions weight lost weight gain weight gain

fraction collisions * weight to capture by fission by (n,xn)

2 2 1001.60c 8.4764E-02 10340 5.9405E-01 2.1850E-03 0.0000E+00 0.0000E+00

8016.60c 6.0409E-01 22198 1.5088E+00 6.1243E-02 0.0000E+00 0.0000E+00

11023.60c 9.4725E-03 468 3.1718E-02 1.1632E-03 0.0000E+00 0.0000E+00

12000.60c 2.9983E-03 137 9.0702E-03 4.5896E-04 0.0000E+00 0.0000E+00

13027.60c 2.4834E-02 783 5.5924E-02 2.6602E-03 0.0000E+00 3.4810E-04

14000.60c 2.4186E-01 7891 5.6185E-01 5.0501E-02 0.0000E+00 5.5828E-03

19000.60c 6.8551E-03 273 1.9129E-02 3.3765E-03 0.0000E+00 0.0000E+00

20000.60c 2.0481E-02 734 5.2808E-02 6.4912E-03 0.0000E+00 6.9295E-05

26054.60c 2.7432E-04 15 9.5659E-04 9.5529E-05 0.0000E+00 0.0000E+00

CHAPTER 5

TEST1 PROBLEM AND OUTPUT

5-26 18 December 2000

26056.60c 4.2645E-03 329 2.1357E-02 1.0886E-03 0.0000E+00 3.4627E-04

26057.60c 9.7640E-05 3 2.4055E-04 1.4268E-06 0.0000E+00 0.0000E+00

26058.60c 1.3019E-05 0 0.0000E+00 0.0000E+00 0.0000E+00 0.0000E+00

SKIP 182 LINES IN OUTPUT

17 17 1001.60c 8.4764E-02 22594 2.8629E-02 2.1287E-04 0.0000E+00 0.0000E+00

8016.60c 6.0409E-01 26465 3.4788E-02 7.6736E-05 0.0000E+00 0.0000E+00

11023.60c 9.4725E-03 507 6.7990E-04 3.9230E-05 0.0000E+00 0.0000E+00

12000.60c 2.9983E-03 158 2.1435E-04 2.4640E-06 0.0000E+00 0.0000E+00

13027.60c 2.4834E-02 543 7.3032E-04 4.3007E-05 0.0000E+00 0.0000E+00

14000.60c 2.4186E-01 6684 8.9450E-03 3.4309E-04 0.0000E+00 8.0602E-06

19000.60c 6.8551E-03 295 3.8275E-04 1.3559E-04 0.0000E+00 0.0000E+00

20000.60c 2.0481E-02 758 9.9921E-04 8.1763E-05 0.0000E+00 0.0000E+00

26054.60c 2.7432E-04 14 1.8735E-05 4.4469E-06 0.0000E+00 0.0000E+00

26056.60c 4.2645E-03 613 7.7358E-04 9.2645E-05 0.0000E+00 0.0000E+00

26057.60c 9.7640E-05 6 7.1280E-06 2.3103E-06 0.0000E+00 0.0000E+00

26058.60c 1.3019E-05 0 0.0000E+00 0.0000E+00 0.0000E+00 0.0000E+00

total 1143962 3.1309E+01 6.0858E-01 0.0000E+00 1.4960E-02

N48total over all cells for each nuclide total collisions weight lost weight gain weight gain

collisions * weight to capture by fission by (n,xn)

1001.60c 408622 9.7833E+00 5.5216E-02 0.0000E+00 0.0000E+00

8016.60c 532245 1.5216E+01 1.9888E-01 0.0000E+00 0.0000E+00

11023.60c 10922 3.4009E-01 1.2149E-02 0.0000E+00 1.7272E-04

12000.60c 2698 8.2127E-02 1.6235E-03 0.0000E+00 8.0808E-05

13027.60c 13111 4.4408E-01 1.9264E-02 0.0000E+00 5.1297E-04

14000.60c 144029 4.5450E+00 2.2217E-01 0.0000E+00 1.2428E-02

19000.60c 5627 1.5700E-01 3.6066E-02 0.0000E+00 6.6908E-05

20000.60c 15525 4.5637E-01 3.8979E-02 0.0000E+00 6.9295E-05

26054.60c 329 1.0683E-02 1.2161E-03 0.0000E+00 0.0000E+00

26056.60c 10728 2.6969E-01 2.2624E-02 0.0000E+00 1.5756E-03

26057.60c 113 3.3625E-03 3.5985E-04 0.0000E+00 5.4288E-05

26058.60c 13 6.5846E-04 3.8437E-05 0.0000E+00 0.0000E+00

X 1photon activity of each nuclide in each cell, per source particle print table 140

cell nuclides atom total collisions weight lost total from weight from avg photon

fraction collisions * weight to capture neutrons neutrons energy

2 2 1000.02p 8.4764E-02 15 1.1098E-02 1.8345E-08 2 2.0000E-03 2.2246E+00

8000.02p 6.0409E-01 1285 8.5473E-01 3.0457E-02 174 1.7400E-01 4.4156E+00

11000.02p 9.4725E-03 42 2.4661E-02 2.3057E-03 6 6.0000E-03 1.3193E+00

12000.02p 2.9983E-03 19 1.3428E-02 1.8033E-03 3 3.0000E-03 1.9604E+00

13000.02p 2.4834E-02 118 7.0548E-02 1.0100E-02 14 1.4000E-02 2.8540E+00

14000.02p 2.4186E-01 1244 7.2981E-01 1.2357E-01 231 2.3076E-01 2.6378E+00

19000.02p 6.8551E-03 62 2.9799E-02 9.6794E-03 9 9.0000E-03 2.6985E+00

20000.02p 2.0481E-02 252 1.2116E-01 4.6341E-02 16 1.6000E-02 2.2538E+00

18 December 2000 5-27

CHAPTER 5

TEST1 PROBLEM AND OUTPUT

26000.02p 2.7432E-04 10 4.0221E-03 1.0467E-03 1 1.0000E-03 2.0484E+00

26000.02p 4.2645E-03 116 5.0704E-02 3.2799E-02 10 9.9948E-03 3.3209E+00

26000.02p 9.7640E-05 2 2.5624E-04 2.5467E-04 0 0.0000E+00 0.0000E+00

26000.02p 1.3019E-05 0 0.0000E+00 0.0000E+00 0 0.0000E+00 0.0000E+00

SKIP 182 LINES IN OUTPUT

17 17 1000.02p 8.4764E-02 175 3.7967E-04 1.1179E-09 362 2.2625E-04 2.2246E+00

8000.02p 6.0409E-01 12302 2.2518E-02 8.4831E-04 209 1.4640E-04 4.4367E+00

11000.02p 9.4725E-03 272 5.0251E-04 3.2896E-05 229 1.6335E-04 1.9765E+00

12000.02p 2.9983E-03 99 1.5427E-04 1.2841E-05 13 1.0031E-05 2.1827E+00

13000.02p 2.4834E-02 1001 1.5484E-03 2.0988E-04 173 1.2242E-04 3.2435E+00

14000.02p 2.4186E-01 10719 1.8202E-02 2.9834E-03 1329 9.4460E-04 3.3940E+00

19000.02p 6.8551E-03 505 8.1839E-04 2.8755E-04 252 2.8214E-04 3.0306E+00

20000.02p 2.0481E-02 1847 2.8742E-03 1.1173E-03 296 2.0028E-04 3.1209E+00

26000.02p 2.7432E-04 49 5.8109E-05 3.5484E-05 9 1.2335E-05 4.1360E+00

26000.02p 4.2645E-03 783 1.2441E-03 7.8711E-04 333 2.3524E-04 3.0109E+00

26000.02p 9.7640E-05 17 1.7974E-05 1.2444E-05 3 6.4338E-06 6.2223E-01

26000.02p 1.3019E-05 1 1.7165E-06 1.6131E-06 0 0.0000E+00 0.0000E+00

total 170585 1.7674E+01 2.4540E+00 14595 1.9430E+00 3.1865E+00

total over all cells for each nuclide total collisions weight lost total from weight from avg photon

N49 collisions * weight to capture neutrons neutrons energy

1000.02p 1190 1.3776E-01 4.8485E-07 1298 5.8148E-02 2.2246E+00

8000.02p 73520 8.1008E+00 3.0358E-01 1480 5.1081E-01 4.3640E+00

11000.02p 1857 2.1492E-01 1.7868E-02 954 5.8290E-02 1.5248E+00

12000.02p 585 5.6857E-02 5.8439E-03 54 7.4416E-03 2.7424E+00

13000.02p 6033 6.1133E-01 8.6939E-02 752 9.0933E-02 3.2459E+00

14000.02p 65585 6.6356E+00 1.1714E+00 6255 9.5711E-01 2.8011E+00

19000.02p 3490 3.0481E-01 1.0940E-01 1248 8.4874E-02 2.9592E+00

20000.02p 12398 1.1039E+00 4.5537E-01 1199 9.0651E-02 2.7107E+00

26000.02p 5927 5.0785E-01 3.0363E-01 1355 8.4797E-02 2.9565E+00

N501summary of photons produced in neutron collisions

cell number of weight per energy per avg photon mev/gm per weight/neut energy/neut

photons source neut source neut energy source neut collision collision

1 1 0 0.00000E+00 0.00000E+00 0.00000E+00 0.00000E+00 0.00000E+00 0.00000E+00

2 2 466 4.65757E-01 1.53080E+00 3.28668E+00 6.15865E-06 1.63086E-01 5.36011E-01

3 3 437 3.93094E-01 1.22727E+00 3.12206E+00 4.93750E-06 1.01845E-01 3.17965E-01

4 4 348 2.78202E-01 8.53895E-01 3.06934E+00 3.43537E-06 6.64563E-02 2.03977E-01

5 5 317 2.21749E-01 7.08568E-01 3.19536E+00 2.85069E-06 5.40855E-02 1.72823E-01

6 6 261 1.56515E-01 4.62941E-01 2.95781E+00 1.86249E-06 4.29571E-02 1.27059E-01

7 7 239 1.19427E-01 4.20886E-01 3.52423E+00 1.69330E-06 3.86872E-02 1.36342E-01

8 8 450 8.99374E-02 2.93122E-01 3.25918E+00 1.17928E-06 3.61670E-02 1.17875E-01

9 9 430 6.44581E-02 2.00949E-01 3.11751E+00 8.08451E-07 3.30002E-02 1.02878E-01

10 10 463 4.62759E-02 1.45762E-01 3.14986E+00 5.86427E-07 3.03195E-02 9.55022E-02

CHAPTER 5

TEST1 PROBLEM AND OUTPUT

5-28 18 December 2000

11 11 818 3.47471E-02 1.13936E-01 3.27900E+00 4.58383E-07 2.97751E-02 9.76326E-02

12 12 755 2.64048E-02 8.84458E-02 3.34961E+00 3.55833E-07 3.05754E-02 1.02416E-01

13 13 694 1.73507E-02 5.18621E-02 2.98905E+00 2.08650E-07 2.80586E-02 8.38685E-02

14 14 1510 1.32609E-02 4.28907E-02 3.23436E+00 1.72556E-07 3.06414E-02 9.91053E-02

15 15 1619 8.36792E-03 2.64074E-02 3.15579E+00 1.06241E-07 2.97350E-02 9.38374E-02

16 16 2580 5.15099E-03 1.64214E-02 3.18801E+00 6.60661E-08 3.01936E-02 9.62574E-02

17 17 3208 2.34947E-03 7.34408E-03 3.12584E+00 2.95465E-08 3.08457E-02 9.64187E-02

total 14595 1.94305E+00 6.19149E+00 3.18649E+00

energy number of number cum number weight of weight cum weight

interval photons frequency distribution photons frequency distribution

20.000 0 0.00000E+00 0.00000E+00 0.00000E+00 0.00000E+00 0.00000E+00

15.000 0 0.00000E+00 0.00000E+00 0.00000E+00 0.00000E+00 0.00000E+00

10.000 22 1.50737E-03 1.50737E-03 6.13839E-03 3.15916E-03 3.15916E-03

9.000 40 2.74066E-03 4.24803E-03 5.44764E-03 2.80366E-03 5.96282E-03

8.000 93 6.37205E-03 1.06201E-02 1.89398E-02 9.74749E-03 1.57103E-02

7.000 758 5.19356E-02 6.25557E-02 9.52411E-02 4.90163E-02 6.47266E-02

6.000 1265 8.66735E-02 1.49229E-01 2.56124E-01 1.31816E-01 1.96543E-01

5.000 475 3.25454E-02 1.81775E-01 5.94310E-02 3.05865E-02 2.27129E-01

4.000 1752 1.20041E-01 3.01816E-01 1.48707E-01 7.65330E-02 3.03662E-01

3.000 2345 1.60671E-01 4.62487E-01 2.67643E-01 1.37744E-01 4.41406E-01

2.000 3014 2.06509E-01 6.68996E-01 3.02696E-01 1.55784E-01 5.97190E-01

1.000 2646 1.81295E-01 8.50291E-01 4.24335E-01 2.18387E-01 8.15577E-01

0.500 903 6.18705E-02 9.12162E-01 1.55504E-01 8.00312E-02 8.95608E-01

0.100 631 4.32340E-02 9.55396E-01 1.22427E-01 6.30077E-02 9.58616E-01

0.010 571 3.91230E-02 9.94519E-01 7.22862E-02 3.72025E-02 9.95818E-01

0.000 80 5.48133E-03 1.00000E+00 8.12513E-03 4.18165E-03 1.00000E+00

total 14595 1.00000E+00 1.94305E+00 1.00000E+00

N511tally 11 nps = 10000

tally type 1 number of particles crossing a surface.

tally for neutrons

surface 18

energy

1.0000E-04 4.73427E-03 0.0428

1.0000E-03 3.41844E-04 0.0949

1.0000E-02 2.66169E-04 0.1057

5.0000E-02 2.08038E-04 0.1153

1.0000E-01 1.43064E-04 0.1545

5.0000E-01 3.19056E-04 0.1002

1.0000E+00 2.80542E-04 0.0978

2.0000E+00 4.25783E-04 0.0941

3.0000E+00 4.56545E-04 0.0992

4.0000E+00 1.52545E-04 0.1326

5.0000E+00 1.50381E-04 0.1501

18 December 2000 5-29

CHAPTER 5

TEST1 PROBLEM AND OUTPUT

6.0000E+00 7.57808E-05 0.2022

7.0000E+00 9.91571E-05 0.1866

8.0000E+00 4.22977E-05 0.2764

9.0000E+00 3.08416E-05 0.3153

1.0000E+01 2.63550E-05 0.3440

1.1000E+01 4.98461E-05 0.2830

1.2000E+01 1.77223E-05 0.3324

1.3000E+01 3.75968E-05 0.2468

1.4000E+01 1.67052E-04 0.1568

1.5000E+01 9.08942E-05 0.3073

total 8.11579E-03 0.0393

1analysis of the results in the tally fluctuation chart bin (tfc) for tally 11 with nps = 10000 print table 160

N52normed average tally per history = 8.11579E-03 N53unnormed average tally per history = 8.11579E-03

estimated tally relative error = 0.0393 N54estimated variance of the variance = 0.0050

N55relative error from zero tallies = 0.0278 N56relative error from nonzero scores = 0.0278

number of nonzero history tallies = 1143 N57efficiency for the nonzero tallies = 0.1143

history number of largest tally = 9766 N58largest unnormalized history tally = 5.04052E-01

(largest tally)/(average tally) = 6.21076E+01 (largest tally)/(avg nonzero tally)= 7.09889E+00

N59(confidence interval shift)/mean = 0.0012 shifted confidence interval center = 8.12539E-03

N60if the largest history score sampled so far were to occur on the next history, the tfc bin quantities would change as follows:

estimated quantities value at nps value at nps+1 value(nps+1)/value(nps)-1.

mean 8.11579E-03 8.16537E-03 0.006110

relative error 3.93120E-02 3.95385E-02 0.005761

variance of the variance 4.95966E-03 5.27658E-03 0.063899

shifted center 8.12539E-03 8.12574E-03 0.000043

figure of merit 1.96707E+02 1.94460E+02 -0.011424

N61the estimated slope of the 57 largest tallies starting at 1.99571E-01 appears to be decreasing at least exponentially.

the large score tail of the empirical history score probability density function appears to have no unsampled regions.

===================================================================================================================================

N62 results of 10 statistical checks for the estimated answer for the tally fluctuation chart (tfc) bin of tally 11

tfc bin --mean-- ---------relative error--------- ----variance of the variance---- --figure of merit-- -pdf-

behavior behavior value decrease decrease rate value decrease decrease rate value behavior slope

desired random <0.10 yes 1/sqrt(nps) <0.10 yes 1/nps constant random >3.00

observed random 0.04 yes yes 0.00 yes yes constant random 10.00

passed? yes yes yes yes yes yes yes yes yes yes

N63 N64 N65 N66 N67 N68 N69 N70 N71 N72

=================================================================================================================================

N73this tally meets the statistical criteria used to form confidence intervals: check the tally fluctuation chart to verify.

CHAPTER 5

TEST1 PROBLEM AND OUTPUT

5-30 18 December 2000

the results in other bins associated with this tally may not meet these statistical criteria.

estimated asymmetric confidence interval(1,2,3 sigma): 7.8060E-03 to 8.4448E-03; 7.4865E-03 to 8.7642E-03; 7.1671E-03 to 9.0837E-03

estimated symmetric confidence interval(1,2,3 sigma): 7.7967E-03 to 8.4348E-03; 7.4777E-03 to 8.7539E-03; 7.1586E-03 to 9.0729E-03

fom = (histories/minute)*(f(x) signal-to-noise ratio)**2 = (3.040E+03)*( 2.544E-01)**2 = (3.040E+03)*(6.471E-02) = 1.967E+02

N741unnormed tally density for tally 11 nonzero tally mean(m) = 7.100E-02 nps = 10000 print table 161

abscissa ordinate log plot of tally probability density function in tally fluctuation chart bin(d=decade,slope=10.0)

tally number num den log den:d--------------------------d---------------------------d----------------------------d---------------

7.94-03 5 3.06-01 -0.514 ***************************|***************************|****************** |

1.00-02 51 2.48+00 0.394 ***************************|***************************|****************************|***************

1.26-02 46 1.78+00 0.250 ***************************|***************************|****************************|***********

1.58-02 50 1.53+00 0.186 ***************************|***************************|****************************|*********

2.00-02 54 1.32+00 0.119 ***************************|***************************|****************************|*******

2.51-02 80 1.55+00 0.190 ***************************|***************************|****************************|*********

3.16-02 81 1.25+00 0.095 ***************************|***************************|****************************|*******

3.98-02 110 1.34+00 0.128 ***************************|***************************|****************************|*******

5.01-02 102 9.90-01 -0.005 ***************************|***************************|****************************|****

6.31-02 100 7.71-01 -0.113 ***************************|***************************|****************************|*

7.94-02 106 6.49-01 -0.188 mmmmmmmmmmmmmmmmmmmmmmmmmmm|mmmmmmmmmmmmmmmmmmmmmmmmmmm|mmmmmmmmmmmmmmmmmmmmmmmmmmmm|

1.00-01 98 4.76-01 -0.322 ***************************|***************************|************************ |

1.26-01 78 3.01-01 -0.521 ***************************|***************************|****************** |

1.58-01 71 2.18-01 -0.662 ***************************|***************************|************** |

2.00-01 54 1.32-01 -0.881 ***************************|***************************|******** |

2.51-01 27 5.23-02 -1.282 ***************************|************************* | |

3.16-01 15 2.31-02 -1.637 ***************************|*************** | |

3.98-01 11 1.34-02 -1.872 ***************************|******** | |

5.01-01 3 2.91-03 -2.536 ***************** | | |

6.31-01 1 7.71-04 -3.113 * | | |

total 1143 1.14-01 d--------------------------d---------------------------d----------------------------d---------------

N75 cumulative tally number for tally 11 nonzero tally mean(m) = 7.100E-02 nps = 10000 print table 162

abscissa cum ordinate plot of the cumulative number of tallies in the tally fluctuation chart bin from 0 to 100 percent

tally number cum pct:--------10--------20--------30--------40--------50--------60--------70--------80--------90-------100

7.94328E-03 5 0.437| | | | | | | | | | |

1.00000E-02 56 4.899|***** | | | | | | | | | |

1.25893E-02 102 8.924|*********| | | | | | | | | |

1.58490E-02 152 13.298|*********|*** | | | | | | | | |

1.99527E-02 206 18.023|*********|******** | | | | | | | | |

2.51188E-02 286 25.022|*********|*********|***** | | | | | | | |

3.16228E-02 367 32.108|*********|*********|*********|** | | | | | | |

3.98108E-02 477 41.732|*********|*********|*********|*********|** | | | | | |

5.01188E-02 579 50.656|*********|*********|*********|*********|*********|* | | | | |

6.30959E-02 679 59.405|*********|*********|*********|*********|*********|*********| | | | |

1.00000E-01 883 77.253|*********|*********|*********|*********|*********|*********|*********|******* | | |

1.25893E-01 961 84.077|*********|*********|*********|*********|*********|*********|*********|*********|**** | |

1.58490E-01 1032 90.289|*********|*********|*********|*********|*********|*********|*********|*********|*********| |

18 December 2000 5-31

CHAPTER 5

TEST1 PROBLEM AND OUTPUT

1.99527E-01 1086 95.013|*********|*********|*********|*********|*********|*********|*********|*********|*********|***** |

2.51188E-01 1113 97.375|*********|*********|*********|*********|*********|*********|*********|*********|*********|******* |

3.16228E-01 1128 98.688|*********|*********|*********|*********|*********|*********|*********|*********|*********|*********|

3.98108E-01 1139 99.650|*********|*********|*********|*********|*********|*********|*********|*********|*********|*********|

5.01188E-01 1142 99.913|*********|*********|*********|*********|*********|*********|*********|*********|*********|*********|

6.30959E-01 1143 100.000|*********|*********|*********|*********|*********|*********|*********|*********|*********|*********|

total 1143 100.000:--------10--------20--------30--------40--------50--------60--------70--------80--------90-------100

1cumulative unnormed tally for tally 11 nonzero tally mean(m) = 7.100E-02 nps = 10000 print table 162

abscissa cum ordinate plot of the cumulative tally in the tally fluctuation chart bin from 0 to 100 percent

tally tally/nps cum pct:--------10--------20--------30--------40--------50--------60--------70--------80--------90-------100

7.943E-03 3.930E-06 0.048| | | | | | | | | | |

1.000E-02 4.894E-05 0.603|* | | | | | | | | | |

1.259E-02 1.004E-04 1.237|* | | | | | | | | | |

1.585E-02 1.713E-04 2.111|** | | | | | | | | | |

1.995E-02 2.686E-04 3.310|*** | | | | | | | | | |

2.512E-02 4.505E-04 5.551|****** | | | | | | | | | |

3.162E-02 6.771E-04 8.343|******** | | | | | | | | | |

3.981E-02 1.069E-03 13.170|*********|*** | | | | | | | | |

5.012E-02 1.530E-03 18.848|*********|*********| | | | | | | | |

6.310E-02 2.093E-03 25.783|*********|*********|****** | | | | | | | |

1.000E-01 3.712E-03 45.742|*********|*********|*********|*********|****** | | | | | |

1.259E-01 4.587E-03 56.525|*********|*********|*********|*********|*********|******* | | | | |

1.585E-01 5.594E-03 68.922|*********|*********|*********|*********|*********|*********|*********| | | |

1.995E-01 6.539E-03 80.574|*********|*********|*********|*********|*********|*********|*********|*********|* | |

2.512E-01 7.133E-03 87.889|*********|*********|*********|*********|*********|*********|*********|*********|******** | |

3.162E-01 7.551E-03 93.038|*********|*********|*********|*********|*********|*********|*********|*********|*********|*** |

3.981E-01 7.943E-03 97.867|*********|*********|*********|*********|*********|*********|*********|*********|*********|******** |

5.012E-01 8.065E-03 99.379|*********|*********|*********|*********|*********|*********|*********|*********|*********|*********|

6.310E-01 8.116E-03 100.000|*********|*********|*********|*********|*********|*********|*********|*********|*********|*********|

total 8.11579E-03 100.000:--------10--------20--------30--------40--------50--------60--------70--------80--------90-------100

SKIP 1220 LINES OF OUTPUT

N76tally result of statistical checks for the tfc bin (the first check not passed is listed) and error magnitude check for all bins

11 passed the 10 statistical checks for the tally fluctuation chart bin result

missed all bin error check: 22 tally bins had 0 bins with zeros and 16 bins with relative errors exceeding 0.10

16 missed 1 of 10 tfc bin checks: the estimated mean has a trend during the last half of the problem

missed all bin error check: 22 tally bins had 0 bins with zeros and 13 bins with relative errors exceeding 0.10

34 missed 1 of 10 tfc bin checks: the estimated mean has a trend during the last half of the problem

missed all bin error check: 22 tally bins had 0 bins with zeros and 13 bins with relative errors exceeding 0.10

1 passed the 10 statistical checks for the tally fluctuation chart bin result

missed all bin error check: 22 tally bins had 6 bins with zeros and 9 bins with relative errors exceeding 0.10

12 passed the 10 statistical checks for the tally fluctuation chart bin result

passed all bin error check: 1 tally bins all have relative errors less than 0.10 with no zero bins

CHAPTER 5

TEST1 PROBLEM AND OUTPUT

5-32 18 December 2000

26 missed 1 of 10 tfc bin checks: the slope of decrease of largest tallies is less than the minimum acceptable value of 3.0

missed all bin error check: 22 tally bins had 5 bins with zeros and 8 bins with relative errors exceeding 0.10

6 passed the 10 statistical checks for the tally fluctuation chart bin result

missed all bin error check: 23 tally bins had 0 bins with zeros and 10 bins with relative errors exceeding 0.10

the 10 statistical checks are only for the tally fluctuation chart bin and do not apply to other tally bins.

the tally bins with zeros may or may not be correct: compare the source, cutoffs, multipliers, et cetera with the tally bins.

warning. 3 of the 7 tally fluctuation chart bins did not pass all 10 statistical checks.

warning. 6 of the 7 tallies had bins with relative errors greater than recommended.

N771tally fluctuation charts

tally 11 tally 16 tally 34

nps mean error vov slope fom mean error vov slope fom mean error vov slope fom

1000 7.2180E-03 0.1255 0.0365 0.0 197 7.2179E-09 0.2134 0.0889 0.0 68 1.6244E-08 0.2134 0.0889 0.0 68

2000 6.8028E-03 0.0929 0.0303 0.0 179 6.9752E-09 0.1661 0.0630 0.0 56 1.5698E-08 0.1661 0.0630 0.0 56

3000 7.4265E-03 0.0742 0.0174 0.0 189 8.2071E-09 0.1351 0.0471 0.0 57 1.8470E-08 0.1351 0.0471 0.0 57

4000 7.0511E-03 0.0655 0.0143 0.0 183 7.5738E-09 0.1168 0.0383 0.0 58 1.7045E-08 0.1168 0.0383 0.0 58

5000 7.4809E-03 0.0575 0.0107 10.0 188 8.5002E-09 0.1012 0.0359 10.0 61 1.9130E-08 0.1012 0.0359 10.0 61

6000 7.7297E-03 0.0515 0.0084 10.0 194 9.7427E-09 0.0901 0.0248 10.0 63 2.1926E-08 0.0901 0.0248 10.0 63

7000 7.9401E-03 0.0473 0.0070 10.0 196 9.9476E-09 0.0848 0.0267 10.0 61 2.2387E-08 0.0848 0.0267 10.0 61

8000 8.0837E-03 0.0436 0.0058 10.0 201 1.0507E-08 0.0783 0.0218 10.0 62 2.3646E-08 0.0783 0.0218 10.0 62

9000 8.1393E-03 0.0413 0.0052 10.0 198 1.0646E-08 0.0738 0.0185 10.0 62 2.3960E-08 0.0738 0.0185 10.0 62

10000 8.1158E-03 0.0393 0.0050 10.0 197 1.0908E-08 0.0702 0.0170 10.0 62 2.4549E-08 0.0702 0.0170 10.0 62

tally 1 tally 12 tally 26

nps mean error vov slope fom mean error vov slope fom mean error vov slope fom

1000 1.1542E-02 0.1630 0.1624 0.0 117 1.0229E-06 0.1433 0.0886 0.0 151 5.4916E-08 0.1475 0.1010 0.0 143

2000 1.1783E-02 0.1181 0.0755 0.0 111 1.0670E-06 0.1111 0.0740 0.0 125 5.7742E-08 0.1073 0.0392 0.0 134

3000 1.0625E-02 0.0942 0.0573 0.0 117 9.7872E-07 0.0886 0.0527 0.0 132 5.2554E-08 0.0861 0.0286 0.0 140

4000 1.0378E-02 0.0798 0.0404 0.0 123 9.5219E-07 0.0760 0.0364 0.0 136 5.0781E-08 0.0747 0.0210 0.0 141

5000 1.1485E-02 0.0714 0.0283 4.7 122 1.0385E-06 0.0680 0.0252 3.4 134 5.6158E-08 0.0706 0.0473 3.6 125

6000 1.1698E-02 0.0642 0.0228 5.0 125 1.0537E-06 0.0610 0.0200 3.8 138 5.7865E-08 0.0647 0.0375 3.3 123

7000 1.1569E-02 0.0603 0.0198 10.0 121 1.0470E-06 0.0572 0.0173 3.6 134 5.6676E-08 0.0598 0.0312 3.5 123

8000 1.1624E-02 0.0552 0.0166 8.9 125 1.0481E-06 0.0526 0.0147 5.8 138 5.7311E-08 0.0545 0.0256 2.9 128

9000 1.2111E-02 0.0536 0.0195 4.8 117 1.0966E-06 0.0509 0.0163 4.2 130 5.9588E-08 0.0523 0.0219 2.9 123

10000 1.1784E-02 0.0509 0.0179 4.0 117 1.0654E-06 0.0483 0.0150 5.3 130 5.8153E-08 0.0495 0.0199 2.9 124

tally 6

nps mean error vov slope fom

1000 6.2134E-08 0.1377 0.0837 0.0 164

2000 6.4718E-08 0.1017 0.0349 0.0 149

3000 6.0761E-08 0.0812 0.0236 0.0 158

18 December 2000 5-33

CHAPTER 5

TEST1 PROBLEM AND OUTPUT

4000 5.8355E-08 0.0707 0.0179 10.0 157

5000 6.4658E-08 0.0656 0.0372 4.7 144

6000 6.7607E-08 0.0599 0.0289 4.0 144

7000 6.6623E-08 0.0552 0.0239 4.0 144

8000 6.7818E-08 0.0504 0.0192 4.1 150

9000 7.0235E-08 0.0482 0.0166 4.2 145

10000 6.9061E-08 0.0458 0.0147 4.2 145

N781neutron weight-window lower bounds from the weight-window generator print table 190

energy: 1.000E+02

cell

1 -1.000E+00

2 5.000E-01

3 3.886E-01

4 3.065E-01

5 2.441E-01

6 1.960E-01

7 1.576E-01

8 1.274E-01

9 9.844E-02

10 7.579E-02

11 5.991E-02

12 4.737E-02

13 3.811E-02

14 3.181E-02

15 2.758E-02

16 2.483E-02

17 2.831E-02

1photon weight-window lower bounds from the weight-window generator print table 190

energy: 1.000E+02

cell

1 -1.000E+00

2 1.155E+02

3 3.156E+01

4 1.538E+01

5 6.782E+00

6 4.129E+00

7 2.122E+00

8 1.243E+00

9 7.722E-01

10 4.657E-01

11 2.671E-01

12 1.528E-01

13 9.071E-02

14 5.465E-02

15 3.162E-02

16 1.733E-02

17 9.827E-03

CHAPTER 5

TEST1 PROBLEM AND OUTPUT

5-34 18 December 2000

N791weight-window cards from the weight-window generator print table 200

each card has ten leading blanks that must be removed by a text editor.

wwp:n 5 3 5 0 0 0

wwe:n 1.0000E+02

wwn 1:n -1.0000E+00 5.0000E-01 3.8863E-01 3.0648E-01 2.4414E-01

1.9598E-01 1.5757E-01 1.2742E-01 9.8439E-02 7.5788E-02

5.9910E-02 4.7368E-02 3.8112E-02 3.1809E-02 2.7585E-02

2.4827E-02 2.8306E-02

wwp:p 5 3 5 0 0 0

wwe:p 1.0000E+02

wwn 1:p -1.0000E+00 1.1548E+02 3.1561E+01 1.5376E+01 6.7818E+00

4.1291E+00 2.1218E+00 1.2429E+00 7.7217E-01 4.6575E-01

2.6710E-01 1.5285E-01 9.0710E-02 5.4652E-02 3.1623E-02

1.7327E-02 9.8268E-03

***********************************************************************************************************************

N80dump no. 2 on file test1.r nps = 10000 coll = 1314547 ctm = 3.29 nrn = 20699414

8 warning messages so far.

N80run terminated when 10000 particle histories were done.

computer time = 3.42 minutes

mcnp version 4c 01/20/00 06/23/00 11:34:30 probid = 06/23/00 11:30:40

18 December 2000 5-35

CHAPTER 5

TEST1 PROBLEM AND OUTPUT

Notes:

N1: The ﬁrst line of the output ﬁle indentiﬁes the code name and version. LD=xx identiﬁes

the code version date. The last two entries are the date and time the run was made.

N2: This is an echo of the execution line.

N3: The numbers in this ﬁrst column are sequential line numbers for the input ﬁle. They may

be useful if you make changes to the ﬁle with an editor.

N4: Deﬁnes a source at the point 0,0,0 on surface 2. The particles will enter cell 2. The entry

CEL=2 is not needed, but if you choose to use it and type in the wrong cell number, the

code will give you an error message. The weight of each source particle is 1, the default.

VEC and DIR determine the starting direction. In this problem, the source is

monodirectional in the y direction. All source particles have a starting energy of 14.19

MeV.

N5: The PWT card controls the number and weight of neutron−induced photons produced at

neutron collisions.

N6: Energy bins for all tallies but F6 and F12. 13I means put 13 interpolates between 1 and

15 MeV. These energy bins are printed in PRINT TABLE 30.

−Table 10−

N7: All source variables deﬁned explicitly or by default are printed. The order of sampling of

the source variables is also printed, which is important for sources that are dependent

upon functions.

−Table 30−

N8: This entry identiﬁes which particle type and tally type is used (neutron, photon, or

electron).

N9: This warning is generated because the upper limit of the E0 card of 20 MeV is higher

than the maximum energy speciﬁed on the PHYS:N card.

N10: The energy bins are speciﬁed by the E0, E6, and E12 cards. Tallies F1, F11, F16, F26,

and F34 have energy bins speciﬁed by the E0 card. F6 and F12 have energy bins speciﬁed

by the E6 and E12 cards.

5-36 18 December 2000

CHAPTER 5

TEST1 PROBLEM AND OUTPUT

−Table 50−

N11: A cell can be composed of physically separate regions or pieces joined with the union

operator. Improperly deﬁned cells can be composed unintentionally of more than one

piece (for example, a surface is extended unknowingly and forms a cell). If a cell is

composed of more than one piece, a warning message is given and you should verify that

the number of pieces is correct or incorrect.

−Table 60−

N12: If you know the mass or volume of a geometry or parts of it, you can compare the known

volume or mass with what MCNP calculates to verify the correctness of your geometry.

Be careful, however, that volumes or masses that MCNP cannot calculate (but supplies a

value such as unity) do not affect the totals.

−Table 70−

N13: These entries are the surface coefﬁcients used by the code and are not necessarily the

entries on the surface cards.

−Table 72−

N14: This is the temperature calculated by MCNP for cells 2−17. Because there was not a

TMPn:N card in the input ﬁle, room temperature (2.53E−08 MeV) is assumed. Cell 1 has

zero importance and is therefore not affected. The minimum and maximum source

weights are also printed here because they are sometimes dependent upon cell volumes

and cannot be printed earlier. When the source is biased in any way, there will be a

ﬂuctuation in starting source weights. The minimum source weight is used in the weight

cutoff game when negative weight cutoffs are entered on the CUT cards. By playing the

weight cutoff game relative to the minimum source weight, the weight cutoff in each cell

is the same regardless of starting source wight. Note that if the source weight can go to

zero, the miniumum source weight is set to 1.E-10 times the value of the WGT parameter

on the SDEF card.

−Table 98−

N15: The physical constants used in MCNP and changeable in parameter statements in

COMMON blocks are listed here. The compilation options are also listed. Knowing how

the code was compiled is very useful if it runs slowly (pointer option), runs out of space

(pointer option not used), doesn't plot (plot option wrong for your machine or run−time

libraries for plotting located differently on your machine), or can't ﬁnd the data libraries

(wrong datapath−so you must use “setenv DATAPATH ...” on Unix systems).

18 December 2000 5-37

CHAPTER 5

TEST1 PROBLEM AND OUTPUT

−Table 100−

N16: The cross-section table list shows the nuclear data used in the problem. The C appended

to the neutron data indicates continuous energy. A D would indicate discrete reactions. A

P indicates photon data, and an E indicates electron data. Note that photon and electron

data are all elemental (1000.02P) rather than isotopic (1001.60C). Warnings are printed

in MODE N P problems if the photon production cross sections are unavailable or are in

the less accurate equiprobable bin format. Note that electron data are loaded even though

electrons are not transported in this MODE N P problem. The electron data are used for

the thick target Bremsstrahlung model.

N17: If a neutron is born at an energy greater than Emax as set by the PHYS:N card, that

neutron is rejected and the event (such as ﬁssion) is resampled until an energy below Emax

is obtained.

N18: Any neutron cross sections outside the energy range of the problem as speciﬁed by the

PHYS:N and CUT:N cards are deleted.

N19: The ‘Density Effect Data’ Table contains the material data necesary to correct the

stopping power term for the polarization of the media. If a fast electron passes through

an equal linear density of two materials it will lose more energy in a sparse material than

in a dense material. This effect is very small for heavier particles but for electrons with

relativistic velocities transversing dielectrics media it can be signiﬁcant. For 1 MeV

electrons in water this correction can be as large as 5%.

N20: This is the electron range and straggling table for material 1 (Los Alamos concrete). It

lists 133 electron energies in ascending order (only some are shown in this listing) and

gives the respective stopping powers due to collision and radiation and the range of the

electron in the material. Radiation yield is the fraction of the electron's kinetic energy

which is converted into bremsstrahlung energy. The electron physics is turned on in this

MODE N P problem for the thick target bremsstrahlung model.

N21: The table entitled “Secondary electron production for material 1” contains a list of 133

electron energies in ascending order (only some are shown in this listing) and gives the

respective stopping powers due to collision and radiation and the range of the secondary

electron created in the electron in the material.

N22: At the end of the cross section processing, and before histories are started, the ﬁrst dump

is made to the RUNTPE ﬁle. This dump contains all the ﬁxed information about the

problem, namely the problem speciﬁcation and all nuclear data. Subsequent dumps to

RUNTPE will contain only information that accumulates as histories are run, such as

tally information and particle statistics for summary and ledger tables.

5-38 18 December 2000

CHAPTER 5

TEST1 PROBLEM AND OUTPUT

−Table 110−

N23: This table gives starting information about the ﬁrst 50 source particles. X, Y, and Z tells

the initial position. CELL identiﬁes what cell the particle started in or was directed into.

SURF identiﬁes what surface the particle started on, if any. U, V, and W identiﬁes the

starting direction cosines. The starting time, weight and energy of the particles are also

given.

−Problem Summary−

N24: This is the summary page of the problem. It is a balance sheet with the left side showing

how particle tracks, weight, and energy were created and right side showing how they

were lost. The problem summary is for accounting only, because most entries, such as

“tracks,” have no physical meaning and trying to give physical interpretation to these

numerical quantities may be dangerous. The weight and energy columns contain the

physical results. Because the summary contains net creation and loss, physical

interpretation must be done with care.

N25: 35681 represents the increased number of tracks obtained and banked from cell splitting,

which occurs when the ratio of importances of the cell entered to the cell exited is greater

than one. If the ratio is less than one, Russian roulette is played. If the track survives the

roulette, its increase in weight and energy are recorded as a gain. If it loses, it is recorded

as a loss in all three categories on the loss side of the table.

N26: The creation from weight cutoff represents the weight and energy gained from winning

the weight cutoff Russian roulette game. No tracks are created because the original track

continues with an increased weight.

N27: Any tracks that enter a cell of zero importance are considered to have escaped the

geometry and are recorded here. This is the physical leakage from the system. The

precision of this result is unknown because no relative error is calculated as is with a

tally.

N28: Loss to importance sampling results from losing the Russian roulette game played when

crossing a surface into a cell of lower importance. The weight and energy losses should

agree with gains in N25 with perfect sampling.

N29: Loss to weight cutoff comes from losing the weight cutoff Russian roulette game. With

perfect sampling, the weight and energy lost here should equal the weight and energy

gained in N26. What is accumulated in the three loss entries is the number, weight, and

weight times energy of the tracks lost to weight cutoff. The weight entry in the table is

normalized by the number of source particles and the energy entry by the total weight of

18 December 2000 5-39

CHAPTER 5

TEST1 PROBLEM AND OUTPUT

source particles. Thus the average weight of each track lost to the weight cutoff is: weight

entry ∗NPS/number of tracks lost: 0.080533 * 10000 / 8625 = 0.09337. The small

average track weight is caused by the scaling of the weight cutoff criteria by the ratio of

the source cell importance to the collision cell importance. The average energy of a

particle lost to weight cutoff per source particle is: energy entry/weight entry ∗ average

source starting weight: (0.055477/0.080533) ∗ 1.0 = 0.68889. The same normalizing

procedure applies to all energy entries in both the creation and loss columns of this table.

N30: In a scattering event only energy is changed. Energy difference = energy in − energy out.

If this difference is positive, it is entered as downscattering on the loss side; if negative, as

in a thermal neutron upscatter, it is entered on the creation side as upscattering. Thermal

neutron scatter always results in a small energy gain or loss. (Elastic collisions in the

center−of−mass system gain or lose energy in the laboratory system.) Higher energy

scatter usually is an energy loss mechanism. This energy is only for the track being

followed. If the collision is a ﬁssion or (x,xn), the tracks in addition to one outgoing track

are recorded in the three creation columns of the ﬁssion and (x,xn) rows.

N31: Tracks are lost to capture only if the analog capture option is used (PHYS or CUT card).

In this problem, implicit capture was used to remove a fraction of each particle's weight

at each collision. The energy lost is the incident energy of the particle times the weight

lost to capture. The weight lost to absorption (n,0n) is a physically meaningful quantity.

No relative error is calculated.

N32: Note that the total gain and the total loss of the track quantities balance exactly in all

problems.

N33: Whereas all neutrons in this problem started at time zero, the average time of escape is

also the prompt neutron escape lifespan and the average time of capture is also the

prompt capture (n,0n) lifespan. As there is no ﬁssion in the problem, escape and capture

are the only two physical removal mechanisms; thus the average time to capture or escape

is both the prompt removal lifespan and the prompt removal lifetime. See Chapter 2, page

2-164. These quantities are absorption estimates averaged over all histories; track length

estimates can be calculated with the FM card. The “average time of” is always measured

relative to time zero and is mostly of use in setting a time cutoff, time bins, or getting a

better feel for what is happening in the problem.

N34: The second entry of the net multiplication is the relative error or the multiplication

corresponding to one standard deviation. In this problem, the net multiplication, which is

the sum of the source weight and the weight from (x,xn) reactions, is 1.0150 +/- %. The

net multiplication is not the criticality eigenvalue keff of the system. See page 2-176 for

further discussion of this subject.

5-40 18 December 2000

CHAPTER 5

TEST1 PROBLEM AND OUTPUT

N35: Pair production caused the loss of 999 tracks with a weight of 0.13767. The electron from

pair production is assumed to immediately annihilate and lose all its energy in the cell,

unless it is followed in MODE P E. The positron is annihilated (p−annihilation),

producing two photons (1998 tracks with weight 0.27534), each with energy 0.511 MeV

isotropic in direction.

N36: For a MODE N P problem, the “average time of” for a photon is relative to zero time, and

not the time when the photon was produced. Thus the “average time of” escape or capture

includes the mean time to creation.

−Table 126−

N37: Tracks entering a cell refers to all tracks entering a cell, including source particles. If a

track leaves a cell and later reenters that same cell, it is counted again. Does not include

particles from the bank (from variance reduction events at collisions or physical events at

collisions.)

N38: Population in a cell is the number of tracks entering a cell plus source particles plus

particles from the bank (from variance reduction or physical events at collisions.)

Population does not include reentrant tracks. Comparing N37 to N38 will indicate the

amount of back scattering in the problem. An often successful rule of thumb for choosing

importances is to select them so that population is kept roughly constant in all cells

between the source and tally regions. Information, once lost, cannot be regained. The

13029 particles in cell 8 can contain no more information than the 7005 particles in cell 6

because all particles in cell 8 are progeny of the particles in cell 6. Oversplitting or

undersplitting has occurred between cell 6 and cell 8.

N39: The number of collisions in a cell is important for a detector tally or anything involving

collision rate. A lack of collisions may indicate a need to force them. This quantity is not

normalized by cell volume. In some problems most of the computer time is spent

modeling collisions. Cells with excessive numbers of collisions are possibly oversampled.

This often happens when many thermal neutrons rattle around and contribute little of

signiﬁcance to the problem solution. In such cases energy−dependent weight windows

are most effective, followed by energy roulette, exponential transform, time cutoff, or

energy cutoff. Note that the last two methods may introduce a bias into the problem.

Subdividing the cell into smaller cells with different importances also is effective.

N40: The collision times the weight of the particles having the collisions is an indication of

how important the collisions were.

N41: The next four items are determined from the distance D to the next collision or surface.

The time DT to traverse this distance is determined from DT=D/VEL where VEL is the

18 December 2000 5-41

CHAPTER 5

TEST1 PROBLEM AND OUTPUT

speed of the particle. Furthermore, the ﬂux Φis equal to the number density n(E) times

the speed.

The energy ERG averaged over the number density of particles is determined by

N42: The energy averaged over the ﬂux density is

It is very difﬁcult, and perhaps meaningless, to determine an average energy because a

large spectrum involving several orders of magnitude is frequently involved leading to the

problem of representing this by one number. That is why it has been calculated by the

two methods of items N41 and N42. If the number−averaged energy is signiﬁcantly lower

than the ﬂux−averaged energy (as is true in this problem), it indicates a large number of

low-energy particles. As the energy cutoff in this problem is raised, these two average

energies come into closer agreement.

N43: The relative average track weight is , where Ic and Is are the

importances of the cell and the source cell. By making the average track weight relative

to the cell importance, the weight reduction from importance splitting is removed. For

most problems with proper cell importances, the average track weight is constant from

cell to cell and deviations indicate a poor importance function. The variation in average

track weight for the photons in the following table suggests that the photon importances

(same as neutrons) are poor. With weight windows, the average track weight should be

within the weight window bounds.

N44: The average track mean free path is

where TOTM = Σt(E) is the total macroscopic cross section. The mean free path is

strongly dependent upon energy and so this average mean free path may be meaningless.

A rule of thumb for guessing at importances is that they should double approximately

nEt,()E⋅Edtd

∫∫

nEt,()Edtd

∫∫

------------------------------------------- WGT∗DT∗ERG()

∑WGT∗D()

∑

------------------------------------------------------

ΦEt,()E⋅Edtd

∫∫ ΦEt,()Edtd

∫∫

---------------------------------------------WGT∗D∗ERG()

∑WGT∗D()

∑

-------------------------------------------------

IcΣWGT∗D()IsΣD()⁄

ΦE()Σ

⁄E()Ed

∫ΦE()Ed

∫

----------------------------------------- W

∑GT∗DT⁄OTM

WGT∗D

∑

---------------------------------------------------=

5-42 18 December 2000

CHAPTER 5

TEST1 PROBLEM AND OUTPUT

every mean free path. This is usually a very poor rule, but it is sometimes better than

nothing. The average track mean free path is thus useful for making poor guesses at cell

importances. It is also useful for determining the ﬁcticious radius of point detectors, the

outer radius of DXTRAN spheres, exponential transform stretching parameters, the

necessity of forced collisions, etc. Occasionally this quantity may even provide physical

insight into your problem.

N45: For photons, the number−weighted energy and ﬂux−weighted energy are equal because a

photon has a constant velocity regardless of energy. See N37 −− N44.

−Table 130−

N46: The next six tables (three for neutrons and three for photons) show all possible ways a

particle's weight may be changed in each cell. In addition to telling you what is

happening to the particle and where, this information can be useful in debugging a

problem. The totals agree with the problem summary.

Note that the neutron weight entering cell 17 is 0.018644, whereas in Table 126 the

average relative track weight in cell 17 is 0.41987. This apparent discrepancy is resolved

by realizing that the average weight in Table 126 is for a track, while it is for a history in

Table 130. Furthermore, in Table 126 the weight is relative, whereas it is absolute here in

Table 130. If the average track weight is multiplied by the tracks entering cell 17 (13964)

and then divided by both the number of source particles (10,000) and the importance ratio

(32), the two weights are in close agreement. Most of the totals over the cells can be

compared directly with the weight gain, loss or difference in the Problem Summary. The

average value of in a problem with ﬁssionable material can be obtained by taking the

ratio of ﬁssion neutrons to ﬁssion loss in the neutron physical events table.

−Table 140−

N47: The activity of each nuclide per cell can tell you how important various nuclides, such as

trace elements, are to the problem and may aid in selecting cross-section libraries when

memory is limited. This chapter only shows a partial listing of this table.

N48: This table is the activity summed over all cells in the problem.

N49: This column shows the total number of photons produced by each isotope in the problem.

The earlier entries in this column show photon production per isotope in each cell.

N50: This table is printed only for MODE N P or MODE N P E. It gives you an idea of how

many photons were produced in each cell and the energy spectrum of the photons

averaged over the problem. Because photons are produced only at neutron collisions,

18 December 2000 5-43

CHAPTER 5

TEST1 PROBLEM AND OUTPUT

there is a correlation between the number of collisions in a cell, the PWT card, and this

table. The previous table showing the photon activity for the problem includes isotope−

dependent neutron−induced photon production information.

TALLY AND TALLY FLUCTUATION CHARTS

N51: All tallies here are caused by the F1, F11, F12, F6, F16, F26, and F34 cards in the input

ﬁle. Only the F11 results are shown. The F11 tally gives the neutron current summed in

both directions integrated over a surface. This tally says that between 13 and 14 MeV, the

current is 1.67052E−04 ± 15.68% within one standard deviation.

N52: The normed average tally per history describes the average tally normalized over the tally

surface or volume. It includes energy- and time-dependent mutlipliers and some constant

multipliers,but excludes most constant multipliers. This is always equal to the total tally.

N53: The unnormed average tally per history does not always include all multipliers. It is the

tally used for statistical analysis and is for the same TFC bin as the normed tally.

N54: This is the variance of the variance which checks the tally for any effects of inadequately

sampled problems. It can pick up tally errors due to insufﬁcient sampling of high weight

scores which can cause an underestimated mean and RE. The typical acceptable VOV is

0.1 or less in order to provide a reliable conﬁdence interval.

N55: This is the relative error component from histories which do not contribute to the tally

(zero history scores).

N56: This is the relative error from only the non zero history scores.

N57: This is the fraction of total NPS that resulted in nonzero score tallies.

N58: If there was a great difference between the largest and average tally, the large weight

particles would represent important phenomena that have been undersampled and/or poor

variance reduction technique selection. To understand what causes the large weight

particles, the history number of the largest is printed so that this history can be rerun to

get its event log. When the undersampled event is identiﬁed, the variance reduction

should be modiﬁed and the problem rerun. Improved variance reduction usually causes

fewer source histories to be run per minute because more time is spent sampling the

formerly undersampled important phenomena outside the source. The ﬁnal result will be

an improved (higher) FOM and a lower largest/average tally ratio. As the largest/average

tally ratio approaches unity, the problem approaches an ideal zero variance solution. In

practice, performing the steps discussed above is an art usually beyond all but the most

experienced users and is often difﬁcult, time-consuming, frustrating, and sometimes

5-44 18 December 2000

CHAPTER 5

TEST1 PROBLEM AND OUTPUT

unsuccessful. An alternative is to let MCNP determine the better importance function for

the next run with the weight window generator, as has been done in this problem. Use of

the generated weight windows printed in PRINT TABLE 190 caused a factor of three

improvement in problem efﬁciency when the problem was rerun.

N59: This ratio expresses the conﬁdence interval shift as a fraction of the mean. The

conﬁdence interval is shifted in the case of an asymmetric probability density function.

N60: This table provides the user with the information on how the TFC bins would be effected

by a high magnitude score occurring on the next history. This can reveal the impact of an

infrequent high weight score distorting the TFC bin quantities. The three columns show

the value at the current NPS, the value at the next NPS (which is the value of the highest

past score), and the ratio of the highest value over the previous lower value.

N61: These two lines summarize brieﬂy the behavior of the tail of the probability density

function. MCNP checks the slope of the high score tail in order to discern whether the

problem has been sampled well. If the tail of the probability density function is not

decreasing at a fast enough rate, then MCNP will ﬂag this as an insufﬁciently sampled

problem.

N62: This is the TFC statistical check table which provides the results of ten checks that are

used to test the tally for reliability. MCNP checks the behavior of the mean, relative

error, variance of the variance, ﬁgure of merit and the probability density function. The

table presents the desired, observed and actual results along with the pass/no pass

message for each test.

N63: This column shows the desired, observed, and actual behavior of the mean. Random

behavior of the mean is desired because an ideal random quantity should exhibit a normal

distribution of values around an average value. MCNP checks for non-monotonic (no

increasing or decreasing trend) behavior of the mean for the last half of the problem. If

the behavior of the mean meets this criteria, then it passes this test. The tally was random

over the last half of the problem so it passed this check.

N64: This column checks if the relative error is below the limit required to provide a reliable

conﬁdence interval.

N65: This column checks if the relative error is decreasing over the length of the problem.

N66: This column checks for the decrease rate of the relative error as a function of the number

of histories(NPS). If the relative error is decreasing at the desired rate for the last half of

the problem, then it passes this check.

18 December 2000 5-45

CHAPTER 5

TEST1 PROBLEM AND OUTPUT

N67: This column checks if the variance of the variance (VOV) is below the prescribed value

of 0.1.

N68: This column checks for a monotonically decreasing VOV for the last half of the problem.

N69: This column is the check for the rate of decrease of the VOV for the last half of the

problem.

N70: This column checks for a statistically constant value of the ﬁgure of merit (FOM) for the

last half of the problem.

N71: This column checks the FOM for random behavior.

N72: This column checks the probability density function (PDF) for the slope of the 25 to 200

largest history scores. If the slope is greater than 3 then the second moment of the PDF

exists and the central limit theorem is satisﬁed. Basically, this means that as the slope

increases, a more reliable conﬁdence interval is formed because the problem is sampled

more.

N73: All of the statistical checks were passed, therefore a range of conﬁdence intervals for the

unshifted asymmetric distribution is provided. Three ranges are given for the conﬁdence

intervals of 1, 2, and 3 standard deviations. The second line displays the ranges for the

shifted symmetric conﬁdence intervals. If the checks had not been satisﬁed, a warning

would have been provided.

N74: This plot is the unnormed probability density for the tally ﬂucuation chart bin of tally 11.

The probability density is the number of tallies plotted (horizontal) against the value of

the tally (vertical). The central mean is denoted by the line of m's. If a problem has been

undersampled, this plot will often show “holes,” or unsampled regions of the PDF. If the

slope is less than 10, this plot will also show a curve of S's which represent the Pareto

curve ﬁt to the PDF. This allows the user to visually compare the curve ﬁt to the

calculated distribution. The total 1.14E-1 is 1143 tallies from 10000 histories.

−Table 162−

N75: This plot is the cumulative number of tallies in the tally ﬂuctuation chart bin of tally 11.

It is simply the cumulative version of table 161, or the cumulative probability density

function. The ordinate and abscissa values are printed in the left-hand columns and are

read as, “785 scores were made with a value of 7.94328E-02 or less and these 785 scores

accounted for 68.679% of the total tally.” This plot is followed by a plot of the cumulative

tally in the tally ﬂuctuation chart bin. These entries are read as “Of the total tally value

8.11579E-03, 2.841E-3 (or 35.007%) was from tallies with values less than 7.943E-02.”

5-46 18 December 2000

CHAPTER 5

TEST1 PROBLEM AND OUTPUT

N76: Only tally 11 was shown to save space. After all the tallies are printed a summary of

statistical checks for all of the tallies is given.

N77: The tally ﬂuctuation charts always should be studied to see how stable or reliable the tally

mean, relative error, variance of the variance, slope, and FOM are, indicating how the

problem is converging as a function of history number, NPS. The FOM is deﬁned as 1/σ2t,

where σ is the relative error and t is the computer time in minutes. In a well-behaved

problem, tis proportional to the number of histories run, N, and σ is proportional to

. Thus the FOM should rapidly approach a constant value as it does in this problem.

Big changes in the FOM indicate sampling problems that need attention.

The order of printing tallies is: neutron, photon, combined neutron/photon, electron, and

combined photon/electron. Notice that the combined heating tally F6 is exactly the sum

of the neutron, F16, and photon, F26, heating tallies.

−Table 190−

N78: This table is a list of the lower weight window bounds generated by the WWG card.

These window bounds are themselves estimated quantities and must be well converged or

they can cause more harm than good. When well converged, they can improve efﬁciency

dramatically. Use of these printed weight windows results in an increase of three in the

FOM for tally 12 when the problem was rerun. Note that the number of histories per

minute is often lower in the more efﬁcient problem because more time is spent sampling

important regions of the problem phase space.

These weight windows were chosen to optimize tally 12 as speciﬁed on the WWG card.

In the subsequent run using these weight windows, the FOM of tally 12 improved by three

as did the other photon tallies, and tallies 16 and 34 were slightly degraded. The weight

window generator optimizes the importance function for one tally at the expense of all

others, if necessary.

Sometimes the calculated lower bound for the photon weight window in a cell is zero,

meaning that no photon in that cell ever contributed to the tally of interest in that run. If

the zero is unchanged in the run using these windows, the weight cutoff game will be

played that cell, sometimes with disastrous consequences. Thus a guess should be made

for a lower bound rather than leaving the zero value. A good guess is 10, which is several

times higher than the weight window generated for its nearest neighboring cell.

The generated weight windows may be thought of as a forward adjoint solution and thus

can provide considerable insight into the physics of a problem. Low weight windows

indicate important regions. A low window on a cell bounding the outside world often

1N⁄

18 December 2000 5-47

CHAPTER 5

TEST1 PROBLEM AND OUTPUT

indicates that the geometry was truncated and more cells need to be added outside the

present geometry. Weight windows that differ greatly between adjacent cells indicate

poor weight window convergence or, more likely, a need to subdivide the geometry into

smaller phase space units that will have different importances.

Energy dependent weight windows are also available.

−Table 200−

N79: The weight window cards from the weight window generator can, with some ﬁle editing,

be used instead of the IMP:N and IMP:P cards in the next run of this problem. Zero

windows should be replaced with a good guess. Windows differing greatly from those in

neighboring cells should be replaced (there are no such cases in this problem). The space

between WWN and 1:N must be removed.

We suggest the user read these generated window values from the WWOUT ﬁle rather

than the editing method just discussed (WWG card, WWINP=WWOUT on the execute

line).

N80: With this initial run there are two dumps on the RUNTPE. The ﬁrst dump occurs at the

end of XACT. The second dump is done at the problem end. A continue−run will pick up

from this second dump and add a third dump to the RUNTPE when it ﬁnishes. CTM =

3.29 is the computer time in minutes used in the transport portion of the problem.

N81: One or more reasons are always given as to why the run was terminated. If there are no

errors, most runs terminate after the desired number of particles are run or by a time limit.

Computer time = 3.29 minutes is the total time for the problem, including initiation,

output, etc.

5-48 18 December 2000

CHAPTER 5

CONC PROBLEM AND OUTPUT

III. CONC PROBLEM AND OUTPUT

This simple problem illustrates how to use and interpret results from detectors. It also shows

how the statistical checks can reveal deﬁciencies in the output of an otherwise well−behaved

problem. The problem consists of a spherical shell of concrete with a 390-cm outer radius and a

360−cm inner radius. A 14 MeV point isotropic neutron source is at 0,0,0, the center of the void

region. It is a neutron−only problem (MODE N), with a neutron lower energy cutoff at 12 MeV.

A surface ﬂux tally is used in addition to point and ring detectors.

Even though this is a simple problem, it is difﬁcult, and even inappropriate, for the F65 point

detector. Detectors are inappropriate when particles can be transported readily to the region of

interest and another type of tally, such as the F2 surface ﬂux tally, can be used. Also, detectors do

not work well close to or in scattering regions. A detailed discussion of this problem is presented

in Chapter 2, page 2−150.

The following notes on the output describe the pertinent details dealing with the point detector

results. The notes will provide a description of the TFC bin checks that test the tally for its

reliability. This problem dramatically illustrates the importance of the VOV (variance of the

variance) and the PDF (probability density function) slope checks in determining the reliability

of the results.

The following notes apply to the CONC problem output ﬁle. Only the default print tables appear

because there is no PRINT card.

18 December 2000 5-49

CHAPTER 5

CONC PROBLEM AND OUTPUT

1mcnp version 4c ld=01/20/00 07/18/00 12:56:34

************************************************************************* probid = 07/18/00 12:56:34

N1inp=conc name=conc.

1- conc: 30 concrete shell with a point 14 MeV source in the center

2- C ex=500

3- 1 0 -1

4- 2 1 -2.3 1 -2

5- 3 0 2 -3

6- 4 0 3 -4

7- 5 0 4

9- 1 so 360

10- 2 so 390

11- 3 so 420

12- 4 so 4000

13-

14- sdef

15- imp:n 1 3r 0

16- m1 1001.60c 1.68756E-01

17- 8016.60c 5.62493E-01

18- 11023.60c 1.18366E-02

19- 12000.60c 1.39951E-03

20- 13027.60c 2.14316E-02

21- 14000.60c 2.04076E-01

22- 19000.60c 5.65495E-03

23- 20000.60c 1.86720E-02

24- 26054.60c 2.47295E-04

25- 26056.60c 3.91067E-03

26- 26057.60c 9.38014E-05

27- 26058.60c 1.19384E-05

28- 6012.50c 1.41730E-03

29- c

30- c surface, point, and ring tallies

31- f2:n 4

32- e2 12.5 2i 14. c

33- f12:n 3

34- f22:n 2

35- f25:n 0 -4000 0 0

36- f35:n 0 4000 0 0

37- f45:n 0 -420 0 0

38- f55:n 0 420 0 0

N239- f65:n 0 -390 0 -0.5

40- f75:n 0 390 0 -0.5

41- f85y:n 0 4000 0

N342- f95y:n 0 420 0

43- f105y:n 0 390 -0.5

44- e0 12.5 2i 14.

CHAPTER 5

CONC PROBLEM AND OUTPUT

5-50 18 December 2000

N445- dd 0.1 1e100

46- c

47- c cutoff the neutrons at 12 MeV

N548- cut:n j 12.0

49- nps 14000

1cells print table 60

atom gram neutron

cell mat density density volume mass pieces importance

1 1 0 0.00000E+00 0.00000E+00 1.95432E+08 0.00000E+00 1 1.0000E+00

2 2 1 8.14382E-02 2.30000E+00 5.30427E+07 1.21998E+08 1 1.0000E+00

3 3 0 0.00000E+00 0.00000E+00 6.18642E+07 0.00000E+00 1 1.0000E+00

4 4 0 0.00000E+00 0.00000E+00 2.67772E+11 0.00000E+00 1 1.0000E+00

5 5 0 0.00000E+00 0.00000E+00 0.00000E+00 0.00000E+00 0 0.0000E+00

total 2.68083E+11 1.21998E+08

minimum source weight = 1.0000E+00 maximum source weight = 1.0000E+00

1cross-section tables print table 100

table length

tables from file endf602

1001.60c 488 1-h-1 from endf-vi.1 mat 125 11/25/93

8016.60c 7725 8-o-16 from endf/b-vi mat 8 25 11/25/93

11023.60c 1433 11-na-23 from endf/b-vi.1 mat1125 11/25/93

12000.60c 5743 12-mg-nat from endf/b-vi mat1200 11/25/93

13027.60c 5790 13-al-27 from endf/b-vi mat1325 11/25/93

14000.60c 7846 14-si-nat from endf/b-vi mat1400 11/25/93

19000.60c 1981 19-k-nat from endf/b-vi mat1900 11/25/93

20000.60c 8534 20-ca-nat from endf/b-vi mat2000 11/25/93

26054.60c 6047 endf/b-vi.1 fe54a mat2625 11/25/93

26056.60c 17615 endf/b-vi.1 fe56a mat2631 11/25/93

26057.60c 7045 endf/b-vi.1 fe57a mat2634 11/25/93

26058.60c 5702 endf/b-vi.1 fe58a mat2637 11/25/93

tables from file rmccs2

6012.50c 3844 njoy ( 1306) 79/07/31.

total 79793

neutron cross sections outside the range from 1.2000E+01 to 1.0000E+37 mev are expunged.

18 December 2000 5-51

CHAPTER 5

CONC PROBLEM AND OUTPUT

decimal words of dynamically allocated storage

general 10092

tallies 79288

bank 46724

cross sections 159586

total 249322 = 997288 bytes

***********************************************************************************************************************

dump no. 1 on file conc.r nps = 0 coll = 0 ctm = 0.00 nrn = 0

N61problem summary

run terminated when 14000 particle histories were done.

+ 07/18/00 12:57:14

conc: 30 concrete shell with a point 14 MeV source in the center probid = 07/18/00 12:56:34

neutron creation tracks weight energy neutron loss tracks weight energy

(per source particle) (per source particle)

source 14000 1.0000E+00 1.4000E+01 escape 1568 8.6876E-02 1.1939E+00

energy cutoff 12673 6.7294E-01 3.6320E+00

time cutoff 0 0. 0.

weight window 0 0. 0. weight window 0 0. 0.

cell importance 0 0. 0. cell importance 0 0. 0.

weight cutoff 0 1.5341E-04 1.9648E-03 weight cutoff 16 2.6504E-04 3.3844E-03

energy importance 0 0. 0. energy importance 0 0. 0.

dxtran 0 0. 0. dxtran 0 0. 0.

forced collisions 0 0. 0. forced collisions 0 0. 0.

exp. transform 0 0. 0. exp. transform 0 0. 0.

upscattering 0 0. 0. downscattering 0 0. 5.5765E+00

delayed fission 0 0. 0. capture 0 2.5297E-01 3.4795E+00

(n,xn) 514 2.5795E-02 6.2067E-02 loss to (n,xn) 257 1.2898E-02 1.7868E-01

prompt fission 0 0. 0. loss to fission 0 0. 0.

total 14514 1.0259E+00 1.4064E+01 total 14514 1.0259E+00 1.4064E+01

number of neutrons banked 0 average time of (shakes) cutoffs

neutron tracks per source particle 1.0367E+00 escape 7.9538E+01 tco 1.0000E+34

neutron collisions per source particle 1.8890E+00 capture 7.2934E+00 eco 1.2000E+01

total neutron collisions 26446 capture or escape 2.5762E+01 wc1 -5.0000E-01

net multiplication 1.0129E+00 0.0008 any termination 1.3250E+01 wc2 -2.5000E-01

computer time so far in this run 0.61 minutes maximum number ever in bank 0

computer time in mcrun 0.28 minutes bank overflows to backup file 0

source particles per minute 4.9296E+04 dynamic storage 249326 words, 997304 bytes.

random numbers generated 421786 most random numbers used was 160 in history 4334

CHAPTER 5

CONC PROBLEM AND OUTPUT

5-52 18 December 2000

range of sampled source weights = 1.0000E+00 to 1.0000E+00

1neutron activity in each cell print table 126

tracks population collisions collisions number flux average average

cell entering * weight weighted weighted track weight track mfp

(per history) energy energy (relative) (cm)

1 1 14145 14000 0 0.0000E+00 1.3989E+01 1.3989E+01 9.9715E-01 0.0000E+00

2 2 14145 14000 26446 1.6648E+00 1.3745E+01 1.3755E+01 8.8157E-01 8.0847E+00

3 3 1568 1568 0 0.0000E+00 1.3679E+01 1.3687E+01 7.5956E-01 0.0000E+00

4 4 1568 1568 0 0.0000E+00 1.3733E+01 1.3740E+01 7.7460E-01 0.0000E+00

total 31426 31136 26446 1.6648E+00

1tally 2 nps = 14000

tally type 2 particle flux averaged over a surface. units 1/cm**2

tally for neutrons

N7 energy bins are cumulative.

areas

surface: 4

2.01062E+08

surface 4

energy

1.2500E+01 1.43092E-11 0.1277

1.3000E+01 3.32372E-11 0.0848

1.3500E+01 6.70291E-11 0.0593

1.4000E+01 4.32412E-10 0.0244

SKIP 703 LINES OF OUTPUT

1tally 65 nps = 14000

tally type 5 particle flux at a point detector. units 1/cm**2

tally for neutrons

detector located at x,y,z = 0.00000E+00-3.90000E+02 0.00000E+00

energy

1.2500E+01 6.95657E-10 0.7722

1.3000E+01 9.43523E-10 0.6869

1.3500E+01 2.27344E-10 0.4054

1.4000E+01 1.31533E-08 0.0096

total 1.50198E-08 0.0620

detector located at x,y,z = 0.00000E+00-3.90000E+02 0.00000E+00

uncollided neutron flux

energy

1.2500E+01 0.00000E+00 0.0000

1.3000E+01 0.00000E+00 0.0000

1.3500E+01 0.00000E+00 0.0000

1.4000E+01 1.29226E-08 0.0000

18 December 2000 5-53

CHAPTER 5

CONC PROBLEM AND OUTPUT

total 1.29226E-08 0.0000

N8 detector score diagnostics cumulative tally cumulative

fraction of per fraction of

times average score transmissions transmissions history total tally

*a* 1.00000E-01 4071 0.22336 1.91294E-11 0.00127

*b* 1.00000E+00 13841 0.98277 1.27537E-08 0.85040

2.00000E+00 12 0.98343 1.78407E-11 0.85158

5.00000E+00 6 0.98376 2.03013E-11 0.85294

1.00000E+01 7 0.98414 5.12260E-11 0.85635

1.00000E+02 23 0.98541 8.76880E-10 0.91473

1.00000E+03 2 0.98552 1.09590E-09 0.98769

1.00000E+38 0 0.98552 0.00000E+00 0.98769

1st 200 histories 264 1.00000 1.84875E-10 1.00000

average tally per history = 1.50198E-08 largest score = 7.92170E-06

(largest score)/(average tally) = 5.27417E+02 nps of largest score = 6698

score contributions by cell

cell misses hits tally per history weight per hit

1 1 0 *c* 14000 1.29226E-08 1.29226E-08

2 2 19611 *d* 4226 2.09721E-09 6.94769E-09

total 19611 *e* 18226 1.50198E-08 1.15372E-08

score misses

russian roulette on pd 0

psc=0. 1929

russian roulette in transmission 562

underflow in transmission 0

hit a zero-importance cell 0

energy cutoff 17120

1analysis of the results in the tally fluctuation chart bin (tfc) for tally 65 with nps = 14000 print table 160

normed average tally per history = 1.50198E-08 unnormed average tally per history = 1.50198E-08

estimated tally relative error = 0.0620 estimated variance of the variance = 0.4479

relative error from zero tallies = 0.0000 relative error from nonzero scores = 0.0620

number of nonzero history tallies = 14000 efficiency for the nonzero tallies = 1.0000

history number of largest tally = 6698 largest unnormalized history tally = 9.96753E-06

*f*(largest tally)/(average tally) = 6.63626E+02 (largest tally)/(avg nonzero tally)= 6.63626E+02

(confidence interval shift)/mean = 0.0199 shifted confidence interval center = 1.53193E-08

N9if the largest history score sampled so far were to occur on the next history, the tfc bin quantities would change as follows:

estimated quantities value at nps value at nps+1 value(nps+1)/value(nps)-1.

CHAPTER 5

CONC PROBLEM AND OUTPUT

5-54 18 December 2000

mean 1.50198E-08 1.57306E-08 0.047327

relative error 6.19530E-02 7.44344E-02 0.201465

variance of the variance 4.47855E-01 3.14281E-01 -0.298254

shifted center 1.53193E-08 1.53395E-08 0.001322

figure of merit 9.17397E+02 6.35528E+02 -0.307248

the estimated inverse power slope of the 193 largest tallies starting at 1.32362E-08 is 1.3691

the history score probability density function appears to have an unsampled region at the largest history scores: please examine.

===================================================================================================================================

results of 10 statistical checks for the estimated answer for the tally fluctuation chart (tfc) bin of tally 65

N10tfc bin --mean-- ---------relative error--------- ----variance of the variance---- --figure of merit-- -pdf-

behavior behavior value decrease decrease rate value decrease decrease rate value behavior slope

desired random <0.05 yes 1/sqrt(nps) <0.10 yes 1/nps constant random >3.00

observed random 0.06 yes yes 0.45 yes no constant increase 1.37

passed? yes no yes yes no yes no yes no no

===================================================================================================================================

warning. the tally in the tally fluctuation chart bin did not pass 5 of the 10 statistical checks.

fom = (histories/minute)*(f(x) signal-to-noise ratio)**2 = (4.930E+04)*( 1.364E-01)**2 = (4.930E+04)*(1.861E-02) = 9.174E+02

N111unnormed tally density for tally 65 nonzero tally mean(m) = 1.502E-08 nps = 14000 print table 161

abscissa ordinate log plot of tally probability density function in tally fluctuation chart bin(d=decade,slope= 1.4)

tally number num den log den:d------------d--------------d-------------d-------------d-------------d--------------d-------------d

2.00-08 12 2.09+05 5.320 *************|**************|*************|************ | | | s |

2.51-08 8 1.11+05 5.044 *************|**************|*************|******** | | | s |

3.16-08 5 5.49+04 4.740 *************|**************|*************|**** | | | s |

3.98-08 5 4.36+04 4.640 *************|**************|*************|** | | | s |

5.01-08 2 1.39+04 4.142 *************|**************|********* | | | | s |

6.31-08 1 5.50+03 3.741 *************|**************|*** | | | |s |

7.94-08 0 0.00+00 0.000 | | | | | s| |

1.00-07 2 6.95+03 3.842 *************|**************|***** | | | s | |

1.26-07 2 5.52+03 3.742 *************|**************|*** | | | s | |

1.58-07 1 2.19+03 3.341 *************|************* | | | | s | |

2.00-07 1 1.74+03 3.241 *************|*********** | | | | s | |

2.51-07 4 5.53+03 3.743 *************|**************|*** | | | s | |

3.16-07 1 1.10+03 3.041 *************|******** | | | | s | |

3.98-07 0 0.00+00 0.000 | | | | |s | |

5.01-07 3 2.08+03 3.318 *************|************ | | | s| | |

18 December 2000 5-55

CHAPTER 5

CONC PROBLEM AND OUTPUT

6.31-07 0 0.00+00 0.000 | | | | s | | |

7.94-07 0 0.00+00 0.000 | | | | s | | |

1.00-06 0 0.00+00 0.000 | | | | s | | |

1.26-06 2 5.52+02 2.742 *************|**** | | | s | | |

1.58-06 1 2.19+02 2.341 ************ | | | | s | | |

2.00-06 1 1.74+02 2.241 *********** | | | |s | | |

2.51-06 0 0.00+00 0.000 | | | s| | | |

3.16-06 1 1.10+02 2.041 ******** | | | s | | | |

3.98-06 0 0.00+00 0.000 | | | s | | | |

5.01-06 0 0.00+00 0.000 | | | s | | | |

6.31-06 0 0.00+00 0.000 | | | s | | | |

7.94-06 1 4.37+01 1.641 ** | | | s | | | |

1.00-05 1 3.47+01 1.541 * | | |s | | | |

total 14000 1.00+00 d------------d--------------d-------------d-------------d-------------d--------------d-------------d

SKIP 554 LINES OF OUTPUT

N121tally fluctuation charts

tally 2 tally 12 tally 22

nps mean error vov slope fom mean error vov slope fom mean error vov slope fom

1000 4.7068E-10 0.0876 0.0068 0.0 7313 4.7567E-08 0.0876 0.0075 0.0 7308 5.8922E-08 0.0971 0.0685 0.0 5947

2000 4.2275E-10 0.0653 0.0039 0.0 6375 4.2930E-08 0.0656 0.0046 0.0 6316 5.3176E-08 0.0715 0.0290 0.0 5314

3000 4.3409E-10 0.0525 0.0025 0.0 6470 4.3773E-08 0.0526 0.0028 0.0 6451 5.3670E-08 0.0562 0.0154 0.0 5657

4000 4.4451E-10 0.0448 0.0018 0.0 6589 4.5086E-08 0.0449 0.0020 0.0 6555 5.5571E-08 0.0481 0.0109 0.0 5701

5000 4.3402E-10 0.0407 0.0015 10.0 6179 4.3893E-08 0.0408 0.0017 10.0 6149 5.3928E-08 0.0434 0.0081 6.1 5419

6000 4.3257E-10 0.0373 0.0013 10.0 6111 4.3604E-08 0.0373 0.0014 10.0 6084 5.3513E-08 0.0397 0.0064 4.5 5376

7000 4.3106E-10 0.0346 0.0011 10.0 5997 4.3644E-08 0.0347 0.0012 10.0 5957 5.3608E-08 0.0368 0.0051 5.8 5297

8000 4.2402E-10 0.0327 0.0010 10.0 5858 4.2792E-08 0.0328 0.0011 10.0 5818 5.2434E-08 0.0346 0.0043 6.8 5207

9000 4.2702E-10 0.0307 0.0009 10.0 5905 4.3025E-08 0.0308 0.0010 10.0 5873 5.2555E-08 0.0324 0.0036 4.9 5306

10000 4.2944E-10 0.0290 0.0008 10.0 5886 4.3157E-08 0.0291 0.0008 10.0 5864 5.2550E-08 0.0305 0.0030 3.6 5344

11000 4.3427E-10 0.0275 0.0007 10.0 5953 4.3540E-08 0.0276 0.0007 10.0 5942 5.2867E-08 0.0288 0.0026 4.3 5458

12000 4.3401E-10 0.0264 0.0006 10.0 5953 4.3602E-08 0.0264 0.0007 10.0 5945 5.2961E-08 0.0275 0.0023 5.3 5481

13000 4.3445E-10 0.0253 0.0006 10.0 5953 4.3746E-08 0.0253 0.0006 10.0 5938 5.3617E-08 0.0276 0.0109 3.2 5015

14000 4.3241E-10 0.0244 0.0005 10.0 5899 4.3519E-08 0.0244 0.0006 10.0 5891 5.3284E-08 0.0265 0.0098 3.5 5012

tally 25 tally 35 tally 45

nps mean error vov slope fom mean error vov slope fom mean error vov slope fom

1000 4.3003E-10 0.1894 0.3080 2.2 1563 4.4792E-10 0.1596 0.1612 2.6 2201 1.5987E-08 0.1572 0.4035 1.3 2269

2000 4.9422E-10 0.1342 0.1158 2.0 1508 3.9840E-10 0.1174 0.1139 2.3 1971 2.2806E-08 0.3296 0.8828 1.3 250

3000 4.6236E-10 0.1055 0.0852 2.0 1604 4.3304E-10 0.0968 0.0813 2.3 1905 5.1530E-08 0.6212 0.9507 1.2 46

4000 4.6662E-10 0.0921 0.0610 2.1 1557 4.5852E-10 0.0905 0.0804 2.2 1614 4.2554E-08 0.5646 0.9482 1.2 41

5000 4.4901E-10 0.0802 0.0520 2.0 1589 4.5054E-10 0.0778 0.0664 2.4 1688 3.8499E-08 0.5004 0.9401 1.2 41

6000 4.3176E-10 0.0711 0.0479 2.4 1681 4.3363E-10 0.0689 0.0611 2.7 1790 3.4036E-08 0.4717 0.9401 1.3 38

7000 4.5114E-10 0.0658 0.0388 2.7 1653 4.3277E-10 0.0621 0.0514 2.9 1856 4.6830E-08 0.4339 0.4898 1.3 38

8000 4.3486E-10 0.0610 0.0362 2.9 1681 4.3543E-10 0.0587 0.0403 3.4 1813 4.2681E-08 0.4166 0.4898 1.4 36

9000 4.2029E-10 0.0572 0.0339 3.1 1699 4.3790E-10 0.0553 0.0347 3.7 1816 3.9306E-08 0.4021 0.4898 1.4 34

10000 4.1498E-10 0.0541 0.0308 3.1 1699 4.2707E-10 0.0520 0.0326 4.1 1838 3.6630E-08 0.3884 0.4898 1.4 33

11000 4.1164E-10 0.0515 0.0284 3.1 1703 4.4293E-10 0.0494 0.0268 4.4 1851 3.4458E-08 0.3753 0.4897 1.5 32

12000 4.1067E-10 0.0487 0.0257 3.9 1744 4.4056E-10 0.0468 0.0244 4.6 1884 3.3080E-08 0.3587 0.4883 1.4 32

CHAPTER 5

CONC PROBLEM AND OUTPUT

5-56 18 December 2000

13000 4.0983E-10 0.0459 0.0239 4.5 1804 4.3922E-10 0.0458 0.0241 5.1 1815 3.2244E-08 0.3400 0.4862 1.3 33

14000 4.1441E-10 0.0444 0.0210 4.6 1788 4.3043E-10 0.0438 0.0232 5.2 1833 3.1520E-08 0.3237 0.4823 1.4 34

tally 55 tally 65 tally 75

nps mean error vov slope fom mean error vov slope fom mean error vov slope fom

1000 1.2948E-08 0.0640 0.4349 1.3 13703 1.3031E-08 0.0062 0.9837 1.8 1458748 1.2970E-08 0.0018 0.9031 2.0 1.7E+07

2000 1.6116E-08 0.1796 0.5871 1.3 842 1.3893E-08 0.0567 0.9588 1.7 8444 1.3262E-08 0.0162 0.5148 1.9 103449

3000 2.0729E-08 0.1568 0.1859 1.3 727 1.3747E-08 0.0391 0.8777 1.7 11696 1.3798E-08 0.0351 0.7769 1.6 14454

4000 1.9257E-08 0.1297 0.1702 1.4 786 1.3659E-08 0.0306 0.7616 1.7 14103 1.5296E-08 0.1119 0.9120 1.6 1055

5000 1.8678E-08 0.1139 0.1421 1.4 787 1.3993E-08 0.0351 0.4095 1.6 8298 1.4827E-08 0.0924 0.9121 1.5 1198

6000 1.9004E-08 0.1117 0.1478 1.4 680 1.3822E-08 0.0296 0.4095 1.6 9678 1.4793E-08 0.0789 0.8353 1.5 1362

7000 1.8810E-08 0.1009 0.1282 1.5 703 1.6194E-08 0.1116 0.5009 1.5 575 1.4715E-08 0.0684 0.8145 1.5 1529

8000 2.1095E-08 0.1475 0.5038 1.3 287 1.5802E-08 0.1001 0.5010 1.5 624 1.5333E-08 0.0791 0.4496 1.4 999

9000 2.2176E-08 0.1323 0.4022 1.3 318 1.5509E-08 0.0907 0.5007 1.4 676 1.5320E-08 0.0713 0.4263 1.4 1093

10000 2.1091E-08 0.1252 0.4022 1.2 317 1.5300E-08 0.0828 0.4994 1.4 725 1.5084E-08 0.0652 0.4263 1.4 1169

11000 2.2194E-08 0.1247 0.2687 1.2 290 1.5096E-08 0.0763 0.4994 1.4 776 1.5189E-08 0.0616 0.3624 1.4 1189

12000 2.2296E-08 0.1184 0.2341 1.3 295 1.5018E-08 0.0706 0.4906 1.4 830 1.5033E-08 0.0571 0.3618 1.4 1268

13000 2.4274E-08 0.1530 0.3679 1.3 163 1.5158E-08 0.0661 0.4480 1.4 872 1.6616E-08 0.1152 0.6972 1.4 287

14000 2.4154E-08 0.1456 0.3420 1.4 166 1.5020E-08 0.0620 0.4479 1.4 917 1.6359E-08 0.1087 0.6972 1.4 298

1tally fluctuation charts

tally 85 tally 95 tally 105

nps mean error vov slope fom mean error vov slope fom mean error vov slope fom

1000 4.6893E-10 0.1150 0.0940 3.1 4237 3.7894E-08 0.3062 0.6184 1.7 598 3.1655E-08 0.2324 0.5702 1.3 1038

2000 4.5072E-10 0.0754 0.0503 3.3 4773 3.1910E-08 0.1915 0.5058 1.8 741 7.0026E-08 0.5600 0.9385 1.3 87

3000 4.5901E-10 0.0597 0.0296 4.2 5012 2.8192E-08 0.1470 0.4726 1.9 826 5.7102E-08 0.4603 0.9189 1.3 84

4000 4.7243E-10 0.0551 0.0277 2.9 4351 2.8978E-08 0.1196 0.3186 1.9 924 4.8966E-08 0.4030 0.9156 1.4 81

5000 4.6252E-10 0.0503 0.0241 2.9 4034 3.4948E-08 0.1418 0.3532 1.9 508 4.9120E-08 0.3254 0.8720 1.4 97

6000 4.5707E-10 0.0467 0.0221 3.7 3898 3.6619E-08 0.1239 0.2512 2.0 553 4.5316E-08 0.2943 0.8673 1.5 98

7000 4.4496E-10 0.0430 0.0201 3.8 3872 3.5555E-08 0.1108 0.2390 2.2 584 4.7860E-08 0.2480 0.7483 1.6 116

8000 4.3427E-10 0.0397 0.0182 4.3 3969 3.6111E-08 0.1029 0.1883 2.4 590 4.4972E-08 0.2312 0.7456 1.7 117

9000 4.3578E-10 0.0376 0.0157 4.6 3937 3.5384E-08 0.0944 0.1804 2.6 624 5.1731E-08 0.2129 0.4267 1.9 123

10000 4.3122E-10 0.0353 0.0143 4.9 3977 3.8466E-08 0.1031 0.1700 2.4 468 5.0118E-08 0.1987 0.4189 1.9 126

11000 4.3632E-10 0.0339 0.0130 5.6 3919 3.8983E-08 0.0956 0.1499 2.4 494 4.9698E-08 0.1836 0.4060 2.0 134

12000 4.3565E-10 0.0325 0.0119 7.5 3915 3.8267E-08 0.0900 0.1450 2.5 510 4.9341E-08 0.1737 0.3705 2.0 137

13000 4.3031E-10 0.0310 0.0112 8.9 3955 3.8663E-08 0.0851 0.1280 2.6 526 5.0503E-08 0.1675 0.2987 2.0 136

14000 4.3184E-10 0.0297 0.0102 10.0 3980 3.8499E-08 0.0801 0.1235 2.8 549 4.8699E-08 0.1614 0.2976 2.1 135

***********************************************************************************************************************

dump no. 2 on file conc.r nps = 14000 coll = 26446 ctm = 0.28 nrn = 421786

9 warning messages so far.

run terminated when 14000 particle histories were done.

computer time = 0.33 minutes

mcnp version 4c 01/20/00 07/18/00 12:57:14 probid = 07/18/00 12:56:34

18 December 2000 5-57

CHAPTER 5

CONC PROBLEM AND OUTPUT

Notes:

N1: MCNP was run with the name execute line option that renames the output ﬁle conc.o and

the continue−run ﬁle conc.r.

N2: The point detector for tally 65 is placed on surface 2 (at 0,-390,0) with a sphere of

exclusion of .5 mean free paths. This tally is a good example of what NOT to do when

using point detectors. First of all, the point detector (or ring detector) should not be

placed directly on a surface, especially if the cell on one side has a zero importance. As a

rule of thumb, the point detector should lie just inside or outside a surface. Another

signiﬁcant item about this tally is that the radius of the sphere of exclusion is expressed as

0.5 mean free paths. It is generally not recommended to use a radius expressed in mean

free paths because this increases the variance of the tally. However, the radius can be

entered in mean free paths if the user does not know what other value to use. The

ﬁctitious sphere radius of 0.5 mean free paths (approximately 4.3 cm) assumes a uniform

isotropic ﬂux within the sphere. Although this assumption will smooth out the detector

response, it is false. The ﬁctitious sphere should never be in more than one material

medium as it is here because the material is assumed to be uniform throughout the sphere.

This point detector is included in the example to demonstrate how MCNP can sometimes

be fooled into giving supposedly accurate results.

N3: The ring detector for tally 95 is about the y-axis centered at the origin. The radius of the

ring is 420 cm and it is coincident with surface 3. The radius of the sphere of exclusion

for this detector is set to 0. Because the detector lies in a void region, it will not produce

erroneous results if concident with a geometric surface.

N4: The DD card controls the Russian roulette games that are played for all detector

problems unless explicitly turned off. The ﬁrst entry of this card, 0.1, designates the level

at which Russian roulette will be played. For the ﬁrst 200 histories, all contributions to

the detector are counted. The average then is computed and is updated whenever the tally

ﬂuctuation chart entry is computed. Russian roulette is played on all contributions below

0.1 times the computed average. This Russian roulette game is one of the few default

MCNP variance reduction schemes and typically speeds up detector problems by an order

of magnitude. The second entry on the DD card causes a diagnostic message to be

printed if a tally greater than 0.1 * 1e100 is reached (which in this case is never). If this

second entry is too high, the diagnostic messages will never be printed, conversely, if this

number is too low, the output will be cluttered with these messages.

N5: The cutoff card for this problem uses the default time cutoff value and an energy cutoff

of 12.0 MeV. If a neutron time is greater than the time entry or if the neutron energy is

below 12 MeV, the particle is terminated. These cutoff parameters can reduce

computational time, but they should be used with caution. In some applications, ignoring

5-58 18 December 2000

CHAPTER 5

CONC PROBLEM AND OUTPUT

neutrons and photons beneath a certain energy cutoff will not signiﬁcantly affect the tally.

But, if these lower energy interactions are important (ﬁssion and photon interactions) then

the ﬁnal result may be truncated.

N6: The problem summary table provides an accounting of particle track, weight, and energy

creation and loss. For this problem, the largest neutron loss was caused by energy cutoff.

There is a total of 26,446 collisions for 14,000 source histories. The net multiplication of

1.0129 is caused by (n,xn) reactions; the system is clearly not supercritical because there

is no ﬁssionable material. The weight per escaping source particle is 0.086876, meaning

that the ﬂux on the shell of radius 4000 cm is approximately 0.086876/(4∗π∗40002) =

4.321E−10 neutrons/cm2. The energy cutoff terminated 12673 tracks out of 14000

starting particles, making for a very fast problem run time.

N7: The energy bins for tally 2 are cummulative so that any particle with energy less than or

equal to the energy of a bin scores in that bin.

N8: The letters *a*, etc, throughout the diagnostics table correspond to the notes, (a),(b), etc.

There were 18226 detector contributions (e). 14000 were from the source (there were

14000 hits from cell 1)(c) and 4226 from collisions inside cell 2 (d). According to the

problem summary there were 26,446 collisions. Thus the DD card roulette game

eliminated 84% of the collision contributions. Of the 4226 collisions that did contribute

to the tally, 4071 (a) made a tally less than the 1E−1 cutoff (it was conservatively

estimated that their contributions would be higher so that they would not be rouletted).

These 4071 transmissions to the detector contributed only 0.127% of the cumulative

fraction of total tally(a). The majority of the total tally was contributed by transmissions

with an average score of 1.0 or less (b); these scores accounted for 85.04% of the total

tally. The remaining fraction of the tally was contributed by the transmissions with scores

greater than 1.0. The largest tally is 663.63 times larger than the average tally(f).

N9: This section describes how the TFC bins would be affected if the largest previously

sampled score was encountered on the next history. The “value at nps” column shows the

TFC bin values of the current history, while the “value at nps+1” column shows the

results after the largest previous history has been added to the tally. The last column

shows the relative change of the TFC bin values from the NPS value to the NPS+1 value.

The effect of having a very large score on the next history appears to have an overall

detrimental effect on these TFC values. The relative error increased by 20% while the

ﬁgure of merit decreased by 31%. One positive effect is that the VOV decreased by

29.8% (to 0.314281), however, it was still not beneath the required value of 0.1.

N10: This problem passed only ﬁve of the ten TFC bin statistical checks, clearly a bad sign.

The relative error (RE) was more than 5%. The VOV was not below the required 0.1

maximum and is not decreasing as 1/NPS. The probability density function (PDF) slope

18 December 2000 5-59

CHAPTER 5

CONC PROBLEM AND OUTPUT

was not greater than 3. Both indicate that the problem was not sampled adequately.

Undersampling of infrequent high scoring tallies gives a result with an underpredicted RE

and variance. The VOV is more sensitive to large tally score ﬂuctuations than the RE, and

is one good indicator of conﬁdence interval reliability. The PDF slope check conﬁrms

whether the PDF function's high score tail is decreasing with at least a 1/x3 dependence.

If the high score tail follows this criteria, then the Central Limit Theorem is satisﬁed and

the distribution should converge to a normal distribution if enough histories are run. It can

be seen that a low relative error and variance do not always guarantee a reliable result.

These ten statistical checks do not ensure a totally reliable result; they just provide a more

rigorous check of the tally reliability.

N11: This plot is the unnormed probability density for tally 65. It is a log−log plot of the PDF

that is shown by asterisks, along with the central mean (denoted by the line of m's). The

curve of S's denotes the Pareto curve ﬁt to the PDF distribution. This curve is included so

that the user can see if the ﬁt is fairly accurate when compared to the calculated

distribution. To the left of the plot are the columns that show the abscissa, number,

number density and the ordinate of the PDF.

N12: These are the TFC bin results all of the tallies. For the tally 65 point detector, the RE is

just above 5%, the FOM is decent, and the answer is wrong. To ensure a reliable

conﬁdence interval, the acceptable value of the VOV is 0.1. As mentioned previously, the

VOV checks the higher moments (3rd and 4th) of the PDF because they are more

sensitive to any aberrations in the PDF caused by insufﬁcient sampling. For this tally, the

VOV of 0.4479 clearly does not fall below the acceptable limit of 0.1. To achieve a

reliable conﬁdence interval, the slope of the PDF must be greater than or equal to three in

order to produce a distribution that has a 1/x3 behavior. The tally also fails this criterion,

indicating that the Central Limit Theorem is not satisﬁed. Tally 65 appears to have

converged to a ﬂux of 1.5020E-08. However, surface tally 22 at 390 cm is 5.3284E-08

and the still−unconverged ring detector tally 105 at 390 cm is 4.8699E-08. Tally 65

appears from its relative error to be close to convergence but it is actually low by a factor

of 4! Tallies 25 and 35 at 4000 cm agree with to the ﬂux extracted from the problem

summary (see note N6), namely 4.321E-10.

COMMENTS:

How should the CONC problem be better speciﬁed? First, detectors are inappropriate for

this problem and should not be used. The shell should be divided into four spherically

concentric geometrical regions with outwardly increasing importances of 1, 2, 4 and 8.

Then for every source particle, approximately one particle would cross the outer surface

of the shell and score, instead of the present 14381 out of 100000.

5-60 18 December 2000

CHAPTER 5

CONC PROBLEM AND OUTPUT

How could detectors be made to work better in this problem? In any problem with

symmetry, a ring detector rather than a point detector should be used to at least take

advantage of the symmetry. The ﬁctitious sphere radius could be made smaller so that the

1/r2singularity made about as much difference as the ﬂuctuation in PSC value. Perhaps

this ﬁctitious sphere radius would be 1 cm. Most importantly, the source direction could

be biased to direct particles at the ring, causing a lot more collisions in the vicinity of the

detector.

18 December 2000 5-61

CHAPTER 5

KCODE

IV. KCODE

The problem selected to illustrate the output from a criticality calculation is the one−dimensional

model of the GODIVA critical assembly, composed of about 94% 235U. This assembly is one of

several fast neutron critical assemblies discussed in LA-4208 entitled “Reevaluated Critical

Speciﬁcations of Some Los Alamos Fast−Neutron Systems” by G. E. Hansen and H. C. Paxton

(September 1964).

An MCNP input ﬁle that models GODIVA and performs only the criticality calculation with no

separate tallies would be only 11 lines long. The KCODE card indicates that the problem is a

criticality calculation for the keff eigenvalue. To perform this same calculation with neutron-

induced photon production, add the MODE N P card. Any tallies that are made in a criticality

problem are normalized to the starting weight (default) or number of particles as deﬁned by the

user (see Chapter 2, section VIII for details). Tallies should be scaled for the appropriate steady

state neutron generation rate.

Following is a partial listing of the output from a KCODE calculation. The pages selected

emphasize the criticality aspects of the problem.

CHAPTER 5

KCODE

5-62 18 December 2000

N11mcnp version 4c ld=01/20/00 07/31/00 12:11:37

************************************************************************* probid = 07/31/00 12:11:37

inp=kcode name=kcode.

1- bare u(94) sphere ref. la-4208, g. e. hansen and h. c. paxton, 1969, page 4

2- 1 10 -18.74 -1

3- 2 0 1

5- 1 so 8.7037

7- imp:n 1 0

N2 8- m10 92235.61c 0.045217 92238.61c 0.0024355 92234.61c 0.0004935

N3 9- kcode 3000 1. 5 35

N410- ksrc 0 0 0

11- print

12- C

13- C Pertubations

N514- pert1:n cell=1 rho=-20.0 method=-1 $ perterb density and give changes

15- c

16- c tallies

17- c

18- f1:n 1

19- f14:n 1

20- fc14 total total fission neutrons (track-lenght Keff), total loss to (n,xn)

21- total neutron absorptions,total fission,and neutron heating (mev/gram)

22- fq14 e m

N623- fm14 (132.534 10 (-6 -7) (16:17) (-2) (-6)) (0.002560689 10 1 -4)

24- f6:n 1

25- f7:n 1

26- c

27- c use the sixteen group hansen-roach energy structure as the default

28- c

N729- e0 1-7 4-7 1-6 3-6 1-5 3-5 1-4 5.5-4 3-3 1.7-2 0.1 0.4 0.9 1.4 3 20

N830- f34:n 1

31- sd34 1

32- fq34 m f

33- fm34 (-1 10 -6 -7) (-1 10 16:17) (-1 10 -2) (-1 10 -6) (-0.000019321 10 1 -4)

34- e34 20 nt

35-

1 initial source from ksrc card. print table 90

original number of points 1

points not in any cell 0

points in cells of zero importance 0

points in void cells 0

points in ambiguous cells 0

18 December 2000 5-63

CHAPTER 5

KCODE

total points rejected 0

points remaining 1

points after expansion or contraction 3000

nominal source size 3000

initial guess for k(eff.) 1.000000

cycles to skip before tallying 5

number of keff cycles that can be stored 6500

total fission nubar data are being used.

SKIP 69 LINES in OUTPUT

1tally 14 print table 30

+ total total fission neutrons (track-lenght Keff), total loss to (n,xn)

total neutron absorptions,total fission,and neutron heating (mev/gram)

tally type 4 track length estimate of particle flux.

tally for neutrons

N10 warning. perturbation may require negative fm constant. tally 14

warning. perturbation may require negative fm constant. tally 14

SKIP 47 LINES IN OUTPUT

1material composition print table 40

the sum of the fractions of material 10 was 4.814600E-02

material

number component nuclide, atom fraction

10 92235, 9.39164E-01 92238, 5.05857E-02 92234, 1.02501E-02

material

number component nuclide, mass fraction

10 92235, 9.38598E-01 92238, 5.12020E-02 92234, 1.02002E-02

N11 warning. 1 materials had unnormalized fractions. print table 40.

N12 warning. perturbation correction not applied to tally 6

warning. perturbation correction not applied to tally 7

CHAPTER 5

KCODE

5-64 18 December 2000

1cell volumes and masses print table 50

N13

cell atom gram input calculated reason volume

density density volume volume mass pieces not calculated

1 1 4.79847E-02 1.87400E+01 0.00000E+00 2.76185E+03 5.17571E+04 1

2 2 0.00000E+00 0.00000E+00 0.00000E+00 0.00000E+00 0.00000E+00 0 infinite

SKIP 70 LINES IN OUTPUT

1cross-section tables print table 100

N14

table length

tables from file endf6dn2

92234.61c 82031 92-u-234 from endf-vi total nu mat9225 11/27/93

92235.61c 234221 92-u-235 from lanl proposed endf-vi.2 total nu mat9228 11/27/93

92238.61c 184612 92-u-238 from endf-vi.2 total nu mat9237 11/27/93

total 500864

decimal words of dynamically allocated storage

N15

general 532058

tallies 35996

bank 58244

cross sections 1001728

total 1621886 = 6487544 bytes

***********************************************************************************************************************

dump no. 1 on file kcode.r nps = 0 coll = 0 ctm = 0.00 nrn = 0

N16

source distribution written to file kcode.s cycle = 0

8 warning messages so far.

1 starting mcrun. dynamic storage = 1621890 words, 6487560 bytes. cp0 = 0.03 print table 110

bare u(94) sphere ref. la-4208, g. e. hansen and h. c. paxton, 1969, page 4

N17

nps x y z cell surf u v w energy weight time

1 0.000E+00 0.000E+00 0.000E+00 1 0 5.085E-01 4.733E-01 7.193E-01 2.209E+00 1.000E+00 0.000E+00

2 0.000E+00 0.000E+00 0.000E+00 1 0 8.952E-01 -4.447E-01 -2.944E-02 4.904E+00 1.000E+00 0.000E+00

3 0.000E+00 0.000E+00 0.000E+00 1 0 -6.184E-01 -4.495E-01 6.446E-01 3.809E-01 1.000E+00 0.000E+00

4 0.000E+00 0.000E+00 0.000E+00 1 0 9.710E-01 -5.665E-02 -2.323E-01 1.331E+00 1.000E+00 0.000E+00

5 0.000E+00 0.000E+00 0.000E+00 1 0 5.861E-01 1.496E-01 -7.963E-01 1.902E+00 1.000E+00 0.000E+00

18 December 2000 5-65

CHAPTER 5

KCODE

6 0.000E+00 0.000E+00 0.000E+00 1 0 -6.489E-02 -1.626E-01 9.845E-01 4.410E-01 1.000E+00 0.000E+00

7 0.000E+00 0.000E+00 0.000E+00 1 0 -7.068E-02 3.263E-02 -9.970E-01 4.750E-01 1.000E+00 0.000E+00

8 0.000E+00 0.000E+00 0.000E+00 1 0 -3.915E-01 4.664E-01 -7.932E-01 4.136E+00 1.000E+00 0.000E+00

9 0.000E+00 0.000E+00 0.000E+00 1 0 -2.368E-01 9.215E-01 -3.079E-01 7.453E-02 1.000E+00 0.000E+00

10 0.000E+00 0.000E+00 0.000E+00 1 0 1.946E-01 -3.204E-01 9.271E-01 3.128E+00 1.000E+00 0.000E+00

11 0.000E+00 0.000E+00 0.000E+00 1 0 -6.698E-01 -7.177E-01 -1.905E-01 1.014E+00 1.000E+00 0.000E+00

12 0.000E+00 0.000E+00 0.000E+00 1 0 -8.398E-01 -4.129E-01 3.524E-01 1.395E+00 1.000E+00 0.000E+00

13 0.000E+00 0.000E+00 0.000E+00 1 0 -1.714E-01 -8.572E-01 4.857E-01 7.748E-01 1.000E+00 0.000E+00

14 0.000E+00 0.000E+00 0.000E+00 1 0 -2.489E-01 -5.118E-01 -8.222E-01 1.101E+00 1.000E+00 0.000E+00

15 0.000E+00 0.000E+00 0.000E+00 1 0 -2.959E-01 2.119E-01 9.314E-01 1.951E+00 1.000E+00 0.000E+00

16 0.000E+00 0.000E+00 0.000E+00 1 0 1.395E-01 -9.829E-01 1.202E-01 2.186E+00 1.000E+00 0.000E+00

17 0.000E+00 0.000E+00 0.000E+00 1 0 6.909E-01 -7.110E-01 1.307E-01 1.865E+00 1.000E+00 0.000E+00

18 0.000E+00 0.000E+00 0.000E+00 1 0 -6.580E-01 5.320E-01 -5.329E-01 1.229E+00 1.000E+00 0.000E+00

19 0.000E+00 0.000E+00 0.000E+00 1 0 -9.903E-01 -1.380E-01 1.353E-02 1.305E+00 1.000E+00 0.000E+00

20 0.000E+00 0.000E+00 0.000E+00 1 0 7.462E-01 4.859E-01 -4.551E-01 1.000E+00 1.000E+00 0.000E+00

21 0.000E+00 0.000E+00 0.000E+00 1 0 -1.977E-01 9.797E-01 3.360E-02 3.990E+00 1.000E+00 0.000E+00

22 0.000E+00 0.000E+00 0.000E+00 1 0 -9.117E-01 -3.647E-01 -1.891E-01 2.665E-01 1.000E+00 0.000E+00

23 0.000E+00 0.000E+00 0.000E+00 1 0 -4.287E-01 8.361E-01 -3.423E-01 1.156E+00 1.000E+00 0.000E+00

24 0.000E+00 0.000E+00 0.000E+00 1 0 1.080E-01 3.412E-01 -9.338E-01 2.669E+00 1.000E+00 0.000E+00

25 0.000E+00 0.000E+00 0.000E+00 1 0 -9.111E-01 -9.012E-03 -4.122E-01 2.185E+00 1.000E+00 0.000E+00

26 0.000E+00 0.000E+00 0.000E+00 1 0 -2.568E-01 -6.391E-01 -7.249E-01 4.225E+00 1.000E+00 0.000E+00

27 0.000E+00 0.000E+00 0.000E+00 1 0 -2.912E-01 8.086E-01 5.113E-01 1.079E+00 1.000E+00 0.000E+00

28 0.000E+00 0.000E+00 0.000E+00 1 0 1.472E-01 -9.514E-01 2.705E-01 3.461E+00 1.000E+00 0.000E+00

29 0.000E+00 0.000E+00 0.000E+00 1 0 -6.135E-01 -7.645E-01 -1.978E-01 1.836E+00 1.000E+00 0.000E+00

30 0.000E+00 0.000E+00 0.000E+00 1 0 -5.702E-01 5.651E-01 -5.963E-01 4.556E-01 1.000E+00 0.000E+00

31 0.000E+00 0.000E+00 0.000E+00 1 0 -6.607E-01 5.373E-01 -5.242E-01 6.415E-01 1.000E+00 0.000E+00

32 0.000E+00 0.000E+00 0.000E+00 1 0 -9.742E-02 -3.639E-01 -9.263E-01 2.764E+00 1.000E+00 0.000E+00

33 0.000E+00 0.000E+00 0.000E+00 1 0 -1.965E-01 -3.145E-01 -9.287E-01 2.785E-01 1.000E+00 0.000E+00

34 0.000E+00 0.000E+00 0.000E+00 1 0 4.097E-01 8.465E-01 -3.399E-01 9.097E-01 1.000E+00 0.000E+00

35 0.000E+00 0.000E+00 0.000E+00 1 0 -4.048E-02 8.831E-01 4.675E-01 3.360E-01 1.000E+00 0.000E+00

36 0.000E+00 0.000E+00 0.000E+00 1 0 3.371E-01 -9.269E-01 -1.652E-01 6.376E-01 1.000E+00 0.000E+00

37 0.000E+00 0.000E+00 0.000E+00 1 0 -1.867E-01 9.756E-01 -1.155E-01 2.186E+00 1.000E+00 0.000E+00

38 0.000E+00 0.000E+00 0.000E+00 1 0 -2.616E-01 2.336E-01 -9.365E-01 7.314E-01 1.000E+00 0.000E+00

39 0.000E+00 0.000E+00 0.000E+00 1 0 9.780E-01 -7.641E-02 -1.939E-01 2.997E-01 1.000E+00 0.000E+00

40 0.000E+00 0.000E+00 0.000E+00 1 0 2.580E-01 -7.076E-01 6.578E-01 1.444E+00 1.000E+00 0.000E+00

41 0.000E+00 0.000E+00 0.000E+00 1 0 -3.212E-01 -7.678E-01 -5.543E-01 1.914E+00 1.000E+00 0.000E+00

42 0.000E+00 0.000E+00 0.000E+00 1 0 5.039E-01 -1.460E-01 8.513E-01 1.502E+00 1.000E+00 0.000E+00

43 0.000E+00 0.000E+00 0.000E+00 1 0 6.080E-01 5.487E-01 5.738E-01 5.971E+00 1.000E+00 0.000E+00

44 0.000E+00 0.000E+00 0.000E+00 1 0 -2.932E-01 9.304E-01 -2.199E-01 1.827E+00 1.000E+00 0.000E+00

45 0.000E+00 0.000E+00 0.000E+00 1 0 -8.475E-01 -3.993E-01 -3.497E-01 1.928E+00 1.000E+00 0.000E+00

46 0.000E+00 0.000E+00 0.000E+00 1 0 1.200E-01 -9.195E-01 -3.743E-01 1.351E+00 1.000E+00 0.000E+00

47 0.000E+00 0.000E+00 0.000E+00 1 0 7.085E-01 5.879E-01 3.904E-01 2.288E+00 1.000E+00 0.000E+00

48 0.000E+00 0.000E+00 0.000E+00 1 0 4.261E-01 9.046E-01 9.254E-03 1.230E+00 1.000E+00 0.000E+00

49 0.000E+00 0.000E+00 0.000E+00 1 0 5.431E-01 4.270E-01 -7.230E-01 1.433E+00 1.000E+00 0.000E+00

50 0.000E+00 0.000E+00 0.000E+00 1 0 -1.053E-01 -9.805E-01 1.658E-01 6.572E-01 1.000E+00 0.000E+00

1estimated keff results by cycle print table 175

N18

cycle 1 k(collision) 1.358125 prompt removal lifetime(abs) 9.1005E-01 source points generated 4119

CHAPTER 5

KCODE

5-66 18 December 2000

cycle 2 k(collision) 1.154061 prompt removal lifetime(abs) 7.1727E-01 source points generated 2562

cycle 3 k(collision) 1.064506 prompt removal lifetime(abs) 6.6305E-01 source points generated 2784

cycle 4 k(collision) 1.021321 prompt removal lifetime(abs) 6.4986E-01 source points generated 2867

cycle 5 k(collision) 1.023708 prompt removal lifetime(abs) 6.5835E-01 source points generated 2971

cycle 6 k(collision) 0.986612 prompt removal lifetime(abs) 6.2340E-01 source points generated 2887

estimator cycle 7 ave of 2 cycles combination simple average combined average corr

k(collision) 1.017630 1.002121 0.0155 k(col/abs) 0.000000 0.0000 0.000000 0.0000 0.0000

k(absorption) 1.016863 1.000145 0.0167 k(abs/tk ln) 0.000000 0.0000 0.000000 0.0000 0.0000

k(trk length) 1.000350 0.996914 0.0034 k(tk ln/col) 0.000000 0.0000 0.000000 0.0000 0.0000

rem life(col) 6.2033E-01 6.2210E-01 0.0029

rem life(abs) 6.2043E-01 6.2191E-01 0.0024 life(col/abs) 0.0000E+00 0.0000 0.0000E+00 0.0000 0.0000

source points generated 3110

estimator cycle 8 ave of 3 cycles combination simple average combined average corr

k(collision) 0.984670 0.996304 0.0107 k(col/abs) 0.995820 0.0108 0.996554 0.0173 0.9935

k(absorption) 0.985717 0.995336 0.0108 k(abs/tk ln) 0.995518 0.0066 0.995799 0.0004 0.9962

k(trk length) 0.993270 0.995699 0.0023 k(tk ln/col) 0.996002 0.0065 0.995531 0.0001 0.9996

rem life(col) 6.1084E-01 6.1835E-01 0.0063

rem life(abs) 6.1184E-01 6.1856E-01 0.0056 life(col/abs) 6.1845E-01 0.0059 6.2020E-01 0.0012 0.9999

source points generated 2891

estimator cycle 9 ave of 4 cycles combination simple average combined average corr

k(collision) 0.978535 0.991862 0.0088 k(col/abs) 0.991791 0.0087 0.991518 0.0102 0.9910

k(absorption) 0.980872 0.991720 0.0085 k(abs/tk ln) 0.990160 0.0072 0.989571 0.0089 0.6206

k(trk length) 0.967305 0.988601 0.0074 k(tk ln/col) 0.990231 0.0075 0.989312 0.0092 0.6885

rem life(col) 6.0996E-01 6.1625E-01 0.0056 k(col/abs/tk ln) 0.990728 0.0076 0.988013 0.0101

rem life(abs) 6.0986E-01 6.1638E-01 0.0053 life(col/abs/tl) 6.1588E-01 0.0072 6.1774E-01 0.0011

source points generated 3042

estimator cycle 10 ave of 5 cycles combination simple average combined average corr

k(collision) 1.011025 0.995694 0.0078 k(col/abs) 0.995962 0.0079 0.995489 0.0093 0.9895

k(absorption) 1.014266 0.996229 0.0080 k(abs/tk ln) 0.994777 0.0072 0.994330 0.0086 0.7562

k(trk length) 1.012225 0.993326 0.0074 k(tk ln/col) 0.994510 0.0072 0.994211 0.0085 0.7753

rem life(col) 6.4614E-01 6.2223E-01 0.0105 k(col/abs/tk ln) 0.995083 0.0073 0.994205 0.0106

rem life(abs) 6.4614E-01 6.2233E-01 0.0104 life(col/abs/tl) 6.2165E-01 0.0108 6.2336E-01 0.0148

source points generated 3073

SKIP 185 LINES IN OUTPUT

estimator cycle 34 ave of 29 cycles combination simple average combined average corr

k(collision) 1.012300 0.992941 0.0032 k(col/abs) 0.992787 0.0032 0.992883 0.0034 0.9945

k(absorption) 1.013670 0.992633 0.0033 k(abs/tk ln) 0.993246 0.0027 0.994279 0.0023 0.8487

k(trk length) 1.006845 0.993858 0.0023 k(tk ln/col) 0.993399 0.0027 0.994190 0.0022 0.8539

rem life(col) 6.2892E-01 6.1948E-01 0.0045 k(col/abs/tk ln) 0.993144 0.0028 0.994148 0.0023

rem life(abs) 6.2883E-01 6.1972E-01 0.0044 life(col/abs/tl) 6.1933E-01 0.0041 6.1870E-01 0.0040

source points generated 3073

18 December 2000 5-67

CHAPTER 5

KCODE

N19

estimator cycle 35 ave of 30 cycles combination simple average combined average corr

k(collision) 1.014312 0.993653 0.0032 k(col/abs) 0.993520 0.0032 0.993675 0.0033 0.9947

k(absorption) 1.015224 0.993386 0.0032 k(abs/tk ln) 0.993694 0.0026 0.994208 0.0022 0.8387

k(trk length) 0.998200 0.994003 0.0022 k(tk ln/col) 0.993828 0.0026 0.994125 0.0022 0.8453

rem life(col) 6.1576E-01 6.1935E-01 0.0043 k(col/abs/tk ln) 0.993681 0.0028 0.994084 0.0022

rem life(abs) 6.1562E-01 6.1959E-01 0.0043 life(col/abs/tl) 6.1924E-01 0.0040 6.1875E-01 0.0038

source points generated 2856

source distribution written to file kcode.s cycle = 35

1problem summary (active cycles only) source particle weight for summary table normalization = 90000.00

N20

run terminated when 35 kcode cycles were done.

+ 07/31/00 12:16:01

bare u(94) sphere ref. la-4208, g. e. hansen and h. c. paxton, 1969, page 4 probid = 07/31/00 12:11:37

neutron creation tracks weight energy neutron loss tracks weight energy

(per source particle) (per source particle)

source 89903 1.0000E+00 2.0594E+00 escape 77458 5.7633E-01 9.2165E-01

energy cutoff 0 0. 0.

time cutoff 0 0. 0.

weight window 0 0. 0. weight window 0 0. 0.

cell importance 0 0. 0. cell importance 0 0. 0.

weight cutoff 0 3.3230E-02 1.2081E-02 weight cutoff 12845 3.2561E-02 1.1439E-02

energy importance 0 0. 0. energy importance 0 0. 0.

dxtran 0 0. 0. dxtran 0 0. 0.

forced collisions 0 0. 0. forced collisions 0 0. 0.

exp. transform 0 0. 0. exp. transform 0 0. 0.

upscattering 0 0. 0. downscattering 0 0. 5.2490E-01

delayed fission 0 0. 0. capture 0 4.4700E-02 2.7392E-02

(n,xn) 798 5.5379E-03 3.7998E-03 loss to (n,xn) 398 2.7615E-03 2.1751E-02

prompt fission 0 0. 0. loss to fission 0 3.8242E-01 5.6810E-01

total 90701 1.0388E+00 2.0752E+00 total 90701 1.0388E+00 2.0752E+00

number of neutrons banked 471 average time of (shakes) cutoffs

neutron tracks per source particle 1.0078E+00 escape 6.0468E-01 tco 1.0000E+34

neutron collisions per source particle 4.0471E+00 capture 1.0174E+00 eco 0.0000E+00

total neutron collisions 364239 capture or escape 6.3438E-01 wc1 -5.0000E-01

net multiplication 1.0028E+00 0.0002 any termination 6.7341E-01 wc2 -2.5000E-01

computer time so far in this run 7.66 minutes maximum number ever in bank 2

computer time in mcrun 3.81 minutes bank overflows to backup file 0

source particles per minute 2.7621E+04 dynamic storage 1621890 words, 6487560 bytes.

random numbers generated 5379507 most random numbers used was 445 in history 98676

range of sampled source weights = 7.2833E-01 to 1.1710E+00

1neutron activity in each cell print table 126

CHAPTER 5

KCODE

5-68 18 December 2000

tracks population collisions collisions number flux average average

cell entering * weight weighted weighted track weight track mfp

(per history) energy energy (relative) (cm)

1 1 89903 90303 364239 2.6395E+00 8.5689E-01 1.4906E+00 6.7470E-01 2.6346E+00

total 89903 90303 364239 2.6395E+00

1neutron weight balance in each cell -- external events print table 130

cell entering source energy time exiting other total

cutoff cutoff

1 1 0.0000E+00 1.0000E+00 0.0000E+00 0.0000E+00 -5.7633E-01 0.0000E+00 4.2367E-01

total 0.0000E+00 1.0000E+00 0.0000E+00 0.0000E+00 -5.7633E-01 0.0000E+00 4.2367E-01

1neutron weight balance in each cell -- variance reduction events print table 130

cell weight cell weight energy dxtran forced exponential total

window importance cutoff importance collision transform

1 1 0.0000E+00 0.0000E+00 6.6914E-04 0.0000E+00 0.0000E+00 0.0000E+00 0.0000E+00 6.6914E-04

total 0.0000E+00 0.0000E+00 6.6914E-04 0.0000E+00 0.0000E+00 0.0000E+00 0.0000E+00 6.6914E-04

1neutron weight balance in each cell -- physical events print table 130

cell (n,xn) fission capture loss to loss to total

(n,xn) fission

1 1 5.5379E-03 0.0000E+00 -4.4700E-02 -2.7615E-03 -3.8242E-01 -4.2434E-01

total 5.5379E-03 0.0000E+00 -4.4700E-02 -2.7615E-03 -3.8242E-01 -4.2434E-01

1neutron activity of each nuclide in each cell, per source particle print table 140

cell nuclides atom total collisions weight lost weight loss weight gain

fraction collisions * weight to capture to fission by (n,xn)

1 1 92235.61c 9.3916E-01 341484 2.4732E+00 4.2463E-02 3.7584E-01 2.6519E-03

92238.61c 5.0586E-02 18718 1.3668E-01 1.5079E-03 3.3677E-03 1.1715E-04

92234.61c 1.0250E-02 4037 2.9556E-02 7.2938E-04 3.2123E-03 7.3822E-06

total 364239 2.6395E+00 4.4700E-02 3.8242E-01 2.7764E-03

total over all cells for each nuclide total collisions weight lost weight loss weight gain

collisions * weight to capture to fission by (n,xn)

92234.61c 4037 2.9556E-02 7.2938E-04 3.2123E-03 7.3822E-06

92235.61c 341484 2.4732E+00 4.2463E-02 3.7584E-01 2.6519E-03

18 December 2000 5-69

CHAPTER 5

KCODE

92238.61c 18718 1.3668E-01 1.5079E-03 3.3677E-03 1.1715E-04

1keff results for: bare u(94) sphere ref. la-4208, g. e. hansen and h. c. paxton, 1969, page 4 probid = 07/31/00 12:11:37

the initial fission neutron source distribution used the 1 source points that were input on the ksrc card.

the criticality problem was scheduled to skip 5 cycles and run a total of 35 cycles with nominally 3000 neutrons per cycle.

this problem has run 5 inactive cycles with 15332 neutron histories and 30 active cycles with 89903 neutron histories.

this calculation has completed the requested number of keff cycles using a total of 105235 fission neutron source histories.

all cells with fissionable material were sampled and had fission neutron source points.

the results of the w test for normality applied to the individual collision, absorption, and track-length keff cycle values are:

the k( collision) cycle values appear normally distributed at the 95 percent confidence level

the k(absorption) cycle values appear normally distributed at the 95 percent confidence level

the k(trk length) cycle values appear normally distributed at the 95 percent confidence level

N21

-----------------------------------------------------------------------------------------------------------------------------------

| |

| the final estimated combined collision/absorption/track-length keff = 0.99408 with an estimated standard deviation of 0.00220 |

| |

| the estimated 68, 95, & 99 percent keff confidence intervals are 0.99185 to 0.99632, 0.98956 to 0.99861, and 0.98797 to 1.00019 |

| |

| the final combined (col/abs/tl) prompt removal lifetime = 6.1875E-09 seconds with an estimated standard deviation of 2.3789E-11 |

| |

-----------------------------------------------------------------------------------------------------------------------------------

the estimated average keffs, one standard deviations, and 68, 95, and 99 percent confidence intervals are:

keff estimator keff standard deviation 68% confidence 95% confidence 99% confidence corr

collision 0.99365 0.00319 0.99042 to 0.99689 0.98712 to 1.00019 0.98485 to 1.00246

absorption 0.99339 0.00321 0.99013 to 0.99664 0.98681 to 0.99996 0.98452 to 1.00225

track length 0.99400 0.00219 0.99178 to 0.99622 0.98951 to 0.99849 0.98795 to 1.00005

col/absorp 0.99367 0.00329 0.99034 to 0.99700 0.98694 to 1.00041 0.98459 to 1.00276 0.9947

abs/trk len 0.99421 0.00215 0.99203 to 0.99639 0.98980 to 0.99861 0.98826 to 1.00015 0.8387

col/trk len 0.99413 0.00214 0.99196 to 0.99630 0.98974 to 0.99851 0.98820 to 1.00005 0.8453

col/abs/trk len 0.99408 0.00220 0.99185 to 0.99632 0.98956 to 0.99861 0.98797 to 1.00019

N22

if the largest of each keff occurred on the next cycle, the keff results and 68, 95, and 99 percent confidence intervals would be:

keff estimator keff standard deviation 68% confidence 95% confidence 99% confidence

collision 0.99452 0.00321 0.99128 to 0.99777 0.98797 to 1.00108 0.98570 to 1.00335

CHAPTER 5

KCODE

5-70 18 December 2000

absorption 0.99416 0.00320 0.99092 to 0.99741 0.98762 to 1.00071 0.98535 to 1.00298

track length 0.99476 0.00225 0.99248 to 0.99705 0.99016 to 0.99937 0.98856 to 1.00097

col/abs/trk len 0.99459 0.00226 0.99229 to 0.99688 0.98995 to 0.99921 0.98833 to 1.00081

N23

the estimated average prompt removal lifetimes, one standard deviations, and 68, 95, and 99 percent confidence intervals are (sec):

estimator lifetime std. dev. 68% confidence 95% confidence 99% confidence corr

collision 6.19351E-09 2.66771E-11 6.1665E-09 to 6.2205E-09 6.1390E-09 to 6.2480E-09 6.1201E-09 to 6.2669E-09

absorption 6.19586E-09 2.66582E-11 6.1689E-09 to 6.2228E-09 6.1414E-09 to 6.2503E-09 6.1226E-09 to 6.2692E-09

track length 6.18776E-09 2.16289E-11 6.1659E-09 to 6.2096E-09 6.1436E-09 to 6.2319E-09 6.1283E-09 to 6.2472E-09

col/absorp 6.19578E-09 2.94182E-11 6.1660E-09 to 6.2256E-09 6.1355E-09 to 6.2561E-09 6.1145E-09 to 6.2771E-09 0.9992

abs/trk len 6.18454E-09 2.16832E-11 6.1626E-09 to 6.2065E-09 6.1401E-09 to 6.2290E-09 6.1246E-09 to 6.2445E-09 0.9045

col/trk len 6.18542E-09 2.15725E-11 6.1636E-09 to 6.2073E-09 6.1412E-09 to 6.2296E-09 6.1258E-09 to 6.2450E-09 0.9056

col/abs/trk len 6.18747E-09 2.37889E-11 6.1634E-09 to 6.2116E-09 6.1387E-09 to 6.2362E-09 6.1217E-09 to 6.2532E-09

absorption estimates of prompt lifetimes (sec):

escape capture fission removal

lifespan 6.04675E-09 1.01744E-08 5.95603E-09 6.19605E-09

fraction 5.74348E-01 4.45467E-02 3.81105E-01 1.00000E+00

lifetime(abs) 1.07876E-08 1.39087E-07 1.62576E-08 6.19586E-09

lifetime(c/a/t) 1.07730E-08 1.38899E-07 1.62356E-08 6.18747E-09

N24

=========================================================================================================================

= =

= the following output gives the predicted changes in keff (track length estimator) for the perturbations. =

= the differential operator method was used to obtain these results (1st and/or 2nd order). =

= warning: fundamental eigenvector (fission distribution) approximated as unperturbed. =

= =

=========================================================================================================================

perturbation k(trk ln) std. dev.

1 1.04614 0.00234

1average keff results summed over 2 cycles each to form 15 batch values of keff print table 178

N25

batch start end keff estimators by batch average keff estimators and deviations col/abs/tl keff

number cycle cycle k(coll) k(abs) k(track) k(coll) st dev k(abs) st dev k(track) st dev k(c/a/t) st dev

1 6 7 1.00212 1.00015 0.99691

2 8 9 0.98160 0.98329 0.98029 0.99186 0.01026 0.99172 0.00843 0.98860 0.00831

3 10 11 0.99780 0.99946 1.00115 0.99384 0.00624 0.99430 0.00551 0.99278 0.00637

4 12 13 0.98606 0.98403 0.99690 0.99190 0.00483 0.99173 0.00466 0.99381 0.00462 0.99291 0.00744

5 14 15 0.98197 0.98098 0.99300 0.98991 0.00423 0.98958 0.00420 0.99365 0.00358 0.99255 0.00536

18 December 2000 5-71

CHAPTER 5

KCODE

6 16 17 0.99533 0.99559 0.99768 0.99081 0.00357 0.99058 0.00358 0.99432 0.00300 0.99350 0.00421

7 18 19 0.98969 0.98927 0.99566 0.99065 0.00302 0.99040 0.00303 0.99451 0.00254 0.99358 0.00354

8 20 21 0.99881 1.00126 0.99601 0.99167 0.00281 0.99175 0.00295 0.99470 0.00221 0.99449 0.00298

9 22 23 1.00451 1.00256 1.00154 0.99310 0.00286 0.99295 0.00287 0.99546 0.00209 0.99542 0.00246

10 24 25 1.00098 0.99824 0.99327 0.99389 0.00268 0.99348 0.00262 0.99524 0.00188 0.99480 0.00217

-----------------------------------------------------------------------------------------------------------------------------------

11 26 27 0.99197 0.99192 0.99288 0.99371 0.00243 0.99334 0.00237 0.99503 0.00172 0.99461 0.00194

12 28 29 0.97730 0.97666 0.98363 0.99235 0.00260 0.99195 0.00257 0.99408 0.00183 0.99389 0.00213

13 30 31 1.00089 1.00059 0.99503 0.99300 0.00248 0.99262 0.00246 0.99415 0.00169 0.99395 0.00190

14 32 33 0.98246 0.98234 0.98356 0.99225 0.00242 0.99188 0.00239 0.99339 0.00174 0.99317 0.00194

15 34 35 1.01331 1.01445 1.00252 0.99365 0.00265 0.99339 0.00269 0.99400 0.00173 0.99396 0.00184

average keff results summed over 3 cycles each to form 10 batch values of keff

batch start end keff estimators by batch average keff estimators and deviations col/abs/tl keff

number cycle cycle k(coll) k(abs) k(track) k(coll) st dev k(abs) st dev k(track) st dev k(c/a/t) st dev

1 6 8 0.99630 0.99534 0.99570

2 9 11 0.99138 0.99326 0.98987 0.99384 0.00246 0.99430 0.00104 0.99278 0.00292

3 12 14 0.97881 0.97657 0.99144 0.98883 0.00521 0.98839 0.00594 0.99234 0.00174

4 15 17 0.99677 0.99717 1.00028 0.99081 0.00418 0.99058 0.00474 0.99432 0.00234 0.99494 0.00506

5 18 20 0.99579 0.99651 0.99666 0.99181 0.00339 0.99177 0.00386 0.99479 0.00187 0.99544 0.00343

6 21 23 0.99955 0.99888 0.99881 0.99310 0.00305 0.99295 0.00337 0.99546 0.00167 0.99621 0.00229

7 24 26 0.99835 0.99620 0.99345 0.99385 0.00269 0.99342 0.00288 0.99517 0.00144 0.99531 0.00186

8 27 29 0.98182 0.98168 0.98640 0.99235 0.00277 0.99195 0.00290 0.99408 0.00166 0.99437 0.00206

9 30 32 0.99038 0.99041 0.99133 0.99213 0.00245 0.99178 0.00256 0.99377 0.00149 0.99402 0.00181

10 33 35 1.00739 1.00784 0.99608 0.99365 0.00267 0.99339 0.00280 0.99400 0.00136 0.99400 0.00151

average keff results summed over 5 cycles each to form 6 batch values of keff

batch start end keff estimators by batch average keff estimators and deviations col/abs/tl keff

number cycle cycle k(coll) k(abs) k(track) k(coll) st dev k(abs) st dev k(track) st dev k(c/a/t) st dev

1 6 10 0.99569 0.99623 0.99333

2 11 15 0.98413 0.98293 0.99397 0.98991 0.00578 0.98958 0.00665 0.99365 0.00032

3 16 20 0.99561 0.99614 0.99707 0.99181 0.00384 0.99177 0.00442 0.99479 0.00116

4 21 25 1.00012 0.99863 0.99659 0.99389 0.00342 0.99348 0.00356 0.99524 0.00093 0.99526 0.00167

5 26 30 0.98972 0.98902 0.99012 0.99305 0.00278 0.99259 0.00290 0.99422 0.00125 0.99433 0.00202

6 31 35 0.99665 0.99736 0.99293 0.99365 0.00235 0.99339 0.00250 0.99400 0.00105 0.99397 0.00141

average keff results summed over 6 cycles each to form 5 batch values of keff

batch start end keff estimators by batch average keff estimators and deviations col/abs/tl keff

CHAPTER 5

KCODE

5-72 18 December 2000

number cycle cycle k(coll) k(abs) k(track) k(coll) st dev k(abs) st dev k(track) st dev k(c/a/t) st dev

1 6 11 0.99384 0.99430 0.99278

2 12 17 0.98779 0.98687 0.99586 0.99081 0.00303 0.99058 0.00372 0.99432 0.00154

3 18 23 0.99767 0.99770 0.99774 0.99310 0.00288 0.99295 0.00320 0.99546 0.00144

4 24 29 0.99008 0.98894 0.98993 0.99235 0.00217 0.99195 0.00247 0.99408 0.00172 0.99430 0.00243

5 30 35 0.99889 0.99912 0.99371 0.99365 0.00213 0.99339 0.00239 0.99400 0.00133 0.99469 0.00150

average keff results summed over 10 cycles each to form 3 batch values of keff

batch start end keff estimators by batch average keff estimators and deviations

number cycle cycle k(coll) k(abs) k(track) k(coll) st dev k(abs) st dev k(track) st dev

1 6 15 0.98991 0.98958 0.99365

2 16 25 0.99786 0.99739 0.99683 0.99389 0.00398 0.99348 0.00390 0.99524 0.00159

3 26 35 0.99318 0.99319 0.99153 0.99365 0.00231 0.99339 0.00226 0.99400 0.00154

average keff results summed over 15 cycles each to form 2 batch values of keff

batch start end keff estimators by batch average keff estimators and deviations

number cycle cycle k(coll) k(abs) k(track) k(coll) st dev k(abs) st dev k(track) st dev

1 6 20 0.99181 0.99177 0.99479

2 21 35 0.99550 0.99500 0.99322 0.99365 0.00184 0.99339 0.00162 0.99400 0.00079

1average individual and combined collision/absorption/track-length keff results for 5 different batch sizes

N26

cycles per number of average keff estimators and deviations normality average k(c/a/t) k(c/a/t) confidence intervals

keff batch k batches k(col) st dev k(abs) st dev k(trk) st dev co/ab/trk k(c/a/t) st dev 95% confidence 99% confidence

1 30 | 0.9937 0.0032 0.9934 0.0032 0.9940 0.0022 |95/95/95| 0.99408 0.00220 0.98956-0.99861 0.98797-1.00019

2 15 | 0.9937 0.0027 0.9934 0.0027 0.9940 0.0017 |95/95/95| 0.99396 0.00184 0.98995-0.99797 0.98834-0.99958

3 10 | 0.9937 0.0027 0.9934 0.0028 0.9940 0.0014 |95/95/95| 0.99400 0.00151 0.99042-0.99758 0.98871-0.99930

5 6 | 0.9937 0.0023 0.9934 0.0025 0.9940 0.0010 |95/95/95| 0.99397 0.00141 0.98949-0.99844 0.98575-1.00218

6 5 | 0.9937 0.0021 0.9934 0.0024 0.9940 0.0013 |95/95/95| 0.99469 0.00150 0.98823-1.00116 0.97979-1.00960

1individual and average keff estimator results by cycle

N27

keff neutron keff estimators by cycle average keff estimators and deviations average k(c/a/t)

cycle histories k(coll) k(abs) k(track) k(coll) st dev k(abs) st dev k(track) st dev k(c/a/t) st dev fom

1 3000 | 1.35813 1.35567 1.34299 |

2 4119 | 1.15406 1.15263 1.14618 |

3 2562 | 1.06451 1.06650 1.06327 |

4 2784 | 1.02132 1.02362 1.01860 |

5 2867 | 1.02371 1.02286 1.03020 |

18 December 2000 5-73

CHAPTER 5

KCODE

------------------- begin active keff cycles ------------------------------------------------------------------------------------

6 2971 | 0.98661 0.98343 0.99348 |

7 2887 | 1.01763 1.01686 1.00035 | 1.00212 0.01551 1.00015 0.01672 0.99691 0.00344 |

8 3110 | 0.98467 0.98572 0.99327 | 0.99630 0.01068 0.99534 0.01078 0.99570 0.00233 |

9 2891 | 0.97854 0.98087 0.96731 | 0.99186 0.00876 0.99172 0.00844 0.98860 0.00729 | 0.98801 0.00996 19883

10 3042 | 1.01102 1.01427 1.01223 | 0.99569 0.00779 0.99623 0.00794 0.99333 0.00736 | 0.99420 0.01052 14421

-----------------------------------------------------------------------------------------------------------------------------------

11 3073 | 0.98457 0.98465 0.99008 | 0.99384 0.00663 0.99430 0.00677 0.99278 0.00603 | 0.99308 0.00780 21812

12 2925 | 0.95715 0.95549 0.97617 | 0.98860 0.00767 0.98876 0.00796 0.99041 0.00563 | 0.99045 0.00696 23358

13 2886 | 1.01497 1.01256 1.01762 | 0.99190 0.00742 0.99173 0.00751 0.99381 0.00594 | 0.99395 0.00703 20390

14 3148 | 0.96430 0.96165 0.98053 | 0.98883 0.00722 0.98839 0.00742 0.99234 0.00544 | 0.99305 0.00648 21229

15 2812 | 0.99964 1.00031 1.00547 | 0.98991 0.00655 0.98958 0.00674 0.99365 0.00504 | 0.99443 0.00600 22558

16 3090 | 1.01165 1.01124 1.00505 | 0.99189 0.00625 0.99155 0.00641 0.99469 0.00468 | 0.99542 0.00531 26204

17 3078 | 0.97902 0.97995 0.99031 | 0.99081 0.00580 0.99058 0.00593 0.99432 0.00429 | 0.99535 0.00488 28459

18 2831 | 1.00428 1.00388 1.00553 | 0.99185 0.00544 0.99161 0.00555 0.99518 0.00404 | 0.99620 0.00454 30555

19 3049 | 0.97510 0.97467 0.98579 | 0.99065 0.00517 0.99040 0.00528 0.99451 0.00380 | 0.99584 0.00429 31745

20 2899 | 1.00799 1.01099 0.99867 | 0.99181 0.00495 0.99177 0.00510 0.99479 0.00354 | 0.99594 0.00387 36437

-----------------------------------------------------------------------------------------------------------------------------------

21 3105 | 0.98962 0.99153 0.99335 | 0.99167 0.00464 0.99175 0.00477 0.99470 0.00332 | 0.99594 0.00365 38456

22 2974 | 1.00901 1.00760 1.00932 | 0.99269 0.00447 0.99269 0.00458 0.99556 0.00323 | 0.99683 0.00347 40099

23 3073 | 1.00001 0.99752 0.99377 | 0.99310 0.00424 0.99295 0.00433 0.99546 0.00305 | 0.99626 0.00322 43706

24 2971 | 1.02064 1.01748 1.01019 | 0.99455 0.00426 0.99425 0.00429 0.99624 0.00299 | 0.99670 0.00304 46545

25 2976 | 0.98132 0.97900 0.97634 | 0.99389 0.00410 0.99348 0.00414 0.99524 0.00300 | 0.99558 0.00316 40777

26 2917 | 0.99309 0.99212 0.99382 | 0.99385 0.00390 0.99342 0.00394 0.99517 0.00286 | 0.99550 0.00301 42849

27 3084 | 0.99086 0.99172 0.99193 | 0.99371 0.00372 0.99334 0.00376 0.99503 0.00273 | 0.99535 0.00286 45487

28 3002 | 0.98990 0.98813 0.99055 | 0.99355 0.00356 0.99311 0.00360 0.99483 0.00261 | 0.99516 0.00275 46746

29 3123 | 0.96469 0.96520 0.97671 | 0.99235 0.00361 0.99195 0.00363 0.99408 0.00261 | 0.99463 0.00271 45816

30 2908 | 1.01006 1.00794 0.99760 | 0.99305 0.00353 0.99259 0.00354 0.99422 0.00251 | 0.99451 0.00260 47825

-----------------------------------------------------------------------------------------------------------------------------------

31 3169 | 0.99172 0.99324 0.99247 | 0.99300 0.00340 0.99262 0.00341 0.99415 0.00241 | 0.99446 0.00247 50814

32 2839 | 0.96936 0.97006 0.98392 | 0.99213 0.00338 0.99178 0.00338 0.99377 0.00235 | 0.99431 0.00239 52343

33 2919 | 0.99556 0.99462 0.98321 | 0.99225 0.00326 0.99188 0.00326 0.99339 0.00230 | 0.99364 0.00237 51345

34 3078 | 1.01230 1.01367 1.00685 | 0.99294 0.00322 0.99263 0.00323 0.99386 0.00227 | 0.99415 0.00230 52575

35 3073 | 1.01431 1.01522 0.99820 | 0.99365 0.00319 0.99339 0.00321 0.99400 0.00219 | 0.99408 0.00220 55321

N28

the largest active cycle keffs by estimator are: the smallest active cycle keffs by estimator are:

collision 1.02064 on cycle 24 collision 0.95715 on cycle 12

absorption 1.01748 on cycle 24 absorption 0.95549 on cycle 12

track length 1.01762 on cycle 13 track length 0.96731 on cycle 9

1plot of the estimated col/abs/track-length keff one standard deviation interval versus cycle number (| = final keff = 0.99408)

N29

cycle active 0.98 0.99 1.00 1.01

number cycles |-------------------------------|--------------------------------|--------------------------------|

11 6 | (-------------------------k--|----------------------) |

12 7 | (----------------------k-----------|-----------) |

13 8 | (----------------------k----------------------) |

CHAPTER 5

KCODE

5-74 18 December 2000

14 9 | (---------------------k--|------------------) |

15 10 | (-----------------|-k-------------------) |

16 11 | (------------|----k-----------------) |

17 12 | (-----------|----k---------------) |

18 13 | (-------|-------k--------------) |

19 14 | (-------|-----k--------------) |

20 15 + (-----|------k------------) +

21 16 | (-----|------k-----------) |

22 17 | (-|---------k----------) |

23 18 | (--|-------k---------) |

24 19 | (|--------k---------) |

25 20 | (----|----k----------) |

26 21 | (----|----k---------) |

27 22 | (----|----k--------) |

28 23 | (----|---k--------) |

29 24 | (------|-k--------) |

30 25 + (------|-k--------) +

31 26 | (------|-k-------) |

32 27 | (------|k-------) |

33 28 | (-------k|------) |

34 29 | (------|k------) |

35 30 | (------k-------) |

|-------------------------------|--------------------------------|--------------------------------|

0.98 0.99 1.00 1.01

1individual and collision/absorption/track-length keffs for different numbers of inactive cycles skipped for fission source settling

N30

skip active active average keff estimators and deviations normality average k(c/a/t) k(c/a/t) confidence intervals

cycles cycles neutrons k(col) st dev k(abs) st dev k(trk) st dev co/ab/tl k(c/a/t) st dev 95% confidence 99% confidence

0 35 105235| 1.0123 0.0117 1.0121 0.0116 1.0120 0.0110 |no/no/no| 1.00981 0.01058 0.98826-1.03135 0.98083-1.03878

1 34 102235| 1.0022 0.0059 1.0020 0.0059 1.0023 0.0053 |no/no/no| 1.00237 0.00537 0.99141-1.01333 0.98762-1.01712

2 33 98116| 0.9975 0.0038 0.9974 0.0038 0.9979 0.0031 |95/95/no| 0.99828 0.00316 0.99182-1.00474 0.98958-1.00698

3 32 95554| 0.9955 0.0032 0.9953 0.0033 0.9959 0.0025 |95/95/95| 0.99608 0.00251 0.99095-1.00120 0.98917-1.00298

4 31 92770| 0.9946 0.0032 0.9943 0.0033 0.9952 0.0024 |95/95/95| 0.99522 0.00249 0.99011-1.00032 0.98833-1.00211

5 30* 89903| 0.9937 0.0032 0.9934 0.0032 0.9940 0.0022 |95/95/95| 0.99408 0.00220 0.98956-0.99861 0.98797-1.00019

6 29 86932| 0.9939 0.0033 0.9937 0.0033 0.9940 0.0023 |95/95/95| 0.99405 0.00227 0.98938-0.99871 0.98774-1.00035

7 28 84045| 0.9930 0.0033 0.9929 0.0033 0.9938 0.0023 |95/95/95| 0.99404 0.00236 0.98918-0.99890 0.98747-1.00062

8 27 80935| 0.9934 0.0034 0.9932 0.0034 0.9938 0.0024 |95/95/95| 0.99395 0.00245 0.98889-0.99901 0.98710-1.00081

9 26 78044| 0.9939 0.0035 0.9936 0.0035 0.9948 0.0023 |95/95/95| 0.99558 0.00218 0.99108-1.00009 0.98947-1.00170

10 25 75002| 0.9932 0.0036 0.9928 0.0036 0.9941 0.0023 |95/95/95| 0.99466 0.00221 0.99008-0.99923 0.98844-1.00088

-----------------------------------------------------------------------------------------------------------------------------------

11 24 71929| 0.9936 0.0037 0.9932 0.0037 0.9943 0.0024 |95/95/95| 0.99473 0.00230 0.98994-0.99952 0.98821-1.00125

12 23 69004| 0.9952 0.0035 0.9948 0.0035 0.9951 0.0023 |95/95/95| 0.99501 0.00234 0.99013-0.99990 0.98835-1.00168

13 22 66118| 0.9943 0.0035 0.9940 0.0035 0.9941 0.0022 |95/95/95| 0.99420 0.00209 0.98983-0.99857 0.98823-1.00017

14 21 62970| 0.9957 0.0034 0.9955 0.0033 0.9947 0.0022 |95/95/95| 0.99438 0.00217 0.98982-0.99893 0.98813-1.00062

15 20 60158| 0.9955 0.0036 0.9953 0.0035 0.9942 0.0022 |95/95/95| 0.99360 0.00217 0.98902-0.99818 0.98730-0.99989

16 19 57068| 0.9947 0.0036 0.9944 0.0036 0.9936 0.0023 |95/95/95| 0.99319 0.00223 0.98846-0.99792 0.98668-0.99970

17 18 53990| 0.9955 0.0037 0.9953 0.0037 0.9938 0.0024 |95/95/95| 0.99303 0.00240 0.98792-0.99815 0.98596-1.00010

18 17 51159| 0.9950 0.0039 0.9947 0.0039 0.9931 0.0024 |95/95/95| 0.99220 0.00239 0.98707-0.99733 0.98509-0.99932

18 December 2000 5-75

CHAPTER 5

KCODE

19 16 48110| 0.9963 0.0040 0.9960 0.0039 0.9936 0.0025 |95/95/95| 0.99226 0.00262 0.98661-0.99791 0.98438-1.00014

20 15 45211| 0.9955 0.0042 0.9950 0.0040 0.9932 0.0027 |95/95/95| 0.99223 0.00282 0.98608-0.99837 0.98361-1.00084

-----------------------------------------------------------------------------------------------------------------------------------

21 14 42106| 0.9959 0.0044 0.9953 0.0043 0.9932 0.0029 |95/95/95| 0.99193 0.00317 0.98496-0.99891 0.98209-1.00178

22 13 39132| 0.9949 0.0047 0.9943 0.0046 0.9920 0.0028 |95/95/95| 0.99066 0.00289 0.98422-0.99711 0.98149-0.99983

23 12 36059| 0.9945 0.0051 0.9940 0.0050 0.9918 0.0031 |95/95/95| 0.99059 0.00308 0.98361-0.99757 0.98057-1.00061

24 11 33088| 0.9921 0.0049 0.9919 0.0049 0.9901 0.0028 |95/95/95| 0.98968 0.00250 0.98392-0.99544 0.98130-0.99805

25 10 30112| 0.9932 0.0053 0.9932 0.0052 0.9915 0.0027 |95/95/95| 0.99061 0.00244 0.98483-0.99639 0.98206-0.99916

26 9 27195| 0.9932 0.0059 0.9933 0.0059 0.9913 0.0030 |95/95/95| 0.98990 0.00269 0.98331-0.99649 0.97991-0.99989

27 8 24111| 0.9935 0.0067 0.9935 0.0066 0.9912 0.0034 |95/95/95| 0.98981 0.00312 0.98179-0.99782 0.97723-1.00238

28 7 21109| 0.9940 0.0077 0.9943 0.0076 0.9913 0.0040 |95/95/95| 0.98855 0.00371 0.97826-0.99885 0.97148-1.00562

29 6 17986| 0.9989 0.0070 0.9991 0.0069 0.9937 0.0037 |95/95/95| 0.99051 0.00465 0.97572-1.00531 0.96336-1.01766

30 5 15078| 0.9967 0.0081 0.9974 0.0082 0.9929 0.0045 |95/95/95| 0.98499 0.00423 0.96680-1.00318 0.94304-1.02694

-----------------------------------------------------------------------------------------------------------------------------------

31 4 11909| 0.9979 0.0104 0.9984 0.0105 0.9930 0.0058 |95/95/95| 0.98556 0.00438 0.92987-1.04126 0.70661-1.26452

32 3 9070| 1.0074 0.0059 1.0078 0.0066 0.9961 0.0069 |

33 2 6151| 1.0133 0.0010 1.0144 0.0008 1.0025 0.0043 |

N31

the minimum estimated standard deviation for the col/abs/tl keff estimator occurs with 4 inactive cycles and 31 active cycles.

N32

the first active half of the problem skips 5 cycles and uses 15 active cycles; the second half skips 20 and uses 15 cycles.

the col/abs/trk-len keff, one standard deviation, and 68, 95, and 99 percent intervals for each active half of the problem are:

problem keff standard deviation 68% confidence 95% confidence 99% confidence

first half 0.99594 0.00387 0.99192 to 0.99995 0.98751 to 1.00437 0.98412 to 1.00776

second half 0.99223 0.00282 0.98930 to 0.99515 0.98608 to 0.99837 0.98361 to 1.00084

final result 0.99408 0.00220 0.99185 to 0.99632 0.98956 to 0.99861 0.98797 to 1.00019

the first and second half values of k(collision/absorption/track length) appear to be the same at the 68 percent confidence level.

1plot of the estimated col/abs/track-length keff one standard deviation interval by active cycle number (| = final keff = 0.99408)

N33

inactive active 0.98 0.99 1.00 1.01 1.02 1.03

cycles cycles |------------------|-------------------|-------------------|-------------------|-------------------|

0 35 | | (--------------------k--------------------) |

1 34 | | (----------k----------) |

2 33 | | (-----k-----) |

3 32 | (|---k----) |

4 31 | (--|-k----) |

5 30 * (----k---) *

6 29 | (----k---) |

7 28 | (----k---) |

8 27 | (---k|---) |

9 26 + (-|--k---) +

10 25 | (---|k---) |

11 24 | (---|k----) |

CHAPTER 5

KCODE

5-76 18 December 2000

12 23 | (--|-k---) |

13 22 | (---k---) |

14 21 | (---k----) |

15 20 | (----k|--) |

16 19 | (----k-|-) |

17 18 | (----k-|-) |

18 17 | (----k---|) |

19 16 + (----k---|) +

20 15 | (-----k---|-) |

21 14 | (-----k----|-) |

22 13 | (-----k-----)| |

23 12 | (-----k-----)| |

24 11 | (----k----) | |

25 10 | (----k----) | |

|------------------|-------------------|-------------------|-------------------|-------------------|

0.98 0.99 1.00 1.01 1.02 1.03

N34 1tally 1 nps = 105235

tally type 1 number of particles crossing a surface.

tally for neutrons

number of histories used for normalizing tallies = 90000.00

surface 1

energy

1.0000E-07 0.00000E+00 0.0000

4.0000E-07 0.00000E+00 0.0000

1.0000E-06 0.00000E+00 0.0000

3.0000E-06 0.00000E+00 0.0000

1.0000E-05 0.00000E+00 0.0000

3.0000E-05 0.00000E+00 0.0000

1.0000E-04 0.00000E+00 0.0000

5.5000E-04 0.00000E+00 0.0000

3.0000E-03 1.93456E-05 0.5056

1.7000E-02 5.56838E-04 0.1069

1.0000E-01 1.43341E-02 0.0202

4.0000E-01 9.50148E-02 0.0077

9.0000E-01 1.41255E-01 0.0067

1.4000E+00 9.03271E-02 0.0093

3.0000E+00 1.44701E-01 0.0073

2.0000E+01 9.01202E-02 0.0097

total 5.76327E-01 0.0019

1analysis of the results in the tally fluctuation chart bin (tfc) for tally 1 with nps = 105235 print table 160

normed average tally per history = 5.76327E-01 unnormed average tally per history = 5.76327E-01

estimated tally relative error = 0.0019 estimated variance of the variance = 0.0000

relative error from zero tallies = 0.0014 relative error from nonzero scores = 0.0013

number of nonzero history tallies = 77248 efficiency for the nonzero tallies = 0.8583

history number of largest tally = 27863 largest unnormalized history tally = 1.69041E+00

18 December 2000 5-77

CHAPTER 5

KCODE

(largest tally)/(average tally) = 2.93307E+00 (largest tally)/(avg nonzero tally)= 2.51749E+00

(confidence interval shift)/mean = 0.0000 shifted confidence interval center = 5.76327E-01

if the largest history score sampled so far were to occur on the next history, the tfc bin quantities would change as follows:

nps = 89903 for this table because 5 keff cycles and 15332 histories were skipped before tally accumulation.

estimated quantities value at nps value at nps+1 value(nps+1)/value(nps)-1.

mean 5.76327E-01 5.76340E-01 0.000022

relative error 1.86279E-03 1.85963E-03 -0.001695

variance of the variance 1.30178E-05 1.30942E-05 0.005862

shifted center 5.76327E-01 5.76327E-01 0.000000

figure of merit 7.93646E+04 7.96343E+04 0.003399

the estimated slope of the 70 largest tallies starting at 1.06982E+00 appears to be decreasing at least exponentially.

the empirical history score probability density function appears to be increasing at the largest history scores: please examine.

the large score tail of the empirical history score probability density function appears to have no unsampled regions.

===================================================================================================================================

results of 10 statistical checks for the estimated answer for the tally fluctuation chart (tfc) bin of tally 1

tfc bin --mean-- ---------relative error--------- ----variance of the variance---- --figure of merit-- -pdf-

behavior behavior value decrease decrease rate value decrease decrease rate value behavior slope

desired random <0.10 yes 1/sqrt(nps) <0.10 yes 1/nps constant random >3.00

observed random 0.00 yes yes 0.00 yes yes constant random 10.00

passed? yes yes yes yes yes yes yes yes yes yes

===================================================================================================================================

this tally meets the statistical criteria used to form confidence intervals: check the tally fluctuation chart to verify.

the results in other bins associated with this tally may not meet these statistical criteria.

estimated asymmetric confidence interval(1,2,3 sigma): 5.7525E-01 to 5.7740E-01; 5.7418E-01 to 5.7847E-01; 5.7311E-01 to 5.7955E-01

estimated symmetric confidence interval(1,2,3 sigma): 5.7525E-01 to 5.7740E-01; 5.7418E-01 to 5.7847E-01; 5.7311E-01 to 5.7955E-01

fom = (histories/minute)*(f(x) signal-to-noise ratio)**2 = (2.479E+04)*( 1.789E+00)**2 = (2.479E+04)*(3.202E+00) = 7.936E+04

SKIP 49 LINES IN OUTPUT

N35 1tally 6 nps = 105235

tally type 6 track length estimate of heating. units mev/gram

tally for neutrons

number of histories used for normalizing tallies = 90000.00

CHAPTER 5

KCODE

5-78 18 December 2000

masses

cell: 1

5.17571E+04

cell 1

energy

1.0000E-07 0.00000E+00 0.0000

4.0000E-07 0.00000E+00 0.0000

1.0000E-06 0.00000E+00 0.0000

3.0000E-06 0.00000E+00 0.0000

1.0000E-05 0.00000E+00 0.0000

3.0000E-05 0.00000E+00 0.0000

1.0000E-04 4.21331E-08 1.0000

5.5000E-04 7.99328E-08 0.5976

3.0000E-03 5.02019E-07 0.2414

1.7000E-02 4.40209E-06 0.0678

1.0000E-01 6.29932E-05 0.0168

4.0000E-01 2.56615E-04 0.0071

9.0000E-01 2.77559E-04 0.0060

1.4000E+00 1.74875E-04 0.0079

3.0000E+00 3.00692E-04 0.0058

2.0000E+01 1.69072E-04 0.0081

total 1.24683E-03 0.0020

1analysis of the results in the tally fluctuation chart bin (tfc) for tally 6 with nps = 105235 print table 160

normed average tally per history = 1.24683E-03 unnormed average tally per history = 6.45325E+01

estimated tally relative error = 0.0020 estimated variance of the variance = 0.0000

relative error from zero tallies = 0.0001 relative error from nonzero scores = 0.0020

number of nonzero history tallies = 89903 efficiency for the nonzero tallies = 0.9989

history number of largest tally = 101153 largest unnormalized history tally = 3.31676E+02

(largest tally)/(average tally) = 5.13968E+00 (largest tally)/(avg nonzero tally)= 5.13414E+00

(confidence interval shift)/mean = 0.0000 shifted confidence interval center = 1.24684E-03

if the largest history score sampled so far were to occur on the next history, the tfc bin quantities would change as follows:

nps = 89903 for this table because 5 keff cycles and 15332 histories were skipped before tally accumulation.

estimated quantities value at nps value at nps+1 value(nps+1)/value(nps)-1.

mean 1.24683E-03 1.24689E-03 0.000046

relative error 1.99134E-03 1.98875E-03 -0.001303

variance of the variance 3.81875E-05 3.85391E-05 0.009208

shifted center 1.24684E-03 1.24684E-03 0.000000

figure of merit 6.94483E+04 6.96296E+04 0.002611

18 December 2000 5-79

CHAPTER 5

KCODE

the estimated slope of the 198 largest tallies starting at 2.21201E+02 appears to be decreasing at least exponentially.

the large score tail of the empirical history score probability density function appears to have no unsampled regions.

===================================================================================================================================

results of 10 statistical checks for the estimated answer for the tally fluctuation chart (tfc) bin of tally 6

tfc bin --mean-- ---------relative error--------- ----variance of the variance---- --figure of merit-- -pdf-

behavior behavior value decrease decrease rate value decrease decrease rate value behavior slope

desired random <0.10 yes 1/sqrt(nps) <0.10 yes 1/nps constant random >3.00

observed random 0.00 yes yes 0.00 yes yes constant random 10.00

passed? yes yes yes yes yes yes yes yes yes yes

===================================================================================================================================

this tally meets the statistical criteria used to form confidence intervals: check the tally fluctuation chart to verify.

the results in other bins associated with this tally may not meet these statistical criteria.

estimated asymmetric confidence interval(1,2,3 sigma): 1.2444E-03 to 1.2493E-03; 1.2419E-03 to 1.2518E-03; 1.2394E-03 to 1.2543E-03

estimated symmetric confidence interval(1,2,3 sigma): 1.2444E-03 to 1.2493E-03; 1.2419E-03 to 1.2518E-03; 1.2394E-03 to 1.2543E-03

fom = (histories/minute)*(f(x) signal-to-noise ratio)**2 = (2.479E+04)*( 1.674E+00)**2 = (2.479E+04)*(2.802E+00) = 6.945E+04

SKIP TABLES 161 AND 162 IN OUTPUT

N36 1tally 7 nps = 105235

tally type 7 track length estimate of fission heating. units mev/gram

tally for neutrons

number of histories used for normalizing tallies = 90000.00

masses

cell: 1

5.17571E+04

cell 1

energy

1.0000E-07 0.00000E+00 0.0000

4.0000E-07 0.00000E+00 0.0000

1.0000E-06 0.00000E+00 0.0000

3.0000E-06 0.00000E+00 0.0000

1.0000E-05 0.00000E+00 0.0000

3.0000E-05 0.00000E+00 0.0000

1.0000E-04 4.50318E-08 1.0000

5.5000E-04 8.55183E-08 0.5978

3.0000E-03 5.36601E-07 0.2413

1.7000E-02 4.70495E-06 0.0678

CHAPTER 5

KCODE

5-80 18 December 2000

1.0000E-01 6.72758E-05 0.0168

4.0000E-01 2.74136E-04 0.0071

9.0000E-01 2.96092E-04 0.0060

1.4000E+00 1.87357E-04 0.0079

3.0000E+00 3.23609E-04 0.0058

2.0000E+01 1.83378E-04 0.0081

total 1.33722E-03 0.0020

1analysis of the results in the tally fluctuation chart bin (tfc) for tally 7 with nps = 105235 print table 160

normed average tally per history = 1.33722E-03 unnormed average tally per history = 6.92107E+01

estimated tally relative error = 0.0020 estimated variance of the variance = 0.0000

relative error from zero tallies = 0.0001 relative error from nonzero scores = 0.0020

number of nonzero history tallies = 89903 efficiency for the nonzero tallies = 0.9989

history number of largest tally = 101153 largest unnormalized history tally = 3.54596E+02

(largest tally)/(average tally) = 5.12343E+00 (largest tally)/(avg nonzero tally)= 5.11791E+00

(confidence interval shift)/mean = 0.0000 shifted confidence interval center = 1.33722E-03

if the largest history score sampled so far were to occur on the next history, the tfc bin quantities would change as follows:

nps = 89903 for this table because 5 keff cycles and 15332 histories were skipped before tally accumulation.

estimated quantities value at nps value at nps+1 value(nps+1)/value(nps)-1.

mean 1.33722E-03 1.33728E-03 0.000046

relative error 1.98857E-03 1.98597E-03 -0.001308

variance of the variance 3.80138E-05 3.83625E-05 0.009174

shifted center 1.33722E-03 1.33722E-03 0.000000

figure of merit 6.96421E+04 6.98246E+04 0.002621

the estimated slope of the 200 largest tallies starting at 2.36318E+02 appears to be decreasing at least exponentially.

the large score tail of the empirical history score probability density function appears to have no unsampled regions.

===================================================================================================================================

results of 10 statistical checks for the estimated answer for the tally fluctuation chart (tfc) bin of tally 7

tfc bin --mean-- ---------relative error--------- ----variance of the variance---- --figure of merit-- -pdf-

behavior behavior value decrease decrease rate value decrease decrease rate value behavior slope

desired random <0.10 yes 1/sqrt(nps) <0.10 yes 1/nps constant random >3.00

observed random 0.00 yes yes 0.00 yes yes constant random 10.00

passed? yes yes yes yes yes yes yes yes yes yes

===================================================================================================================================

18 December 2000 5-81

CHAPTER 5

KCODE

this tally meets the statistical criteria used to form confidence intervals: check the tally fluctuation chart to verify.

the results in other bins associated with this tally may not meet these statistical criteria.

estimated asymmetric confidence interval(1,2,3 sigma): 1.3346E-03 to 1.3399E-03; 1.3319E-03 to 1.3425E-03; 1.3292E-03 to 1.3452E-03

estimated symmetric confidence interval(1,2,3 sigma): 1.3346E-03 to 1.3399E-03; 1.3319E-03 to 1.3425E-03; 1.3292E-03 to 1.3452E-03

fom = (histories/minute)*(f(x) signal-to-noise ratio)**2 = (2.479E+04)*( 1.676E+00)**2 = (2.479E+04)*(2.810E+00) = 6.964E+04

SKIP TABLES 161 AND 162 IN OUTPUT

N37 1tally 14 nps = 105235

+ total total fission neutrons (track-lenght Keff), total loss to (n,xn)

total neutron absorptions,total fission,and neutron heating (mev/gram)

tally type 4 track length estimate of particle flux.

tally for neutrons

number of histories used for normalizing tallies = 90000.00

multiplier bin 1: 1.32534E+02 10 -6 -7

multiplier bin 2: 1.32534E+02 10 16 : 17

multiplier bin 3: 1.32534E+02 10 -2

multiplier bin 4: 1.32534E+02 10 -6

multiplier bin 5: 2.56069E-03 10 1 -4

volumes

cell: 1

2.76185E+03

cell 1

mult bin: 1 2 3 4 5

energy

1.0000E-07 0.00000E+00 0.0000 0.00000E+00 0.0000 0.00000E+00 0.0000 0.00000E+00 0.0000 0.00000E+00 0.0000

4.0000E-07 0.00000E+00 0.0000 0.00000E+00 0.0000 0.00000E+00 0.0000 0.00000E+00 0.0000 0.00000E+00 0.0000

1.0000E-06 0.00000E+00 0.0000 0.00000E+00 0.0000 0.00000E+00 0.0000 0.00000E+00 0.0000 0.00000E+00 0.0000

3.0000E-06 0.00000E+00 0.0000 0.00000E+00 0.0000 0.00000E+00 0.0000 0.00000E+00 0.0000 0.00000E+00 0.0000

1.0000E-05 0.00000E+00 0.0000 0.00000E+00 0.0000 0.00000E+00 0.0000 0.00000E+00 0.0000 0.00000E+00 0.0000

3.0000E-05 0.00000E+00 0.0000 0.00000E+00 0.0000 0.00000E+00 0.0000 0.00000E+00 0.0000 0.00000E+00 0.0000

1.0000E-04 3.13391E-05 1.0000 0.00000E+00 0.0000 9.43974E-06 1.0000 1.28862E-05 1.0000 4.21354E-08 1.0000

5.5000E-04 5.95150E-05 0.5978 0.00000E+00 0.0000 8.30220E-06 0.6257 2.44716E-05 0.5978 7.99372E-08 0.5976

3.0000E-03 3.73438E-04 0.2413 0.00000E+00 0.0000 7.82420E-05 0.2710 1.53552E-04 0.2413 5.02047E-07 0.2414

1.7000E-02 3.27337E-03 0.0678 0.00000E+00 0.0000 4.94682E-04 0.0675 1.34635E-03 0.0678 4.40234E-06 0.0678

1.0000E-01 4.67088E-02 0.0168 0.00000E+00 0.0000 6.04415E-03 0.0170 1.92515E-02 0.0168 6.29967E-05 0.0168

4.0000E-01 1.92544E-01 0.0071 0.00000E+00 0.0000 1.67116E-02 0.0073 7.84466E-02 0.0071 2.56629E-04 0.0071

9.0000E-01 2.11200E-01 0.0060 0.00000E+00 0.0000 1.14261E-02 0.0061 8.47343E-02 0.0060 2.77575E-04 0.0060

1.4000E+00 1.36490E-01 0.0079 0.00000E+00 0.0000 4.89812E-03 0.0079 5.36179E-02 0.0079 1.74885E-04 0.0079

3.0000E+00 2.45846E-01 0.0058 0.00000E+00 0.0000 4.47338E-03 0.0061 9.26079E-02 0.0058 3.00709E-04 0.0058

2.0000E+01 1.57533E-01 0.0082 2.64782E-03 0.0239 5.58023E-04 0.0095 5.24771E-02 0.0081 1.69082E-04 0.0081

total 9.94059E-01 0.0020 2.64782E-03 0.0239 4.47020E-02 0.0037 3.82672E-01 0.0020 1.24690E-03 0.0020

1analysis of the results in the tally fluctuation chart bin (tfc) for tally 14 with nps = 105235 print table 160

CHAPTER 5

KCODE

5-82 18 December 2000

normed average tally per history = 9.94059E-01 unnormed average tally per history = 2.74544E+03

estimated tally relative error = 0.0020 estimated variance of the variance = 0.0000

relative error from zero tallies = 0.0001 relative error from nonzero scores = 0.0020

number of nonzero history tallies = 89903 efficiency for the nonzero tallies = 0.9989

history number of largest tally = 84240 largest unnormalized history tally = 1.58138E+04

(largest tally)/(average tally) = 5.76003E+00 (largest tally)/(avg nonzero tally)= 5.75382E+00

(confidence interval shift)/mean = 0.0000 shifted confidence interval center = 9.94062E-01

if the largest history score sampled so far were to occur on the next history, the tfc bin quantities would change as follows:

nps = 89903 for this table because 5 keff cycles and 15332 histories were skipped before tally accumulation.

estimated quantities value at nps value at nps+1 value(nps+1)/value(nps)-1.

mean 9.94059E-01 9.94112E-01 0.000053

relative error 1.97345E-03 1.97099E-03 -0.001245

variance of the variance 3.80437E-05 3.86107E-05 0.014903

shifted center 9.94062E-01 9.94062E-01 0.000000

figure of merit 7.07134E+04 7.08898E+04 0.002494

the estimated slope of the 200 largest tallies starting at 9.32151E+03 appears to be decreasing at least exponentially.

the large score tail of the empirical history score probability density function appears to have no unsampled regions.

===================================================================================================================================

results of 10 statistical checks for the estimated answer for the tally fluctuation chart (tfc) bin of tally 14

tfc bin --mean-- ---------relative error--------- ----variance of the variance---- --figure of merit-- -pdf-

behavior behavior value decrease decrease rate value decrease decrease rate value behavior slope

desired random <0.10 yes 1/sqrt(nps) <0.10 yes 1/nps constant random >3.00

observed random 0.00 yes yes 0.00 yes yes constant random 10.00

passed? yes yes yes yes yes yes yes yes yes yes

===================================================================================================================================

this tally meets the statistical criteria used to form confidence intervals: check the tally fluctuation chart to verify.

the results in other bins associated with this tally may not meet these statistical criteria.

estimated asymmetric confidence interval(1,2,3 sigma): 9.9210E-01 to 9.9602E-01; 9.9014E-01 to 9.9799E-01; 9.8818E-01 to 9.9995E-01

estimated symmetric confidence interval(1,2,3 sigma): 9.9210E-01 to 9.9602E-01; 9.9014E-01 to 9.9798E-01; 9.8817E-01 to 9.9994E-01

18 December 2000 5-83

CHAPTER 5

KCODE

fom = (histories/minute)*(f(x) signal-to-noise ratio)**2 = (2.479E+04)*( 1.689E+00)**2 = (2.479E+04)*(2.853E+00) = 7.071E+04

SKIP TABLES 161 AND 162 IN OUTPUT

N38 1tally 34 nps = 105235

tally type 4 track length estimate of particle flux.

tally for neutrons

number of histories used for normalizing tallies = 90000.00

multiplier bin 1: -1.00000E+00 10 -6 -7

multiplier bin 2: -1.00000E+00 10 16 : 17

multiplier bin 3: -1.00000E+00 10 -2

multiplier bin 4: -1.00000E+00 10 -6

multiplier bin 5: -1.93210E-05 10 1 -4

volumes

cell: 1

1.00000E+00

energy bin: 0. to 2.00000E+01

cell: 1

mult bin

1 9.94003E-01 0.0020

2 2.64767E-03 0.0239

3 4.46995E-02 0.0037

4 3.82651E-01 0.0020

5 1.24683E-03 0.0020

1analysis of the results in the tally fluctuation chart bin (tfc) for tally 34 with nps = 105235 print table 160

normed average tally per history = 9.94003E-01 unnormed average tally per history = 9.94003E-01

estimated tally relative error = 0.0020 estimated variance of the variance = 0.0000

relative error from zero tallies = 0.0001 relative error from nonzero scores = 0.0020

number of nonzero history tallies = 89903 efficiency for the nonzero tallies = 0.9989

history number of largest tally = 84240 largest unnormalized history tally = 5.72548E+00

(largest tally)/(average tally) = 5.76003E+00 (largest tally)/(avg nonzero tally)= 5.75382E+00

(confidence interval shift)/mean = 0.0000 shifted confidence interval center = 9.94006E-01

if the largest history score sampled so far were to occur on the next history, the tfc bin quantities would change as follows:

nps = 89903 for this table because 5 keff cycles and 15332 histories were skipped before tally accumulation.

estimated quantities value at nps value at nps+1 value(nps+1)/value(nps)-1.

mean 9.94003E-01 9.94055E-01 0.000053

relative error 1.97345E-03 1.97099E-03 -0.001245

variance of the variance 3.80437E-05 3.86107E-05 0.014903

shifted center 9.94006E-01 9.94006E-01 0.000000

figure of merit 7.07134E+04 7.08898E+04 0.002494

CHAPTER 5

KCODE

5-84 18 December 2000

the estimated slope of the 200 largest tallies starting at 3.37490E+00 appears to be decreasing at least exponentially.

the large score tail of the empirical history score probability density function appears to have no unsampled regions.

===================================================================================================================================

results of 10 statistical checks for the estimated answer for the tally fluctuation chart (tfc) bin of tally 34

tfc bin --mean-- ---------relative error--------- ----variance of the variance---- --figure of merit-- -pdf-

behavior behavior value decrease decrease rate value decrease decrease rate value behavior slope

desired random <0.10 yes 1/sqrt(nps) <0.10 yes 1/nps constant random >3.00

observed random 0.00 yes yes 0.00 yes yes constant random 10.00

passed? yes yes yes yes yes yes yes yes yes yes

===================================================================================================================================

this tally meets the statistical criteria used to form confidence intervals: check the tally fluctuation chart to verify.

the results in other bins associated with this tally may not meet these statistical criteria.

estimated asymmetric confidence interval(1,2,3 sigma): 9.9204E-01 to 9.9597E-01; 9.9008E-01 to 9.9793E-01; 9.8812E-01 to 9.9989E-01

estimated symmetric confidence interval(1,2,3 sigma): 9.9204E-01 to 9.9596E-01; 9.9008E-01 to 9.9793E-01; 9.8812E-01 to 9.9989E-01

fom = (histories/minute)*(f(x) signal-to-noise ratio)**2 = (2.479E+04)*( 1.689E+00)**2 = (2.479E+04)*(2.853E+00) = 7.071E+04

SKIP TABLES 161 AND 162 IN OUTPUT

=========================================================================================================================

= N39 =

= the following output gives the predicted change in a tally for perturbation 1. =

= the differential operator method was used to obtain these results (1st and/or 2nd order). =

= =

=========================================================================================================================

N40

1tally 1 nps = 105235

tally type 1 number of particles crossing a surface.

tally for neutrons

number of histories used for normalizing tallies = 90000.00

surface 1

energy

1.0000E-07 0.00000E+00 0.0000

4.0000E-07 0.00000E+00 0.0000

1.0000E-06 0.00000E+00 0.0000

3.0000E-06 0.00000E+00 0.0000

1.0000E-05 0.00000E+00 0.0000

3.0000E-05 0.00000E+00 0.0000

18 December 2000 5-85

CHAPTER 5

KCODE

1.0000E-04 0.00000E+00 0.0000

5.5000E-04 0.00000E+00 0.0000

3.0000E-03 1.65701E-05 0.5047

1.7000E-02 5.53326E-04 0.1064

1.0000E-01 1.40325E-02 0.0202

4.0000E-01 9.27035E-02 0.0076

9.0000E-01 1.36687E-01 0.0066

1.4000E+00 8.63582E-02 0.0092

3.0000E+00 1.37120E-01 0.0073

2.0000E+01 8.55312E-02 0.0096

total 5.53003E-01 0.0018

SKIP 102 LINES IN OUTPUT

1tally 6 nps = 105235

tally type 6 track length estimate of heating. units mev/gram

tally for neutrons

number of histories used for normalizing tallies = 90000.00

masses

cell: 1

5.17571E+04

cell 1

energy

1.0000E-07 0.00000E+00 0.0000

4.0000E-07 0.00000E+00 0.0000

1.0000E-06 0.00000E+00 0.0000

3.0000E-06 0.00000E+00 0.0000

1.0000E-05 0.00000E+00 0.0000

3.0000E-05 0.00000E+00 0.0000

1.0000E-04 3.78522E-08 1.0000

5.5000E-04 7.88833E-08 0.5905

3.0000E-03 4.77198E-07 0.2340

1.7000E-02 4.49400E-06 0.0661

1.0000E-01 6.37347E-05 0.0165

4.0000E-01 2.58753E-04 0.0069

9.0000E-01 2.75932E-04 0.0059

1.4000E+00 1.71299E-04 0.0077

3.0000E+00 2.91938E-04 0.0057

2.0000E+01 1.63937E-04 0.0079

total 1.23068E-03 0.0019

SKIP 214 LINES IN OUTPUT

1tally 7 nps = 105235

tally type 7 track length estimate of fission heating. units mev/gram

tally for neutrons

number of histories used for normalizing tallies = 90000.00

masses

cell: 1

5.17571E+04

CHAPTER 5

KCODE

5-86 18 December 2000

cell 1

energy

1.0000E-07 0.00000E+00 0.0000

4.0000E-07 0.00000E+00 0.0000

1.0000E-06 0.00000E+00 0.0000

3.0000E-06 0.00000E+00 0.0000

1.0000E-05 0.00000E+00 0.0000

3.0000E-05 0.00000E+00 0.0000

1.0000E-04 4.04563E-08 1.0000

5.5000E-04 8.43930E-08 0.5907

3.0000E-03 5.10073E-07 0.2340

1.7000E-02 4.80318E-06 0.0661

1.0000E-01 6.80677E-05 0.0165

4.0000E-01 2.76420E-04 0.0069

9.0000E-01 2.94357E-04 0.0059

1.4000E+00 1.83525E-04 0.0077

3.0000E+00 3.14188E-04 0.0057

2.0000E+01 1.77808E-04 0.0079

total 1.31980E-03 0.0019

SKIP 217 LINES IN OUTPUT

N41 1tally 14 nps = 105235

+ total total fission neutrons (track-lenght Keff), total loss to (n,xn)

total neutron absorptions,total fission,and neutron heating (mev/gram)

tally type 4 track length estimate of particle flux.

tally for neutrons

number of histories used for normalizing tallies = 90000.00

multiplier bin 1: 1.32534E+02 10 -6 -7

multiplier bin 2: 1.32534E+02 10 16 : 17

multiplier bin 3: 1.32534E+02 10 -2

multiplier bin 4: 1.32534E+02 10 -6

multiplier bin 5: 2.56069E-03 10 1 -4

volumes

cell: 1

2.76185E+03

cell 1

mult bin: 1 2 3 4 5

energy

1.0000E-07 0.00000E+00 0.0000 0.00000E+00 0.0000 0.00000E+00 0.0000 0.00000E+00 0.0000 0.00000E+00 0.0000

4.0000E-07 0.00000E+00 0.0000 0.00000E+00 0.0000 0.00000E+00 0.0000 0.00000E+00 0.0000 0.00000E+00 0.0000

1.0000E-06 0.00000E+00 0.0000 0.00000E+00 0.0000 0.00000E+00 0.0000 0.00000E+00 0.0000 0.00000E+00 0.0000

3.0000E-06 0.00000E+00 0.0000 0.00000E+00 0.0000 0.00000E+00 0.0000 0.00000E+00 0.0000 0.00000E+00 0.0000

1.0000E-05 0.00000E+00 0.0000 0.00000E+00 0.0000 0.00000E+00 0.0000 0.00000E+00 0.0000 0.00000E+00 0.0000

3.0000E-05 0.00000E+00 0.0000 0.00000E+00 0.0000 0.00000E+00 0.0000 0.00000E+00 0.0000 0.00000E+00 0.0000

1.0000E-04 2.81549E-05 1.0000 0.00000E+00 0.0000 8.48061E-06 1.0000 1.15769E-05 1.0000 3.78543E-08 1.0000

5.5000E-04 5.87319E-05 0.5907 0.00000E+00 0.0000 8.17629E-06 0.6181 2.41496E-05 0.5907 7.88876E-08 0.5905

3.0000E-03 3.54977E-04 0.2340 0.00000E+00 0.0000 7.44051E-05 0.2606 1.45961E-04 0.2340 4.77224E-07 0.2340

18 December 2000 5-87

CHAPTER 5

KCODE

1.7000E-02 3.34170E-03 0.0661 0.00000E+00 0.0000 5.05434E-04 0.0660 1.37446E-03 0.0661 4.49425E-06 0.0661

1.0000E-01 4.72587E-02 0.0165 0.00000E+00 0.0000 6.11329E-03 0.0166 1.94780E-02 0.0165 6.37382E-05 0.0165

4.0000E-01 1.94145E-01 0.0069 0.00000E+00 0.0000 1.68603E-02 0.0071 7.91001E-02 0.0069 2.58767E-04 0.0069

9.0000E-01 2.09957E-01 0.0059 0.00000E+00 0.0000 1.13657E-02 0.0060 8.42378E-02 0.0059 2.75947E-04 0.0059

1.4000E+00 1.33697E-01 0.0077 0.00000E+00 0.0000 4.79891E-03 0.0077 5.25212E-02 0.0077 1.71309E-04 0.0077

3.0000E+00 2.38680E-01 0.0057 0.00000E+00 0.0000 4.34611E-03 0.0060 8.99118E-02 0.0057 2.91954E-04 0.0057

2.0000E+01 1.52744E-01 0.0080 2.56445E-03 0.0234 5.41123E-04 0.0093 5.08832E-02 0.0079 1.63946E-04 0.0079

total 9.80264E-01 0.0019 2.56445E-03 0.0234 4.46220E-02 0.0036 3.77688E-01 0.0019 1.23075E-03 0.0019

SKIP 214 LINES IN OUTPUT

N42 1tally 34 nps = 105235

tally type 4 track length estimate of particle flux.

tally for neutrons

number of histories used for normalizing tallies = 90000.00

multiplier bin 1: -1.00000E+00 10 -6 -7

multiplier bin 2: -1.00000E+00 10 16 : 17

multiplier bin 3: -1.00000E+00 10 -2

multiplier bin 4: -1.00000E+00 10 -6

multiplier bin 5: -1.93210E-05 10 1 -4

volumes

cell: 1

1.00000E+00

energy bin: 0. to 2.00000E+01

cell: 1

mult bin

1 1.04614E+00 0.0019

2 2.73727E-03 0.0234

3 4.76183E-02 0.0036

4 4.03067E-01 0.0019

5 1.31345E-03 0.0019

SKIP 214 LINES IN OUTPUT

1status of the statistical checks used to form confidence intervals for the mean for each tally bin

N43

tally result of statistical checks for the tfc bin (the first check not passed is listed) and error magnitude check for all bins

1 passed the 10 statistical checks for the tally fluctuation chart bin result

missed all bin error check: 17 tally bins had 8 bins with zeros and 2 bins with relative errors exceeding 0.10

6 passed the 10 statistical checks for the tally fluctuation chart bin result

missed all bin error check: 17 tally bins had 6 bins with zeros and 3 bins with relative errors exceeding 0.10

7 passed the 10 statistical checks for the tally fluctuation chart bin result

missed all bin error check: 17 tally bins had 6 bins with zeros and 3 bins with relative errors exceeding 0.10

14 passed the 10 statistical checks for the tally fluctuation chart bin result

missed all bin error check: 85 tally bins had 39 bins with zeros and 12 bins with relative errors exceeding 0.10

34 passed the 10 statistical checks for the tally fluctuation chart bin result

passed all bin error check: 5 tally bins all have relative errors less than 0.10 with no zero bins

for perturbation 1

1 missed 1 of 10 tfc bin checks: the slope of decrease of largest tallies is less than the minimum acceptable value of 3.0

missed all bin error check: 17 tally bins had 8 bins with zeros and 2 bins with relative errors exceeding 0.10

CHAPTER 5

KCODE

5-88 18 December 2000