Brookhaven2009 ENDF 6 Formats Manual

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CSEWG Document ENDF-102
BNL-90365-2009
Report BNL-XXXXX-2009

ENDF-6 Formats Manual
Data Formats and Procedures for the Evaluated Nuclear Data File
ENDF/B-VI and ENDF/B-VII

Written by the Members of the Cross Sections Evaluation Working Group

Last Revision Edited by
M. Herman and A. Trkov

June 2009

National Nuclear Data Center
Brookhaven National Laboratory
Upton, NY 11973-5000
www.nndc.bnl.gov
Notice: This manuscript has been authored by employees of Brookhaven Science Associates, LLC under Contract No. DEAC02-98CH10886 with the U.S. Department of Energy. The publisher by accepting the manuscript for publication acknowledges
that the United States Government retains a non-exclusive, paid-up, irrevocable, world-wide license to publish or reproduce
the published form of this manuscript, or allow others to do so, for United States Government purposes.

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, nor any of their contractors, subcontractors, or their employees, makes
any warranty, express or implied, or assumes any legal liability or responsibility for the accuracy, completeness, or any third party’s use or the results of such use 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 or its contractors or subcontractors. The views and opinions of authors
expressed herein do not necessarily state or reflect those of the United States Government
or any agency thereof.

PREFACE
In December 2006, the Cross Section Evaluation Working Group (CSEWG) of the United
States released the new ENDF/B-VII.0 library. This represented considerable achievement
as it was the 1st major release since 1990 when ENDF/B-VI has been made publicly
available. The two libraries have been released in the same format, ENDF-6, which has
been originally developed for the ENDF/B-VI library. In the early stage of work on the
VII-th generation of the library CSEWG made important decision to use the same formats.
This decision was adopted even though it was argued that it would be timely to modernize
the formats and several interesting ideas were proposed. After careful deliberation CSEWG
concluded that actual implementation would require considerable resources needed to
modify processing codes and to guarantee high quality of the files processed by these codes.
In view of this the idea of format modernization has been postponed and ENDF-6 format
was adopted for the new ENDF/B-VII library.
In several other areas related to ENDF we made our best to move beyond established
tradition and achieve maximum modernization. Thus, the “Big Paper” on ENDF/B-VII.0
has been published, also in December 2006, as the Special Issue of Nuclear Data Sheets
107 (1996) 2931-3060. The new web retrieval and plotting system for ENDF-6 formatted
data, Sigma, was developed by the NNDC and released in 2007. Extensive paper has been
published on the advanced tool for nuclear reaction data evaluation, EMPIRE, in 2007.
This effort was complemented with release of updated set of ENDF checking codes in 2009.
As the final item on this list, major revision of ENDF-6 Formats Manual was made. This
work started in 2006 and came to fruition in 2009 as documented in the present report.
I would like to thank Mike Herman, the ENDF library data manager at the NNDC, as
well as Andrej Trkov, IJS Ljubljana, for their dedicated effort on the revised manual. They
converted it into LaTeX, carefully checked and implemented updates since the latest 2005
release and consolidated Manual into fully updated and revised version. Many thanks are
due to numerous CSEWG members and international colleagues for their useful comments
and observations on several draft versions. It should be stressed that even though the
Manual is maintained by the United States, active contribution from the international data
community is essential for achieving its high quality.
It is my expectation that this major revision will be embraced by the data evaluators as
well as members of numerous user communities worldwide.

Upton, June 2009

Pavel Obložinský
CSEWG chair until January 2009

Contents
Preface

I

0 INTRODUCTION
0.1 Introduction to the ENDF-6 Format . . . . . . . . . . . . . .
0.2 Philosophy of the ENDF System . . . . . . . . . . . . . . . .
0.2.1 Evaluated Data . . . . . . . . . . . . . . . . . . . . .
0.2.2 ENDF/B Library . . . . . . . . . . . . . . . . . . . .
0.2.3 Choices of Data . . . . . . . . . . . . . . . . . . . . .
0.2.4 Experimental Data Libraries . . . . . . . . . . . . . .
0.2.5 Processing Codes . . . . . . . . . . . . . . . . . . . . .
0.2.6 Testing . . . . . . . . . . . . . . . . . . . . . . . . . .
0.2.7 Documentation . . . . . . . . . . . . . . . . . . . . . .
0.3 General Description of the ENDF System . . . . . . . . . . .
0.3.1 Library (NLIB, NVER, LREL, NFOR) . . . . . . . .
0.3.2 Incident Particles and Data Types (NSUB) . . . . . .
0.3.2.1 Incident-Neutron Data (NSUB=10) . . . . .
0.3.2.2 Thermal Neutron Scattering (NSUB=12) . .
0.3.2.3 Fission Product Yield Data . . . . . . . . . .
0.3.2.4 Radioactive Decay Data (NSUB=4) . . . . .
0.3.2.5 Charged-Particle (NSUB≥10010) and
Photo-Nuclear (NSUB=0) Sub-libraries . . .
0.3.2.6 Photo-Atomic Interaction Data (NSUB=3) .
0.3.2.7 Electro-Atomic Interaction Data (NSUB=113)
0.3.2.8 Atomic Relaxation Data (NSUB=6) . . . . .
0.4 Contents of an ENDF Evaluation . . . . . . . . . . . . . . . .
0.4.1 Material (MAT, MOD) . . . . . . . . . . . . . . . . .
0.4.2 ENDF Data Blocks (Files - MF) . . . . . . . . . . . . .
0.4.3 Reaction Nomenclature (MT) . . . . . . . . . . . . . .
0.4.3.1 Elastic Scattering . . . . . . . . . . . . . . .
0.4.3.2 Simple Single Particle Reactions . . . . . . .
0.4.3.3 Simple Multi-Particle Reactions . . . . . . .
0.4.3.4 Breakup Reactions . . . . . . . . . . . . . . .
0.4.3.5 Complex Reactions . . . . . . . . . . . . . .
0.4.3.6 Radiative Capture . . . . . . . . . . . . . . .

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CONTENTS

0.5

0.6

0.7

0.4.3.7 Fission . . . . . . . . . . . . . . . . . . . .
0.4.3.8 Nonelastic Reaction for Photon Production
0.4.3.9 Special Production Cross Sections . . . . .
0.4.3.10 Auxiliary MT Numbers . . . . . . . . . . .
0.4.3.11 Sum Rules for ENDF . . . . . . . . . . . .
Representation of Data . . . . . . . . . . . . . . . . . . . .
0.5.1 Definitions and Conventions . . . . . . . . . . . . . .
0.5.1.1 Atomic Masses Versus Nuclear Masses . . .
0.5.2 Interpolation Laws . . . . . . . . . . . . . . . . . . .
0.5.2.1 One-dimensional Interpolation Schemes . .
0.5.2.2 Two-Dimensional Interpolation Schemes . .
General Description of Data Formats . . . . . . . . . . . . .
0.6.1 Structure of an ENDF Data Tape . . . . . . . . . .
0.6.2 Symbol Nomenclature . . . . . . . . . . . . . . . . .
0.6.3 Types of Records . . . . . . . . . . . . . . . . . . . .
0.6.3.1 TEXT Records . . . . . . . . . . . . . . . .
0.6.3.2 CONT Records . . . . . . . . . . . . . . . .
0.6.3.3 HEAD Records . . . . . . . . . . . . . . . .
0.6.3.4 END Records . . . . . . . . . . . . . . . . .
0.6.3.5 DIR Records . . . . . . . . . . . . . . . . .
0.6.3.6 LIST Records . . . . . . . . . . . . . . . .
0.6.3.7 TAB1 Records . . . . . . . . . . . . . . . .
0.6.3.8 TAB2 Records . . . . . . . . . . . . . . . .
0.6.3.9 INTG records . . . . . . . . . . . . . . . .
ENDF Documentation . . . . . . . . . . . . . . . . . . . . .

1 File 1: GENERAL INFORMATION
1.1 Descriptive Data and Directory (MT=451) . . . . . . . .
1.1.1 Formats . . . . . . . . . . . . . . . . . . . . . . . .
1.1.2 Procedures . . . . . . . . . . . . . . . . . . . . . .
1.2 Number of Neutrons per Fission, ν, (MT=452) . . . . . .
1.2.1 Formats . . . . . . . . . . . . . . . . . . . . . . . .
1.2.2 Procedures . . . . . . . . . . . . . . . . . . . . . .
1.3 Delayed Neutron Data, ν d , (MT=455) . . . . . . . . . . .
1.3.1 Formats . . . . . . . . . . . . . . . . . . . . . . . .
1.3.2 Procedures . . . . . . . . . . . . . . . . . . . . . .
1.4 Number of Prompt Neutrons per Fission, ν p , (MT=456) .
1.4.1 Formats . . . . . . . . . . . . . . . . . . . . . . . .
1.4.2 Procedures . . . . . . . . . . . . . . . . . . . . . .
1.5 Components of Energy Release Due to Fission (MT=458)
1.5.1 Formats . . . . . . . . . . . . . . . . . . . . . . . .
1.5.2 Procedures . . . . . . . . . . . . . . . . . . . . . .
1.6 Delayed Photon Data (MT=460) . . . . . . . . . . . . . .
1.6.1 Formats . . . . . . . . . . . . . . . . . . . . . . . .
1.6.1.1 Discrete Representation (LO=1) . . . . .
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CONTENTS
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2 File 2: RESONANCE PARAMETERS
2.1 General Description . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
2.2 Resolved Resonance Parameters (LRU=1) . . . . . . . . . . . . . . . . . . .
2.2.1 Formats . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
2.2.1.1 SLBW and MLBW (LRF=1 or 2) . . . . . . . . . . . . . .
2.2.1.2 Reich-Moore (LRF=3) . . . . . . . . . . . . . . . . . . . .
2.2.1.3 Adler-Adler (LRF=4) . . . . . . . . . . . . . . . . . . . . .
2.2.1.4 General R-Matrix (LRF=5) . . . . . . . . . . . . . . . . . .
2.2.1.5 Hybrid R-function (LRF=6) . . . . . . . . . . . . . . . . .
2.2.1.6 R-Matrix Limited Format (LRF=7) . . . . . . . . . . . . .
2.3 Unresolved Resonance Parameters (LRU=2) . . . . . . . . . . . . . . . . . .
2.3.1 Formats . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
2.4 Procedures for the Resolved and Unresolved Resonance Regions . . . . . . .
2.4.1 Abbreviations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
2.4.2 Interpolation in the Unresolved Resonance Region (URR) . . . . . .
2.4.3 Unresolved Resonances in the Resolved Resonance Range . . . . . .
2.4.4 Energy Range Boundary Problems . . . . . . . . . . . . . . . . . . .
2.4.5 Numerical Integration Procedures in the URR . . . . . . . . . . . . .
2.4.6 Doppler-broadening of File 3 Background Cross Sections . . . . . . .
2.4.7 Assignment of Unknown J-Values . . . . . . . . . . . . . . . . . . .
2.4.8 Competitive Width in the Resonance Region . . . . . . . . . . . . .
2.4.8.1 Resolved Region . . . . . . . . . . . . . . . . . . . . . . . .
2.4.8.2 Unresolved Region . . . . . . . . . . . . . . . . . . . . . . .
2.4.9 Negative Cross Sections in the Resolved Resonance Region . . . . . .
2.4.9.1 In the SLBW Formalism: . . . . . . . . . . . . . . . . . . .
2.4.9.2 In the MLBW Formalism: . . . . . . . . . . . . . . . . . .
2.4.9.3 In the R-matrix, Reich-Moore, and R-function Formalisms:
2.4.9.4 In the Adler-Adler Formalism: . . . . . . . . . . . . . . . .
2.4.10 Negative Cross Sections in the Unresolved Resonance Region . . . .
2.4.11 Use of Two Nuclear Radii . . . . . . . . . . . . . . . . . . . . . . . .
2.4.12 The Multilevel Adler-Gauss Formula for MLBW . . . . . . . . . . .
2.4.13 Notes on the Adler-Adler Formalism . . . . . . . . . . . . . . . . . .
2.4.14 Multi-Level Versus Single-Level Formalisms in the Resolved and Unresolved Resonance Regions . . . . . . . . . . . . . . . . . . . . . . .
2.4.14.1 In the Resolved Resonance Region: . . . . . . . . . . . . .
2.4.14.2 In the Unresolved Resonance Region: . . . . . . . . . . . .
2.4.15 Preferred Formalisms for Evaluating Data . . . . . . . . . . . . . . .
2.4.16 Degrees of Freedom for Unresolved Resonance Parameters . . . . . .
2.4.17 Procedures for the Unresolved Resonance Region . . . . . . . . . . .

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1.6.2

1.6.1.2 Continuous Representation (LO=2)
Procedures . . . . . . . . . . . . . . . . . . .
1.6.2.1 Discrete Representation (LO=1) .
1.6.2.2 Continuous Representation (LO=2)

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CONTENTS
2.4.18 Procedures for Computing Angular Distributions in the Resolved Resonance Range . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
2.4.18.1 Background . . . . . . . . . . . . . . . . . . . . . . . . . .
2.4.18.2 Further Considerations . . . . . . . . . . . . . . . . . . . .
2.4.18.3 Summary of Recommendations for Evaluation Procedures .
2.4.19 Completeness and Convergence of Channel Sums . . . . . . . . . . .
2.4.20 Channel Spin and Other Considerations . . . . . . . . . . . . . . . .
3 File
3.1
3.2
3.3

3: REACTION CROSS SECTIONS
General Description . . . . . . . . . . . . . . . . . . . . . . . . . .
Formats . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
General Procedures . . . . . . . . . . . . . . . . . . . . . . . . . .
3.3.1 Cross Sections, Energy Ranges, and Thresholds . . . . . . .
3.3.2 Q-Values . . . . . . . . . . . . . . . . . . . . . . . . . . . .
3.3.3 Relationship Between File 3 and File 2 . . . . . . . . . . . .
3.4 Procedures for Incident Neutrons . . . . . . . . . . . . . . . . . . .
3.4.1 Total Cross Section (MT=1) . . . . . . . . . . . . . . . . .
3.4.2 Elastic Scattering Cross Section (MT=2) . . . . . . . . . .
3.4.3 Nonelastic Cross Section (MT=3) . . . . . . . . . . . . . .
3.4.4 Inelastic Scattering Cross Sections (MT=4,51-91) . . . . . .
3.4.5 Fission (MT=18,19-21,38) . . . . . . . . . . . . . . . . . . .
3.4.6 Charged-Particle Emission to Discrete and Continuum
(MT=600-849) . . . . . . . . . . . . . . . . . . . . . . . . .
3.5 Procedures for Incident Charged Particles and Photons . . . . . . .
3.5.1 Total Cross Sections . . . . . . . . . . . . . . . . . . . . . .
3.5.2 Elastic Scattering Cross Sections . . . . . . . . . . . . . . .
3.5.3 Inelastic Scattering Cross Sections . . . . . . . . . . . . . .
3.5.4 Stopping Power . . . . . . . . . . . . . . . . . . . . . . . .

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4 File 4: ANGULAR DISTRIBUTIONS
4.1 General Description . . . . . . . . . . . . . . . . . . . . . . . . . . . .
4.2 Formats . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
4.2.1 Legendre Polynomial Coefficients (LTT=1, LI=0) . . . . . . . .
4.2.2 Tabulated Probability Distributions (LTT=2, LI=0) . . . . . .
4.2.3 Purely Isotropic Angular Distributions (LTT=0, LI=1) . . . . .
4.2.4 Angular Distribution Over Two Energy Ranges (LTT=3, LI=0)
4.3 Procedures . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
4.4 Procedures for Specific Reactions . . . . . . . . . . . . . . . . . . . . .
4.4.1 Elastic Scattering (MT=2) . . . . . . . . . . . . . . . . . . . .
4.4.2 Discrete Channel Two-Body Reactions . . . . . . . . . . . . . .
4.4.3 Other Particle-Producing Reactions . . . . . . . . . . . . . . .

iv

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87
87
88
89
89
91
93
93
93
94
94
95
95
96
97
97
98
98
99
99
100
100
100
100
101
102
102
104
105
105
105
106
106
107
107
108
109

CONTENTS
5 File 5: ENERGY DISTRIBUTIONS
5.1 General Description . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
5.1.1 Secondary Energy Distribution Laws . . . . . . . . . . . . . . . . . .
5.1.1.1 Arbitrary Tabulated Function (LF=1) . . . . . . . . . . . .
5.1.1.2 General Evaporation Spectrum (LF=5) . . . . . . . . . . . .
5.1.1.3 Simple Maxwellian Fission Spectrum (LF=7) . . . . . . . .
5.1.1.4 Evaporation Spectrum (LF=9) . . . . . . . . . . . . . . . .
5.1.1.5 Energy-Dependent Watt Spectrum (LF=11) . . . . . . . . .
5.1.1.6 Energy-Dependent Fission Neutron Spectrum (Madland and
Nix) (LF=12) . . . . . . . . . . . . . . . . . . . . . . . . . .
5.2 Formats . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
5.2.1 Arbitrary Tabulated Function (LF=1) . . . . . . . . . . . . . . . . .
5.2.2 General Evaporation Spectrum (LF=5) . . . . . . . . . . . . . . . .
5.2.3 Simple Maxwellian Fission Spectrum (LF=7) . . . . . . . . . . . . .
5.2.4 Evaporation Spectrum (LF=9) . . . . . . . . . . . . . . . . . . . . .
5.2.5 Energy-Dependent Watt Spectrum (LF=11) . . . . . . . . . . . . . .
5.2.6 Energy-Dependent Fission Neutron Spectrum
(Madland and Nix) (LF=12) . . . . . . . . . . . . . . . . . . . . . .
5.3 Procedures . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
5.4 Additional Procedures . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
5.4.1 General Comments . . . . . . . . . . . . . . . . . . . . . . . . . . . .
5.4.2 Tabulated Distributions (LF=1) . . . . . . . . . . . . . . . . . . . .
5.4.3 Maxwellian Spectrum (LF=7) . . . . . . . . . . . . . . . . . . . . .
5.4.4 Evaporation Spectrum (LF=9) . . . . . . . . . . . . . . . . . . . . .
5.4.5 Watt Spectrum (LF=11) . . . . . . . . . . . . . . . . . . . . . . . .
5.4.6 Madland-Nix Spectrum (LF=12) . . . . . . . . . . . . . . . . . . . .
5.4.7 Selection of the Integration Constant, U . . . . . . . . . . . . . . . .
5.4.8 Multiple Nuclear Temperatures . . . . . . . . . . . . . . . . . . . . .
5.4.9 Average Energy for a Distribution . . . . . . . . . . . . . . . . . . .
5.4.10 Additional Procedures for Energy-Dependent
Fission Neutron Spectrum (Madland and Nix) . . . . . . . . . . . .
5.4.10.1 Region I (a > EF , b > EF ) . . . . . . . . . . . . . . . . . . .
5.4.10.2 Region II (a < EF , b < EF ) . . . . . . . . . . . . . . . . . .
5.4.10.3 Region III (a < EF , b > EF ) . . . . . . . . . . . . . . . . . .

110
110
111
112
112
112
112
113

6 File 6: PRODUCT ENERGY-ANGLE DISTRIBUTIONS
6.1 General Description . . . . . . . . . . . . . . . . . . . . . . . . . . . .
6.2 Formats . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
6.2.1 Unknown Distribution (LAW=0) . . . . . . . . . . . . . . . . .
6.2.2 Continuum Energy-Angle Distributions (LAW=1) . . . . . . .
6.2.2.1 Legendre Coefficients Representation (LANG=1) . . .
6.2.2.2 Kalbach-Mann Systematics Representation (LANG=2)
6.2.2.3 Tabulated Function Representation (LANG=11-15) . .
6.2.3 Discrete Two-Body Scattering (LAW=2) . . . . . . . . . . . .
6.2.4 Isotropic Discrete Emission (LAW=3) . . . . . . . . . . . . . .

123
123
124
125
126
127
127
130
131
132

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113
114
115
115
116
116
116
116
116
117
117
118
118
118
118
119
119
120
120
121
121
122
122

CONTENTS

6.3

6.2.5 Discrete Two-Body Recoils (LAW=4) . . . .
6.2.6 Charged-Particle Elastic Scattering (LAW=5)
6.2.7 N-Body Phase-Space Distributions (LAW=6)
6.2.8 Laboratory Angle-Energy Law (LAW=7) . .
Procedures . . . . . . . . . . . . . . . . . . . . . . .
6.3.1 Elastic Scattering . . . . . . . . . . . . . . .
6.3.2 Photons . . . . . . . . . . . . . . . . . . . . .
6.3.3 Particles . . . . . . . . . . . . . . . . . . . .
6.3.4 Neutron Emission . . . . . . . . . . . . . . .
6.3.5 Recoil Distributions . . . . . . . . . . . . . .
6.3.6 Elements as Targets . . . . . . . . . . . . . .
6.3.7 CM versus LAB Coordinate System . . . . .
6.3.8 Phase Space . . . . . . . . . . . . . . . . . .

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7 File 7: THERMAL NEUTRON SCATTERING LAW DATA
7.1 General Description . . . . . . . . . . . . . . . . . . . . . . . .
7.2 Coherent Elastic Scattering . . . . . . . . . . . . . . . . . . . .
7.2.1 Formats for Coherent Elastic Scattering . . . . . . . . .
7.2.2 Procedures for Coherent Elastic Scattering . . . . . . . .
7.3 Incoherent Elastic Scattering . . . . . . . . . . . . . . . . . . .
7.3.1 Format for Incoherent Elastic Scattering . . . . . . . . .
7.3.2 Procedures for Incoherent Elastic Scattering . . . . . . .
7.4 Incoherent Inelastic Scattering . . . . . . . . . . . . . . . . . .
7.4.1 Formats for Incoherent Inelastic Scattering . . . . . . .
7.4.2 Procedures for Incoherent Inelastic Scattering . . . . . .
8 File 8: DECAY AND FISSION PRODUCT
8.1 General Description . . . . . . . . . . . . .
8.2 Radioactive Nuclide Production . . . . . . .
8.2.1 Formats . . . . . . . . . . . . . . . .
8.2.2 Procedures . . . . . . . . . . . . . .
8.3 Fission Product Yield Data
(MT=454, MT=459) . . . . . . . . . . . . .
8.3.1 Formats . . . . . . . . . . . . . . . .
8.3.2 Procedures . . . . . . . . . . . . . .
8.4 Radioactive Decay Data (MT=457) . . . .
8.4.1 Formats . . . . . . . . . . . . . . . .
8.4.2 Procedures . . . . . . . . . . . . . .
9 File
9.1
9.2
9.3

9: MULTIPLICITIES
General Description . .
Formats . . . . . . . . .
Procedures . . . . . . .

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132
132
136
137
138
138
138
141
141
141
142
142
143

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144
144
144
145
146
146
146
147
147
150
151

YIELDS
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153
153
153
155
156

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157
158
158
159
163
164

OF RADIOACTIVE PRODUCTS
. . . . . . . . . . . . . . . . . . . . . . . . . . . . .
. . . . . . . . . . . . . . . . . . . . . . . . . . . . .
. . . . . . . . . . . . . . . . . . . . . . . . . . . . .

168
168
168
169

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CONTENTS
10 File
10.1
10.2
10.3

10: PRODUCTION CROSS
General Description . . . . . .
Formats . . . . . . . . . . . . .
Procedures . . . . . . . . . . .

SECTIONS
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FOR RADIONUCLIDES
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. . . . . . . . . . . . . . . . .
. . . . . . . . . . . . . . . . .

171
171
171
172

11 File 11: GENERAL COMMENTS ON PHOTON PRODUCTION
174
11.1 General Description . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 174
12 File 12: PHOTON PRODUCTION YIELD DATA
12.1 General Description . . . . . . . . . . . . . . . . . . . .
12.2 Formats . . . . . . . . . . . . . . . . . . . . . . . . . . .
12.2.1 Option 1: Multiplicities (LO=1) . . . . . . . . .
12.2.2 Option 2: Transition Probability Arrays (LO=2)
12.3 Procedures . . . . . . . . . . . . . . . . . . . . . . . . .
13 File
13.1
13.2
13.3
13.4

13: PHOTON PRODUCTION CROSS
General Description . . . . . . . . . . . . .
Formats . . . . . . . . . . . . . . . . . . . .
Procedures . . . . . . . . . . . . . . . . . .
Preferred Representations . . . . . . . . . .

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178
178
178
179
181
182

SECTIONS
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184
184
184
186
187

14 File 14: PHOTON ANGULAR DISTRIBUTIONS
14.1 General Description . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
14.2 Formats . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
14.2.1 Isotropic Distribution (LI=1) . . . . . . . . . . . . . . . . . . . . . .
14.2.2 Anisotropic Distribution with Legendre Coefficient Representation
(LI=0, LTT=1) . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
14.2.3 Anisotropic Distribution with Tabulated Angular Distributions (LI=0,
LTT=2) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
14.3 Procedures . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

188
188
189
190

15 File
15.1
15.2
15.3

15: CONTINUOUS PHOTON ENERGY SPECTRA
General Description . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
Formats . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
Procedures . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

193
193
194
195

23 File
23.1
23.2
23.3
23.4

23: PHOTON INTERACTION
General Comments on Photon Production
General Description . . . . . . . . . . . .
Formats . . . . . . . . . . . . . . . . . . .
Procedures . . . . . . . . . . . . . . . . .

196
196
196
197
197

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190
191
191

26 File 26:
SECONDARY DISTRIBUTIONS FOR PHOTO- AND
ELECTRO-ATOMIC DATA
198
26.1 General Description . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 198
26.2 Formats . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 199
vii

CONTENTS
26.2.1 Continuum Distribution (LAW=1) . . . . . . . . . . . . . . . . . . . 200
26.2.2 Two-Body Angular Distribution (LAW=2) . . . . . . . . . . . . . . 200
26.2.3 Energy Transfer for Excitation (LAW=8) . . . . . . . . . . . . . . . 200
27 File 27: ATOMIC FORM FACTORS
27.1 General Description . . . . . . . . . .
27.1.1 Incoherent Scattering . . . .
27.1.2 Cooherent Scattering . . . .
27.2 Formats . . . . . . . . . . . . . . . .
27.3 Procedures . . . . . . . . . . . . . .
28 File
28.1
28.2
28.3

28: ATOMIC RELAXATION
General Description . . . . . . .
Formats . . . . . . . . . . . . . .
Procedures . . . . . . . . . . . .

OR SCATTERING FUNCTIONS
. . . . . . . . . . . . . . . . . . . . . .
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201
201
201
202
202
203

DATA
204
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. . . . . . . . . . . . . . . . . . . . . . . . 205
. . . . . . . . . . . . . . . . . . . . . . . . 206

29 INTRODUCTION TO COVARIANCE FILES
210
29.1 General Comments . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 210
30 File 30. COVARIANCES OF MODEL PARAMETERS
30.1 General Comments . . . . . . . . . . . . . . . . . . . . . .
30.1.1 Definitions . . . . . . . . . . . . . . . . . . . . . .
30.1.2 Treatment of Various Data Types . . . . . . . . .
30.1.3 Multigrouped Sensitivities . . . . . . . . . . . . . .
30.2 Formats . . . . . . . . . . . . . . . . . . . . . . . . . . . .
30.2.1 Directory and Correspondence Table (MT=1) . . .
30.2.2 Covariance Matrix (MT=2) . . . . . . . . . . . . .
30.2.3 Sensitivities (MT=11-999) . . . . . . . . . . . . . .
30.3 Additional Procedures . . . . . . . . . . . . . . . . . . . .
30.3.1 Relation of MP-values to Physical Parameters . . .
30.3.2 Parameter Values . . . . . . . . . . . . . . . . . .
30.3.3 Eigenvalue Representation . . . . . . . . . . . . . .
30.3.4 Thinning of Sensitivity Information . . . . . . . .
30.3.5 Cross-file Correlations . . . . . . . . . . . . . . . .
30.4 Multigroup Applications of Parameter Covariances . . . .
31 File
31.1
31.2
31.3

31. COVARIANCES OF
General Comments . . . . .
Formats . . . . . . . . . . .
Procedures . . . . . . . . .

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214
214
215
216
217
217
218
220
220
221
221
221
222
222
222
223

FISSION ν
225
. . . . . . . . . . . . . . . . . . . . . . . . . . . 225
. . . . . . . . . . . . . . . . . . . . . . . . . . . 225
. . . . . . . . . . . . . . . . . . . . . . . . . . . 226

32 File 32. COVARIANCES OF RESONANCE PARAMETERS
32.1 General Comments . . . . . . . . . . . . . . . . . . . . . . . . . .
32.2 Formats . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
32.2.1 Compatible Resolved Resonance Format (LCOMP=0) . .
32.2.2 General Resolved Resonance Formats (LCOMP=1) . . . .
viii

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228
228
230
231
232

CONTENTS
32.2.2.1
32.2.2.2
32.2.2.3
32.2.2.4
32.2.2.5

SLBW and MLBW (LRF=1 or 2) . . . . . . . . . . . . . .
Reich-Moore (LRF=3) . . . . . . . . . . . . . . . . . . . .
Adler-Adler (LRF=4) . . . . . . . . . . . . . . . . . . . . .
R-Matrix Limited Format (LRF=7) . . . . . . . . . . . . .
Format for Long-Range Covariance Subsubsections
(LRU=1) . . . . . . . . . . . . . . . . . . . . . . . . . . . .
32.2.3 Resolved Resonance Compact Covariance Format (LCOMP=2) . . .
32.2.3.1 SLBW and MLBW (LRF=1 or 2) . . . . . . . . . . . . . .
32.2.3.2 Reich-Moore (LRF=3) . . . . . . . . . . . . . . . . . . . .
32.2.3.3 R-Matrix Limited Format (LRF=7) . . . . . . . . . . . . .
32.2.4 Unresolved Resonance Format (LRU=2) . . . . . . . . . . . . . . . .
32.3 Procedures . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
33 File 33, COVARIANCES OF NEUTRON CROSS SECTIONS
33.1 General Comments . . . . . . . . . . . . . . . . . . . . . . . . . .
33.2 Formats . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
33.2.1 Subsections . . . . . . . . . . . . . . . . . . . . . . . . . .
33.2.2 Sub-Subsections . . . . . . . . . . . . . . . . . . . . . . .
33.2.2.1 NC-type Sub-Subsections . . . . . . . . . . . . .
33.2.2.2 NI-type Sub-Subsections . . . . . . . . . . . . . .
33.2.3 Lumped Reaction Covariances . . . . . . . . . . . . . . .
33.3 Procedures . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
33.3.1 Ordering of Sections, Subsections and Sub-Subsections . .
33.3.2 Completeness . . . . . . . . . . . . . . . . . . . . . . . . .
33.3.3 Other Procedures . . . . . . . . . . . . . . . . . . . . . .
33.3.4 Examples . . . . . . . . . . . . . . . . . . . . . . . . . . .
33.3.4.1 Use of LTY=1 and LTY=2 NC-type Subsections
33.3.4.2 Use of LTY=0, NC-type Sub-Subsections . . . .

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232
234
234
234
235
237
241
242
243
244
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249
249
250
251
252
252
257
263
263
263
265
266
269
269
272

34 File
34.1
34.2
34.3

34. COVARIANCES FOR
General Comments . . . . . .
Formats . . . . . . . . . . . .
Procedures . . . . . . . . . .

ANGULAR
. . . . . . . .
. . . . . . . .
. . . . . . . .

35 File
35.1
35.2
35.3

35. COVARIANCES FOR
General Comments . . . . . .
Formats . . . . . . . . . . . .
Procedures . . . . . . . . . .

ENERGY DISTRIBUTIONS
280
. . . . . . . . . . . . . . . . . . . . . . . . . . 280
. . . . . . . . . . . . . . . . . . . . . . . . . . 280
. . . . . . . . . . . . . . . . . . . . . . . . . . 282

40 File
40.1
40.2
40.3
40.4
40.5

40. COVARIANCES FOR RADIONUCLIDE PRODUCTION
General Comments . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
Formats . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
Procedures . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
Ordering of Sections, Subsections, Sub-subsections, and Sub-sub-subsections
Completeness . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
ix

DISTRIBUTIONS
276
. . . . . . . . . . . . . . . . . . 276
. . . . . . . . . . . . . . . . . . 276
. . . . . . . . . . . . . . . . . . 279

283
283
283
285
285
286

CONTENTS
Acknowledgments

287

APPENDIX

287

A Glossary

288

B Definition of Reaction Types
B.1 Reaction Type Numbers MT . . . . . . . . . . . . . . . . . . . . . . . . . . .
B.2 Residual Breakup Flags LR . . . . . . . . . . . . . . . . . . . . . . . . . . .
B.3 Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

303
303
313
314

C ZA Designations of Materials and MAT Numbers

315

D Resonance Region Formulae
D.1 The Resolved Resonance Region (LRU=1) . . . . . . . . . . . . . . . . . . .
D.1.1 Single-Level Breit-Wigner (SLBW, LRF=1) . . . . . . . . . . . . . .
D.1.1.1 Elastic Scattering Cross Sections . . . . . . . . . . . . . . .
D.1.1.2 Radiative Capture Cross Section . . . . . . . . . . . . . . .
D.1.1.3 Fission Cross Section . . . . . . . . . . . . . . . . . . . . .
D.1.1.4 The Competitive Reaction Cross Section . . . . . . . . . .
D.1.2 Multilevel Breit-Wigner (MLBW, LRF=2) . . . . . . . . . . . . . .
D.1.3 Reich-Moore (R-M, LRF=3) . . . . . . . . . . . . . . . . . . . . . .
D.1.4 Adler-Adler (AA, LRF=4) . . . . . . . . . . . . . . . . . . . . . . .
D.1.5 General R-Matrix (GRM, LRF=5) . . . . . . . . . . . . . . . . . . .
D.1.6 Hybrid R-Function (HRF, LRF=6) . . . . . . . . . . . . . . . . . . .
D.1.7 R-Matrix Limited Format (RML, LRF=7) . . . . . . . . . . . . . . .
D.1.7.1 Energy-Differential (Angle-Integrated) Cross Sections (NonCoulomb Channels) . . . . . . . . . . . . . . . . . . . . . .
D.1.7.2 Angular Distributions . . . . . . . . . . . . . . . . . . . . .
D.1.7.3 Kinematics for Angular Distributions of Elastic Scattering .
D.1.7.4 Spin and Angular Momentum Conventions . . . . . . . . .
D.1.7.5 Extensions to R-Matrix Theory . . . . . . . . . . . . . . .
D.1.7.6 Modifications for Charged Particles . . . . . . . . . . . . . .
D.2 The Unresolved Resonance Region (LRU=2) . . . . . . . . . . . . . . . . .
D.2.1 Cross Sections in the Unresolved Region . . . . . . . . . . . . . . . .
D.2.2 Definitions for the Unresolved Resonance Region . . . . . . . . . . .
D.2.2.1 Sums and Averages . . . . . . . . . . . . . . . . . . . . . .
D.2.2.2 Reduced Widths . . . . . . . . . . . . . . . . . . . . . . . .
D.2.2.3 Strength Function . . . . . . . . . . . . . . . . . . . . . . .
D.2.2.4 Level Spacings . . . . . . . . . . . . . . . . . . . . . . . . .
D.2.2.5 Gamma Widths . . . . . . . . . . . . . . . . . . . . . . . .
D.2.2.6 Degrees of Freedom . . . . . . . . . . . . . . . . . . . . . .
D.2.3 Equivalent Quantities in Sections D.1 and D.2 . . . . . . . . . . . .
D.3 The Competitive Width . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

317
317
317
317
318
318
318
322
323
325
326
326
326

x

329
331
333
334
334
335
337
337
339
339
339
340
343
344
344
345
345

CONTENTS
D.3.1 Penetrability Factor for the Competitive Width in the Resolved Resonance Region . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 345
D.3.2 Penetrability Factor for the Competitive Width in the Unresolved Resonance Region . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 347
D.3.3 Calculation of the Total Cross Section when a Competitive Reaction
is Specified . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 347
E Kinematic Formulas

349

F Summary of Important ENDF Rules
F.1 General . . . . . . . . . . . . . . . . .
F.2 File 2 - Resonance Parameters . . . .
F.3 File 3 - Tabulated Cross Sections . . .
F.4 Relation Between Files 2 and 3 . . . .
F.5 File 4 - Angular Distributions . . . . .
F.6 File 5 - Secondary Energy Distribution

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353
353
353
354
354
355
355

G Maximum Dimensions of ENDF Parameters

357

H Recommended Values of Physical Constants
H.1 Sources for Fundamental Constants . . . . . . . . .
H.2 Fundamental Constants and Derived Data . . . . .
H.3 Use of Fundamental Constants by Code Developers
H.4 Use of Fundamental Constants by Evaluators . . .

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359
359
359
360
360

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362
364
364
366
367
367
368
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369
370
370

I

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Historic perspective by Norman E. Holden
I.1 Introduction . . . . . . . . . . . . . . . . . . . . . . .
I.1.1 ENDF/B-I . . . . . . . . . . . . . . . . . . . .
I.1.2 ENDF/B-II . . . . . . . . . . . . . . . . . . .
I.1.3 ENDF/B-III . . . . . . . . . . . . . . . . . . .
I.1.4 ENDF/B-IV . . . . . . . . . . . . . . . . . . .
I.1.5 ENDF/B-V . . . . . . . . . . . . . . . . . . .
I.1.6 ENDF/B-VI . . . . . . . . . . . . . . . . . . .
I.1.7 ENDF/B-VII . . . . . . . . . . . . . . . . . .
I.2 Status of ENDF Versions and Formats . . . . . . . .
I.3 CSEWG . . . . . . . . . . . . . . . . . . . . . . . . .
I.3.1 Codes and Formats Subcommittee Leadership

xi

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List of Figures
1
2
3

Interpolation of a tabulated one-dimensional function for a case with NP=10,
NR=3. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
Interpolation between Two-Dimensional Panels. . . . . . . . . . . . . . . . .
Structure of an ENDF data tape. . . . . . . . . . . . . . . . . . . . . . . . .

6.1
6.2
6.3
6.4

Example of Charged-Particle Elastic Scattering Cross Section .
Example of Residual Cross Section for Elastic Scattering . . . .
Typical Level Structure for Proton-Induced Photon Production
Typical Energy Spectrum Showing Levels on a Continuum . . .

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24
25
28
139
139
140
143

11.1 Schematic diagram of a level scheme to illustrate gamma production. . . . . 176
E.1 Kinematics variables for two-particle reactions in the center-of-mass system. 350
E.2 Kinematics variables for two-particle reactions in the laboratory system. . . . 351

xii

List of Tables
1
2
3
4
5
6
7
8
9
10
11
12
13
14
15
16

Key parameters defining the hierarchy of entries in an ENDF file.
Currently defined NLIB numbers . . . . . . . . . . . . . . . . . .
Sub-library Numbers and Names . . . . . . . . . . . . . . . . . .
Definitions of File Types (MF) . . . . . . . . . . . . . . . . . . . .
Definitions of MT numbers of Simple Single Particle Reactions . .
Examples of simple single-particle reactions . . . . . . . . . . . .
Examples of simple multi-particle reactions . . . . . . . . . . . . .
Examples of breakup reactions . . . . . . . . . . . . . . . . . . . .
Identification of additional particles in File 6 . . . . . . . . . . . .
Some examples of LR values . . . . . . . . . . . . . . . . . . . . .
Definition of MT numbers related to fission . . . . . . . . . . . . .
Special MT numbers for particle production . . . . . . . . . . . .
Auxiliary MT numbers . . . . . . . . . . . . . . . . . . . . . . . .
ENDF sum rules for cross sections . . . . . . . . . . . . . . . . . .
Summary of ENDF units . . . . . . . . . . . . . . . . . . . . . . .
Definition of Interpolation Types . . . . . . . . . . . . . . . . . .

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6
7
7
13
14
15
15
17
18
18
19
20
21
21
22
23

33.1 Analogies Between File 33 Covariances Within One Section and Uncertainties
in a Hypothetical Experiment. . . . . . . . . . . . . . . . . . . . . . . . . . . 249
D.1 Hard sphere penetrability (penetration factor) P , level shift factor S, and
potential scattering phase shift φ for orbital angular momentum l, center of
mass momentum k, and channel radius ac , with ρ = kac . . . . . . . . . . . . 328
D.2 Spin and angular momentum conventions . . . . . . . . . . . . . . . . . . . . 334

xiii

Chapter 0
INTRODUCTION
This revision of ”ENDF-6 Formats Manual” pertains to version 6 of the ENDF formats.
The seventh version of the ENDF/B library, namely ENDF/B-VII, uses ENDF-6 formats.
The manual is based on the previous version of the document, revised in June 2005,
In some cases the older version from October 1991 was consulted in order to resolve some
ambiguities and errors that were introduced when the reference document was translated
from WordPerfect in Word format.
Below is a list of changes to the formats and procedures that appear in this edition. In
addition, many typographical error corrections and improvements to the text for clarity are
included. Users of this manual who note deficiencies or have suggestions are encouraged to
contact the National Nuclear Data Center.
Major updates to Manual for Revision June 2009

• Detailed review of the entire manual.
• Conversion to LATEX.
• Inclusion of recommendations from previous CSEWG Meetings that were not incorporated into the text.
• Inclusion of recommendations from the November 2008 CSEWG Meeting.

1

CHAPTER 0. INTRODUCTION

0.1

Introduction to the ENDF-6 Format

The ENDF formats and libraries are decided by the Cross Section Evaluation Working
Group (CSEWG), a cooperative effort of national laboratories, industry, and universities in
the U.S. and Canada, and are maintained by the National Nuclear Data Center (NNDC).
Earlier versions of the ENDF format provided representations for neutron cross sections
and distributions, photon production from neutron reactions, a limited amount of chargedparticle production from neutron reactions, photo-atomic interaction data, thermal neutron
scattering data, and radionuclide production and decay data (including fission products).
Version 6 of the formats (ENDF-6) allows higher incident energies, adds more complete
descriptions of the distributions of emitted particles, and provides for incident charged particles and photonuclear data by partitioning the ENDF library into sub-libraries. Decay
data, fission product yield data, thermal scattering data, and photo-atomic data have also
been formally placed in sub-libraries.

0.2

Philosophy of the ENDF System

The ENDF system was developed for the storage and retrieval of evaluated nuclear data to
be used for applications of nuclear technology. These applications control many features of
the system including the choice of materials to be included, the data used, the formats used,
and the testing required before a library is released. An important consequence of this is
that each evaluation must be complete for its intended application. If the required data
are not available for some particular reactions, the evaluator should supply them by using
systematics or nuclear models.
The ENDF system is logically divided into formats and procedures. Formats describe
how the data are arranged in the libraries and give the formulas needed to reconstruct
physical quantities such as cross sections and angular distributions from the parameters in
the library. Procedures are the more restrictive rules that specify what data types must
be included, which format can be used in particular circumstances, and so on. Procedures
are, generally, imposed by a particular organization, and the library sanctioned by the Cross
Section Evaluation Working Group (CSEWG) is referred to as ENDF/B. Other organizations
may use somewhat different procedures, if necessary, but they face the risk that their libraries
will not work with processing codes sanctioned by CSEWG.

0.2.1

Evaluated Data

An evaluation is the process of analyzing experimentally measured physical parameters
(such as cross sections), combining them with the predictions of nuclear model calculations,
and attempting to extract the true values of such parameters. Parameterization and reduction of the data to tabular form produces an evaluated data set. If a written description
of the preparation of a unique data set from the data sources is available, the data set is
referred to as a documented evaluation.

2

CHAPTER 0. INTRODUCTION

0.2.2

ENDF/B Library

The ENDF/B library maintained at the National Nuclear Data Center (NNDC) contains
the recommended evaluation for each material. Each material is as complete as possible;
however, completeness depends on the intended application. For example, when a user is
interested in performing a reactor physics calculation or in doing a shielding analysis, he
needs evaluated data for all neutron-induced reactions, covering the full range of incident
neutron energies, for each material in the system that he is analyzing. Also, the user
expects that the file will contain information such as the angular and energy distributions
for secondary neutrons. For another calculation, the user might only be interested in some
minor isotope activation, and would then be satisfied by an evaluation that contains only
reaction cross sections.
ENDF/B data sets are revised or replaced only after extensive review and testing. This
allows them to be used as standard reference data during the lifetime of the particular
ENDF/B library version.
There may be other libraries not endorsed by CSEWG that adopt the ENDF-6 format.
In the text that follows, any library in ENDF-6 format will be referred to as an ENDF
library.

0.2.3

Choices of Data

The data sets contained in the ENDF/B library are those chosen by CSEWG from evaluations submitted for review. The choice is made on the basis of requirements for applications,
conformance of the evaluation to the formats and procedures, and performance in testing.
The data set that represents a particular material may change when:
• new significant experimental results become available,
• integral tests show that the data give erroneous results, or
• users’ requirements indicate a need for more accurate data and/or better representations of the data for a particular material.
New or revised data sets are included in new releases of the ENDF/B library.

0.2.4

Experimental Data Libraries

The CSISRS/EXFOR library of experimentally measured nuclear reaction data is maintained internationally by the Nuclear Reaction Data Centers Network (NRDC)1 . In addition
to numerical data, the CSISRS/EXFOR library contains bibliographic information, as well
as details about the experiment (standards, renormalization, corrections, etc.).
At the beginning of the evaluation process the evaluator may retrieve the available experimental data for a particular material by direct access to the CSISRS/EXFOR database
via the World Wide Web2
1
2

”http://www-nds.iaea.org/nrdc.html”
”http://www.nndc.bnl.gov/exfor/” or ”http://www-nds.iaea.org/exfor/”.

3

CHAPTER 0. INTRODUCTION

0.2.5

Processing Codes

Once the evaluated data sets have been prepared in ENDF format, they can be converted
into forms appropriate for testing and actual applications using processing codes. Processing codes that generate pointwise and group-averaged cross sections for use in neutronics
calculations from an ENDF library are available. These codes3 4 include functions such as
resonance reconstruction, Doppler broadening, multigroup averaging, and/or rearrangement
into specified interface formats.
The basic data formats for an ENDF library have been developed in such a manner that
few constraints are placed on using the data as input to the codes that generate any of the
secondary libraries.

0.2.6

Testing

All ENDF/B evaluations go through at least some testing before being released as a part of
the library. Phase 1 testing uses a set of utility codes5 maintained by NNDC for checking and
visual inspection by a reviewer to assure that the evaluation conforms to the current formats
and procedures, takes advantage of the best recent data, and chooses format options suited
to the physics being represented. Phase 2 uses calculations of data testing ”benchmarks,”
when available, to evaluate the usefulness of the evaluation for actual applications.6 7 This
checking and testing process is a critically important part of the ENDF system.

0.2.7

Documentation

The system is documented by a set of ENDF reports (see Section 0.7) published by the
National Nuclear Data Center at Brookhaven National Laboratory. In addition, the current
status of the formats, procedures, evaluation process, and testing program is contained in
the Summary of the Meetings of the Cross Section Evaluation Working Group.

0.3

General Description of the ENDF System

The ENDF libraries are a collection of documented data evaluations stored in a defined
computer-readable format that can be used as the main input to nuclear data processing
programs. For this reason, the ENDF format has been constructed with the processing codes
in mind. The ENDF format uses 80-character records. For historic reasons the parameters
are defined in the old-fashioned form of FORTRAN variables (that is, integers start with
3

D.E. Cullen, PREPRO 2007 - 2007 ENDF/B Pre-processing Codes, report IAEA-NDS-39, Rev. 13,
March 2007
4
R.E. MacFarlane, D.W. Muir, The NJOY Nuclear Data Processing System, Version 91, LA-12740-M
(1994)
5
C. L. Dunford, ENDF Utility Codes, ”http://www.nndc.bnl.gov/nndcscr/endf/”
6
Cross Section Evaluation Working Group Benchmark Specifications, ENDF-202, 1974 (last updated
1991).
7
OECD/NEA: The International Criticality Safety Benchmark Evaluation Project (ICSBEP),
”http://icsbep.inel.gov/”.

4

CHAPTER 0. INTRODUCTION
the letters I, J, K, L, M, or N, and parameters starting with other letters represent real
numbers). A complete list of all the parameters defined for the ENDF-6 format can be
found in Appendix A (Glossary).
Each ENDF evaluation is identified by a set of key parameters organized into a hierarchy.
Following is a list of these parameters and their definitions.

0.3.1

Library (NLIB, NVER, LREL, NFOR)

A library is a collection of material evaluations from a recognized evaluation group. Each
of these collections is identified by an NLIB number. Currently defined NLIB numbers are
given in Table 2. The NVER, LREL and NFOR parameters that describe the version, release
number and format of the library are defined in Table 16.

0.3.2

Incident Particles and Data Types (NSUB)

The sub-library distinguishes between different incident particles and types of data using
NSUB = 10*IPART+ITYPE. In this formula, IPART=1000*Z+A defines the incident particle; use IPART=0 for incident photons or no incident particle (decay data), use IPART=11
for incident electrons, and IPART=0 for photo-atomic or electro-atomic data. The sublibraries allowed in ENDF-6 are listed in Table 3.
Sublibraries contain the data for different materials identified by MAT numbers. Each
material evaluation contains data blocks referred to as ”Files”. For the precise definition of
the material numbers and Files see Section 0.4.
0.3.2.1

Incident-Neutron Data (NSUB=10)

The procedures for describing neutron-induced reactions for ENDF-6 have been kept similar
to the procedures used for previous versions so that current evaluations can be carried over,
and in order to protect existing processing capabilities. The new features have most of
their impact at high energies (above 5-10 MeV) or low atomic weight (2 H, 9 Be), and include
improved energy-angle distributions, improved nuclear heating and damage capabilities,
improved charged-particle spectral data, and the use of R-matrix or R-function resonance
parameterization.
Each evaluation starts with a descriptive data and directory, File 1 (see Section 1.1). For
fissionable isotopes, sections of File 1 may be present that describe the number of neutrons
produced per fission and the energy release from fission.
A File 2 with resonance parameters is always given. For some materials, it may contain
only the effective scattering radius, and for other materials, it may contain complete sets of
resolved and/or unresolved resonance parameters.
A File 3 with tabulated cross sections is always given. The minimum required energy
range is from the threshold or from 10−5 eV up to 20 MeV, but higher energies are allowed.
There is a section for each important reaction or sum of reactions. The reaction MT-numbers
for these sections are chosen based on the emitted particles as described in Section 0.4.3
(Reaction Nomenclature). For resonance materials in the resolved resonance energy range,
the cross sections for the elastic, fission, and capture reactions are normally the sums of
5

CHAPTER 0. INTRODUCTION

Table 1: Key parameters defining the hierarchy of entries in an ENDF file.
Library
NLIB
a collection of evaluations from a specific evaluation
group (e.g., NLIB 0=ENDF/B, see Table 2 for details).
Version
NVER
one of the periodic updates to a library in ENDF format
(e.g., NVER=7 for ENDF/B-VII). A change of version
may imply a change in format, standards, and/or procedures.
Release
LREL
library release number of an intermediate release containing minor updates and corrections of errors after a
general release of a library. A release number is appended to the library/version name for each succeeding
revision of the data set(e.g., ENDF/B-VI.2 for Release 2
of the ENDF/B-VI library).
Sublibrary NSUB
a set of evaluations for a particular data type, (e.g.,
NSUB=4 for radioactive decay data, 10 for incidentneutron data, 12 for thermal neutron scattering data,
etc.). (See Table 3 for the complete list of sub-libraries).
Format
NFOR
format in which the data is tabulated; tells the processing codes how to read the subsequent data records (e.g.,
NFOR=6 for ENDF-6).
Material MAT
integer designation for the target in a reaction sublibrary, or the radioactive (parent) nuclide in a decay
sub-library; see Section 0.4.1.
Mod
NMOD
”modification” flag; see Section 0.4.1.
File
MF
subdivision of material identified by (MAT) into logical units referred to as Files; each File contains data
for a certain class of information (e.g., MF=3 contains
reaction cross sections, MF=4 contains angular distributions, etc.). MF runs from 1 to 99. (See Table 4 for
a complete list of assigned MF numbers).
Section
MT
subdivision of a File identified by (MF) into sections;
each section describes a particular reaction or a particular type of auxiliary data (e.g., MT=102 contains
capture data, etc.). MT runs from 1 to 999. (See Appendix B for a complete list of assigned MT numbers).

6

CHAPTER 0. INTRODUCTION

NLIB
0
1
2
3
4
5
6
21
31
32
33
34
35
36
37
41

Table 2: Currently defined NLIB numbers
Library Definition
ENDF/B - United States Evaluated Nuclear Data File
ENDF/A - United States Evaluated Nuclear Data File
JEFF - NEA Joint Evaluated Fission and Fusion File (formerly
JEF)
EFF - European Fusion File (now part of JEFF)
ENDF/B High Energy File
CENDL - China Evaluated Nuclear Data Library
JENDL - Japan Evaluated Nuclear Data Library
SG-23 - Fission product library of the Working Party on Evaluation
Cooperation Subgroup-23 (WPEC-SG23)
INDL/V - IAEA Evaluated Neutron Data Library
INDL/A - IAEA Nuclear Data Activation Library
FENDL - IAEA Fusion Evaluated Nuclear Data Library
IRDF - IAEA International Reactor Dosimetry File
BROND - Russian Evaluated Nuclear Data File (IAEA version)
INGDB-90 - Geophysics Data
FENDL/A - FENDL activation evaluations
BROND - Russian Evaluated Nuclear Data File (original version)

Table 3: Sub-library Numbers and Names
NSUB IPART ITYPE Sub-library Names
0
0
0
Photo-Nuclear Data
1
0
1
Photo-Induced Fission Product Yields
3
0
3
Photo-Atomic Interaction Data
4
0
4
Radioactive Decay Data
5
0
5
Spontaneous Fission Product Yields
6
0
6
Atomic Relaxation Data
10
1
0
Incident-Neutron Data
11
1
1
Neutron-Induced Fission Product Yields
12
1
2
Thermal Neutron Scattering Data
113
11
3
Electro-Atomic Interaction Data
10010 1001
0
Incident-Proton Data
10011 1001
1
Proton-Induced Fission Product Yields
10020 1002
0
Incident-Deuteron Data
10030 1003
0
Incident-Triton Data
20030 2003
0
Incident-Helion (3 He) Data
20040 2004
0
Incident-Alpha data
...

7

CHAPTER 0. INTRODUCTION
the values given in File 3 and the resonance contributions computed from the parameters
given in File 2. An exception to this rule is allowed for certain derived evaluations (see
LRP=2 in Section 1.1). In the unresolved resonance range, the self-shielded cross sections
will either be sums of File 2 and File 3 contributions, as above, or File 3 values multiplied
by a self-shielding factor computed from File 2 (see Sections 2.3.1, 2.4.17).
Distributions for emitted neutrons and other particles or nuclei are given using File 4,
a combination of Files 4 and 5, or File 6. As described in more detail in Chapter 4, File 4
is used for simple two-body reactions (elastic, discrete inelastic). Files 4 and 5 are used
for simple continuum reactions, which are nearly isotropic, have minimal pre-equilibrium
component, and emit only one important particle. File 6 is used for more complex reactions
that require energy-angle correlation, that are important for heating or damage, or that
have several important products, which must be tallied.
If any of the reaction products are radioactive, they should be described further in File 8.
This file indicates how the production cross section is to be determined (from File 3, 6, 9,
or 10) and gives minimal information on the further decay of the product. Additional decay
information can be retrieved from the decay data sub-library when required.
Note that yields of particles and residual nuclei are sometimes implicit; for example,
the neutron yield from reaction A(n,2n) is two and the yield of the product A-1 is one. If
File 6 is used, all yields are given explicitly. This is convenient for computing gas production
and transmutation cross sections. Branching ratios (or relative yields) for the production of
different isomeric states of a radionuclide may be given in File 9. Alternatively, radionuclide
isomer-production cross sections can be given in File 10. In the latter case, it is possible
to determine the yield by dividing by the corresponding cross section from File 3. File 9 is
used in preference to File 10 for reactions described by resonance parameters (e.g., radiative
capture).
For compatibility with earlier versions, photon production and photon distributions can
be described using File 12 (photon production yields), File 13 (photon production cross
sections), File 14 (photon angular distributions), and File 15 (photon energy distributions).
Note that File 12 is preferred over File 13 when strong resonances are present (capture,
fission). Whenever possible, photons should be given with the individual reaction that
produced them using File 12. When this cannot be done, summation MT numbers can be
used in Files 12 or 13 as described in Section 0.4.3.8.
When File 6 is used to represent neutron and charged-particle distributions for a reaction,
it should also be used for all other emitted particles like photons and recoils. This makes an
accurate energy-balance check possible for the reaction. When emitted photons cannot be
assigned to a particular reaction, they can be represented using summation MT numbers as
described in Section 0.4.3.8.
Finally, covariance data are given in Files 30-40. Procedures for these files are given in
Chpaters 30-40.

8

CHAPTER 0. INTRODUCTION
0.3.2.2

Thermal Neutron Scattering (NSUB=12)

Thermal neutron scattering data8 are kept in a separate sub-library because the targets are
influenced by their binding to surrounding atoms and their thermal motion; therefore, the
physics represented9 requires different formats than other neutron data. The data extend
to a few eV for several molecules, liquids, solids, and gases. As usual, each evaluation starts
with descriptive data and a directory file (see Section 1.1). The remaining data is included
in File 7. Either the cross sections for elastic coherent scattering, if important, are derived
from Bragg edges and structure factors, or cross sections for incoherent elastic scattering
are derived from the bound cross section and Debye-Waller integral. Finally, scattering
law data for inelastic incoherent scattering are given, using the S(α,β) formalism and the
short-collision-time approximation.
0.3.2.3

Fission Product Yield Data

Data for the production of fission products are given in different sub-libraries according to
the mechanism inducing fission. Currently, sub-libraries are defined for yields from spontaneous fission (NSUB=5), neutron-induced fission product yields (NSUB=11), photo-induced
fission product yields (NSUB=1), proton-induced fission product yields (NSUB=10011) and
other charged-particle induced fission product yields, where the NSUB value is defined as
described in Section 0.3.2. Each material starts with a descriptive data and directory file
(see Section 1.1). The remaining data is given in File 8, which contains two sections: independent yields, and cumulative yields. As described in Section 8.3, the format for these two
sections is identical. Uncertainty data are self-contained in File 8.
0.3.2.4

Radioactive Decay Data (NSUB=4)

Evaluations of decay data for radioactive nuclides are grouped together into a sub-library.
This sub-library contains decay data for all radioactive products (e.g., fission products and
activation products). Fission product yields and activation cross sections will be found
elsewhere. Each material contains two, three, or four files, and starts with a descriptive
data and directory file (see Section 1.1). For materials undergoing spontaneous fission,
additional sections in File 1 give the total, delayed, and prompt fission neutron yields. In
addition, the spectra of the delayed and prompt neutrons from spontaneous fission are given
in File 5. The File 5 formats are the same as for particle-induced fission (see Section 5),
and the distributions are assumed to be isotropic in the laboratory system. File 8 contains
half-lives, decay modes, decay energies, and radiation spectra (see Section 8.4). Finally,
covariance data for the spectra in File 5 may be given in File 35; covariance data of other
parameters are self-contained in File 8.
8

Used for incident neutrons, IPART=1 only.
J.U. Koppel and D.H. Houston, Reference Manual for ENDF Thermal Neutron Scattering Data, General
Atomic report GA-8774 (ENDF-269) (Revised and reissued by NNDC, July 1978).
9

9

CHAPTER 0. INTRODUCTION
0.3.2.5

Charged-Particle (NSUB≥10010) and
Photo-Nuclear (NSUB=0) Sub-libraries

Evaluations for incident charged-particle and photo-nuclear reactions are grouped together
into sub-libraries by projectile. As usual, each evaluation starts with a descriptive data and
directory file (see Section 1.1). For particle-induced fission or photo-fission, File 1 can also
contain sections giving the total, delayed, and prompt number of neutrons per fission, and
the energy released in fission. Resonance parameter data (File 2) may be omitted entirely
(see LRP=-1 in Section 1.1).
Cross sections are given in File 3. The MT numbers used are based upon the particles
emitted in the reaction as described in Section 0.4.3. A special case is the elastic scattering
of charged particles, which is discussed separately in Section 0.4.3.1. Explicit yields for
all products (including photons) must be given in File 6. In addition, the charged-particle
stopping power should be given. If any of the products described by a section of File 6
are radioactive, they should be described further in a corresponding section of File 8. This
section gives the half-life, minimum information about the decay chain, and decay energies
for the radioactive product. Further details, if required, can be found in the decay data
sub-library.
Angular distributions or correlated energy-angle distributions can be given for all particles, recoil nuclei, and photons in File 6. It is also possible to give only the average
particle energy for less important reactions, or even to mark the distribution ”unknown”
(see Section 6.2.1).
Finally, Files 30 through 40 may be used to describe the covariances for charged-particle
and photo-nuclear reactions.
0.3.2.6

Photo-Atomic Interaction Data (NSUB=3)

Incident photon reactions with the atomic electrons10 are kept in a separate sub-library.
These data are associated with elements rather than isotopes. Each material starts with a
descriptive data and directory file (see Section 1.1), as usual. In addition, the material may
contain File 23 for photon interaction cross sections, and File 27 for atomic form factors.
0.3.2.7

Electro-Atomic Interaction Data (NSUB=113)

Incident electron reactions with the atomic electrons are also kept in a separate sublibrary.
These data are again associated with elements rather than isotopes. Each material starts
with a descriptive data and directory file (see Section 1.1), as usual. In addition, File 23 is
given for the elastic, ionization, bremsstrahlung, and excitation cross sections, and File 26
is given for the elastic angular distribution, the bremsstrahlung photon spectra and energy
loss, the excitation energy transfer, and the spectra of the scattered and recoil electrons
associated with subshell ionization.
10

D.E. Cullen, et al., EPDL97: the Evaluated Photon Data Library, ’97 Version, UCRL–50400, Vol. 6,
Rev. 5, September 1997.

10

CHAPTER 0. INTRODUCTION
0.3.2.8

Atomic Relaxation Data (NSUB=6)

The target atom can be left in an ionized state due to a variety of different types of interactions, such as photon or electron induced ionization, internal conversion, etc. This section
provides the data needed to describe the relaxation of an ionized atom back to neutrality. This includes subshell energies, transition energies, transition probabilities, and other
parameters needed to compute the X-ray and electron spectra due to atomic relaxation.
The materials are elements. Each material starts with a descriptive data and directory
file (see Section 1.1), as usual. In addition, File 28 is given containing the relaxation data
for all the subshells defined in the photo-atomic or electro-atomic sublibraries.

0.4

Contents of an ENDF Evaluation

For a given sub-library (NSUB) that defines the projectile and the type of data, the target
material for a reaction evaluation or the radioactive nuclide for a decay evaluation is specified
by the material (MAT) number. An evaluation for a material is further subdivided into
data blocks called ”Files”, identified by the MF-number. Sections within individual Files
are identified by the MT-numbers, which indicate the type of data represented by a section
and the products resulting from the reaction.

0.4.1

Material (MAT, MOD)

A material may be a single nuclide, a natural element containing several isotopes, or a
mixture of several elements (compound, alloy, molecule, etc.). A single isotope can be in
the ground state or an excited (or isomeric) state. Each material in an ENDF library is
assigned a unique identification number, designated by the symbol MAT, which ranges from
1 to 9999.11
The assignment of MAT numbers for ENDF/B libraries is made on a systematic basis
assuming uniqueness of the four digit MAT number for a material. A material will have
the same MAT number in each sub-library (decay data, incident neutrons, incident charged
particles, etc.).
One hundred MAT numbers (Z01-Z99) have been allocated to each element Z, through
Z = 98. Natural elements have MAT numbers Z00. The MAT numbers for isotopes of an
element are assigned on the basis of increasing mass in steps of three, allowing for the ground
state and two metastable states.12 In the ENDF/B files, which are application oriented, the
evaluations of neutron excess nuclides are of importance, since this category of nuclide is
required for decay heat applications. Therefore, the lightest stable isotope is assigned the
MAT number Z25 so that the formulation can easily accommodate all the neutron excess
nuclides.
11

The strategy for assigning MAT numbers for ENDF/B libraries is described here; other libraries may
have different schemes.
12
This procedure leads to difficulty for the nuclides of xenon, cesium, osmium, platinum, etc., where more
than 100 MAT numbers could be needed and some decay data where more than two isomeric states might
be present.

11

CHAPTER 0. INTRODUCTION
For the special cases of elements from einsteinium to lawrencium (Z≥ 99) MAT numbers
99xx are assigned, where xx = 30, 25, 20, 15, and 12 for elements 99 to 103 respectively;
such a scheme covers all known nuclides with allowance for expansion.
For mixtures, compounds, alloys, and molecules, MAT numbers between 0001 and 0099
are assigned on a special basis (see Appendix C).
The above conventions are adopted in ENDF/B libraries and are recommended (but not
mandatory) in other libraries in ENDF-6 format.
All versions of a data set (i.e., the initial release, revisions, or total re-evaluations) are
indicated using the material ”modification” MOD number. For example, for the initial
release of ENDF/B-VI, the modification flag for each material (MAT) and section (MT)
carried over from previous versions is set to zero (MOD=0); for new evaluations they are set
to one (MOD=1). Each time a change is made to a material, the modification flag for the
material is incremented by one. The modification flag for each section changed in the revised
evaluation is set equal to the new material modification number. If a complete re-evaluation
is performed, the modification flag for every section is changed to equal the new material
”modification” number.
As an example, consider the following. Evaluator X evaluates a set of data for 235 Uand
transmits it to the NNDC. The Center assigns the data set a MAT number of 9228 subject
to CSEWG’s approval of the evaluation. This evaluation has ”modification” flags equal to
1 for the material and for all sections. Should the evaluation of material 9228 subsequently
be revised and released with CSEWG’s approval, the material will be assigned MOD flag
of 2. This material would have MOD flags of 2 on each revised section, but the unchanged
sections will have MOD flags of 1.

0.4.2

ENDF Data Blocks (Files - MF)

A ”File” in the ENDF nomenclature is a block of data in an evaluation that describes a
certain data type. The list of allowed Files (MF) and a description of their usage in different
sub-libraries is given in Table 4.
With respect to the earlier versions of the ENDF formats the following MF numbers
have been retired: 16, 17, 18, 19, 20, 21, 22, 24, and 25.
The structure and the contents of the Files are described in details in the Chapters that
follow.

0.4.3

Reaction Nomenclature (MT)

The following paragraphs explain how to choose MT numbers for particle-induced and photonuclear reactions in ENDF-6. A complete list of the definitions of the MT numbers can be
found in Appendix B.
0.4.3.1

Elastic Scattering

Elastic scattering is a two-body reaction that obeys the kinematic equations given in Appendix E. The sections are labeled by MT=2 (except for photo-atomic data, see Chapter 23).
For incident neutrons, the elastic scattering cross section is determined from File 3 together
12

CHAPTER 0. INTRODUCTION

Table 4: Definitions of File Types (MF)
MF
1
2
3
4
5
6
7
8
9
10
12
13
14
15
23
26
27
28
30
31
32
33
34
35
39
40

Description
General information
Resonance parameter data
Reaction cross sections
Angular distributions for emitted particles
Energy distributions for emitted particles
Energy-angle distributions for emitted particles
Thermal neutron scattering law data
Radioactivity and fission-product yield data
Multiplicities for radioactive nuclide production
Cross sections for radioactive nuclide production
Multiplicities for photon production
Cross sections for photon production
Angular distributions for photon production
Energy distributions for photon production
Photo- or electro-atomic interaction cross sections
Electro-atomic angle and energy distribution
Atomic form factors or scattering functions for photo-atomic interactions
Atomic relaxation data
Data covariances obtained from parameter covariances and sensitivities
Data covariances for nu(bar)
Data covariances for resonance parameters
Data covariances for reaction cross sections
Data covariances for angular distributions
Data covariances for energy distributions
Data covariances for radionuclide production yields
Data covariances for radionuclide production cross sections

with resonance contributions, if any, from File 2. The angular distribution of scattered
neutrons is given in File 4.
For incident charged particles, the Coulomb scattering makes it impossible to define
an integrated cross section, and File 3, MT=2 contains either a dummy value of 1.0 or a
”nuclear plus interference” cross section defined by a particular cutoff angle. The rest of
the differential cross section for the scattered particle is computed from parameters given in
File 6, MT=2 (see Section 6.2.6).
0.4.3.2

Simple Single Particle Reactions

Many reactions have only a single particle and a residual nucleus (and possibly photons)
in the final state. These reactions are associated with well-defined discrete states or a
continuum of levels in the residual nucleus, or they may proceed through a set of broad
levels that may be treated as a continuum. The MT numbers to be used are given in
Table 5.
13

CHAPTER 0. INTRODUCTION
Table 5: Definitions of MT numbers of Simple Single Particle Reactions
Discrete
Continuum Discrete +
Emitted
Continuum
Particle
50-90
91
4
n
600-648
649
103
p
650-698
699
104
d
700-748
749
105
t
3
750-798
799
106
He
800-848
849
107
α
By definition, the emitted particle is the lighter of the two particles in the final state.
If the reaction is associated with a discrete state in the residual nucleus, use the first
column of numbers in Table 5. For example, neutron emission MT=50 leaves the residual
nucleus in the ground state, MT=51 leaves it in the first excited state, MT=52 in the
second, and so on. A similar convention applies to all other reactions. The elastic reaction
uses MT=2 as described above and the emitted particle is of the same type as the incident
particle; therefore, do not use MT=50 for incident neutrons, do not use MT=600 for incident
protons, and so on. For incident neutrons, the discrete reactions are assumed to obey twobody kinematics (see Appendix E), and the angular distribution for the particle is given in
File 4 or File 6 (except for MT=2). If possible, the emitted photons associated with discrete
levels should be represented in full detail using the corresponding MT numbers in File 6 or
File 12. For incident charged particles, the emitted particle must be described in File 6. A
two-body law can be used for narrow levels, but broader levels can also be represented using
energy-angle correlation. Photons associated with the particle should be given in the same
section (MT) of File 6 when possible.
If the reaction is associated with a range of levels in the residual nucleus (i.e., continuum),
use the second column of MT numbers. For incident neutrons, Files 4 and 5 are allowed for
compatibility with previous versions, but it may be necessary to use File 6 to obtain the
desired accuracy. When Files 4 and 5 are used, photons should be given in File 12 using the
same MT number if possible. For more complicated neutron reactions or incident charged
particles, File 6 must be used for the particle and the photons.
The ”sum” MT numbers (3rd column in Table 5) are used in File 3 for the sum of all
the other reactions in that row, but they are not allowed for describing particle distributions in Files 4, 5, or 6. As an example, a neutron evaluation might contain sections with
MF/MT=3/4, 3/51, 3/91, 4/51, and 6/91. A deuteron evaluation might contain sections
with 3/103, 3/600, and 6/600 (the two sections in File 3 would be identical). For a neutron evaluation with no 600-series distributions or partial reactions given, MT=103-107 can
appear by themselves; they are simply components of the absorption cross section.
In some cases, it is difficult to assign all the photons associated with a particular particle
to the reactions used to describe the particle. In such cases, these photons can be described
using the ”sum” MT numbers in File 12 or 13 (for neutrons) or in File 6 (for other projectiles).
Some examples of simple single-particle reactions are listed in Table 6. Nomenclature zn
for the emitted particle implies particle ”z” emission with residual in the n-th discrete level

14

CHAPTER 0. INTRODUCTION
state.
Table 6: Examples of simple single-particle reactions
Reaction
MT
9
12
Be(α,n0 ) C
50
Fe(n,nc )Fe
91
2
3
H(d,p0 ) He
600
6
7
Li(t,d0 ) Li
650
6
Li(t,d1 )7 Li
651
For the purposes of this manual, reactions are written as if all prompt photons have
been emitted; that is, the photons do not appear explicitly in the reaction nomenclature.
Therefore, no ”*” is given on Li in the last example above that would signify the excited
state of the nucleus.
0.4.3.3

Simple Multi-Particle Reactions

If a reaction has only two to four particles, a residual nucleus, and photons in the final
state, and if the residual nucleus does not break up, it will be called a ”simple multi-particle
reaction.” The MT numbers that can be used are listed in Table 7.

MT
11
16
17
22
23
24
25
28
29
30
32
33
34
35
36

Table 7: Examples of simple multi-particle reactions
Emitted Particles
MT
Emitted Particles
2nd
37
4n
2n
41
2np
3n
42
3np
nα
44
n2p
n3α
45
npα
2nα
108
2α
3nα
109
3α
np
111
2p
n2α
112
pα
2n2α
113
t2α
nd
114
d2α
nt
115
pd
n3 He
116
pt
nd2α
117
dα
nt2α

For naming purposes, particles are always arranged in ZA order; thus, (n,np) and (n,pn)
are summed together under MT=28. In addition, there must always be a residual particle.
By definition, it is the particle or nucleus in the final state with the largest ZA. This means
that the reaction d+t→ n+α must be classified as the reaction 3 H(d,n)4 He (MT=50) rather

15

CHAPTER 0. INTRODUCTION
than the reaction 3 H(d,nα) (MT=22). The cross sections for these reactions will be found
in File 3, as usual.
This list is not exhaustive, and new MT numbers can be added if necessary. However,
some reactions are more naturally defined as ”breakup” or ”complex” reactions (see below).
For compatibility with previous versions, Files 4 and 5 are allowed in the incident-neutron
sub-library. In this case, the particle described in Files 4 and 5 is the first one given under
”Emitted Particles” above. At high neutron energies, the use of File 6 is preferred because
it allows to describe energy-angle correlated distributions resulting from pre-equilibrium
effects for more than one kind of particle. Using File 6 also makes it possible to give an
energy distribution for the recoil nucleus and photons. These distributions are needed in
calculating nuclear heating and radiation damage. If Files 4/5 are used, photons should be
given in File 12 or 13 using the same MT number when possible. In some cases the photons
cannot be assigned to individual reactions, in which case their lumped contribution can be
described as the nonelastic cross section with MT=3, as described below.
For charged-particle sub-libraries, File 6 must be used for these reactions and should
include recoils and photons. If the photons cannot be assigned to a particular reaction, the
nonelastic MT=3 can be used as described below.
0.4.3.4

Breakup Reactions

A number of important reactions can be described as proceeding in two steps: first, one or
several particles are emitted as in the simple reactions described above, then the remaining
nuclear system either breaks up or emits another particle. In the nomenclature of ENDF-6,
these are both called ”breakup reactions.” For ENDF/B-V, these reactions were represented
using special MT numbers or ”LR flags”. For ENDF/B-VI, the preferred representation
uses File 3 and File 6. The same MT numbers are used as for the simple reactions described
above. The cross section goes in File 3 as usual, but a special LR flag is used to indicate
that this is a breakup reaction (see below). The yield and angular distribution or energyangle distribution for each particle emitted before breakup is put into File 6. In addition,
yields and distributions for all the breakup products are allowed in File 6. For photonuclear and charged-particle sub-libraries, the photons are also given in File 6; but for
neutron sub-libraries, the photons may be given in Files 6 or 12-15. The approach of using
File 6 provides complete accounting of particle and recoil spectra for transport, heating, and
damage calculations. It also provides a complete accounting of products for gas production
and activation calculations. Finally, it does all of this without requiring a large list of new
MT numbers.
Some examples of breakup reactions are listed in Table 8. Nomenclature zn for the
emitted particle implies particle ”z” emission with residual in the n-th discrete level state.
By convention, the particles are arranged in Z, A order in each set of parentheses. This
leads to ambiguity in the choice of the intermediate state. For example,two possibilities of
representation exist for neutrons incident on a 12 C target:
12
C(n,n’) 12 C → 3α
12
C(n,α) 9 Be → n+2α
and similarly, there are four possible representations for tritons incident on a 7 Li target:
16

CHAPTER 0. INTRODUCTION
Table 8: Examples of breakup
Reaction
3
H(t,n0 ) 5 He → n+α
6
Li(d,n3 ) 7 Be → 3 He+α
7
Li(n,nc ) 7 Li → t+α
7
Li(t,2n) 8 Be → 2α
7
Li(p,d1 ) 6 Li → d+α
9
Be(a,n3 ) 12 C → 3α
16
O(n,n6 ) 16 O → α+12 C

reactions
MT
50
53
91
16
651
53
56

7

Li(t,2n) 8 Be → 2α
7
Li(t,n) 9 Be → n+2α
7
Li(t,α) 6 He → 2n+α
7
Li(t,nα) 5 He → n+α

The evaluator must either choose one channel, partition the reaction between several channels, or use the ”complex reaction” notation (see below). Care must be taken to avoid
double counting.
In some cases, a particular intermediate state can break up by more than one path; for
example:
6
Li(d,p4 ) 7 Li → t+α
Ex = 7.47 MeV,
6
7
6
Li(d,p4 ) Li → n+ Li.

If two channels are both given under the same MT number, File 6 is used to list the
emitted particles and to give their fractional yields. The notation to be used for this type
of reaction is:
6
Li(d,p4 )7 Li → X.
where ”X” designates all reaction products.
Note that the Q-value calculated for the entire reaction is not well defined. Another
option is to split the reaction up and use two consecutive MT numbers as follows:
6
Li(d,p4 ) 7 Li → t+α
Ex=7.47 MeV, MT=604,
6
Li(d,p5 ) 7 Li → n+6 Li
Ex=7.4701 MeV, MT=605.

The same proton distribution would be given for MT=604 and 605. The mass-difference
Q-value is well defined for both reactions, but the level index no longer corresponds to real
levels.
The choice between the ”simple multi-particle” and ”breakup” representations should
be based on the physics of the process. As an example, an emission spectrum may show
several peaks superimposed on a smooth background. If the peaks can be identified with
known levels in one or more intermediate systems, they can be extracted and represented
by breakup MT numbers. The remaining smooth background can often be represented as a
simple multi-particle reaction.
As described above, the MT number for a simple reaction indicates which particles are
emitted. However, complex breakup reactions emit additional particles. The identity of
these additional particles can be determined from LR or File 6.
17

CHAPTER 0. INTRODUCTION

LR
0
1
22
23
24
25
28
29
30
32
33
34
35
36
39
40

Table 9: Identification of additional particles in File 6
Meaning
Simple reaction. Identity of the product is implicit in MT.
Complex or breakup reaction. The identity of all products is given explicitly
in File 6.
α emitted (plus residual, if any)
3α emitted (plus residual, if any)
nα emitted (plus residual, if any)
2nα emitted (plus residual, if any)
p emitted (plus residual, if any)
2α emitted (plus residual, if any)
n2α emitted (plus residual, if any)
d emitted (plus residual, if any)
t emitted (plus residual, if any)
3
He emitted (plus residual, if any)
d2α emitted (plus residual, if any)
t2α emitted (plus residual, if any)
internal conversion
electron-positron pair formation

The values LR=22-36 are provided for compatibility with ENDF/B-V. Some examples
of their use are given in Table 10
Table 10: Some examples of LR values
Reaction
MT
LR
6
6
Li(n,n1 ) Li → d+α
51
32
7
7
Li(n,nc ) Li → t+α
91
33
10
B(n,n12 ) 10 B → d+2α
62
35
12
C(n,n2 ) 12 C → 3α
52
23
16
O(n,n1 ) 16 O → e+ +e− +16 O 51
40
16
16
12
O(n,n6 ) O → α+ C
56
22
Note that the identity of the residual must be deduced from MT and LR. Only the first
particle is described in File 4 and/or File 5; the only information available for the breakup
products is the net energy that can be deduced from kinematics.
The use of LR=1 and File 6 is preferred for new evaluations because explicit yields and
distributions can be given for all reaction products.
0.4.3.5

Complex Reactions

At high energies, there are typically many reaction channels open, and it is difficult to
decompose the cross section into simple reactions. In such cases, the evaluation should use
MT=5. This complex reaction identifier is defined as the sum of all reactions not given
explicitly elsewhere in this evaluation. As an example, an evaluation might use only MT=2
18

CHAPTER 0. INTRODUCTION
and 5. Sections of File 6 with MT=5 and the correct energy-dependent yields would then
represent the entire nonelastic neutron spectrum, the entire proton spectrum, and so on. A
slightly more refined evaluation might use MT=2, 5, 51-66, and 600-609. In this case, MT=5
would represent all the continuum neutron and proton emission. The discrete levels would
be given separately to represent the detailed angular distribution and two-body kinematics
correctly. The notation used for complex reactions is, for example, 6 Li(d,X).
0.4.3.6

Radiative Capture

The radiative capture reaction is identified by MT=102. For neutron sublibraries, the only
products are usually photons, and they are represented in Files 6 or 12-15. Note that File 6 or
12 must be used for materials with strong resonances. For charged-particle libraries, simple
radiative capture reactions must be represented using File 3 and File 6. In addition, radiative
capture followed by breakup is common for light targets; an example is d+t→ γ+n+α,
which is written as a breakup reaction 3 H(d,γ)5 He(nα) for the purposes of this format. This
reaction is represented using MT=102 with the special breakup flag set in File 3. The
gamma, neutron, and alpha distributions are all given in File 6.
0.4.3.7

Fission

The nomenclature used for fission is identical to that used in previous versions of the ENDF
format.
Table 11: Definition of MT numbers related to fission
MT Meaning
Description
18
(z,xf)
total prompt fission
19
(z,f)
first chance fission
20
(z,nf)
second chance fission
21
(z,2nf)
third chance fission
38
(z,3nf)
fourth chance fission
452 νT
total number of neutrons per fission
455 νd
number of delayed neutrons per fission
456 νp
number of prompt neutrons per fission
458
components of energy release in fission
Cross sections (File 3) can be given using either MT=18 or the combination of MT=19,
20, 21, and 38. In the latter case, MT=18 may be given and must contain the sum of the
partial reactions.
0.4.3.8

Nonelastic Reaction for Photon Production

Whenever possible, the same MT number should be used to describe both the emitted
particle and the photons. However, this is usually only possible for discrete photons from
low-lying levels, radiative capture, or for photons generated from nuclear models. Any
photons that cannot be assigned to a particular level or particle distribution can be given in
19

CHAPTER 0. INTRODUCTION
a section with the nonelastic summation reaction MT=3 in File 6, 12, or 13 (for neutrons)
or in File 6 (for other projectiles). As described in Section 0.4.3.2, MT=4, 103, 104, 105,
106, and 107 can also be used as summation reactions for photon production in Files 12
and 13.
0.4.3.9

Special Production Cross Sections

A special set of production cross sections are provided, mostly for use in derived libraries.
The list is given in Table 12.
Table 12: Special MT numbers for particle production
MT Meaning
201 neutron production
202 photon production
203 proton production
204 deuteron production
205 triton production
206 3 He production
207 α production
Each one is defined as the sum of the cross section times the particle yield over all
reactions (except elastic scattering) with that particle in the final state. The yields counted
must include implicit yields from reaction names, LR flags, or residual nuclei in addition to
explicit yields from File 6. As an example, for an evaluation containing the reactions (n,α)
(MT=107), and (n,n′ 3α) (MT=91, LR=23), the helium production cross section would be
calculated using
MT207 = MT107 + 3×MT91.
The cross section in File 3 is barns per particle (or photon). A corresponding distribution
can be given using Files 4 and 5, or the distribution can be given using File 6 with the
particle yield of 1.0. These MT numbers will ordinarily be used in File 3 of special gas
production libraries.
0.4.3.10

Auxiliary MT Numbers

Several MT numbers are used to represent auxiliary quantities instead of cross sections.
The values 151, 451, 452, 454, 455, 456, 457, 458, and 459 have already been mentioned.
Additional values are defined in Table 13
The continuous-slowing-down parameters (MT=251-253) and the heat production cross
sections (MT=301-450) are usually used in derived libraries only. A complete list of reaction
MT numbers and auxiliary MT numbers is given in Appendix B.
0.4.3.11

Sum Rules for ENDF

A number of ENDF reaction types can be calculated from other reactions. Whenever one
or more constituting reactions are present, the reaction defined by the summation rules is
20

CHAPTER 0. INTRODUCTION

MT
251
252
253
301-450

851-870

Table 13: Auxiliary MT numbers
Meaning
µL , average cosine of the angle for elastic scattering (laboratory
system). Derived files only.
ξ, average logarithmic energy decrement for elastic scattering.
Derived files only.
γ, average of the square of the logarithmic energy decrement,
divided by 2 × ξ. Derived files only.
Energy release rate parameters (eV-barns) for the reaction obtained by subtracting 300 from this MT; e.g., 301 is total kerma,
407 is kerma for (n,α), etc. Derived files only.
Special series used only in covariance files (MF=31-40) to give
covariances for groups of reactions considered together (lumped
partials). See Chpater 30.

redundant and must be consistent with the sum of the constituting reactions. The rules for
these summations follow in Table 14.
Table 14: ENDF sum rules for cross sections
P
MT
Meaning
2, 3
Total cross sections (incident neutrons only)
4-5, 11, 16-18, 22-37,
Non-elastic
41-42, 44-45
4
50-91
Total of neutron level cross sections (z,n)
18
19-21, 38
Total fission
27
18, 101
Total absorption
101
102-117
Neutron disappearance
103
600-649
Total of proton level cross sections (z,p)
104
650-699
Total of deuteron level cross sections (z,d)
105
700-749
Total of triton level cross sections (z,t)
106
750-799
Total of 3 He level cross sections (z,3 He)
107
800-849
Total of alpha level cross sections (z,α)
Note: Reactions corresponding to MT 3, 4, 18, 27, 101, 103-107 are redundant
when one or more of their constituting components are present. They must be
computed and accounted for internally, when reconstructing cross sections by
summation, even if they do not appear in the file explicitly.
MT
1
3

0.5
0.5.1

Representation of Data
Definitions and Conventions

The data given in all sections always use the same set of units. These are summarized in
Table 15.
21

CHAPTER 0. INTRODUCTION
Table 15: Summary of ENDF units
Quantity
Units
energies
electron-volts (eV)
angles
dimensionless cosines of the angle
cross sections
barns
temperatures
Kelvin
mass
units of the neutron mass
angular distributions
probability per unit-cosine
energy distributions
probability per electron-volt
energy-angle distributions
probability per unit-cosine per electron-volt
half life
seconds
The first record of every section contains a ZA number that identifies the specific material.
ZA variants are also employed to identify projectiles and reaction products. In most cases,
ZA is constructed by
ZA = 1000.0 × Z + A,
where Z is the atomic number and A is the mass number for the material. If the material is
an element containing two or more naturally occurring isotopes in significant concentrations,
A is taken to be 0.0. For mixtures, compounds, alloys, or molecules, special ZA numbers
between 1 and 99 can be defined (see Appendix C).
A material, incident particle (projectile), or reaction product is also characterized by a
quantity that is proportional to its mass relative to that of the neutron. Typically, these
quantities are denoted as AWR, AWI, or AWP for a material, projectile, or product, respectively. For example, the symbol AWR is defined as the ratio of the mass of the material to
that of the neutron.13 Another way to say this is that ”all masses are expressed in neutron
units.” For materials which are mixtures of isotopes, the abundance weighted average mass
is used.
0.5.1.1

Atomic Masses Versus Nuclear Masses

Mass quantities for materials (AWR for all Z) and ”heavy” reaction products (AWP for
Z > 2) should be expressed in atomic units, i.e., the mass of the electrons should be
included. Mass quantities for incident particles (AWI) and ”light” reaction products (AWP
for Z ≤ 2) should be expressed in nuclear mass units. For neutrons, this ratio is 1.00000.
For charged particles likely to appear in ENDF files, see Appendix H.

0.5.2

Interpolation Laws

Many types of ENDF data are given as a table of values on a defined grid with an interpolation law to define the values between the grid points. Simple one-dimensional ”graph paper”
13

See Appendix H for neutron mass.

22

CHAPTER 0. INTRODUCTION
interpolation schemes, a special Gamow interpolation law for charged-particle cross sections, simple Cartesian interpolation for two-dimensional functions, and two non-Cartesian
schemes for two-dimensional distributions are allowed.
0.5.2.1

One-dimensional Interpolation Schemes

Consider how a simple function y(x), which might be a cross section, σ(E), is represented.
y(x) is represented by a series of tabulated values, pairs of x and y(x), plus a method for
interpolating between input values. The pairs are ordered by increasing values of x. There
will be NP values of the pair, x and y(x) given. The complete region over which x is defined
is broken into NR interpolation ranges. An interpolation range is defined as a range of the
independent variable x in which a specified interpolation scheme can be used; i.e., the same
scheme gives interpolated values of y(x) for any value of x within this range. To illustrate
this, see Figure 1 and the definitions, below:
x(n) is the nth value of x,
y(n) is the nth value of y,
NP is the number of pairs (x and y) given,
INT(m) is the interpolation scheme identification number used in the mth range,
NBT(m) is the value of n separating the mth and the (m + 1)th interpolation ranges.
The list of allowed interpolation schemes is given in Table 16.

INT
1
2
3
4
5
6
11-15
21-25

Table 16: Definition of Interpolation Types
Interpolation Scheme
y is constant in x (constant, histogram)
y is linear in x (linear-linear)
y is linear in ln(x) (linear-log)
ln(y) is linear in x (log-linear)
ln(y) is linear in ln(x) (log-log)
special one-dimensional interpolation law, used for charged-particle cross sections only
method of corresponding points (follow interpolation laws of 1-5)
unit base interpolation (follow interpolation laws of 1-5)

Interpolation code, INT=1 (constant), implies that the function is constant and equal
to the value given at the lower limit of the interval.
Note that where a function is discontinuous (for example, when resonance parameters
are used to specify the cross section in one range), the value of x is repeated and a pair
(x, y) is given for each of the two values at the discontinuity (see Figure 1).
A one-dimensional interpolation law, INT=6, is defined for charged-particle cross sections
and is based on the limiting forms of the Coulomb penetrabilities for exothermic reactions
23

CHAPTER 0. INTRODUCTION
NBT(1)=3
RANGE 1
INT(1)

NBT(2)=7
RANGE 2
INT(2)

NBT(3)=10
RANGE 3
INT(3)

y(6)

b

y(7)
b

b

y(2)
b

y(3)
b

b

b

y(5)

y(4)

x(2)

b

y(9)

y(8)

b

y(10)

y(1)

x(1)

b

x(3)
x(4)

x(5)
x(6)

x(7) x(8) x(9) x(10)

Figure 1: Interpolation of a tabulated one-dimensional function for a case with NP=10,
NR=3.
at low energies and for endothermic reactions near the threshold. The expected energy
dependence is:


A
B
σ = exp − √
(1)
E
E−T

where

T = zero for exothermic reactions with Q >0 and
equal to kinematic threshold energy for endothermic reactions with Q ≤ 0.
Note that this formula gives a concave upward energy dependence near E = T that is quite
different from the behavior of the neutron cross sections.
This formula can be converted into a two-point interpolation scheme using
B=
and

E2
ln σσ12 E
1
√ 1
E1 −T



−

√ 1
E2 −T


B
A = exp √
σ1 E1
E1 − T

where E1 , σ1 and E2 , σ2 are two consecutive points in the cross-section tabulation.
24

(2)

(3)

CHAPTER 0. INTRODUCTION
This interpolation method should only be used for E close to T . At higher energies,
non-exponential behavior will normally begin to appear, and linear-linear interpolation is
more suitable.
0.5.2.2

Two-Dimensional Interpolation Schemes

Three schemes are provided for two-dimensional interpolation:
1. simple Cartesian interpolation, wherein one simply interpolates the function values
along constant lines of initial and final energy values,
2. a method called unit-base transform, and
3. the method of corresponding energies.

Figure 2: Interpolation between Two-Dimensional Panels.
Consider Figure 2. Here E is the initial energy and there are panels at and Ei and Ei+1 .
The panels describe the probabilities of scattering from these energies to other energies;
′
e.g., f (Ei , Ei′ ) and f (Ei+1 , Ei+1
) are generally probability distributions that will integrate to
unity, when they are integrated over all E ′ . These panels will be presented using the usual
tabular schemes for arrays, or may be given in the form of an analytic expression.
For the case of simple Cartesian interpolation, intermediate values are determined by
interpolating along lines of constant E and E ′ as noted earlier. Assume that all interpolation
schemes are linear-linear in energy and that one wants to determine a value for a distribution
at E ∈ (Ei , Ei+1 ). The equation for this is:
f (E, E ′ ) = f (Ei , E ′ ) +

E − Ei
[f (Ei+1 , E ′ ) − f (Ei , E ′ )] .
Ei+1 − Ei
25

(4)

CHAPTER 0. INTRODUCTION
An examination of the above figure illustrates the major problem with Cartesian interpolation; viz., that the panel at E will have features from the lower panel at the low end,
and from the upper panel at the high end. This, of course, is reasonable, but the resulting
function will tend to have artificial peaks when the distributions shift as a function of energy,
as is usually the case.
The unit-base transform was devised to try to reduce the non-physical characteristics
of Cartesian interpolation. In this case, the data at the two panels surrounding E are
transformed to a unit base where the new functions vary according to a variable x that
ranges from 0 to 1.
E ′ − Ei′
.
Ei′ (N ) − Ei′ (1)

(5)

f (E ′ ) dE ′ = g(x) dx,

(7)

dE ′
.
dx

(8)

x≡

In this case, Ei′ (1) is the energy of the first sink energy in the panel at Ei and Ei′ (N ) is the
last point. An exactly analogous equation is used to define the value of x at Ei+1 and at
E. From here, the interpolation is made using an expression such as shown in equation (4)
except that it is made at a constant value of x. Special care must be taken to properly
account for the integrals of the panel; i.e.,
Z
Z
′
′
dE f (E ) = dx g(x),
(6)
which requires:

or
g(x) = f (E ′ )

The latter radical is the Jacobian that is required for the transformation to the unit-base
space and is determined from equation (5). In other words, when equation (4) is used, the
interpolation is made at constant values of x and the function values must be multiplied by
the Jacobians for the respective panels.
Two things can be noted about unit-base transform interpolation
(i) if the end points of two panels are the same, unit-base transform is exactly equivalent
to Cartesian interpolation, and
(ii) the same interpolations can be made without transforming to unit-base space and
transforming to the energy space at E.
The low energy value of the intermediate panel is simply:
′
Elow
(E) = Ei′ (1) +

E − Ei
[E ′ (1) − Ei′ (1)].
Ei+1 − Ei i+1

(9)

A similar expression gives the high energy of the intermediate panel, and simply substitutes
the top energies for the two panels in place of the bottom energies:

26

CHAPTER 0. INTRODUCTION

′
Ehigh
(E) = Ei′ (N ) +

E − Ei
[E ′ (M ) − Ei′ (N )].
Ei+1 − Ei i+1

(10)

Here we have assumed the upper panel has M points. The Jacobian that should be used
with the value from the bottom panel is
dE ′ (Ei )
dE ′ (E)
and is determined from equation (11) shown below. The secondary energy at Ei that corresponds to an E ′ at E is calculated from:
Ei′ (E ′ ) = Ei′ (1) +

′
E ′ − Elow
[Ei′ (N ) − Ei′ (1)] .
′
′
Ehigh
− Elow

(11)

Here we have dropped the (E) arguments from the E ′ values at E for clarity. An analogous
expression determines the secondary energy at the upper panel, and also the Jacobian for this
panel. When the two values are interpolated at these two secondary energies and multiplied
by their respective Jacobians, the values are simply interpolated using equation (4), or
another appropriate expression, if the interpolation scheme is not linear in energy).
The Method of Corresponding Energies (MCE) is a scheme that was designed to circumvent one of the major problems associated with the unit-base transform approach; viz.,
that the unit-base transform depends directly on the way the end points of the successive
distributions are taken. The MCE approach splits the integrals of distributions into equal
integral bins and then interpolates linearly between corresponding bins. (The limits of these
bins are the ”corresponding energies”.) This is a more physical approach than either Cartesian or unit-base transform interpolation and tends to emphasize the significant portions of
the distributions. Perhaps a better way of saying this is that it tends to de-emphasize the
insignificant portions of the distributions. For example, if 10 equal integral bins are to be
used for the interpolation, the energies where the panels integrate to a tenth of the total
integral are determined. Then the energies where the panels integrate to two-tenths of the
total integral are determined, etc., until 10 sets of energy boundaries are defined for the bins
in all panels. The interpolations between corresponding bins of successive panels are then
performed using the unit-base transform approach.
It is important to note that that unit-base transform and MCE will require Jacobians
to multiply the function values at successive panels, because a variable transformation is
involved, while Cartesian interpolation is all done in real energy space, so that the unmodified
function values are used.

0.6

General Description of Data Formats

An ENDF ”tape” is built up from a small number of basic structures called ”records,” such
as TPID, TEND, CONT, TAB1, and so on. These ”records” normally consist of one or
more 80-character FORTRAN records. It is also possible to use binary mode, where each of
the basic structures is implemented as a FORTRAN logical record. The advantage of using
27

CHAPTER 0. INTRODUCTION
these basic ENDF ”records” is that a small library of utility subroutines can be used to read
and write the records in a uniform way.

0.6.1

Structure of an ENDF Data Tape

The structure of an ENDF data tape (file) is illustrated schematically in Fig. 0.3. The tape
contains a single record at the beginning that identifies the tape. The major subdivision
between these records is by material (identified by the MAT number). The data for a
material is divided into files, and each file (identified by the MF number) contains the data
for a certain class of information. A file is subdivided into sections, each one containing
data for a particular reaction type (identified by the MT number). Finally, a section is
divided into records. Every record on a tape contains three identification numbers: MAT,
MF, and MT. These numbers are always in increasing numerical order, and the hierarchy is
MAT, MF, MT. The end of a section, file, or material is signaled by special records called
SEND, FEND, and MEND, respectively.

Figure 3: Structure of an ENDF data tape.

0.6.2

Symbol Nomenclature

An attempt has been made to use an internally consistent notation based on the following
rules.
1. Symbols starting with the letter I, J, K, L, M, or N are integers. All other symbols
refer to floating-point (real numbers).

28

CHAPTER 0. INTRODUCTION
2. The letter I or a symbol starting with I refers to an interpolation code (see Section 0.5.2).
3. Letters J, K, L, M, or N when used alone are indices.
4. A symbol starting with M is a control number. Examples are MAT, MF, MT.
5. A symbol starting with L is a test number.
6. A symbol starting with N is a count of items.
All numbers are given in fields of 11 columns. In character mode, floating-point numbers
should be entered in one of the following forms:
±1.234567±n
±1.23456±nn,
where nn ≤ 38
depending on the size of the exponent. Both of these forms can be read by the ”E11.0” format
specification of FORTRAN. However, a special subroutine must be used to output numbers
in the above format. If evaluations are produced using numbers written by ”1PE11.5” (that
is, 1.2345Enn, the numbers will be standardized into 6 or 7 digit form, but the real precision
will remain at the 5 digit level.

0.6.3

Types of Records

All records on an ENDF tape are one of six possible types, denoted by TEXT, CONT,
LIST, TAB1, TAB2, and INTG. The CONT record has six special cases called DIR, HEAD,
SEND, FEND, MEND, and TEND. The TEXT record has the special case TPID. Every
record contains the basic control numbers MAT, MF, and MT, as well as the sequence
number NS. The definitions of the other fields in each record will depend on its usage as
described below.
The sequence number of a record is a remnant of the past when the data were physically
stored on paper cards that could get shuffled by accident. Nowadays NS serves merely as
an additional parameter for checking the integrity of an evaluated library. The following
conventions apply:
• the counter NS is reset for every section in the file. It starts with 1 and is incremented
by 1 on every record up to the SEND record.
• SEND records (identified by MT=0) have NS=99999 by definition.
• FEND, MEND and TEND records have NS=0.
0.6.3.1

TEXT Records

This record is used either as the first entry on an ENDF tape (TPID), or to give the
comments in File 1. It is indicated by the following shorthand notation
[MAT, MF, MT/ HL] TEXT
where HL is 66 characters of text information. The TEXT record can be read with the
following FORTRAN statements
29

CHAPTER 0. INTRODUCTION

10

READ(LIB,10)HL,MAT,MF,MT,NS
FORMAT(A66,I4,I2,I3,I5)

0.6.3.2

CONT Records

The smallest possible record is a control (CONT) record. For convenience, a CONT record
is denoted by
[MAT,MF,MT/C1,C2,L1,L2,N1,N2]CONT
The CONT record can be read with the following FORTRAN statements
10

READ(LIB,10)C1,C2,L1,L2,N1,N2,MAT,MF,MT,NS
FORMAT(2E11.0,4I11,I4,I2,I3,I5)

The actual parameters stored in the six fields C1, C2, L1, L2, N1, and N2 will depend on
the application for the CONT record.
0.6.3.3

HEAD Records

The HEAD record is the first in a section and has the same form as CONT, except that the
C1 and C2 fields always contain ZA and AWR, respectively.
0.6.3.4

END Records

The SEND, FEND, MEND, and TEND records use only the three control integers, which
signal the end of a section, file, material, or tape, respectively. In binary mode, the six
standard fields are all zero. In character mode, the six are all zero as follows
[MAT,MF, 0/ 0.0, 0.0,
0,
0,
0,
0] SEND
[MAT, 0, 0/ 0.0, 0.0,
0,
0,
0,
0] FEND
[ 0, 0, 0/ 0.0, 0.0,
0,
0,
0,
0] MEND
[ -1, 0, 0/ 0.0, 0.0,
0,
0,
0,
0] TEND

0.6.3.5

DIR Records

The DIR records are described in more detail in Section 1.1.1. The only difference between
a DIR record and a standard CONT record is that the first two fields in the DIR record are
blank in character mode.
0.6.3.6

LIST Records

This type of record is used to list a series of numbers B1, B2, B3, etc. The values are given
in an array B(n), and there are NPL of them. The shorthand notation for the LIST record
is
[MAT,MF,MT/ C1, C2, L1, L2, NPL, N2/ Bn ] LIST
The LIST record can be read with the following FORTRAN statements
30

CHAPTER 0. INTRODUCTION

10
20

READ(LIB,10)C1,C2,L1,L2,NPL,N2,MAT,MF,MT,NS
FORMAT(2E11.0,4I11,I4,I2,I3,I5)
READ(LIB,20)(B(N),N=1,NPL)
FORMAT(6E11.0)

The maximum for NPL varies with use (see Appendix G).
0.6.3.7

TAB1 Records

These records are used for one-dimensional tabulated functions such as y(x). The data
needed to specify a one-dimensional tabulated function are the interpolation tables NBT(N)
and INT(N) for each of the NR ranges, and the NP tabulated pairs of x(n) and y(n). The
shorthand representation is:
[MAT,MF,MT/ C1, C2, L1, L2, NR, NP/xint /y(x)]TAB1
The TAB1 record can be read with the following FORTRAN statements

10
20
30

READ(LIB,10)C1,C2,L1,L2,NR,NP,MAT,MF,MT,NS
FORMAT(2E11.0,4I11,I4,I2,I3,I5)
READ(LIB,20)(NBT(N),INT(N),N=1,NR)
FORMAT(6I11)
READ(LIB,30)(X(N),Y(N),N=1,NP)
FORMAT(6E11.0)

The limits on NR and NP vary with use (see Appendix G). The limits must be strictly
observed in primary evaluations in order to protect processing codes that use the simple
binary format. However, these limits can be relaxed in derived libraries in which resonance
parameters have been converted into detailed tabulations of cross section versus energy.
Such derived libraries can be written in character mode or a non-standard blocked-binary
mode.
0.6.3.8

TAB2 Records

The next record type is the TAB2 record, which is used to control the tabulation of a twodimensional function y(x, z). It specifies how many values of z are to be given and how to
interpolate between the successive values of z. Tabulated values of yl (x) at each value of zl
are given in TAB1 or LIST records following the TAB2 record, with the appropriate value
of z in the field designated as C2. The shorthand notation for TAB2 is
[MAT,MF,MT/ C1, C2, L1, L2, NR, NZ/ Zint ]TAB2,
The TAB2 record can be read with the following FORTRAN statements

10
20

READ(LIB,10)C1,C2,L1,L2,NR,NZ,MAT,MF,MT,NS
FORMAT(2E11.0,4I11,I4,I2,I3,I5)
READ(LIB,20)(NBT(N),INT(N),N=1,NR)
FORMAT(6I11)
31

CHAPTER 0. INTRODUCTION
For example, a TAB2 record is used in specifying angular distribution data in File 4. In
this case, NZ in the TAB2 record specifies the number of incident energies at which angular
distributions are given. Each distribution is given in a LIST or TAB1 record.
0.6.3.9

INTG records

INTG, or INTeGer, records are used to store a correlation matrix in integer format. The
shorthand notation is
[MAT, MF, MT / II, JJ, KIJ ] INTG
where II and JJ are position locators, and KIJ is an array whose dimension is specified by
the number of digits NDIGIT to be used for representing the values. NDIGIT can have
any value from 2 to 6; the corresponding dimensions (NROW) are 18, 13, 11, 9, and 8
respectively. The INTG record can be read with the following FORTRAN statements

20
30
40
50
60

PARAMETER (NROW=18)
DIMENSION KIJ(NROW)
IF(NDIGIT.EQ.2) READ(LIB,20) II,JJ,(KIJ(K),K=1,18),MAT,MF,MT,NS
IF(NDIGIT.EQ.3) READ(LIB,30) II,JJ,(KIJ(K),K=1,13),MAT,MF,MT,NS
IF(NDIGIT.EQ.4) READ(LIB,40) II,JJ,(KIJ(K),K=1,11),MAT,MF,MT,NS
IF(NDIGIT.EQ.5) READ(LIB,50) II,JJ,(KIJ(K),K=1, 9),MAT,MF,MT,NS
IF(NDIGIT.EQ.6) READ(LIB,60) II,JJ,(KIJ(K),K=1, 8),MAT,MF,MT,NS
FORMAT (2I5,1X,18I3,1X,I4,I2,I3,I5)
FORMAT (2I5,1X,13I4,3X,I4,I2,I3,I5)
FORMAT (2I5,1X,11I5,
I4,I2,I3,I5)
FORMAT (2I5,1X, 9I6,1X,I4,I2,I3,I5)
FORMAT (2I5,
8I7,
I4,I2,I3,I5)

See File 32, LCOMP=2, for details regarding the use of this format.

0.7

ENDF Documentation

1. BNL 8381, ENDF - Evaluated Nuclear Data File Description and Specifications,
January 1965, H.C. Honeck.
2. BNL 50066 (ENDF 102), ENDF/B - Specifications for an Evaluated Nuclear Data
File for Reactor Applications, May 1966, H.C. Honeck. Revised July 1967 by S. Pearlstein.
3. BNL 50274 (ENDF 102), Vol. I - Data Formats and Procedures for the ENDF
Neutron Cross Section Library, October 1970, M.K. Drake, Editor.
4. LA 4549 (ENDF 102), Vol. II - ENDF Formats and Procedures for Photon Production and Interaction Data, October 1970, D.J. Dudziak.

32

CHAPTER 0. INTRODUCTION
5. BNL-NCS-50496 (ENDF 102), ENDF102 Data Formats and Procedures for the
Evaluated Nuclear Data File, ENDF, October 1975, Revised by D. Garber, C. Dunford,
and S. Pearlstein.
6. ORNL/TM-5938 (ENDF-249), The Data Covariance Files for ENDF/B-V, July
1977, F. Perey.
7. BNL-NCS-50496 (ENDF 102), Second Edition, ENDF-102 Data Formats and
Procedures for the Evaluated Nuclear Data File, ENDF/B-V, October 1979, Edited by
R. Kinsey. Revised by B.A. Magurno, November 1983.
8. BNL-NCS-28949 (Supplement ENDF 102), Second Edition, Supplement to the
ENDF/B-V Formats and Procedures Manual for Using ENDF/B-IV Data, November
1980, S. Pearlstein.
9. BNL-NCS-44945 (ENDF-102), Revision 10/91, ENDF-102 Data Formats and Procedures for the Evaluated Nuclear Data File ENDF-6, October 1991, Edited by P. F.
Rose and C. L. Dunford.

33

Chapter 1
File 1: GENERAL INFORMATION
File 1 is the first part of any set of evaluated cross-section data for a material. Each material
must have a File 1 that contains at least one section. This required section provides a brief
documentation of how the data were evaluated and a directory that summarizes the files and
sections contained in the material. In the case of fissionable materials, File 1 may contain
up to five additional sections giving fission neutron and photon yields and energy release
information. Each section has been assigned an MT number (see below), and the sections
are arranged in order of increasing MT number. A section always starts with a HEAD
record and ends with a SEND record. The end of File 1 (and all other files) is indicated by
a FEND record. These record types are defined in detail in Section 0.6.

1.1

Descriptive Data and Directory (MT=451)

This section is always the first section of any material and has two parts:
1. a brief documentation of the cross-section data, and
2. a directory of the files and sections used for this material.
In the first part, a brief description of the evaluated data set is given. This information
should include the significant experimental results used to obtain the evaluated data, descriptions of any nuclear models used, a clear specification of all the MT numbers defined to
identify reactions, the history of the evaluation, and references. The descriptive information
is given as a series of records, each record containing up to 66 characters.
The first three records of the descriptive information contain a standardized presentation
of information on the material, projectile, evaluators, and modification status. The following
quantities are defined for MF=1, MT=451:
ZA,AWR Standard material charge and mass parameters (see Section 0.5.1).
LRP Flag indicating whether resolved and/or unresolved resonance parameters are
given in File 2:
LRP=−1, no File 2 is given (not allowed for incident neutrons);
34

CHAPTER 1. FILE 1: GENERAL INFORMATION
LRP=0, no resonance parameter data are given, but a File 2 is present containing the effective scattering radius;
LRP=1, resolved and/or unresolved parameter data are given in File 2 and
cross sections computed from them must be added1 to background cross sections given in File 3;
LRP=2, parameters are given in File 2, but cross sections derived from them
are not to be added to the cross sections in File 3. The option LRP=2 is
to be used for derived files only and is typical in the so-called PENDF files,
in which the cross sections are already reconstructed from the resonances
parameters and written in File 3.
LFI Flag indicating whether this material fissions:
LFI=0, this material does not fission;
LFI=1, this material fissions.
NLIB Library identifier (e.g. NLIB= 0 for ENDF/B). Additional values have been
assigned to identify other libraries using ENDF format. See Section 0.3.1 for
details.
NMOD Modification number for this material:
NMOD=0, evaluation converted from a previous version;
NMOD=1, new or revised evaluation for the current library version;
NMOD≥ 2, for successive modifications.
ELIS Excitation energy of the target nucleus relative to 0.0 for the ground state.
STA Target stability flag:
STA=0, stable nucleus;
STA=1 unstable nucleus. If the target is unstable, radioactive decay data
should be given in the decay data sub-library (NSUB=4).
LIS State number of the target nucleus. The ground state is indicated by LIS=0.
LISO Isomeric state number. The ground state is indicated by LISO=0. LIS is
greater than or equal to LISO.
NFOR Library format.
NFOR=6 for all libraries prepared according to the specifications given in
this manual.
AWI Mass of the projectile in neutron mass units. For incident photons or decay
data sub-libraries, use AWI=0.
EMAX Upper limit of the energy range for evaluation.
LREL Library release number; for example, LREL=2 for the ENDF/B-VI.2 library.
1

In the unresolved region, it is also possible to compute self-shielding factors from File 2 and multiply
them by complete unshielded cross section given in File 3. See Chapter 2 for details

35

CHAPTER 1. FILE 1: GENERAL INFORMATION
NSUB Sub-library number. See Section 0.4 for a description of sub-libraries.
NVER Library version number; for example, NVER=7 for version ENDF/B-VII.
TEMP Target temperature (Kelvin) for data that have been generated by Doppler
broadening. For derived data only; use TEMP=0.0 for all primary evaluations.
LDRV Special derived material flag that distinguishes between different evaluations
with the same material keys (i.e., MAT, NMOD, NSUB):
LDRV=0, primary evaluation:
LDRV≥ 1, special derived evaluation (for example, a dosimetry evaluation
using sections (MT) extracted from the primary evaluation).
NWD Number of records with descriptive text for this material. Each record contains up to 66 characters.
NXC Number of records in the directory for this material. Each section (MT) in
the material has a corresponding line in the directory that contains MF, MT,
NC, and MOD. NC is a count of the number of records in the section (not
including SEND), and MOD is the modification flag (see below).
ZSYMAM Character representation of the material defined by the atomic number,
chemical symbol, atomic mass number, and metastable state designation in
the form Z-cc-AM with
Z, right justified in col. 1 to 3,
- (hyphen) in col. 4,
cc, two-character chemical name left justified in col. 5 and 6,
- (hyphen) in col. 7,
A, right justified in col. 8 to 10 or blank,
M for the indication of a metastable state in col. 11,
for example, 1-H - 2, 40-Zr- 90, 95-Am-242M, etc.
ALAB Mnemonic for the originating laboratory(s) left adjusted in col. 12-22.
EDATE Date of evaluation given in the form ”EVAL-DEC74” in col. 23-32.
AUTH Author(s) name(s) left adjusted in col. 34-66.
REF Primary reference for the evaluation left adjusted in col. 2-22.
DDATE Original distribution date given in the form ”DIST-DEC74” in col. 23-32.
RDATE Date and number of the last revision to this evaluation in col. 34-43 in the
form ”REV2-DEC74”, where ”2” in column 37 is the release number and must
be equal to LREL. The term ”revision” (IREV), which appeared in earlier
editions of this manual, was synonymous with ”release” (LREL) and has
been dropped in favor of the more-frequently used term ”release”.
36

CHAPTER 1. FILE 1: GENERAL INFORMATION
ENDATE Master File entry date in the form yyyymmdd right adjusted in col. 56-63.
The Master File entry date is assigned by NNDC for ENDF/B libraries.
HSUB Identifier for the library contained on three successive records is useful for
visual inspection of the files and is followed strictly in the ENDF/B libraries.
The first record contains four dashes starting in col. 1, directly followed by
the library type (NLIB) and version (NVER). For example,
”---- ENDF/B-VI”, followed by MATERIAL XXXX starting in col. 23 where
XXXX is the MAT number, and REVISION 2 (starting in col. 45 only if
required) where ”2” is the revision number IREV.
The second record contains five dashes starting in col. 1, and followed by the
sub-library identifier (see Table 3). For example,
”----- DECAY DATA”,
”----- PHOTO-ATOMIC INTERACTION DATA” or
”----- INCIDENT NEUTRON DATA”
The third record contains six dashes starting in col. 1 and followed by ENDF-6
where ”6” is the library format type (NFOR).
Note: the three HSUB records can be generated by the utility program,
STANEF.
MFn ENDF file number of the nth section.
MTn ENDF reaction designation of the nth section.
NCn Number of records in the nth section. This count does not include the SEND
record.
MODn Modification indicator for the nth section. The value of MODn is equal to
NMOD if the corresponding section was changed in this revision. MODn
must always be less than or equal to NMOD.

1.1.1

Formats

The structure of this section is

37

CHAPTER 1. FILE 1: GENERAL INFORMATION
[MAT,
[MAT,
[MAT,
[MAT,
[MAT,
[MAT,
[MAT,

[MAT,
[MAT,

[MAT,
[MAT,

1.1.2

1,451/
ZA,
AWR,
LRP,
LFI, NLIB, NMOD]HEAD
1,451/ ELIS,
STA,
LIS,
LISO,
0, NFOR]CONT
1,451/
AWI, EMAX, LREL,
0, NSUB, NVER]CONT
1,451/ TEMP,
0.0, LDRV,
0, NWD,
NXC]CONT
1,451/ZSYMAM, ALAB, EDATE,
AUTH
]TEXT
1,451/
REF, DDATE, RDATE, ENDATE
]TEXT
1,451/ HSUB
]TEXT
-----------------------------------continue for the rest of the NWD descriptive records
-----------------------------------1,451/ blank, blank,
MF1,
MT1, NC1, MOD1]CONT
1,451/ blank, blank,
MF2,
MT2, NC2, MOD2]CONT
----------------------------------------------------------------------1,451/ blank, blank, MFNXC, MTNXC,NCNXC,MODNXC]CONT
1, 0/
0.0, 0.0,
0,
0,
0,
0]SEND

Procedures

Note that the parameters NLIB, LREL, NVER, NSUB, MAT, NMOD, LDRV, and sometimes TEMP define a unique set of ”keys” that identifies a particular evaluation or ”material” in the ENDF system. These keys can be used to access materials in a formal data base
management system if desired.
The flag LRP indicates whether resolved and/or unresolved resonance parameter data
are to be found in File 2 (Resonance Parameters) and how these data are to be used with
File 3 to compute the net cross section. For incident neutrons, every material will have a
File 2. If LRP=0, the file contains only the effective scattering radius; the potential cross
section corresponding to this scattering radius has already been included in the File 3 cross
sections. If LRP=1, File 2 contains resolved and/or unresolved resonance parameters. Cross
sections or self-shielding factors computed from these parameters are to be combined with
any cross sections found in File 3 to obtain the correct net cross section. For other sublibraries (decay data, incident photons, incident charged particles, fission product yields),
File 2 can be omitted (use LRP=-1). A number of processing codes exist which reconstruct
resonance-region cross sections from the parameters in File 2 and output the results in
ENDF format. Such a code can set LRP=2 and copy the original File 2 to its output ENDF
tape. Other processing codes using such a tape will know that resonance reconstruction has
already been performed, but the codes will still have easy access to the resonance parameters
if needed. The LRP=2 option is not allowed in primary evaluations.
The flag LFI indicates that this material fissions in the context of the present sub-library.
In this case, a section specifying the total number of neutrons emitted per fission, ν(E),
must be given as MF=1, MT=452. Sections may also be given that specify the number
of delayed neutrons per fission (MT=455) and the number of prompt neutrons per fission
(MT=456), and that specify the components of energy release in fission (MT=458).
The flag LDRV indicates that this material was derived in some way from another evalu38

CHAPTER 1. FILE 1: GENERAL INFORMATION
ation; for example, it could represent an activation reaction extracted from a more complete
evaluation, it could be part of a gas production library containing production cross sections
computed from more fundamental reactions, it could represent a reconstructed library with
resonance parameters expanded into detailed point-wise cross sections, and so on.
The data in the descriptive section must be given for every material. The first three
records are used to construct titles for listings, plots, etc., and the format should be followed
closely. The remaining records give a verbal description of the evaluated data set for the
material. The description should mention the important experimental results upon which the
recommended cross sections are based, the evaluation procedures and nuclear models used,
a brief history and origin of the evaluation, important limitations of the data set, estimated
uncertainties and covariances, references, and any other remarks that will assist the user
in understanding the data. For incident neutron evaluations, the 2200 m/s cross sections
contained in the data should be tabulated, along with the infinite dilution resonance integrals
for capture and fission (if applicable). For charged-particle and high-energy reactions, the
meaning of each MT should be carefully explained using the notation of Section 0.4.3.

1.2

Number of Neutrons per Fission, ν, (MT=452)

If the material fissions (LFI=1), then a section specifying the average total number of neutrons per fission, ν (MT=452), must be given. This format applies to both particle induced
and spontaneous fission, each in its designated sub-library. Values of ν may be tabulated
as a function of energy or in the form of coefficients provided for the following polynomial
expansion:
ν(E) =

NC
X

Cn E n−1

(1.1)

n=1

where

ν(E) the average total (prompt plus delayed) number of neutrons per fission produced by neutrons of incident energy E (eV),
Cn the nth coefficient, and
NC the number of terms in the polynomial.
MT=452 for an energy-dependent neutron multiplicity cannot be represented by a polynomial expansion when MT=455 and MT=456 are utilized in the file.

1.2.1

Formats

The structure of this section depends on whether values of ν(E) are tabulated as a function
of energy or represented by a polynomial. The following quantities are defined:

39

CHAPTER 1. FILE 1: GENERAL INFORMATION
LNU Test that indicates what representation of ν(E) has been used:
LNU=1, polynomial representation;
LNU=2, tabulated representation.
NC Count of the number of terms used in the polynomial expansion. (NC≤ 4).
Cn Coefficients of the polynomial. There are NC coefficients given.
NR Number of interpolation ranges used to tabulate values of ν(E). (See Section 0.5.2)
NP Total number of energy points used to tabulate ν(E).
Eint Interpolation scheme (see Section 0.5.2 for details).
ν(E) Average number of neutrons per fission.
If LNU=1, the structure of the
[MAT, 1, 452/ ZA, AWR,
[MAT, 1, 452/ 0.0, 0.0,
[MAT, 1,
0/ 0.0, 0.0,

section is
0, LNU, 0,
0,
0, NC,
0,
0, 0,

If LNU=2, the structure of the
[MAT, 1, 452/ ZA, AWR,
[MAT, 1, 452/ 0.0, 0.0,
[MAT, 1,
0/ 0.0, 0.0,

section is
0, LNU, 0, 0]HEAD
(LNU=2)
0,
0, NR, NP/Eint ,ν(E)]TAB1
0,
0, 0, 0]SEND

1.2.2

0]HEAD
(LNU=1)
0/ C1, C2, ...CNC]LIST
0]SEND

Procedures

If a polynomial representation (LNU=1) is used to specify ν(E), this representation is valid
over any range in which the fission cross section is specified (as given in Files 2 and 3). The
polynomial fit of ν(E) must be limited to a third-degree polynomial (NC≤4). If such a fit
does not reproduce the recommended values of ν(E), a tabulated form (LNU=2) should be
used.
If tabulated values of ν(E) are specified (LNU=2), then pairs of energy-ν values are
given. Values of ν(E) must be given that cover the entire energy range over which the
fission cross section is given in Files 2 and/or 3.
The values of ν(E) given in this section are for the average total number of neutrons
produced per fission event. When the number of delayed neutrons from fission ν d are also
present in the file (MT=455), the presence of the number of prompt neutrons per fission ν p
is mandatory (MT=456) and the sum of the two must be consistent with the total ν(E). In
this case, only LNU=2 representation is allowed for MT=452.
For spontaneous fission, the polynomial representation (LNU=1) with NC=1 is used to
describe the total number of neutrons per fission and C1 = ν. There is no energy dependence.

40

CHAPTER 1. FILE 1: GENERAL INFORMATION

1.3

Delayed Neutron Data, ν d, (MT=455)

This section describes the delayed neutrons resulting from either particle induced or spontaneous fission. The average total number of delayed neutron precursors emitted per fission,
ν d , is given, along with the decay constants, λi , for each precursor family. The fraction
of ν d generated for each family is given in File 5 (Chapter 5 of this report). The energy
distributions of the neutrons associated with each precursor family are also given in File 5.
For particle-induced fission, the total number of delayed neutrons is given as a function
of energy in tabulated form (LNU=2). The energy dependence is specified by tabulating
ν d (E) at a series of neutron energies using the same format as for MT=452. For spontaneous
fission LNU=1 is used with NC=1 and C1 = ν d as for MT=452.
The total number of delayed neutron precursors emitted per fission event at incident
energy E is given in this file and is defined as the sum of the number of neutrons emitted
for each of the precursor families,
νd =

NNF
X

ν i (E),

(1.2)

i=1

where NNF is the number of precursor families. The fraction of the total, Pi (E), emitted
for each family is given in File 5 (see Chapter 5) and is defined as:
Pi (E) =

1.3.1

ν i (E)
ν d (E)

Formats

The following quantities are defined.
LNU Test indicating which representation is used:
LNU=1 means that polynomial expansion is used;
LNU=2 means that a tabulated representation is used.
LDG Flag indicating energy dependence of delayed-group constants:
LDG=0 means that decay constants are energy-independent;
LDG=1 means that decay constants are energy-dependent.
NE

Number of energies at which the delayed-group constants are given.

NC Number of terms in the polynomial expansion (NC≤ 4).
NR Number of interpolation ranges used (NR≤ 20).
NP Total number of energy points used in the tabulation of ν(E)
Eint Interpolation scheme (see Section 0.5.2)
ν d (E) Total average number of delayed neutrons formed per fission event.
41

(1.3)

CHAPTER 1. FILE 1: GENERAL INFORMATION
NNF Number of precursor families considered.
λi (E) Decay constant (sec−1) for the ith precursor. (May be constant)
αi (E) Delayed-group abundances.
The structure when values of ν d are tabulated (LNU=2) and the delayed-group constants
are energy-independent (LDG=0) is
[MAT, 1,455/ ZA, AWR, LDG, LNU,
0, 0]HEAD
(LDG=0, LNU=2)
[MAT, 1,455/ 0.0, 0.0,
0,
0, NNF, 0/ λ1 ,λ2 ,...λNNF ]LIST
[MAT, 1,455/ 0.0, 0.0,
0,
0, NR, NP/ Eint , ν d (E)]TAB1
[MAT, 1, 0/ 0.0, 0.0,
0,
0,
0, 0]SEND
The structure when values of ν d are tabulated (LNU=2) and the delayed-group constants
are energy-dependent (LDG=1) is
[MAT, 1,455/ ZA, AWR, LDG, LNU, 0, 0]HEAD (LDG=1, LNU=2)
[MAT, 1,455/ 0.0, 0.0,
0,
0, NR, NE/ Eint ]TAB2
[MAT, 1,455/ 0.0, E1 ,
0,
0, NNF*2, 0/
λ1 (E1 ), α1 (E1 ), λ2 (E1 ), α2 (E1 ),

...λNNF (E1 ), αNNF (E1 )]LIST

----------------------------------------------------------------------[MAT, 1,455/ 0.0, ENE ,
0,
0, NNF*2, 0/
λ1 (ENE ), α1 (ENE ), λ2 (ENE ), α2 (ENE ),

...λNNF (ENE ), αNNF (ENE )]LIST

[MAT, 1,455/ 0.0, 0.0, 0, 0, NR, NP/Eint , ν d (E)]TAB1
[MAT, 1, 0/ 0.0, 0.0, 0, 0, 0, 0]SEND
The structure when values of ν d are defined by a polynomial expansion (LNU=1) and the
delayed-group constants are energy-independent (LDG=0), the structure of the section is 2
[MAT, 1,455/ ZA, AWR, LDG, LNU,
0, 0]HEAD (LDG=0, LNU=1)
[MAT, 1,455/ 0.0, 0.0,
0,
0, NNF, 0/
λ1 , λ2 ,...

λNNF ]LIST

[MAT, 1,455/ 0.0, 0.0,
0,
0,
1, 0/ ν d ]TAB1
[MAT, 1, 0/ 0.0, 0.0,
0,
0,
0, 0]SEND
The structure when values of ν d are defined by a polynomial expansion (LNU=1) and the
delayed-group constants are energy dependent (LDG=1), the structure of the section is:
[MAT, 1,455/ ZA, AWR, LDG, LNU, 0, 0]HEAD (LDG=1, LNU=1)
[MAT, 1,455/ 0.0, 0.0,
0,
0, NR, NE/ Eint ]TAB2
[MAT, 1,455/ 0.0, E1 ,
0,
0, NNF*2, 0/
λ1 (E1 ), α1 (E1 ), λ2 (E1 ), α2 (E1 ),

...λNNF (E1 ), αNNF (E1 )]LIST

----------------------------------------------------------------------[MAT, 1,455/ 0.0, ENE ,
0,
0, NNF*2, 0/
λ1 (ENE ), α1 (ENE ), λ2 (ENE ), α2 (ENE ),

[MAT, 1,455/ 0.0, 0.0,
[MAT, 1, 0/ 0.0, 0.0,
2

0,
0,

...λNNF (ENE ), αNNF (ENE )]LIST

0, NR, NP/Eint , ν d (E)]TAB1
0, 0, 0]SEND

Mandatory format option for spontaneous fission.

42

CHAPTER 1. FILE 1: GENERAL INFORMATION

1.3.2

Procedures

When tabulated values of ν d (E) are specified, as is required for particle-induced fission in
Section 1.2, they should be given for the same energy range as that used to specify the
fission cross section.
The probabilities of producing the precursors for each family and the energy distributions
of neutrons produced by each precursor family are given in File 5 (Chapter 5 of this report). If
parameters are incident-particle energy-dependent (LDG=1), the delayed-group abundances
are also given in File 1 (MT=455). It is extremely important that the same precursor families
be given in File 5 as are given in File 1 with the same abundances, and the ordering of the
families should be the same in both files. It is recommended that the families be ordered by
decreasing half-lives (λ1 < λ2 < . . . < λNNF ).
For spontaneous fission, the polynomial form (LNU=1) is used with only one term
(NC=1, C1 =ν d ).
If MT=455 is used, then MT=456 must also be present, in addition to MT=452.

1.4

Number of Prompt Neutrons per Fission, ν p,
(MT=456)

If the material fissions (LFI=1), a section specifying the average number of prompt neutrons
per fission, ν p , (MT=456) can be given using formats identical to MT=452. For particleinduced fission, ν p is given as a function of energy. The prompt ν for spontaneous fission
can also be given using MT=456, but there is no energy dependence.

1.4.1

Formats

The following quantities are defined:
LNU Indicates what representation of ν p (E) has been used:
LNU=1, polynomial representation has been used;
LNU=2, tabulated representation.
NC Count of the number of terms used in the polynomial expansion. (NC≤ 4)
NR Number of interpolation ranges used to tabulate values of ν p (E). (See Section 0.6.3.7)
NP Total number of energy points used to tabulate ν p (E).
Eint Interpolation scheme (see Section 0.5.2)
ν p (E) Average number of prompt neutrons per fission.
If LNU=2, (tabulated values of ν), the structure of the section is:

43

CHAPTER 1. FILE 1: GENERAL INFORMATION
[MAT, 1,456/ ZA,
[MAT, 1,456/ 0.0,
[MAT, 1, 0/ 0.0,
If LNU=1 (spontaneous

AWR, 0, LNU, 0, 0] HEAD
(LNU=2)
0.0, 0,
0, NR, NP/ Eint , /ν p (E)]TAB1
0.0, 0,
0, 0, 0]SEND
fission) the structure of the section is:

[MAT, 1,456/ ZA, AWR,
[MAT, 1,456/ 0.0, 0.0,
[MAT, 1, 0/ 0.0, 0.0,

1.4.2

0, LNU,
0,
0,
0,
0,

0, 0]HEAD
(LNU=1)
1, 0/ ν p ]LIST}
0, 0]SEND

Procedures

If tabulated values of ν p (E) are specified (LNU=2), then pairs of energy-ν values are given.
Values of ν p (E) should be given that cover any energy range in which the fission cross
section is given in File 2 and/or File 3. The values of ν p (E) given in this section are for the
average number of prompt neutrons produced per fission event. The energy independent
ν p for spontaneous fission is given using LNU=1 with NC=1 and C1 =ν p as described for
MT=452.
If MT=456 is specified, then MT=455 must also be specified as well as MT=452.

1.5

Components of Energy Release Due to Fission
(MT=458)

The energy released in fission is carried by fission products, neutrons, gammas, betas
(+ and -), and neutrinos and anti-neutrinos. The term fission products refers to all charged
particles that are emitted promptly, since for energy-deposition calculations, all such particles have short ranges and are usually considered to lose their energy locally. Neutrons and
gammas transport their energy elsewhere and need to be considered separately. In addition,
some gammas and neutrons are delayed, and in a shut-down assembly, one needs to know
the amount of energy tied up in these particles and the rate at which it is released from the
metastable nuclides or precursors. The neutrino energy is lost completely in most applications, but is part of the Q-value. As far as the betas are concerned, prompt betas, being
charged, deposit their energy locally like the fission products, and their prompt energies are
included with the fission product energy.
This format recognises nine specific components for fission energy release. The nomenclature used to define these nine energy release terms is:
ET Sum of all the partial energies that follow. This sum is the total energy
release per fission and equals the Q-value.
EFR Kinetic energy of the fission products (following prompt neutron emission
from the fission fragments).
ENP Kinetic energy of the prompt fission neutrons.
END Kinetic energy of the delayed fission neutrons.
44

CHAPTER 1. FILE 1: GENERAL INFORMATION
EGP Total energy released by the emission of prompt γ rays.
EGD Total energy released by the emission of delayed γ rays.
EB Total energy released by delayed β’s.
ENU Energy carried away by neutrinos.
ER Total energy less the energy of the neutrinos (ET - ENU); equal to the
pseudo-Q-value in File 3 for MT=18.
NPLY Order of the polynomial expansion of the energy-components.
The total energy release due to fission is an incident-energy dependent quantity, as are many,
if not all, of the constituent terms.
There are two formats available to represent these energy dependencies. From the work
of Sher and Beck (Reference 1), the energy dependence is given as:
Ei (Einc ) = Ei (0) − δEi (Einc )

(1.4)

where:
Ei (Einc ) is the fission energy release for each of the nine components;
Ei (0) is a constant, one for each of the nine fission energy release terms. These
constants and their uncertainties are specified via the NPLY=0 format option,
defined in Section 1.5.1;
δEi (Einc ) is a function that allows for the definition of the energy-dependence of this
fission energy component. These functions can not be determined based upon
the data given in the evaluated file; rather, unique functions are defined for
the various fission energy release terms as follows:
δET = −1.057 Einc + 8.07 [ν(Einc ) − ν(0)]
δEB = 0.075Einc

δEGD = 0.075Einc
δENU = 0.100Einc
δEFR = 0
δENP = −1.307Einc + 8.07 [ν(Einc ) − ν(0)]

δEGP = 0

45

CHAPTER 1. FILE 1: GENERAL INFORMATION
However, a recent study of relevant experimental data, supplemented with model calculations, by Madland (Reference 2) has shown that δEFR and δEGP terms are not zero, but
instead have definite dependencies upon the incident neutron energy Einc . This means that
the remaining δ values above are accounting for these dependencies in addition to their
own dependencies. In short, the calculations of individual energy release values are suspect even though the total energy release, ET, is substantially correct. Madland concludes
that the energy dependence for the various fission energy release terms may be accurately
represented with simple polynomial expansions in incident neutron energy. As such, the
energy dependencies are completely defined by specifying the coefficients of NPLYth order
polynomials:
2
Ei (Einc ) = c0 + c1 Einc + c2 Einc
+ ...
The polynomial expansion coefficients and their uncertainties are specified via the NPLY6=0
option in Section 1.5.1 With the Madland representation, the complete energy release may
be determined solely from the data tabulated by the evaluator, without regard to the various
functional forms defined above.

1.5.1

Formats

The structure of this section always starts with a HEAD record and ends with a SEND
record. The section contains no subsections and only one LIST record. The structure of a
section takes one of two forms. When NPLY=0, the format is:
[MAT, 1,458/ ZA, AWR,
0,
0,
0,
0]HEAD
[MAT, 1,458/ 0.0, 0.0,
0, NPLY=0,N1=18, N2=9/
EFR, ∆EFR, ENP, ∆ENP, END, ∆END,
EGP, ∆EGP, EGD, ∆EGD, EB, ∆EB,
ENU, ∆ENU, ER, ∆ER, ET, ∆ET]LIST
[MAT, 1, 0/ 0.0, 0.0,
0,
0,
0,
0]SEND
where the ∆ terms represent the uncertainties on the preceeding quantity. The data are used
in conjunction with the Sher and Beck formulas from Section 1.5 to determine the energy
release.
For the polynomial expansion represntation, NPLY represents the highest polynomial
order needed to define any of the energy release components. The size of the list record is
18×(NPLY+1). The first 18 values are nine pairs of c0 and ∆c0 terms, followed by nine
pairs of c1 and ∆c1 terms, and continuing through the maximum polynomial order.
[MAT, 1,458/ ZA, AWR,
0,
0,
0,
0]HEAD
[MAT, 1,458/ 0.0, 0.0,
0,
NPLY, 18*(NPLY+1), 9*(NPLY+1)/
cEFR
, ∆cEFR
, cENP
, ∆cENP
, cEND
, ∆cEND
0
0
0
0
0
0
cEGP
, ∆cEGP
, cEGD
, ∆cEGD
, cEB
, ∆cEB
0
0
0
0
0
0
ENU
ER
ER
ET
ET
cENU
,
∆c
,
c
,
∆c
,
c
,
∆c
0
0
0
0
0
0
EFR
ENP
ENP
END
END
cEFR
,
∆c
,
c
,
∆c
,
c
,
∆c
1
1
1
1
1
1
, ∆cEB
, cEB
, ∆cEGD
, cEGD
, ∆cEGP
cEGP
1
1
1
1
1
1
ET
ET
ER
ER
ENU
,
∆c
,
c
,
∆c
,
c
,
∆c
cENU
1
1
1
1
1
1
[MAT, 1, 0/ 0.0, 0.0,
0,
0,
0,
0]SEND

46

CHAPTER 1. FILE 1: GENERAL INFORMATION
As with the NPLY=0 option, the ∆ terms represent the uncertainties on the preceeding
quantities, respectively.

1.5.2

Procedures

This section should be used for fertile and fissile isotopes only. Consistency should be
maintained between the Q-values in File 3, the energies calculated from File 5 and 15 and
the energies calculated from the data given here. Note that ER equals the pseudo-Q-value
for fission (MT=18) in File 3.
Other components are not so readily determined or checked. When the NPLY=0 option is
used, the procedure should be that File 5 and File 15 data take precedent, whenever possible.
That is, ”prompt” fission neutron energy calculated from File 5 spectra for MT=18 should
be put into File 1; the same holds true for the delayed neutron spectra given in File 5,
MT=455. The ”prompt” gamma energy calculated from File 15 (MT=18 for fission) should
be put into File 1; these are the prompt gammas due to the fission process. When NPLY6=0,
the nature of the polynomial fitting process may yield small differences between the values
calculated from the polynomial coefficients and the values calculated using the spectral data
from Files 5 and 15.
These quantities should be calculated at the lowest energy given in the Files for MT=18
except for fissile isotopes for which the thermal spectra should be used. For fertile materials,
the spectrum given at threshold would be appropriate. Note that the File 5 spectra for
MT=18 should be used with ν prompt (not ν total) for the fission neutrons. MT=455 in
File 5 contains the delayed fission neutron spectra.
In many reactor applications, the time dependent energy deposition rates are required
rather than the components of the total energy per fission which are the values given in this
MT. Time-dependent energy deposition parameters can be obtained from the group spectra
in File 5 (MT=455) for delayed neutrons. Codes such as CINDER, RIBD, and ORIGEN
must be used, however, to obtain more detailed information on the delayed neutrons and all
time-dependent parameters for the betas and the gammas due to the fission process.
The time-integrated energies for delayed neutrons, delayed gammas, and delayed betas
as calculated from the codes listed above may not always agree with the energy components
given in File 1. The File 1 components must sum to ET (the total energy released per
fission).
In heating calculations, the energy released in all nuclear reactions besides fission, principally the gamma-energy released in neutron radiative capture, enters analogously to the
various fission energy components. Thus the (n,γ) energy-release would be equal to the
Q-value in File 3, MT=102, for the capturing nuclide. The capture gammas can be prompt
or delayed, if branching to isomeric states is involved, and this is relevant to various fissionand burnup-product calculations. The ”sensible energy” in a heating calculation is the sum
of ER, defined previously, plus the energy released in all other reactions.

47

CHAPTER 1. FILE 1: GENERAL INFORMATION

1.6

Delayed Photon Data (MT=460)

This section describes the delayed photon source function resulting from either particle
induced or spontaneous fission. The delayed gamma source function is defined as the number
of gammas emitted per unit time after the fission event, per unit energy for a fixed incident
energy:
d2 nγ
(E, Eγ , t).
(1.5)
Sγ (E, Eγ , t) +
dt dEγ
Here E is the energy of the fission-inducing projectile, Eγ is the energy of the emitted
photon, t is the time following fission at which the photon is emitted, and d2 nγ /dt dEγ is
the number of photons emitted per fission per second per eV. The source function may be
given either in a discrete or continuous representation.
In the discrete representation (LO=1), the source function is given as a series of tables of
photon multiplicities, yi (E), in File 12 with LO=1, with their associated time dependences:
Sγ (E, Eγ , t) =

NG
X
i=1

δ(Eγ − Ei ) yi (E) Ti .

(1.6)

The continuous representation (LO=2) is similar in spirit to the delayed neutron data
(MT=455). In this representation, one must give the photon multiplicities in File 12 and
the fraction of photons emitted at each energy, fi (E ← Eγ ), in an MF=15 table, for each
precursor family. The time dependence of each precursor family is given as a list of time
constants:
NNF
X
Sγ (E, Eγ , t) =
yi (E) fi (E ← Eγ ) λi exp(−tλi )
(1.7)
i=1

Only the precursor family time constants are stored in MF=1.

1.6.1

Formats

1.6.1.1

Discrete Representation (LO=1)

The following quantities are defined:
NG The number of discrete photons.
NR, NP, tint Standard TAB1 parameters.
Ei Energy of the ith photon (eV).
Ti (t) Time dependence of the ith photons multiplicity (sec-1).
The structure of the time dependence data block is:

48

CHAPTER 1. FILE 1: GENERAL INFORMATION
[MAT,1,460/
[MAT,1,460/
[MAT,1,460/

ZA, AWR, LO, 0, NG, 0]HEAD
(LO=1)
E1, 0.0, 1, 0, NR, NP/tint / T1 (t)]TAB1
E2, 0.0, 2, 0, NR, NP/tint / T2 (t)]TAB1
---------------------[MAT,1,460/ ENG , 0.0, NG, 0, NR, NP/tint / TNG (t)]TAB1
[MAT,1, 0/ 0.0, 0.0, 0, 0, 0, 0]SEND

1.6.1.2

Continuous Representation (LO=2)

The following quantities are defined:
NNF The number of precursor families considered.
λi Decay constant (sec−1 ) for the ith precursor.
The structure of this data block is:
[MAT,1,460/ ZA, AWR, LO, 0,
0, 0]HEAD
[MAT,1,460/0.0, 0.0, 0, 0, NNF, 0/
λ1 , λ2 , ---------- λNNF ]LIST
[MAT,1, 0/0.0, 0.0, 0, 0,
0, 0] SEND

1.6.2

Procedures

1.6.2.1

Discrete Representation (LO=1)

(LO=2)

The photon multiplicity is given in File 12 with LO=1 in File 12 set. Each discrete photon in
File 12 must have an associated time dependence table specified in File 1. It is recommended
that the photon multiplicities be given in the same order as the photons that are listed in
File 1.
1.6.2.2

Continuous Representation (LO=2)

The probability of producing precursors for each family and the energy distributions of
photons produced by each precursor family are given in Files 12 and 15. It is extremely
important that the same precursor families be given in Files 12 and 15 as in File 1 (MT=460)
and the ordering of families should be the same in all files. It is recommended that all families
be ordered by decreasing half-lives (λ1 < λ2 < · · · < λNNF ).

References for Chapter 1
1. R. Sher and C. Beck, Electric Power Research Institute report, EPRI-NP-1771 (1981).
2. D.G. Madland, Nucl.Phys. A772 (2006) 113-137.

49

Chapter 2
File 2: RESONANCE
PARAMETERS
2.1

General Description

The primary function of File 2 is to contain data for both resolved and unresolved resonance
parameters. It has only one section, with the reaction type number MT=151. A File 2 is
required for incident-neutron evaluations, but it may be omitted in other cases. The use of
File 2 is controlled by the parameter LRP (see section 1.1):
LRP=-1 No File 2 is given. Not allowed for incident neutrons.
LRP= 0 No resonance parameters are given except for the scattering radius AP.
AP is included for the convenience of users who need an estimate of the
potential scattering cross section. It is not used to calculate a contribution
to the scattering cross section, which in this case is represented entirely in
File 3.
LRP= 1 Resonance contributions for the total, elastic, fission, and radiative capture
cross sections are to be computed from the resonance parameters and added
to the corresponding cross sections in File 31 .
The File 2 resonance contributions should also be added to any lumped reactions included in File 3. For SLBW or MLBW (see below), any other
competing reaction in the resonance range must be given in their entirety in
File 3 under the corresponding MT number or as the background in the total
cross section. The effects of the competing reactions on the resonance reactions are included using a single competitive width, Γx . This width is given
1

In the unresolved resonance region, the evaluator may, optionally, specify a different procedure, which
uses the unresolved resonance parameters in File 2 solely for the purpose of computing an energy-dependent
self-shielding factor. This option is governed by a flag, LSSF, defined in Section 2.3.1, and discussed in
Section 2.4.17. When this option is specified, File 3 is used to specify the entire infinitely-dilute cross
section, and the function of File 2 is to specify the calculation of self-shielding factors for shielded pointwise
or multigroup values.

50

CHAPTER 2. FILE 2: RESONANCE PARAMETERS
explicitly in the unresolved resonance region, and implicitly in the resolved
region; in the latter case, it is permissible for the total width to exceed the
sum of the partials. The difference is the competitive width:
Γx = Γ − (Γn + Γγ + Γf )
For the Reich-Moore or Adler-Adler formalisms, competitive reactions are not
used. For the R-Matrix Limited Format (LRF=7, see below), competitive
reactions are included as separate channels.
LRP= 2 Resonance parameters are given in File 2 but are not to be used in calculating
cross sections, which are assumed to be represented completely in File 3.
This option is usually used in derived libraries to flag that the cross sections
had already been reconstructed, but the source resonance parameters are stil
available, if needed.
The resonance parameters for a material are obtained by specifying the parameters
for each isotope in the material . The data for the various isotopes are ordered by
increasing ZAI values (charge-isotopic mass number). The resonance data for each isotope
may be divided into several incident neutron energy ranges, given in order of increasing
energy. The energy ranges for an isotope should not overlap; each may contain a different
representation of the cross sections.
In addition to these parameterized resonance ranges, the full energy range of the evaluated data file may contain one or two ranges, in which the cross sections are given in
pointwise form. Comments on these ranges follow:
1. The low energy region (LER) is one in which the cross sections are tabulated as smooth
functions of energy. Doppler effects must be small enough so that the values essentially correspond to zero degrees Kelvin. For light elements, i.e., those whose natural
widths far exceed their Doppler widths and hence undergo negligible broadening, the
entire energy range can often be represented in this way. For heavier materials, this
region can sometimes be used below the lowest resolved resonances. With a good
multilevel resonance fit, the LER can often be omitted entirely, and this is preferred.
An important procedure for the LER is described in Section 2.4.6 item 4.
2. The resolved resonance region (RRR) is one in which resonance parameters for individual resonances are given. Usually this implies that experimental resolution is good
enough to ”see” the resonances, and to determine their parameters by area or shape
analysis, but an evaluator may choose to supply fictitious resolved parameters if he
so desires. If the evaluator does this, the resonances must have physically-allowed
quantum numbers, and be in accord with the statistics of level densities (Appendix D,
Section D.2.2). A File 3 background may be given. The essential point is that resonance self-shielding can be accounted for by the user for each resonance individually.
3. The unresolved resonance region (URR) is that region in which the resonances still
do not actually overlap, so that self-shielding is still important, but experimental
51

CHAPTER 2. FILE 2: RESONANCE PARAMETERS
resolution is inadequate to determine the parameters of individual resonances. In this
situation, self-shielding must be handled on a statistical basis. A File 3 may be given.
The interpretation of this cross section depends on the flag LSSF (see Sections 2.3.1
and 2.4.17). It may be interpreted either as a partial background cross section, to
be added to the File 2 contribution, as in the resolved resonance region or it may be
interpreted as the entire dilute cross section, in which case File 2 is to be used solely to
specify the self-shielding appropriate to this energy region. It is important to choose
the boundary between the RRR and the URR so that the statistical assumptions
underlying the unresolved resonance treatments are valid. This problem is discussed
further in Section 2.4.
4. The high-energy region (HER) starts at still higher energies where the resonances
overlap and the cross sections smooth out, subject only to Ericson fluctuations. The
boundary between the URR and HER should be chosen so that self-shielding effects
are small in the HER.
File 3 may contain ”background cross sections” in the resonance ranges resulting from
inadequacies in the resonance representation (e.g., SLBW), the effects of resonances outside
the energy range, the average effects of missed resonances, or competing cross sections, which
are not accounted for explicitly. If these background cross sections are non-zero, there must
be double energy points in File 3 corresponding to each resonance range boundary (except
10−5 eV). See Section 2.4 for a more complete discussion of backgrounds.
Several representations are allowed for specifying resolved resonance parameters. The
flag, LRF, indicates the representation used for a particular energy range:
LRF=1 Single-level Breit-Wigner; (no resonance-resonance interference; one singlechannel inelastic competitive reaction is allowed). Use of this format is discouraged for new evaluations, which should use the Reich-Moore approximation (LRF=3 or 7).
LRF=2 Multilevel Breit-Wigner (resonance-resonance interference effects are included in the elastic scattering and total cross sections; one single-channel inelastic competitive reaction is allowed). Use of this format is discouraged for
new evaluations, which should use the Reich-Moore approximation (LRF=3
or 7).
LRF=3 Reich-Moore (multilevel multi-channel R-matrix; no competitive reactions
allowed).
It is possible to define partial widths Γls1 J and Γls2 J with two different values
of the channel spin, as is required when both the target spin and the orbital
angular momentum are greater than zero. This is accomplished by setting
the resonance spin parameter AJ to a positive value for the larger channel
spin (s = I + 1/2), and negative for the smaller channel spin (s = I − 1/2).
(See definition of AJ in Section 2.2.1.) Older ENDF files have not used this
feature, but instead have only positive AJ; in this case, all resonances of a
given l, J are assumed to have the same channel spin.
52

CHAPTER 2. FILE 2: RESONANCE PARAMETERS
For a given resonance, the only rigorously conserved quantities are J (total
angular momentum) and π (total parity). Nevertheless, this format assumes
that both l (orbital angular momentum) and s (channel spin) are also conserved quantities.
The LRF=3 format permits only a limited subset of Reich-Moore evaluations,
but is adequate for some situations.
LRF=4 Adler-Adler (level-level and channel-channel interference effects are included
in all cross sections via ”effective” resonance parameters; usually applied to
low-energy fissionable materials; no competitive reactions).
LRF=5 This option is no longer available.
LRF=6 This option is no longer available.
LRF=7 R-Matrix Limited format, which contains all the generality of LRF=3 plus
unlimited numbers and types of channels.
Preferred formalisms for evaluation are discussed in Section 2.4.15. Further discussion of
the above formalisms is contained in the Procedures Section 2.4.
Each resonance energy range contains a flag, LRU, that indicates whether it contains
resolved or unresolved resonance parameters. LRU=1 means resolved, LRU=2 means unresolved.
Only one representation is allowed for the unresolved resonance parameters, namely
average single-level Breit-Wigner. However, several options are permitted, designated by
the flag LRF. With the first option, LRF=1, only the average fission width is allowed to
vary as a function of incident neutron energy. The second option, LRF=2, allows the
following average parameters to vary: level spacing, fission width, reduced neutron width,
radiation width, and a width for the sum of all competitive reactions.
The data formats for the various resonance parameter representations are given in Sections 2.2.1 (resolved) and 2.3.1 (unresolved). Formulae for calculating cross sections from
the various formalisms are given in Appendix D.
The following quantities have definitions that are the same for all resonance parameter
representations:
NIS Number of isotopes in the material (NIS<10).
ZAI (Z,A) designation for an isotope.
NER Number of resonance energy ranges for this isotope.
ABN Abundance of an isotope in the material. This is a number fraction, not a
weight fraction, nor a percent.

53

CHAPTER 2. FILE 2: RESONANCE PARAMETERS
LFW Flag indicating whether average fission widths are given in the unresolved
resonance region for this isotope:
LFW=0, average fission widths are not given;
LFW=1, average fission widths are given.
NER Number of resonance energy ranges for isotope.
EL Lower limit for an energy range2 .
EH Upper limit for an energy range3 .
LRU Flag indicating whether this energy range contains data for resolved or unresolved resonance parameters:
LRU=0, only the scattering radius is given (LRF=0, NLS=0, LFW=0);
LRU=1, resolved resonance parameters are given.
LRU=2, unresolved resonance parameters are given.
LRF Flag indicating which representation has been used for the energy range. The
definition of LRF depends on the value of LRU:
If LRU=1 (resolved parameters), then:
LRF=1, single-level Breit-Wigner (SLBW);
LRF=2, multilevel Breit-Wigner (MLBW);
LRF=3, Reich-Moore (RM);
LRF=4, Adler-Adler (AA);
LRF=5, no longer available;
LRF=6, no longer available;
LRF=7, R-Matrix Limited (RML).
If LRU=2 (unresolved parameters), then:
LRF=1, only average fission widths are energy-dependent;
LRF=2, average level spacing, competitive reaction widths, reduced
neutron widths, radiation widths, and fission widths are energydependent.
NRO Flag designating possible energy dependence of the scattering radius:
NRO=0, radius is energy independent;
NRO=1 radius expressed as a table of energy, radius pairs.
2

These energies are the limits to be used in calculating cross sections from the parameters. Some resolved
resonance levels, e.g., bound levels, will have resonance energies outside the limits.

54

CHAPTER 2. FILE 2: RESONANCE PARAMETERS
NAPS Flag controlling the use of the two radii, the channel radius a and the
scattering radius AP.
For NRO=0 (AP energy-independent), if:
NAPS=0 calculate a from Equation (D.0) given in Appendix D, and
read AP as a single energy-independent constant on the subsection’s second CONT record that defines the range; use a
in the penetrabilities Pl and shift factors Sl , and AP in the
hard-sphere phase shifts φl ;
NAPS=1 do not use Equation (D.0); use AP in the penetrabilities and
shift factor as well as in the phase shifts.
For NRO=1 (AP energy-dependent), if:
NAPS=0 calculate a from the above equation and use it in the penetrabilities and shift factors. Read AP(E) as a TAB1 quantity
in each subsection and use it in the phase shifts;
NAPS=1 read AP(E) and use it in all three places, Pl , Sl , φl ;
NAPS=2 read AP(E) and use it in the phase shifts. In addition, read
the single, energy-independent quantity ”AP, see following,
and use it in Pl and Sl , overriding the above equation for a.

File 2 contains a single section (MT=151) containing subsections for each energy range
of each isotope in the material.
Two versions of the File 2 format structure are presented here. The first one (sometimes
denoted as a special case), is merely a particular instance of the general case with particular values NER=1, LRU=0, and NRO=0. No resonance parameters are given, only the
scattering radius is specified (such material is not permitted to have multiple isotopes or an
energy-dependent scattering radius). The structure of File 2, for this special case is
[MAT,
[MAT,
[MAT,
[MAT,
[MAT,
[MAT,

2,151/
2,151/
2,151/
2,151/
2, 0/
0, 0/

ZA,
ZAI,
EL,
SPI,
0.0,
0.0,

AWR, 0, 0,
ABN, 0,LFW,
EH,LRU,LRF,
AP, 0, 0,
0.0, 0, 0,
0.0, 0, 0,

NIS,
0]HEAD
NER,
0]CONT
NRO,NAPS]CONT
NLS,
0]CONT
0,
0]SEND
0,
0]FEND

(NIS=1)
(ZAI=ZA,ABN=1,LFW=0,NER=1)
(LRU=0,LRF=0,NRO=0,NAPS=0)
(NLS=0)}

The second version is a general case with given resonance parameters. The structure of
File 2 for this general case is as follows:

55

CHAPTER 2. FILE 2: RESONANCE PARAMETERS
[MAT, 2,151/ ZA, AWR,
0,
0, NIS,
0]HEAD
[MAT, 2,151/ ZAI, ABN,
0, LFW, NER,
0]CONT (isotope)
[MAT, 2,151/ EL, EH, LRU, LRF, NRO, NAPS]CONT (range)

...
[MAT, 2,151/ EL, EH, LRU, LRF, NRO, NAPS]CONT (range)


...
[MAT, 2,151/ EL, EH, LRU, LRF, NRO, NAPS]CONT (range)


...
[MAT, 2, 0/ 0.0, 0.0,
0,
0,
0,
0]SEND}
The data are given for all ranges for a given isotope, and then for all isotopes. The data for
each range start with a CONT (range) record; those for each isotope, with a CONT (isotope)
record. The specifications for the subsections that include resonance parameters are given
in Sections 2.2.1 for the resolved region and Section 2.3.1 for the unresolved region. A multiisotope material is permitted to have some, but not all, isotopes specified by a scattering
radius only. The structure of a subsection for such an isotope is:
[MAT, 2,151/ SPI,

AP,

0,

0, NLS,

0]CONT

(NLS=0)}

and as above LFW=0, NER=1, LRU=0, LRF=0, NRO=0, and NAPS=0 for this isotope.
In the case that NRO6=0, the ”range” record preceding each subsection is immediately
followed by a record giving the energy dependence of the scattering radius, AP.
[MAT, 2,151/ 0.0, 0.0,

0,

0,

NR,

NP/ Eint / AP(E)]TAB1

If NAPS is 0 or 1 the value of AP on the next record of the subsection should be set to 0.0.
If NAPS is 2, it should be set equal to the desired value of the channel radius.

2.2
2.2.1

Resolved Resonance Parameters (LRU=1)
Formats

Several different resonance formalisms are allowed to represent the resolved resonance parameters. Formulae for the various quantities, and further comments on usage, are given in
Appendix D. The flag LRU=1, given in the second CONT record, indicates that resolved
resonance parameters are given for a particular energy range. Another flag, LRF, in the
same record specifies which resonance formalism has been used.
The following quantities are defined for use with all formalisms:

56

CHAPTER 2. FILE 2: RESONANCE PARAMETERS
SPI Spin, I, of the target nucleus.
AP Scattering radius in units of 10−12 cm. For LRF=1 through 4, it is assumed
to be independent of the channel quantum numbers.
NLS Number of l-values (neutron orbital angular momentum) in this energy region.
LRF=1 through 4, a set of resonance parameters is given for each l-value.
LRF=5 and 6, NLS is the number of l-values required to converge the calculation of the scattering cross section (see Sections 2.4.19 and 2.4.20). Another
cutoff, NLSC, is provided for converging the angular distributions. For the
limit on NLS see Appendix G.
AWRI Ratio of the mass of a particular isotope to that of a neutron.
QX Q-value to be added to the incident particle’s center-of-mass energy to determine the channel energy for use in the penetrability factor. The conversion
to laboratory system energy depends on the reduced mass in the exit channel. For inelastic scattering to a discrete level, the Q-value is minus the level
excitation energy. QX=0.0 if LRX=0.
L Value of l.
LRX Flag indicating whether this energy range contains a competitive width:
LRX=0 no competitive width is given, and Γ = Γn + Γγ + Γf in the
resolved resonance region, while hΓx i = 0 in the unresolved
resonance region; LRX must be 0 for LRF=3, 4 and 7;
LRX=1 a competitive width is given, and is an inelastic process to the
first excited state. In the resolved region, it is determined by
subtraction, Γx = Γ − [Γn + Γγ + Γf ]
This parameter is irrelevant for LRF=7.
NRS Number of resolved resonances for a given l-value. For the limit on NRS see
Appendix G.
ER Resonance energy (in the laboratory system).
AJ The absolute value of AJ is the floating-point value of J (the spin, or total
angular momentum, of the resonance).
When two channel spins are possible, if the sign of AJ is negative, the lower
value for the channel spin is implied; if positive, the higher value is implied.
When AJ is zero, only one value of channel spin is possible so there is no
ambiguity; the channel spin s is equal to the orbital angular momentum l.

57

CHAPTER 2. FILE 2: RESONANCE PARAMETERS
GT Resonance total width, Γ, evaluated at the resonance energy ER.
GN Neutron width Γn evaluated at the resonance energy ER.
GG Radiation width, Γγ , a constant.
GF Fission width, Γf , a constant.
GX Competitive width, Γx , evaluated at the resonance energy ER.
It is not given explicitly for LRF=1 or 2 but is to be obtained by subtraction,
GX = GT –(GN+ GG + GF), if LRX6=0.
a Channel radius, in 10−12 cm. An uppercase symbol is not defined because
it is not an independent library quantity. Depending on the value of NAPS,
it is either calculated from the equation given earlier (and in Appendix D),
or read from the position usually assigned to the scattering radius AP.

2.2.1.1

SLBW and MLBW (LRF=1 or 2)

The structure of a subsection is:
[MAT, 2,151/ 0.0, 0.0, 0, 0, NR, NP/ Eint /AP(E)]TAB1 (if NRO 6=0)
[MAT, 2,151/ SPI, AP, 0, 0, NLS, 0]CONT
Use AP=0.0, if AP(E) is supplied and NAPS=0 or 1.
[MAT, 2,151/ AWRI, QX,
L,
LRX, 6*NRS,
NRS
ER1 ,
AJ1 ,
GT1 ,
GN1 ,
GG1 ,
GF1 ,
ER2 ,
AJ2 ,
GT2 ,
GN2 ,
GG2 ,
GF2 ,
ERN RS , AJN RS , GTN RS , GNN RS , GGN RS , GFN RS ] LIST
The LIST record is repeated until each of the NLS l-values has been specified in order of
increasing l. The values of ER for each l-value are given in increasing order.
The use of either of the Breit-Wigner formats is discouraged for new evaluations. The
recommended format is the R-Matrix Limited Format (LRF=7).
2.2.1.2

Reich-Moore (LRF=3)

The following additional quantities are defined:
LAD Flag indicating whether these parameters can be used to compute angular
distributions.
LAD=0 do not use
LAD=1 can be used if desired. Do not add to file 4.
NLSC Number of l-values which must be used to converge the calculation with
respect to the incident l-value in order to obtain accurate elastic angular
distributions. See Sections D.1.5 and D.1.6 (NLSC≥NLS).
58

CHAPTER 2. FILE 2: RESONANCE PARAMETERS
APL l-dependent scattering radius. If zero, use APL=AP.
GFA First partial fission width, a constant.
GFB Second partial fission width, a constant.
GFA and GFB are signed quantities, their signs being determined by the relative phase of
the width amplitudes in the two fission channels. In this case, the structure of a subsection
is similar to LRF=1 and 2, but the total width is eliminated in favor of an additional partial
fission width. GFA and GFB can both be zero, in which case, Reich-Moore reduces to an
R-function.
The structure for a subsection is:
[MAT, 2,151/ 0.0,0.0,
0, 0,
NR,
NP/Eint /AP(E)]TAB1 (if NRO6= 0)
[MAT, 2,151/ SPI, AP, LAD, 0,
NLS, NLSC]CONT
[MAT, 2,151/AWRI,APL,
L, 0, 6*NRS, NRS/
ER1 , AJ1 , GN1 , GG1 , GFA1 , GFB1 ,
ER2 , AJ2 , GN2 , GG2 , GFA2 , GFB2 ,
-------------------------------ERN RS ,AJN RS ,GNN RS ,GGN RS ,GFAN RS ,GFBN RS ]LIST
The LIST record is repeated until each of the NLS l-values has been specified in order of
increasing l. The values of ER for each l-value are given in increasing order.
2.2.1.3

Adler-Adler (LRF=4)

For the case of LRF=4 additional quantities are defined:
LI Flag to indicate the kind of parameters given:
LI=1, total widths only
LI=2, fission widths only
LI=3, total and fission widths
LI=4, radiative capture widths only
LI=5, total and capture widths
LI=6, fission and capture widths
LI=7, total, fission, and capture widths.
NX Number of sets of background constants given. There are six constants per
set. Each set refers to a particular cross section type. The background
correction for the total cross section is calculated by using the six constants
in the manner following .

59

CHAPTER 2. FILE 2: RESONANCE PARAMETERS
σT

Background
.√
=C
E (AT1 + AT2 /E + AT3 /E 2 + AT4 /E 3 + BT1 E + BT2 E 2 )

where C = πλ̄2 = π/k 2 and k is defined in Appendix D.

The background terms for the fission and radiative capture cross sections
are calculated in a similar manner.
NX=2, background constants are given for the total and capture cross sections.
NX=3, background constants are given for the total, capture, and fission
cross sections.
NJS Number of sets of resolved resonance parameters (each set having its own
J-value) for a specified l.
NLJ Number of resonances for which parameters are given, for a specified AJ and
L.
AT1 , AT2 , AT3 , AT4 , BT1 , BT2 Background constants for the total cross section.
AF1 , AF2 , AF3 , AF4 , BF1 , BF2 Background constants for the fission cross section.
AC1 , AC2 , AC3 , AC4 , BC1, BC2 Background constants for the radiative capture cross
section.
DETr

3

Resonance energy, (µ), for the total cross section. Here and below, the
subscript r denotes the rth resonance.

DEFr Resonance energy, (µ), for the fission cross section.
DECr Resonance energy, (µ), for the radiative capture cross section.
DWTr Value of Γ/2, (v), for the total cross section.
DWFr Value of Γ/2, (v), for the fission cross section.
DWCr Value of Γ/2, (v), for the radiative capture cross section.
GRTr Symmetrical total cross section parameter, GTr .
GITr Asymmetrical total cross section parameter, HTr .
GRFr Symmetrical fission parameter, Gfr .
GIFr Asymmetrical fission parameter, Hfr .
GRCf Symmetrical capture parameter, Gγr .
3

Note: DETr=DEFr=DECr and DWTr=DWFr=DWCr. The redundancy is an historical carryover.

60

CHAPTER 2. FILE 2: RESONANCE PARAMETERS
GICr Asymmetrical capture parameter, Hγr .
The structure of a subsection for LRF=4 depends on the value of NX (the number of
sets of background constants). For the most general case (NX=3) the structure is
[MAT, 2,151/ 0.0, 0.0, 0, 0, NR, NP/ Eint / AP(E)] TAB1

optional record for energy-dependent scattering radius.
[MAT, 2,151/ SPI, AP,
0,
0,
NLS,
0] CONT
[MAT, 2,151/ AWRI, 0.0, LI,
0, 6*NX, NX
AT1 , AT2 , AT3 , AT4 , BT1 , BT2 ,
AF1 , --------------------, BF2 ,
AC1 , --------------------, BC2 ] LIST
[MAT, 2,151/ 0.0, 0.0,
L,
0,
NJS,
0] CONT(l)
[MAT, 2,151/ AJ, 0.0,
0,
0,12*NLJ, NLJ/
DET1 , DWT1 , GRT1 , GIT1 , DEF1 , DWF1 ,
GRF1 , GIF1 , DEC1 , DWC1 , GRC1 , GIC1 ,
DET2 , DWT2 , GIC2 , -------------DET3 ,---------------------------------------------------------------------------------, GICN LJ ] LIST
The last LIST record is repeated for each J-value (there will be NJS such LIST records). A
new CONT (l) record will be given which NJS LIST records will follow. Note that if NX=2
then the quantities AF1 through BF2 will not be given in the first LIST record. Also, if
LI6=7 then certain of the parameters for each level may be set to zero, i.e., the fields for
parameters not given (depending on LI) will be set to zero.
The format has no provision for giving Adler-Adler parameters for the scattering cross
section. The latter is obtained by subtracting the capture and fission cross sections from
the total.
Although the format allows separation of the resonance parameters into J-subsets, no use
is made of J in the Adler-Adler formalism. There is no analog to the resonance-resonance
interference term of the MLBW formalism. Such interference is represented implicitly by
the asymmetric terms in the fission and capture cross sections.
2.2.1.4

General R-Matrix (LRF=5)

The format is no longer available in ENDF-6.
2.2.1.5

Hybrid R-function (LRF=6)

The format is no longer available in ENDF-6.
2.2.1.6

R-Matrix Limited Format (LRF=7)

In the R-matrix scattering theory, a channel is defined by the two particles inhabiting that
channel and by the quantum numbers for the combination. The two particles are hereafter
61

CHAPTER 2. FILE 2: RESONANCE PARAMETERS
referred to as a particle-pair (PP), and are defined by their properties: neutron (or other
particle) plus target nuclide (in ground or excited state), with individual identifiers such
as mass, spin, parity, and charge. The additional quantum numbers defining the channel
include orbital angular momentum l, channel spin s and associated parity, and total spin
and parity J π .
NOTE: This format is NOT restricted to one neutron (entrance) channel and two exit
channels. There may be several entrance channels and a multitude of exit channels. Chargedparticle exit channels are not excluded.
The term ”spin group” may be used to define the set of resonances with the same channels
and quantum numbers. For any given spin group, only total spin and parity are constant;
there may be several entrance channels and/or several reaction channels (and, hence, several
values of l or s, etc.) contributing to the spin group.
The ”R-Matrix Limited” (RML) format was designed to accommodate the features of
R-Matrix theory as implemented in analyses codes being used for current evaluations. In
this format, relevant parameters appear only once. Particle-pairs are given first: the masses,
spins and parities, and charges for the two particles are specified, as well as the Q-value and
the MT value (which defines whether this particle-pair represents elastic scattering, fission,
inelastic, capture, etc.). Two particle-pairs will always be present: gamma + compound
nucleus, and neutron + target nucleus in ground state. Other particle-pairs are included as
needed.
The list of resonance parameters is ordered by J π , which (as stated above) is the only
conserved quantity for any spin group. For each spin group, the channels are first specified in
the order in which they will occur in the list of resonances. For each channel, the particle-pair
number and the values for l and s are given, along with the channel radii.
Formats for the basic RML subsection
Additional quantities are defined (or, in some cases, re-defined):
KRM Flag to specify which formulae for the R-matrix are to be used.
KRM = 1 for single-level Breit-Wigner,
KRM = 2 for multilevel Breit-Wigner,
KRM = 3 for Reich Moore,
KRM = 4 for full R-matrix. (Others may be added at a later date.)
KRL Flag is zero for non-relativistic kinematics, 1 for relativistic.
NJS Number of values of J π to be included.
NPP Total number of particle-pairs.
IA Spin (and parity, if non-zero) of one particle in the pair (the neutron or
projectile, if this is an incident channel).
IB Spin of the other particle in the pair (target nuclide, if this is an incident
channel). IB is set to zero and ignored if the first particle is a photon.
62

CHAPTER 2. FILE 2: RESONANCE PARAMETERS
PA Parity for first particle in the pair, used only in the case where IA is zero and
the parity is negative. (Value = +1.0 if positive, –1.0 if negative.)
PB Parity for second particle, used if IB= 0 and parity is negative.
MA Mass of first particle in the pair (in units of neutron mass).
MB Mass of second particle (in units of neutron mass).
ZA Charge of first particle.
ZB Charge of second particle.
QI Q-value for this particle-pair. (See Section 3.3.2 for details)
PNT Flag if penetrability is to be calculated;
PNT= 1 - calculate penetrability;
PNT=–1 - do not calculate penetrability;
PNT= 0 - assign value depending on the MT number. The default for capture
(MT=102) or fission (MT=19) is PNT=–1; the default for other MT numbers
is PNT=+1.
SHF Flag if shift factor is to be calculated (default = not):
SHF= 1 calculate the shift factor;
SHF=–1 do not calculate the shift factor.
MT Reaction type associated with this particle-pair; see Appendix B.
AJ Floating point value of J (spin); sign indicates parity.
PJ Parity (used only if AJ = 0.0).
NCH Number of channels for the given J π .
IPP Particle-pair number for this channel (written as floating-point number).
L Orbital angular momentum (floating-point value).
CH Channel spin (floating-point value).
BND Boundary condition for this channel (needed when SHF=+1)
APE Effective channel radius (scattering radius), used for calculation of phase shift
only. Units are 10−12 cm.
APT True channel radius (scattering radius), used for calculation of penetrability
and shift factors. Units are 10−12 cm.

63

CHAPTER 2. FILE 2: RESONANCE PARAMETERS
KBK Non-zero if background R-matrix exists; see Subsect. 2.2.1.7.2. (Often set to
zero)
KPS Non-zero if non-hard-sphere phase shift are to be specified. (Often set to
zero)
NRS Number of resonances for the given J π .
NX Number of lines required for all resonances for the given J π , assuming each
resonance starts on a new line; equal to (NCH/6+1)*NRS. If there are no
resonances for a spin group, then NX = 1.
ER Resonance energy in eV.
IFG Flag
IFG=0 - GAM is the channel width in eV,
IFG=1 - GAM is the reduced-width amplitude in eV1/2 .
GAM Channel width in eV or reduced-width amplitude in eV1/2 .
NOTE: For IFG = 0, the input quantity GAM is the width at the energy of the resonance;
it is calculated from reduced width amplitudes through equation (D.45) of Appendix D.1.7,
with E set to Eλ . For negative-energy dummy resonances, the convention is that the input
quantity is the width evaluated at the absolute value of the resonance energy. In all cases, if
the value GAM given in File 2 for the partial width is negative, the standard convention is
assumed: the negative sign is to be associated with the reduced width amplitude γλc rather
than with Γλc (since p
Γλc is always a positive quantity). More specifically, Γλc = |GAM | and
γλc = sign(GAM) × |GAM|/2P, with P evaluated at the energy of the resonance.
If IFG =1, the input quantity is the reduced width amplitude γλc .
The formats are as follows:
[MAT,2,151/ 0.0, 0.0, IFG, KRM, NJS, KRL ]CONT
(The following record provides all particle-pair descriptions. For KRM=1,2, or 3, the first
particle-pair is the gamma plus compound nucleus pair.)
[MAT,2,151/0.0,
0.0,
NPP,
0, 12*NPP, 2*NPP/
MA1 ,
MB1 ,
ZA1 ,
ZB1 ,
IA1 ,
IB1 ,
Q1 ,
PNT1 ,
SHF1 ,
MT1 ,
PA1 ,
PB1 ,
MA2 ,
MB2 ,
ZA2 ,
ZB2 ,
IA2 ,
IB1 ,
Q2 ,
PNT2 ,
SHF2 ,
MT2 ,
PA2 ,
PB1 ,
------------------------------------------MAN P P , MBN P P , ZAN P P , ZBN P P , IAN P P , IBN P P ,
QN P P , PNTN P P , SHFN P P , MTN P P , PAN P P , PBN P P ]LIST
(The following record provides the channel descriptions for one spin group. This record and
the next one are repeated together, once for each of the NJS spin groups.)
64

CHAPTER 2. FILE 2: RESONANCE PARAMETERS
[MAT,2,151/ AJ,
PJ,
KBK,
KPS,
6*NCH,
NCH/
IPP1 ,
L1 ,
SCH1 ,
BND1 ,
APE1 ,
APT1 ,
IPP2 ,
L2 ,
SCH2 ,
BND2 ,
APE2 ,
APT2 ,
----------------------------------------------IPPN CH , LN CH , SCHN CH , BNDN CH , APEN CH , APTN CH ]LIST
(The following record gives the values for resonance energy and widths for each resonance
in this spin group.)
[MAT,2,151/ 0.0,
0.0,
0,
NRS,
6*NX,
NX/
ER1 ,
GAM1,1 ,
GAM2,1 ,
GAM3,1 ,
GAM4,1 ,
GAM5,1 ,
GAM6,1 , ------------------------- GAMN CH,1 ,
ER2 ,
GAM1,2 ,
GAM2,2 ,
GAM3,2 ,
GAM4,2 ,
GAM5,2 ,
GAM6,2 , -------------------------- GAMN CH,2 ,
----------------------------------------------ERN RS , GAM1,N RS , GAM2,N RS , GAM3,N RS , GAM4,N RS , GAM5,N RS ,
GAM6,N RS , ------------------------ GAMN CH,N RS
]LIST
(If the number of resonances is zero for a spin group, then NRS=0 but NX=1 in this record.)
Other records may be included here, as described below. If KBK is greater than zero,
a ”background R-matrix” is given. If KPS is greater than zero, tabulated values exist for
phase shifts. If KBK=0 and KPS=0, no additional records are needed.
The above two records, beginning with ”channel descriptions,” are repeated until each
of the NJS J π spin groups has been fully specified.
Formats for optional extensions to the RML
The formats described above are sufficient for most evaluations currently (2003) available
(using KRM=3, KBK=0, and KPS=0 ). For the sake of generality, and to accommodate
expected future developments in R-matrix analysis codes, additional capabilities are included
in the RML format.
Different R-Matrix formulations (KRM = 1,2,4)
Equations given in Appendix D.1.7 are relevant to the Reich-Moore approximation to RMatrix theory. The format, however, can also be used for single-level Breit-Wigner (KRM
= 1), multilevel Breit-Wigner (KRM = 2), R-Matrix without approximations (KRM = 4).
Equations for KRM = 1 or 2 will be written up if/when the need arises. Equations for KRM
= 4 are identical to those given in Appendix D.1.7 with the elimination of the imaginary
term in the denominator of equation (6), and the inclusion of each gamma-channel on a
equal basis with all other channels.
Background R-matrix (KBK > 0):
As described in Appendix D.1.7.5, a background R-Matrix can be defined in a variety of
different methods.
For KBK=0, Option 0 is used everywhere (that is, for all channels for this spin group)
for the background R-Matrix. No additional formats are required and no additional records
need to be written; the dummy resonances are included along with the physical resonances
in the list record described above.
65

CHAPTER 2. FILE 2: RESONANCE PARAMETERS
For KBK > 0, one LIST record (and two TAB1 records, for tabulated values) is included
for each channel of the current spin group, a total of NCH records. The particular option to
be used for the channel is identified by parameter LBK. The formats for the four options
are as follows:
Option 0. Dummy resonances (LBK=0)
No additional information is conveyed in this record, other than LBK=0. No terms are
added to the R-matrix for this channel.
[MAT,2,151/ 0.0, 0.0,
0,
0, LBK,
1/
0.0, 0.0, 0.0, 0.0, 0.0, 0.0]LIST
Option 1. Tabulated complex function of energy (LBK=1)
Notation:
RBR Value of real part of tabulated function
RBI Value of imaginary part of tabulated function
[MAT,2,151/ 0.0,
0.0,
[MAT,2,151/ 0.0,
[MAT,2,151/ 0.0,

0.0,
0,
0, LBK,
1/
0.0, 0.0, 0.0, 0.0, 0.0]LIST
0.0,
0,
0, NR, NP/ Eint / RBR(E) ]TAB1
0.0,
0,
0, NR, NP/ Eint / RBI(E) ]TAB1

(Recall that NR and NP are parameters, which define the interpolation scheme for TAB1
records, as defined in Section 0.6.3.7. Energy values given by Eint are in units of eV.)
Option 2. SAMMY’s logarithmic parameterization (LBK=2)
Notation (See equation (D.76) of Section D.1.7.5 for the meanings of these quantities):
R0 Rcom,c
S0 scom,c
R1 Rlin,c
S1 slin,c
R2 Rq,c
EU Ecup
ED Ecdown
[MAT,2,151/ ED, EU, 0, 0, LBK,
1/
R0, R1, R2, S0, S1, 0.0]LIST
Option 3. Fröhner’s parameterization (LBK=3)
Notion (See equations (D.77) and D.78) in Section D.1.7.5 for the meanings of these quantities):
66

CHAPTER 2. FILE 2: RESONANCE PARAMETERS
R0 Rc
S0 s
GA Γγ
EU Ecup
ED Ecdown
[MAT,2,151/ ED, EU, 0,
0, LBK,
1/
R0, SO, GA, 0.0, 0.0, 0.0]LIST
Tabulated phase shifts (KPS=1)
When phase shifts are calculated externally (from optical model potentials, for example),
rather than generated from the usual hard-sphere phase shift formulae, then the phase shifts
must be presented in tabular form.
If parameter KPS is equal to 0, all phase shifts are calculated from the hard-sphere
phase shift formulae (see Table D.1 for non-Coulomb and Appendix D.1.7 for a discussion
on Coulomb hard-sphere phase shifts).
For KPS>0, one LIST record (and two TAB1 records, for tabulated values) are included
for each channel of the current spin group, a total of NCH records. The particular option
to be used for the channel is identified by parameter LPS. The formats for the two options
are as follows:
Option 0. Hard-sphere phase shifts (LPS=0)
No additional information is conveyed in this record, other than LPS=0.
[MAT,2,151/ 0.0, 0.0,
0,
0, LPS,
1/
0.0, 0.0, 0.0, 0.0, 0.0, 0.0]LIST
Option 1. Phase shift is a tabulated complex function of energy (LPS=1)
Notation:
PSR Value of the real part of the tabulated phase shift
PSI Value of the imaginary part of the tabulated phase shift

[MAT,2,151/ 0.0,
0.0,
[MAT,2,151/ 0.0,
0.0,

0.0,
0,
0, LPS,
1/
0.0, 0.0, 0.0, 0.0, 0.0]LIST
0.0,
0,
0, NR, NP/Eint / PSR(E) ]TAB1
0.0,
0,
0, NR, NP/Eint / PSI(E) ]TAB1

(Recall that NR and NP are parameters, which define the interpolation scheme for TAB1
records, as defined in Secttion 0.6.3.7. Energy values given by Eint are in units of eV.)

67

CHAPTER 2. FILE 2: RESONANCE PARAMETERS

2.3

Unresolved Resonance Parameters (LRU=2)

2.3.1

Formats

Only the SLBW formalism for unresolved resonance parameters is allowed (see Appendix D
for pertinent formulae). However, several options are available for specifying the energydependence of the parameters, designated by the flag LRF. Since unresolved resonance
parameters are averages of resolved resonance parameters over energy, they are constant
with respect to energy throughout the energy-averaging interval. However, they are allowed
to vary from interval to interval, and it is this energy-dependence, which is referred to above
and in the following paragraphs.
The parameters depend on both l (neutron orbital angular momentum) and J (total
angular momentum). Each width is distributed according to a chi-squared distribution with
a certain number of degrees of freedom. This number may be different for neutron and
fission widths and for different (l,J) channels.
The following quantities are defined for use in specifying unresolved resonance parameters
(LRU=2):
SPI Spin of the target nucleus, I.
AP Scattering radius in units of 10−12 cm. No channel quantum number dependence is permitted by the format.
LSSF Flag governing the interpretation of the File 3 cross sections.
LSSF=0 File 3 contains partial ”background” cross sections, to be
added to the average unresolved cross sections calculated from
the parameters in File 2.
LSSF=1 File 3 contains the entire dilute cross section for the unresolved resonance region. File 2 is to be used solely for the
calculation of the self-shielding factors, as discussed in Section 2.4.17.
NE Number of energy points at which energy-dependent widths are tabulated.
For the limit on NE see Appendix G.
NLS Number of l-values. For the limit on NLS see Appendix G.
ESi Energy of the ith point used to tabulate energy-dependent widths.
L Value of l.
AWRI Ratio of the mass of a particular isotope to that of the neutron.
NJS Number of J-states for a particular l-state. For the limit on NJS see Appendix G.
68

CHAPTER 2. FILE 2: RESONANCE PARAMETERS
AJ Floating-point value of J (the spin, or total angular momentum of the set of
parameters).
D Average level spacing for resonances with spin J.
dependent if LRF=2).

(D may be energy-

AMUX Number of degrees of freedom used in the competitive width distribution.
(Assuming it is inelastic, 1.0≤AMUX≤2.0, determined by whether the spin
of the first excited state is zero or not.)4
AMUN Number of degrees of freedom in the neutron width distribution.
(1.0 ≤AMUN≤ 2.0)
AMUG Number of degrees of freedom in the radiation width distribution. (At present
AMUG = 0.0. This implies a constant value of Γγ .)
AMUF Number of degrees of freedom in the fission width distribution.
(1.0 ≤AMUF≤ 4.0)
MUF Integer value of the number of degrees of freedom for fission widths.
(1 ≤MUF≤ 4)
INT Interpolation scheme to be used for interpolating between the cross sections obtained from average resonance parameters. Parameter interpolation is discussed in the Procedures Section 2.4.2.
GN0 Average reduced neutron width. It may be energy-dependent if LRF=2.
GG Average radiation width. It may be energy-dependent if LRF=2.
GF Average fission width. It may be energy-dependent if LRF=1 or 2.
GX Average competitive reaction width, given only when LRF=2, in which case
it may be energy-dependent.
The structure of a subsection 5 depends on whether LRF=1 or LRF=2. If LRF=1,
only the fission width is given as a function of energy. If LRF=1 and the fission width is
not given (indicated by LFW=0), then the simplest form of a subsection results. If LRF=2,
energy-dependent values may be given for the level density, competitive width, reduced
neutron width, radiation width, and fission width. Three sample formats are shown below
(all LRU=2).
Case A
LFW=0 (fission widths not given)
LRF=1 (all parameters are energy-independent).
The structure of a subsection is:
4

See Appendix D.2.2.6.
The structure of a section was defined previously, and covers both resolved resonance and unresolved
resonance subsections.
5

69

CHAPTER 2. FILE 2: RESONANCE PARAMETERS
[MAT, 2,151/ SPI,
AP,
LSSF,
0,
NLS,
0] CONT
[MAT, 2,151/ AWRI,
0.0,
L,
0, 6*NJS, NJS/
D1 ,
AJ1 ,
AMUN1 ,
GN01 ,
GG1 , 0.0,
D2 ,
AJ2 ,
AMUN2 ,
GN02 ,
GG2 , 0.0,
-------------------------------------DN JS , AJN JS , AMUNN JS , GN0N JS , GGN JS , 0.0] LIST
The LIST record is repeated until data for all l-values have been specified. In this example,
AMUG is assumed to be zero, and there is no competitive width.
Case B
LFW=1 (fission widths given)
LRF=1 (only fission widths are energy-dependent; the rest are energy-independent).
The structure of a subsection is:
[MAT, 2,151/ SPI, AP, LSSF, 0, NE, NLS]CONT
ES1 , ES2 , ES3 , ------------------------------------ ESN E ]LIST
[MAT, 2,151/AWRI, 0.0,
L,
0, NJS,
0]CONT
[MAT, 2,151/ 0.0, 0.0,
L, MUF, NE+6,
0/
D, AJ, AMUN, GN0, GG, 0.0,
GF1 , GF2 , GF3 ,-------------------------------------- GFN E ]LIST
The last LIST record is repeated for each J-value (there will be NJS such LIST records).
A new CONT(l) record will then be given which will be followed by its NJS LIST records
until data for all l-values have been specified (there will be NLS sets of data).
In the above section, no provision was made for INT, and interpolation is assumed to be
linear-linear. AMUG is assumed to be zero, AMUF equals MUF, and there is no competitive
width.
Case C
LFW=0 or 1 (does not depend on LFW).
LRF=2 (all energy-dependent parameters).
The structure of a subsection is:
[MAT, 2,151/ SPI, AP, LSSF,
0,
NLS,
0]CONT
[MAT, 2,151/ AWRI, 0.0,
L,
0,
NJS,
0]CONT
[MAT, 2,151/
AJ, 0.0,
INT,
0, 6*NE+6,
NE/
0.0, 0.0, AMUX, AMUN,
AMUG, AMUF,
ES1 , D1 ,
GX1 , GN01 ,
GG1 , GF1 ,
ES2 , D2 ,
GX2 , GN02 ,
GG2 , GF2 ,
----------------------------ESN E , DN E , GXN E , GN0N E , GGN E , GFN E ] LIST
The LIST record is repeated until all the NJS J-values have been specified for a given lvalue. A new CONT(l) record is then given, and all data for each J-value for that l-value
are given. The structure is repeated until all l-values have been specified. This example
permits the specification of all four degrees of freedom.

70

CHAPTER 2. FILE 2: RESONANCE PARAMETERS

2.4

Procedures for the Resolved and Unresolved Resonance Regions

CONTENTS OF THIS SECTION
2.4.1
2.4.2
2.4.3
2.4.4
2.4.5
2.4.6
2.4.7
2.4.8
2.4.9
2.4.10
2.4.11
2.4.12
2.4.13
2.4.14
2.4.15
2.4.16
2.4.17
2.4.18
2.4.19
2.4.20

Abbreviations
Interpolation in the Unresolved Resonance Region
Unresolved Resonances in the Resolved Resonance Range
Energy Range Boundary Problems
Numerical Integration Procedures in the Unresolved Resonance Region
Doppler-broadening of File 3 Background Cross Sections
Assignment of Unknown J-values
Competitive Width in the Resonance Region
Negative Cross Sections in the Resolved Resonance Region
Negative Cross Sections in the Unresolved Resonance Region
Use of Two Nuclear Radii
The Multilevel Adler-Gauss Formula for MLBW
Notes on the Adler Formalism
Multilevel Versus single-level Formalisms in the Resolved
and Unresolved Resonance Regions
Preferred Formalisms for Evaluating Data
Degrees of Freedom for Unresolved Resonance Parameters
Procedures for the Unresolved Resonance Region
Procedures for Computing Angular Distributions in the
Resolved Resonance Range
Completeness and Convergence of Channel Sums
Channel Spin and Other Considerations

2.4.1
UR(R)
RR(R)
RRP
URP
SLBW
MLBW
MLAG
UCS

2.4.2

Abbreviations
unresolved resonance (region)
resolved resonance (region)
resolved resonance parameter(s)
unresolved resonance parameter(s)
single-level Breit-Wigner
multilevel Breit-Wigner
multilevel Adler-Gauss
unresolved cross section(s)

Interpolation in the Unresolved Resonance Region (URR)

For energy-dependent formats (LRF=2, or LRF=1 with LFW=1), the recommended procedure is to interpolate on the cross sections derived from the unresolved resonance
parameters (URP). This is a change from the ENDF/B-III and IV procedure, which was
to interpolate on the parameters. The energy grid should be fine enough so that the cross
71

CHAPTER 2. FILE 2: RESONANCE PARAMETERS
sections at intermediate energy values can be computed with sufficient accuracy using this
procedure. Normally, three to ten points per decade will be required to obtain reasonable
accuracy. Some evaluations prepared for earlier versions of ENDF/B do not meet these
standards. In such cases, if two adjacent grid points differ by more than a factor of three,
the processing code should add additional intermediate energy points at a spacing of approximately ten-per-decade and compute the cross sections at the intermediate points using
parameter interpolation. Additional cross sections can then be obtained by cross section
interpolation in the normal way.
For many isotopes, there is not sufficient information for a full energy-dependent evaluation. In these cases, the evaluator may provide a single set of unresolved resonance parameters based on systematics or extrapolation from the resolved range (see LRF=1, LFW=0).
Such a set implies a definite energy-dependence of the unresolved cross sections due to the
slowly-varying wave number, penetrability, and phase shift factors in the SLBW formulas.
It is incorrect to calculate cross sections at the ends of the URR, and then to compute intermediate cross sections by cross section interpolation. Instead, the processing code should
generate a set of intermediate energies using a spacing of approximately ten-per-decade and
then compute the cross sections on this grid using the single set of parameters given in the
file. Additional intermediate values are then obtained by linear cross section interpolation
as in the energy-dependent case.
It is recommended that evaluators provide the URP’s on a mesh dense enough that the
difference in results of interpolating on either the parameters or the cross sections be small.
A 1% maximum difference would be ideal, but 5% is probably quite acceptable.
Finally, even if the evaluator provides a dense mesh, the user may end up with different
numbers than the evaluator ”intended”. This is particularly true when genuine structure
exists in the cross section and the user chooses different multigroup breakpoints than those
in the evaluation. There is no solution to this problem, but the dense mesh procedure
minimizes the importance of the discrepancy.
In order to permit the user to determine what ”error” he is incurring, it is recommended
that evaluators state in the documentation what dilute, unbroadened average cross sections
they intended to represent by the parameters in File 2. Note that the self-shielding factor
option specified by the flag LSSF (Sections 2.3.1 and 2.4.17) greatly reduces the impact of
this interpolation ambiguity.

2.4.3

Unresolved Resonances in the Resolved Resonance Range

As discussed in Section 2.4.4, the boundary between the resolved and unresolved resonance
regions should be chosen to make the statistical assumptions used in the URR valid. This
creates problems in evaluating the resonance parameters for the RRR.
Problem l: At the upper end of the resolved range, the smaller resonances will begin to
be missed. An equivalent contribution could be added to the background in File 3. This
contribution will not be self-shielded by the processing codes, so it cannot be allowed to
become ”significant”. A better procedure is to supply fictitious resolved resonance parameters, based on the statistics of the measured ones, checking that the average cross section
agrees with whatever poor-resolution data are available.
72

CHAPTER 2. FILE 2: RESONANCE PARAMETERS
If both procedures are employed, care should be taken not to distort the statistics of the
underlying parameter distributions.
Problem 2: Because d-wave resonances are narrower than p-waves, which are narrower than
s-waves, everything else being equal, the point at which p-waves will be instrumentally
unresolved can be expected to be lower in energy than for s-waves, and lower still for dwaves. Thus the unresolved region for p-waves will usually overlap the resolved region for
s-waves, and similarly for d-waves. Current procedure does not permit representing this
effect explicitly; one cutoff-point must serve for all l-values.
The remedies are the same as above, either putting known or estimated resonances into
the background in the URR, or putting fictitious estimated resonances into the RRR. The
latter is preferred because narrow resonances tend to self-shield more than broad ones,
hence the error incurred by treating them as unshielded File 3 background contributions is
potentially significant.

2.4.4

Energy Range Boundary Problems

There may be as many as four different kinds of boundaries under current procedures which
permit multiple RRR’s:
1. between a low-energy File 3 representation (range 1) and EL for the RRR (range 2),
2. between successive RR ranges,
3. between the highest RRR and the URR,
4. between EH for the URR and the high-energy File 3 representation.
Discontinuities can be expected at each boundary. At item 1, a discontinuity will occur
if range 1 and range 2 are not consistently Doppler-broadened. In general, only an identical
kernel-broadening treatment will produce continuity, i.e., only if the range-1 cross sections
are broadened from the temperature at which they were measured, and range-2 is broadened from absolute zero. A kernel treatment of range 1, or no broadening at all, will be
discontinuous with a ψ-χ treatment of range 2. This effect is not expected to be serious
at normal reactor temperatures and presumably, the CTR and weapons communities are
cognizant of the Doppler problem. In view of these problems, a double energy point will
not usually produce exact continuity in the complete cross section, (file 2 + file 3), unless
evaluator and user employ identical methods throughout.
Discontinuities will occur between successive RRR’s, unless the evaluator takes pains
to adjust the ”outside” resonances for each RRR to produce continuity at absolute zero.
If the unbroadened cross sections in two successive RRR’s are broadened separately, the
discontinuity will be preserved, and possibly enhanced. These discontinuities are not believed
to be technologically significant.
A discontinuity at item 3 is unavoidable, because the basic representation has changed.
However if the RRR cross sections are group-averaged or otherwise smoothed, the discon-

73

CHAPTER 2. FILE 2: RESONANCE PARAMETERS
tinuity6 should be reasonably small. A discontinuity greater than 10 or 15% obtained with
a suitable averaging interval indicates that the evaluator might want to reconsider his parameterization of the poor-resolution data. Some materials have large genuine fluctuations
in the URR, and for these the 10-15% figure is not applicable. A double energy point will
normally occur at this boundary, but will not eliminate the discontinuity.
Discontinuity at item 4 should be small, since both the URR and the high-energy range
represent rather smooth cross sections, and the opportunity for error ought to be small.
Anything over 5% or so should be viewed with suspicion.
The upper and lower energy limits of any energy range indicate the energy range of
validity of the given parameters for calculating cross sections. Outside this energy range the
cross sections must be obtained from the parameters given in another energy range and/or
from data in file 3.
The lower energy limit of the URR should be chosen to make the statistical assumptions
used in this range valid. The basic requirement is that there be ”many” resonances in an
energy-averaging interval, and that the energy-averaging interval be narrow with respect to
slowly-varying functions of E such as wave number and penetrability. As an example, assume
that the energy-averaging interval can extend 10% above and below the energy point, that
the average resonance spacing is 1 eV, and that ”many” is 100. Then the lowest reasonable
energy for the URR would be about 500 eV, as given by 0.2 E=100×1. Some implications
of this choice for the RRR-URR boundary were discussed in Section 2.4.3.
It is sometimes necessary to give parameters whose energies lie outside a specified
energy range in order to compute the cross section for neutron energies that are within the
energy range. For example, the inclusion of bound levels may be required to match the cross
sections at low energies, and resonances will often be needed above EH to compensate the
opposite, positive, bias at the high energy end.
For materials that contain more than one isotope, it is recommended that the lower
energy limit of the resolved resonance region be the same for all isotopes. If resolved and/or
unresolved resonance parameters are given for only some of the naturally occurring isotopes,
then AP should be given for the others.
If more than one energy range is used, the ranges must be contiguous and not overlap.
Overlapping of the resolved and unresolved ranges is not allowed for any one isotope,
but it can occur in an evaluation for an element or other mixture of different isotopes. In
fact, it is difficult to avoid since the average resonance spacing varies widely between eveneven and even-odd isotopes. Such evaluations are difficult to correctly self-shield. A kernel
broadening code must first subtract the infinitely-dilute unresolved cross section, broaden
the pointwise remainder, then add back the unresolved component. A multigroup averaging
code that uses pointwise cross sections must first subtract the infinitely-dilute unresolved
cross section to find the pointwise remainder, and then add back a self-shielded unresolved
cross section computed for a background cross section which includes a contribution from
the pointwise remainder.
6

This refers to the discontinuity between the average cross section in the RRR, and the dilute (unshielded)
pointwise cross section in the URR, which has been generated from the URR parameters. If the self-shielding
factor option has been chosen (LSSF=1, Section 2.3.1), File 3 will contain the entire dilute cross section and
no File 2 unresolved region calculation will be needed to ascertain the discontinuity.

74

CHAPTER 2. FILE 2: RESONANCE PARAMETERS

2.4.5

Numerical Integration Procedures in the URR

The evaluation of effective cross sections in the URR can involve Doppler effects, fluxdepression, and resonance-overlap as well as the statistical distributions of the underlying
resonance parameters for a mixture of materials.
The previous ENDF recommendation for doing the complicated multi-dimensional integrations was the Greebler-Hutchins (GH) scheme [Reference 1], basically a trapezoidal integration. For essentially the same computing effort, a more sophisticated weighted-ordinate
method can be used and it has been shown that the scheme in MC2 -II [Reference 2] produces results differing by up to several percent from GH. The MC2 -II subroutine7 , is the
recommended procedure.
The M. Beer [Reference 3], analytical method has also been suggested, and is quite
elegant, but unfortunately will not treat the general heterogeneous case.

2.4.6

Doppler-broadening of File 3 Background Cross Sections

1. In principle, the contribution to each cross section from File 3 should be Dopplerbroadened, but in practice, many codes ignore it. It is therefore recommended that
the evaluator keep file 3 contributions in the RRR and URR small enough and/or
smooth enough so that omission of Doppler-broadening does not ”significantly” alter
combined File 2 plus File 3 results up to 3000 K. Unfortunately, the diversity of
applications of the data in ENDF files makes the word ”significantly” impossible to
define.
2. A possible source of structured File 3 data is the representation of multilevel or MLBW
cross sections in the SLBW format, the difference being put into File 3. This difference is a series of residual interference blips and dips, which may affect the betweenresonance valleys and possibly the transmission in thick regions or absorption rates
in lumped poisons, shields, blankets, etc. Users of the SLBW formalism should consider estimating these effects for significant regions. A possible remedy is available in
the Multilevel Adler-Gauss form of MLBW. (See Section 2.4.12). If the resonanceresonance interference term in MLBW is expanded in partial fractions, it becomes a
single sum of symmetric and asymmetric SLBW-type terms. Two coefficients occur
which require a single sum over all resonances for each resonance, but these sums are
weakly energy-dependent and lend themselves to approximations that could greatly
facilitate the use of ψ and χ functions with MLBW.
3. An ”in-principle” correct method for constructing resonance cross sections is:
(a) Use a Solbrig kernel [Reference 4] to broaden File 2 to the temperature of File 3,
since the latter may be based on room-temperature or other nonzero 0 K data.
(b) Add File 2 and File 3.
(c) Sollbrig-broaden the result to operating temperature.
7

This subroutine was provided by H. Henryson, II (ANL).

75

CHAPTER 2. FILE 2: RESONANCE PARAMETERS
Using a Gaussian kernel instead of Sollbrig incurs a small error at low energies, unless
it is misused, in which case the error can be large. Using ψ and χ functions introduces
further errors. In fact, the Sollbrig kernel already approximates the true motion of
the target molecules by a free-gas law, but anything more accurate is quite difficult to
handle.
4. Some heavy element evaluations use a File 3 representation below the resolved resonance region. Often these cross sections are room-temperature values, so that if they
are later broadened assuming they are zero-degrees Kelvin, they get broadened twice.
A simple way to reduce the impact of this procedure without altering the representation of the data is to calculate the cross sections from the resonance parameters,
broadened to room temperature, and carry the calculation down through the lowenergy region. Subtract these broadened values from the file 3 values and leave only
the difference in file 3. Then extend the lower boundary of the resonance region to
the bottom of the file. Now the ”double-broadening” problem affects only the (small)
residual file 3 and not the entire cross section.
Note that subtracting off a zero degree resonance contribution would accomplish
nothing.

2.4.7

Assignment of Unknown J-Values

In all multilevel resonance formalisms except Adler-Adler, the J-value determines which
resonances interfere with each other. Usually, J is known only for a few resonances, and
measurers report 2gΓn for the others. If this number is assumed to be Γn , one incurs an
error of uncertain magnitude, depending on how different is the factor g
g=

2J + 1
2 (2I + 1)

from l/2, how large Γn is relative to the other partial widths, and how important resonanceresonance interference is.
It is recommended that evaluators assign J-values to each resonance, in proportion to
the level density factor 2J + l. To reduce the amount of interference, the J-values of strong
neighboring resonances, which would produce the largest interference effects, can be chosen
from different families.
In the past, some evaluations have put J = I, the target nucleus spin, for resonances
with unknown J-values. This corresponds to putting g = 1/2, rather than its true value.
Mixing of the J = I resonances with the physically correct I ± 1/2 families can result in
negative scattering cross sections, or distortions of the potential scattering term, depending
on what formalism is used and how it is evaluated. For this reason, such J = I resonances
must not be used.
In the amplitude-squared form of the MLBW scattering cross section,
X
2
lsJ
σnn =
gJ AlsJ
pot + Aresonance
lsJ

76

CHAPTER 2. FILE 2: RESONANCE PARAMETERS
the use of J = I resonances will destroy the equivalence between this form and the ”squared”
form of MLBW in Appendix D since the sum on lsJ does not go over physically-correct
values.
An exception to the prohibition against J = I is the case where no J-values are known,
since if all resonances are assigned J = I, the MLBW scattering cross section will be
non-negative.

2.4.8

Competitive Width in the Resonance Region

2.4.8.1

Resolved Region

Procedures for the Resolved Resonance Region are contained in Section D.3.1 of Appendix D.
2.4.8.2

Unresolved Region

Procedures for the Unresolved Resonance Region are contained in Section D.3.2 of Appendix D. Users are directed to the discussion of the total cross section in Appendix D,
Section D.3.3, since, as pointed out by H. Henryson, II, in connection with MC2 procedures,
a possibility for erroneous calculations exists.

2.4.9

Negative Cross Sections in the Resolved Resonance Region

To avoid negative cross sections, new evaluations should use only the rigorous formulations
such as Reich-Moore (LRF=3) or R-Matrix Limited (LRF=7). However, resonance parameters coded in other formats may be encountered in some old evaluations.
2.4.9.1

In the SLBW Formalism:

Capture and fission use the positive symmetric Breit-Wigner shape and are never negative.
Scattering involves an asymmetric term which goes negative for E  on whatever energy mesh is needed to
produce agreement with the dilute poor-resolution data.
The creation of SLBW ladders from average parameters can be expected to produce the
same kind of end-effect bias and frequent negative scattering cross sections found in the
resolved resonance region. Again, the scattering cross section per se may not be important,
but the biased total cross section may adversely affect calculated reaction rates.

2.4.11

Use of Two Nuclear Radii

The current ENDF formats defines two different nuclear radii:
a) the scattering radius, AP, and
b) the channel radius, a.
The scattering radius is also referred to as ”the effective scattering radius” and ”the potential
scattering radius”. The channel radius is also referred to as ”the hard-sphere radius”, or
”the nuclear radius”. The former is the quantity defined as AP (for a+ or ˆa) in File 2, which
must be given even if no resonance parameters are given. The nuclear radius is defined in
Appendix D, equation (D.14).
The channel radius is a basic quantity in R-matrix theory, where the internal and external
wave-functions are joined and leads to the appearance of hard-sphere phase shifts defined
in terms of it. The necessity to relax the definition and permit two radii can be thought of
as a ”distant-level effect”, sometimes not explicit in R-matrix discussions.
The original ENDF formats made provision for an AM, or ”A-minus”, although it was
always required that evaluators put AM=0, to signify that it was equal in value to AP.
In the current formats, AM is eliminated, but one can anticipate that more sophisticated
evaluation techniques may eventually force the reinstatement of not only AM, but a more
general dependence of the scattering radius on the channel quantum numbers, especially as
higher energies become important.
In theory, the scattering radius depends on all the channel quantum numbers, and in
practice it is common to find that different optical model parameters are required for different
l-values (s, p, d,...) and for different J-values (p1/2 , p3/2 , ,,,). This implies that one would
require a different scattering radius for each of these states.
For the special case of s-waves, only two J-values are possible, namely I ±1/2, commonly
denoted J+ and J− . This is the origin of the terminology a+ and a− .
79

CHAPTER 2. FILE 2: RESONANCE PARAMETERS
Up through ENDF/B-V, the recommended ENDF procedure was to use the above equation for the channel radius in the penetrabilities Pl (ka) and the shift factors Sl (ka), but to
use the scattering radius to calculate the hard-sphere phase shifts φl (ka).
Since the phase-shifts define the potential scattering cross section, the evaluator had the
freedom to fit AP to a measured cross section while still leaving undisturbed those codes
that use the A1/3 formula to calculate the channel radius.
In the ENDF-6 format, new parameters NRO and NAPS are available to give the evaluator more flexibility for the SLBW, MLBW, and RM formalisms, by allowing the evaluator
to use AP everywhere and to make AP energy-dependent (Section 2.1).
The full flexibility of channel-dependent radii is provided for the RML format.

2.4.12

The Multilevel Adler-Gauss Formula for MLBW

Appendix D gives (implicitly) for the MLBW formalism the equations:
MLBW
SLBW
σn,γ
(E) ≡ σn,γ
(E) ,

MLBW
SLBW
σn,f
(E) ≡ σn,f
(E) ,
MLBW
σn,n
(E)

≡

SLBW
σn,n
(E)

+

(2.1)
(2.2)
NLS−1
X

l
σn,n,RRI
(E)

(2.3)

l=0

where RRI labels the resonance-resonance-interference term for a given l-value:
NR
r−1
XJ X
2Γnr Γns [(E − Er′ )(E − Es′ ) + Γr Γs /4]
π X
g
J
k2 J
[(E − Er′ )2 + (Γr /2)2 ] [(E − Es′ )2 + (Γs /2)2 ]
r=2 s=1

(2.4)

As most users are aware, this double sum over resonances can eat prodigious amounts of
computer time unless handled very tactfully. Thus, for a 200-resonance material, there are
≈ 40,000 cross terms, of which only 20,000 need to be evaluated because the expression is
symmetric in r and s.
It has been noted many times in the past that partial fractions can reduce Equation
(2.3) to a form with only a single Breit-Wigner denominator. DeSaussure, Olsen, and Perez
(Reference 5) have written it compactly as:

l
σn,n,RRI
(E)

where

NR
π X XJ Gr Γr + 2Hr (E − Er′ )
= 2
gJ
k J
(E − Er′ )2 + (Γr /2)2
r=1

Gr =

Hr =

NR
Γnr Γns (Γr + Γs )
1 XJ
2 s=1,s6=r (Er′ − Es′ )2 + 41 (Γr + Γs )2
NR
XJ

Γnr Γns (Er′ − Es′ )
(Er′ − Es′ )2 + 41 (Γr + Γs )2
s=1,s6=r
80

(2.5)

CHAPTER 2. FILE 2: RESONANCE PARAMETERS
The authors give the special case for I=l=0, but it is valid for any set of quantum numbers.
Thus an existing SLBW code can be converted to MLBW by adding Gr Γr to the symmetric
part of the SLBW formula, Γ2nr cos 2φl − 2Γnr (Γr − Γnr ) sin2 φl , and 2Hr to the coefficient
of (E − Er′ ) in the asymmetric part, 2Γnr sin 2φl .
Since Gr and Hr are weakly energy-dependent, via the penetrabilities and shift factors,
they lend themselves to approximations that can sharply reduce computing time compared
to the form with the ”double” Breit-Wigner denominator.
In fact, if the resonances are all
√
treated as s-wave (shifts of zero, penetrabilities of E), and the total widths are taken as
constant, then Gr /k 2 and Hr /k 2 become independent of the neutron energy and consume a
negligible amount of computing time so that MLBW and SLBW become equivalent in that
respect.
The amplitude-squared form of MLBW, defined by equations (D.19) to (D.21), also
reduces computing time.

2.4.13

Notes on the Adler-Adler Formalism

Questions concerning the ENDF treatment of the Adler formalism are enumerated below8 ,
together with recommended procedures for handling them:
1. The resonance energy µ and total half-width ν are the same for each reaction for a
given resonance in the Adler formalism, but, for the October l970 version of ENDFl02, the formulae on page D7, and the format descriptions of pages 7.9 and Nl2 permit
different values for the total, fission, and capture cross sections.
This is a misreading of the formalism; the remedy is to constrain the equalities DETN =
DEFN = DECN and DWTN = DWFN = DWCN . The formulas for capture and fission
should also have the phases eliminated in Appendix D.
2. The Adler formalism, as applied by the Adlers, breaks the resolved resonance region up
into sub-regions, and each is analyzed separately. This avoids problems with contributions from distant resonances, but requires that the polynomial background be tailored
to each sub-region. However, the ENDF formats allow only one resolved resonance
energy region, so this procedure cannot be used.
If a single set of polynomial background constants is insufficient, additional background
can be put into File 3, point-by-point.
3. The ENDF formats formerly permitted incomplete specification of the cross sections.
The allowed values of LI were 5 (total and capture widths); 6 (fission and capture);
and 7 (total, fission, and capture). LI=6 leaves the scattering (and total) undefined
and LI=5 is deficient for fissile elements. LI=6 is now restricted to ENDF/A, and
LI=5 should be used only for non-fissile elements.
4. The nomenclature for the G’s and H’s is not entirely consistent among different authors. The Adlers use for the total cross section the definitions:
8
The following is a condensation and updating of the Appendix in the June, l974, Minutes of the Resonance Region Subcommittee.

81

CHAPTER 2. FILE 2: RESONANCE PARAMETERS

Gt = α cos(2ka) + β sin(2ka);
Ht = β cos(2ka) − α sin(2ka);
and then the combination:
νGt + (µ − E)Ht .
For the reaction cross sections there are no phases, and they write:
νGc + (µ − E)Hc

(capture);

νGf + (µ − E)Hf

(fission);

G and H are properly designated as ”symmetrical” and ”asymmetrical” parameters.
This manual changes α to Gt and β to Ht , viz:
ν[Gt cos(2ka) + Ht sin(2ka)] + (µ − E)[Ht cos(2ka) − Gt sin(2ka)]
These Gt ’s and Ht ’s are no longer symmetrical and asymmetrical, but are referred to
that way. The precedent for this nomenclature is probably Reference 6.
DeSaussure and Perez, in their published tables of G and H, incorporate the Adler’s
constant c into their definition, but otherwise leave the formalism unchanged.
Users and evaluators should adhere to the definitions in this manual.
5. The flag NX, which tells what reactions have polynomial background coefficients given,
should be tied to LI, so that the widths and backgrounds are given for the same
reactions, i.e., use NX=2 with LI=5 (total and capture), and NX=3 with LI=7 (total,
capture, and fission). Since no NX is defined for LI=6 (fission and capture), one is
forced to use NX=3 with the background total coefficients set equal to zero, but this
now occurs only in ENDF/A, if at all.

2.4.14

Multi-Level Versus Single-Level Formalisms in the Resolved and Unresolved Resonance Regions

2.4.14.1

In the Resolved Resonance Region:

The SLBW formalism may be adequate for resonance treatments that do not require actual pointwise scattering cross sections, as, e.g., multigroup slowing-down codes. Because of
the frequent occurrence of negative scattering cross sections, when two or more resonancepotential interference terms overlap, SLBW should not be used to compute pointwise scattering cross sections. Instead, the MLBW formalism may be used, although MLBW is not
a true multilevel formalism, but a limit which is valid if Γ/D is small.
82

CHAPTER 2. FILE 2: RESONANCE PARAMETERS
The Reich-Moore reduced R-matrix formalism is a true multilevel formalism, and is
recommended for all evaluations. All of its cross sections are non-negative, and its only
significant drawbacksare the difficulty of determining a suitable R∞ to represent distantlevel effects, and of determining the parameters of negative-energy resonances.
The Adler form of the Kapur-Peierls formalism is also a true multilevel treatment, but
in actual applications the parameters are determined by fitting data and the theoretical
constraints among them are lost, so that any Adler cross section can be negative.
The simplest true multilevel formalism is the reduced R-function, in which all channels
except elastic scattering have been eliminated. It sometimes makes an adequate evaluation
tool for non-fissile elements up to the threshold for inelastic scattering, since below that the
eliminated channels are (usually) simply radiative capture. It can be corrected for distantlevel effects by substituting optical-model phase shifts for the hard-sphere ones which occur
in the formalism, and by introducing an appropriate R∞ . It can be carried above the
inelastic threshold by augmenting it with the use of SLBW formulas for the reactions other
than elastic scattering, since such reactions often show negligible multilevel effects. For
structural and coolant materials, either Reich-Moore or R-Matrix Limited can be used. The
latter provides more detail in describing competitive reactions, plus angular distributions,
and allows treating resonances with both l >0 and I>0.
Multichannel multilevel fitting is also feasible for light elements, and permits the simultaneous use of non-neutron data leading to the same compound nucleus. Due to the
complexity of such calculations, they may be presented in ENDF libraries as file 3 pointwise
cross sections, although the R-matrix Limited format can handle this case.
2.4.14.2

In the Unresolved Resonance Region:

In principle, if the statistical distributions of the resolved resonance parameters are known,
any formalism can be used to construct fictitious cross sections in the unresolved region.
At the present time, only the SLBW formalism is allowed in ENDF, for the reason that no
significant multilevel effect has been demonstrated, when SLBW is properly handled.
If resolved region statistics are used without adjustment to poor resolution data, then
large multilevel/single-level differences can result, but there is no simple way to determine
which is better. If both are adjusted to yield the same average cross sections, and for
fissile materials, the same capture-to-fission ratio, then the remaining differences are within
the statistical and measurement errors inherent in the method. The above comments on
multilevel effects in the unresolved resonance region are based on the work of DeSaussure
and Perez [Reference 7].
As noted in Section 2.4.l2, the use of SLBW to construct resonance profiles in the unresolved region will result in the defects associated with this formalism elsewhere, and is not
recommended. This application calls for MLBW or better, and the SLBW scheme should be
used only for constructing average cross sections where the negative scattering effects will
combine with the other approximations and presumably be ”normalized out” somewhere
along the line.

83

CHAPTER 2. FILE 2: RESONANCE PARAMETERS

2.4.15

Preferred Formalisms for Evaluating Data

Unless there is strong reason to do otherwise, the R-Matrix Limited format (LRF=7) should
be used for reporting results of new evaluations, as it is the most comprehensive of the current
formats.
1. Light nuclei: Use multilevel, multichannel R-matrix. Present either as pointwise cross
sections in file 3, or as R-matrix parameters using LRF=7.
2. Materials with negligible or moderate multilevel effects, and no multichannel
interference: Reich-Moore or MLBW. These are equivalent in computing time and
all require kernel broadening, although MLBW lends itself to the ψ, χ approximation
discussed in Section 2.4.12. However, RM and RML provide the angular distribution
of elastically-scattered neutrons, which MLBW does not.
3. Materials with strong multilevel effects, but no multichannel interference: Reich-Moore
or R-Matrix Limited. The structural materials do not exhibit channel-channel interference, but have level-level interference that is too strong for an MLBW treatment.
4. Materials with observable channel-channel interference: Reich-Moore or R-matrix Limited. In the past, only low-energy fissionable materials have shown channel-channel
interference, and this is unlikely to change. Reich-Moore evaluations can be converted
to Adler format for presentation in ENDF. The reason why Reich-Moore is preferred
to Adler-Adler as the basic evaluation tool is that it has less flexibility and is therefore better able to distinguish between various grades of experimental data. However,
it requires kernel broadening whereas Adler-Adler uses ψ and χ, making the latter
more convenient to broaden. Unfortunately some of this convenience is lost in practice
because there is no simple equivalence between Adler-Adler and SLBW (see Section
2.4.8). With modern computers and modern computer codes, the slight advantage
offered by kernel broadening is no longer an important issue.
5. Materials with channelchannel interference and one or more competitive reactions:
R-matrix, using the format LRF=7 to present the parameters.

2.4.16

Degrees of Freedom for Unresolved Resonance Parameters

A resonance in the system (neutron plus a target of mass A) corresponds to a quasi-stationary
state in the compound nucleus A + 1. Such a resonance can decay in one or more ways,
each described as a channel. These are labeled by the identity of the emitted particle
(two-body decay), the spins I and i of the residual nucleus and the emitted particle, and
the orbital angular momentum l of the pair. To uniquely specify the channel, two more
quantum numbers are needed, since the magnetic quantum numbers can be eliminated for
unpolarized particles.
It is common to give the channel spin, s, which is the vector sum of I and i, plus
−
→
−
→ −
J =→
s + l , since this facilitates the isolation of the l-dependence of all channel quantities.
84

CHAPTER 2. FILE 2: RESONANCE PARAMETERS
The important point is that the same set of three ingredient angular momenta, I, i, and l,
will give rise to a number of different channels, according to the rules for coupling angular
momenta. The resonance will decay into each of these channels, with a probability that is
governed by a real number γαIiJls , the reduced width amplitude, where α gives the identity
of the emitted particle, the state of excitation of the daughter nucleus, etc. The partial
width for the channel is:
2
ΓαIiJls = 2PαIiJls γαIiJls
.

(2.6)

The penetrabilities depend only on l, and are given in Appendix D for uncharged particles. For charged particles, their
p Coulomb analogs can be found in texts on the subject,
and for gamma rays one uses Γγ rather than Γ and P .
If the collection of channel
P quantum
P numbers (αIiJls) is denoted by c, then the total
width for the level is Γ =
Γ
.
[
c c
c means a sum over all channels]. The argument
from statistical compound nucleus theory is that the Γc ’s are random variables, normally
distributed with zero mean and equal variance. The population referred to is the set of
Γc ’s for a given channel and all the levels (or resonances). It follows that the total width
is distributed as a chi-squared distribution with N degrees of freedom, since this is the
statistical consequence of squaring and adding N normal variates. For N = l, this is the
Porter-Thomas distribution. In determining the behavior of any quantity that is going to be
averaged over resonances, it is necessary to know the way in which the widths are distributed,
hence the inclusion of these degrees of freedom in ENDF.
1. The neutron width is governed by AMUN, which is specified for a particular l value.
Usually, only the lowest allowed l value will be significant in any decay, although the
formats would allow giving both s and d-wave widths for the same resonance. Since
there is only one J value for a given resonance, and we label the widths by one l value,
there can be at most two channels for neutrons (i = 1/2), labeled by the channel spin
values s = I ± 1/2. If I = 0, there is only one channel, s = i = 1/2; hence the
restriction,
1.0 ≤ AMUN ≤ 2.0.

AMUN is the quantity µl,J , discussed in Section D.2.2.2.
Although there is no supporting evidence, it is assumed that the average partial widths
for each channel spin are equal, and that hΓn i is the sum of two equal average partial
widths. In Appendix D this factor of two is absorbed into the definition of hΓn i,
through the use of a multiplicity, which is the number of channel spins, 1 or 2.
2. The competitive width is currently restricted to inelastic scattering, which has the
same behavior as elastic scattering, measured from a different ”zero channel energy,”
hence
1.0 ≤ AMUX ≤ 2.0

Note that one should not set AMUX = 0 out of ignorance of its true value, as suggested
in previous versions of ENDF-l02. This implies a constant from resonance to resonance,
since the chi-squared distribution approaches a delta function as N→ ∞. An inelastic
85

CHAPTER 2. FILE 2: RESONANCE PARAMETERS
reaction can be expected to proceed through a small number of channels and hence to
fluctuate strongly from level to level.
−
→
Specifically, AMUX = µlJ , where J is the spin of the resonance, and l is the orbital
angular momentum of the inelastically scattered neutron. Since the daughter nucleus
−
→
−
→
may have a spin I different from the target spin I, l may be different from l, and
− .
the number of channel spin values µlJ may be different from µ→
lJ
3. For the radiative capture process, AMUG should be set equal to zero. Radiative
capture proceeds through many channels and it is not worthwhile deciding if AMUG
is 30 or 40. (If some nucleus has selection rules that restrict radiative decay to a few
channels, then a different value of AMUG might be appropriate.)
4. The fission value should be given as 1.0≤AMUF≤4.0 and the value zero would be
incorrect. These small values violate the previous discussion of (Wigner-type) channels
and obey instead statistics governed by fission barrier tunneling (Bohr-channels). The
actual value of AMUF is determined by comparison between calculated and measured
cross sections.
The degrees of freedom are constant throughout the unresolved resonance region.

2.4.17

Procedures for the Unresolved Resonance Region

This number of energy points at which the parameters are given must be be sufficient to
reproduce the gross structure in the unresolved cross sections. The limit on the maximum
number of points is given in Appendix G. Within a given isotope the same energy grid must
be used for all J and l-values. The grids may be different for different isotopes. Unresolved
resonance parameters should be provided for neutron energy regions where temperaturebroadening or self-shielding effects are important. It is recommended that the unresolved
resonance region extend up to at least 20 keV.
If the flag LSSF (Section 2.3.1) is set equal to one, the evaluator can specify the gross
structure in the unresolved range on as fine an energy grid as he desires, subject only to the
overall limitation on the number of points in File 3 (see Appendix G). Under this option,
File 3 represents the entire cross section at infinite dilution in the unresolved resonance
region, and no File 2 contribution is to be added to it. Instead, File 2 is to be used
to compute a ”slowly-varying” self-shielding factor that may be applied to the ”rapidlyvarying” File 3 values. The self-shielding factor is defined as the ratio of File 2 average
shielded cross section to the average unshielded value computed from the same parameters.
This ratio is to be applied as a multiplicative factor to the values in File 3.
If LSSF is set equal to zero, File 3 will be interpreted in the same way as a resolved-region
File 3, i.e., it will represent a partial background cross section to be added to the average
cross section, dilute or shielded that is computed from File 2.
The self-shielding factor procedure has certain advantages over the ”additive” procedure:
1. The energy-variation of the dilute cross section in the unresolved region can be more
accurately specified, without the limitation on the number of points imposed in File 2.

86

CHAPTER 2. FILE 2: RESONANCE PARAMETERS
2. The energy grids in File 2 and File 3 are basically uncoupled, so that the File 2 grid
can be made coarser and easier to process.
3. In principle, the results can be more accurate, since File 2 can be devoted entirely to
representing changes in the average parameters that are significant for shielding. The
burden of representing fluctuations in the size of the dilute cross section is taken over
entirely by File 3.
4. The same representation can be used by codes requiring probability tables. For this
application, the average parameters in File 2 can be used to generate random ladders of
resonances, and the resulting cross sections can be used to calculate probability tables
in the usual way. However, instead of using the tables directly, they are normalized by
dividing the various cross section bands by the average cross section in the interval.
These normalized probabilities are then converted back to cross sections by multiplying
them into the File 3 values. The rationale is the same as for the shielding-factors - the
dilute cross section is represented in ”poor-resolution” format in File 3, while the real
fine-structure is established in File 2.
The following caution should be noted by evaluators in choosing this option:
Because File 3 is energy varying, it inherently has the possibility to energy-self-shield itself.
If File 2 also shields it, one may actually ”double-shield”. The problem will probably be
most acute just above the boundary between the resolved and unresolved regions, since the
experimental resolution may still be good enough to see clumps of only a few resonances.
One might consider ”correcting” for this in the choice of File 2 parameters, but this would
be difficult because the degree of shielding is application dependent. A better procedure
would be to insure that each significant structure in File 3 actually represents a statistically
meaningful number of resonances, say ten or more. If the raw data do not satisfy this
criterion, then additional smoothing should be applied by the evaluator to make it a correct
condition on the data. A careful treatment will require the use of statistical level theory to
determine the true widths and spacings underlying the File 3 structures.

2.4.18

Procedures for Computing Angular Distributions in the
Resolved Resonance Range

2.4.18.1

Background

Quantum mechanical scattering theory, which underlies all of the resonance formalisms in
this chapter, describes the angular distribution of exit particles as well as the magnitudes
of the various reactions. When the R-matrix formalism is used to parameterize the collision
matrix, as in the Reich-Moore format (Section D.1.3) or the RML format (Section D.1.7),
then the angular distributions exhibit a resonant behavior, in the sense that they may
change substantially in passing through a resonance. An explicit tabulation of this detailed
resonance behavior will usually imply a very large data file.
Blatt and Biedenharn [Reference 8] simplified the general expression for the angular distribution, which is an absolute square of an angle-dependent amplitude, so that it became a
87

CHAPTER 2. FILE 2: RESONANCE PARAMETERS
single sum over Legendre polynomials. Their expression, particularized to the RML format,
is given in Section D.1.7. In the past, Reich-Moore has been a vehicle for low-energy fissionable isotope evaluations, usually s-waves only, so that the angular distribution is isotropic.
If it were used for higher energies and higher angular momenta, then the angular distributions would become anisotropic. Of course, since the formulae define a center-of-mass
distribution, even the isotropic case generally defines an anisotropic laboratory distribution.
In principle, similar angular distribution formulae underlie the SLBW, MLBW, and
Adler-Adler formalisms, but since these are not formulated in terms of collision matrix
elements (UlsJ ), the Blatt and Biedenharn formulas are not immediately applicable to them.
Although the Blatt and Biedenharn formulas have been around for many years, and
have been much used in the physics literature of scattering theory, they have not been
widely employed in neutron cross section evaluation. ENDF files most often contain either
experimental data or calculated data derived from an optical model. Both of these types
represent a ”smoothing” or ”thinning” of the underlying resonant angular distributions. In
the case of experiment, the smoothing is done by the resolution-broadening of the measuring
apparatus, combined with the necessarily limited number of energies at which data can be
taken. In the optical model case, the smoothing is done in an obscure, highly implicit manner. It seems quite clear that an explicit energy-average over resonant Blatt and Biedenharn
Legendre coefficients will differ from both of the above representations.
This raises the question of whether the Blatt and Biedenharn average will be better or
worse than the others. That question is dealt with in the following paragraphs, which are
somewhat ”theoretical”, since there is not much hard experience in this area.
2.4.18.2

Further Considerations

Firstly, if in some ideal case, all the resonance spins and parities were precisely known,
then the Blatt and Biedenharn values would be exact, and clearly superior to any other
representation. The next step down the accuracy ladder would be a case where the major
resonances, or antiresonances (”windows”) were known, but some minor, narrower ones
were uncertain. For this case, one might find that errors in the ”minor” resonances canceled
each other, again producing a superior result, or one might find an erroneous cooperation,
resulting in spurious values.
Finally, there are evaluations that use compiled resonance parameters, with many guessed
J and l-values, in which case the cancellations and/or cooperations dominate the angular
distributions. In both of the two latter cases, the evaluator either will or will not have
compared with experiment and made a decision on the accuracy of the Blatt and Biedenharn
representation. The flag LAD allows him to inform the processing code whether or not it
is ”safe” to calculate from the Blatt and Biedenharn formulas. Such a flag is necessary
because File 4 is limited in the number of points to represent the angular distributions
(see Appendix G), which is usually not enough for a fully-detailed Blatt and Biedenharn
representation. The recommended ENDF procedure is for the evaluator to provide a low
energy-resolution representation in File 4, and to signal the user with LAD whether he can
independently generate σ(θ) on a finer energy mesh.
For the File 4 representation, the evaluation should smooth the data so as to preserve
significant structure in the first Legendre coefficient, i.e. the average cosine of scattering in
88

CHAPTER 2. FILE 2: RESONANCE PARAMETERS
the laboratory system µ. As always, the word significant is difficult to define exactly but
should be interpreted as accurate enough to meet the target accuracy requirements for most
ordinary reactor engineering applications.
In any case, a user who wishes to examine the implications for his own work of a finer
mesh is free to use the Blatt and Biedenharn formulas. The flag LAD tells him either that
the evaluator has approved this procedure (LAD=1), or that it is either of unknown quality
or known to be poor (LAD=0). In the case of LAD=0, the evaluator should tell which of
these is the case by putting comments into File 1 and the associated documentation.
2.4.18.3

Summary of Recommendations for Evaluation Procedures

1. Supply a coarse-mesh representation of the elastic scattering angular distribution in
File 4 that meets the ENDF-6 limits on the umber of points (see Appendix G). Preserve
significant structure in µ.
2. If the Blatt and Biedenharn angular distributions were not examined, or if they were
examined and found to be inaccurate, supply LAD=0 in File 2. Tell which of these is
the case in File 1 and in the associated documentation.
3. If the Blatt and Biedenharn angular distributions were found, or are believed, to be
accurate, supply LAD=1, and describe the evaluation procedures in the documentation.

2.4.19

Completeness and Convergence of Channel Sums

Two possible errors in the calculation of cross sections from a sum over individual channels
are:
1. omission of channels because they contain no resonances (such ”non-resonant” or
”phase-shift-only” channels must still be included because they contribute to the potential scattering cross section), and
2. failure to include enough non-resonant channels to insure convergence of the potentialscattering cross section with respect to l at high energy.
Avoiding the first is the responsibility of the processing codes for the SLBW, MLBW, AA,
and RM formalisms, since the formats do not allow the evaluator to specify empty channels
explicitly. For the RML formalism, where such specification is explicit, the responsibility is
the evaluator’s. Avoiding the second is always the evaluator’s responsibility, since it would
be awkward for a processing code to decide whether the omission was intentional or not.
In the channel spin representation, the incident spin, i, is coupled to the target spin, I,
to form the channel spin, s, which takes on the values:
|I − i| ≤ s ≤ I + i.
The channel spin couples to the orbital angular momentum to form the total angular
momentum J, with the values:
89

CHAPTER 2. FILE 2: RESONANCE PARAMETERS

|l − s| ≤ J ≤ l + s.
If I > 0 and l > 0, the same J-value may occur for each of the two channel spins, s = I ±1/2,
and each of these J-values must be separately included. A width ΓlJ is a sum of the two
components, Γls1 J and Γls2 J ; and in the SLBW, MLBW, and A-A formalisms, only the sum
is used. In the Reich-Moore format, the specification of J is implicit (via the use of a signed
AJ value). For the RML format, the evaluator should specify two separate channels (for the
two values of s) within the same spin group in this situation.
There is rarely enough information on channel-spin widths to guide the evaluator in
apportioning the total width between the two subchannels, but fortunately, most neutron
reactions are insensitive to the split, so that putting it all in one and none in the other, or
splitting it 50/50 works equally well. Angular distributions are in principle more sensitive,
but it is similarly unusual to find measured data of sufficiently high precision to show an
effect.
The channel sums are infinite,
∞
X

1

I+ 2
X

l+s
X

l=0 s=|I− 1 | J=|l−s|
2

χlsJ

(2.7)

so the question of convergence arises. The simplest case is where the summand is an SLBW
reaction term, in which case one sums only over channels in which there are resonances.
There are then no convergence considerations.
If one is summing scattering cross section terms, there is a potential-scattering amplitude in every channel, independent of whether there are resonances or not. The l=0, or
s-wave amplitudes, are finite at zero energy, but the higher l-waves only come in at higher
energies. The convergence criterion is therefore that the addition of the next higher l-wave
produces a negligible change in the cross section at the highest energy covered by the resonance region. In a conventional R-matrix treatment, the non-resonant channels contain
hard-sphere phase shifts whose behavior has to be compared with experiment.
For the RM format, NLS is defined as that value which converges the cross section
calculation. This is different from the SLBW/MLBW definition, which is the number of lchannels with resonances. The latter is more liable to cause neglect of higher l non-resonant
channels. Such neglect would show up as incorrect between-resonance scattering at high
energies, admittedly not the easiest defect to see.
If angular distributions are to be calculated, besides having more complicated sums,
the range of l-values is much greater, the requirement being that the angular distributions
converge at the highest energies. Because the high l-amplitudes interfere with the low
ones, non-negligible cross terms occur which are absent from the cross section sums. The
different convergence criteria, NLS and NLSC, are compatible because only the B0 -moment
contributes to the cross sections. All the higher moments integrate to zero. Computer codes
which reconstruct such moments should have recursive algorithms for l-dependent quantities
up to l = 20.
For the R-Matrix Limited format, all terms and only those terms specified by the evaluator (i.e., included in the File 2 information) are to be included in every calculation.
90

CHAPTER 2. FILE 2: RESONANCE PARAMETERS

2.4.20

Channel Spin and Other Considerations

For the R-Matrix Limited format, channel spin is explicit and the evaluator must in general
provide partial widths that depend on s as well as l and J.
For the Adler-Adler formalism, the usual area of application is to low-energy fissile
nuclides, with l = 0, so the channel spin is not mentioned in the formulae of Appendix D.
For the Reich-Moore formalism, in those cases where two channel spins are possible, the
channel spin is specified by the sign of the AJ parameter. In older evaluations where the
channel spin is not specified (i.e., where all AJ are positive), all resonances are assumed to
have the same channel spin and the hard-sphere contribution from the second channel spin
must be added separately.
For MLBW the absolute square has been expanded out and all imaginary quantities
eliminated. This has several consequences.
1. Channel spin is effectively eliminated, because the partial widths occur in ”summed”
form:
ΓlJ = Γls1 J + Γls2 J

(2.8)

Since only the sum is required, the evaluator is spared the necessity of specifying the
separate s-values. This converts an (l, s, J) formalism into an (l, J) formalism. The
same effect can be achieved by assuming that I=0, a popular assumption often made
independently of the truth, as in many optical model calculations.
2. The convergence criterion is more transparent, because the potential-scattering cross
section splits off from the resonance and interference terms, as
4π
(2l + 1) sin2 φl .
(2.9)
2
k
Despite the simpler nature of this term than its parent amplitudes, one must still carry
enough terms to make the results physically correct, and if this cannot be done, then
File 3 must be invoked to achieve that goal.
3. The resonance profiles are expressible in terms of symmetric and asymmetric BreitWigner shapes, and thus permit ψ − χ Doppler broadening. The price one pays for
these three advantages is increased computing time, when the number of resonances
is large.
Similar remarks apply to the SLBW formalism, which is MLBW without the resonance
resonance interference terms. The computing time goes way down, but the scattering
cross section is very poor. SLBW has useful applications in certain analytical and
semi-analytical procedures, but should never be used for the calculation of explicit
pointwise scattering cross sections.
The omission of an explicit channel-spin quantum number in the SLBW formalism,
while convenient in the resolved resonance region, has occasioned some difficulty in
the unresolved region. Sections D.2.2 and D.2.3 attempt to clarify the situation with
respect to level densities, strength functions, and spin statistics.
91

CHAPTER 2. FILE 2: RESONANCE PARAMETERS

References for Chapter 2
1. P. Greebler and B. Hutchins. Physics of Fast and Intermediate Reactors,
Vienna, 3-11 August 1961, Vol. III (International Atomic Energy Agency, 1962) p.
121
2. H. Henryson II, B. J. Toppel, and C. G. Stenberg, Argonne National Laboratory report
ANL8144 (1976)
3. M. Beer, Nuc. Sci. Eng . 50 (1973) 171
4. A.W. Solbrig, Am. J. Phys. 24 (1961) 257
5. G. deSaussure, G. Olsen, and R.B. Perez, Nuc. Sci. Eng. 61 (1976) 496
6. D.B. Adler, Brookhaven National Laboratory report BNL 50045 (1967) page 7
7. G. deSaussure and R.B. Perez,, Nuc. Sci. Eng. 52 (1973) 382
8. J.M. Blatt and L.C. Biedenharn, Rev. Mod. Phys. 24, 258 (1952)

92

Chapter 3
File 3: REACTION CROSS
SECTIONS
3.1

General Description

Reaction cross sections and auxiliary quantities are given in File 3 as functions of E, where
E is the incident particle energy in the laboratory system. They are given as energy-cross
section (or auxiliary quantity) pairs. An interpolation scheme is given that specifies the
energy variation of the data for incident energies between a given energy point and the
next higher point. File 3 is divided into sections, each containing the data for a particular
reaction, identified by the MT number; see Section 0.4.3 and Appendix B. The sections are
ordered by increasing MT number. As usual, each section starts with a HEAD record and
ends with a SEND record. The file ends with a FEND record.

3.2

Formats

The following quantities are defined:
ZA,AWR Standard material charge and mass parameters.
QM Mass-difference Q value (eV): defined as the mass of the target and projectile
minus the mass of the residual nucleus in the ground state and masses of all
other reaction products; that is, for the reaction a+A → b+c+...+B,
QM = [(ma + mA ) − (mb + mc + ... + mB )] × (eV /amu)
where the masses are in amu. (See Section 3.3.2 and Appendix H for the
mass conversion constant).
QI Reaction Q value for the (lowest energy) state defined by the given MT
value in a simple two-body reaction or a breakup reaction. Defined as QM
for the ground state of the residual nucleus (or intermediate system before
breakup) minus the energy of the excited level in this system. Use QI=QM
for reactions with no intermediate states in the residual nucleus and without
complex breakup (LR=0). (See Section 3.3.2)
93

CHAPTER 3. FILE 3: REACTION CROSS SECTIONS
LR Complex or ”breakup” reaction flag, which indicates that additional particles
not specified by the MT number will be emitted. See Sections 0.4.3.4 and
3.4.4.
NR,NP,Eint Standard TAB1 parameters.
σ(E) Cross section (barns) for a particular reaction (or the auxiliary quantity)
given as a table of NP energy-cross section pairs.
The structure of a
[MAT, 3, MT/
[MAT, 3, MT/
[MAT, 3, 0/

3.3
3.3.1

section is:
ZA, AWR,
QM,
QI,
0.0, 0.0,

0, 0, 0, 0] HEAD
0, LR, NR, NP/ Eint / σ(E)]TAB1
0, 0, 0, 0] SEND

General Procedures
Cross Sections, Energy Ranges, and Thresholds

For incident neutrons, the cross-section data must extend to the lower energy limit of
10−5 eV, except for threshold reactions, for which tabulation should start at the reaction
threshold energy (with a value of 0.0 barns); the data should cover the energy range up to
the common upper energy limit EMAX (see Section 1.1) of at least 20 MeV.
In case there is a change in the data representation at a certain energy Ex , duplicate
energy points may be entered to allow for the discontinuity. If the tabulated cross section
below Ex is zero (e.g. in the region where resonance parameter representation is used),
the first duplicate point has a cross section value of zero, preceeded by another zero value
at 10−5 eV. If the tabulated cross section above Ex is zero (e.g. in the region where the
cross section is included in the lumped reaction MT=5, or otherwise), the second duplicate
point has a cross section value of zero, followed by another zero value at energy EMAX. The
evaluator should document the change in data representation (if any) in the comments in
File 1.
For some charged-particle emission reactions, the cross section might have a low threshold
(or no threshold at all), but could be very small (< 10−10 barns) due to the Coulomb barrier
up to an effective threshold . The evaluator should be careful to enter a zero value at
the true threshold (or 10−5 eV), as well as at the effective threshold, in order to avoid
interpolation problems.
A cross section (or auxiliary quantity) in File 3 may become zero at and above a certain
energy (e.g. the cross section may become very small for physical reasons or due to a change
in representation). The energy at which the cross section becomes zero and the maximum
energy EMAX must be given in the file with zero cross section values.
The limit on the number of energy points (NP) to be used to represent a particular cross
section is given in Appendix G. The evaluator should not use more points than are necessary
to represent the cross section accurately. When appropriate, resonance parameters can be
used to help reduce the number of points needed. The evaluator should avoid sharp features
such as triangles or steps (except for the required discontinuities discussed above), because
such features my cause unphysical results during processing (e.g. Doppler broadening).
94

CHAPTER 3. FILE 3: REACTION CROSS SECTIONS

3.3.2

Q-Values

Accurate reaction Q-values should be given for all reactions, if possible. If QI is not well
defined (as for a range of levels in MT=91, 649, 699, 749, 799, or 849), use the value of QI
which corresponds to the threshold of the reaction. Similarly, if the value of QM is not well
defined (as in elements or for summation reactions like MT=5), use the value of QM which
gives the threshold. If there is no threshold, use the most positive Q value of the component
reactions. Note that these ill-defined values of QM cannot be relied on for energy-release
calculations.
In general, the masses used in the calculation of QM and QI should correspond the rest
masses of the target and residual nuclei. If the target or residual is in the ground state, the
rest mass is just the nuclear mass. If the target or residual is an isomer, the rest mass is
the nuclear mass plus the excitation energy of the nucleus, which should be specified in the
ELIS field in the MF=1 MT=451 file.
As an example to clarify the use of QM and QI, consider the reaction α +9 Be → n+X.
After the neutron has been emitted, the compound system is 12 C with QM=5.702 MeV and
energy levels (Ex ) at 0.0, 4.439, 7.654, and 9.641 MeV. The ground state is stable against
particle breakup, the first level decays by photon emission, and the higher levels decay with
a high probability of breaking up into three alpha particles (7.275 MeV is required). This
pattern can be represented as follows:
Reaction
9
Be(α,n0 )12 C
9
Be(α,n1 )12 C
9
Be(α,n2 )12 C(3α)
9
Be(α,n3 )12 C(3α)
9
Be(α,nC )12 C(3α)

QM
5.702
5.702
-1.573
-1.573
-1.573

QI
5.702
1.263
-1.952
-3.939
-1.573

EX MT
0.000 50
4.439 51
7.654 52
9.641 53
91

The emitted gamma photons for the second and higher reactions are not represented
explicitly in this notation. The last reaction includes the contributions of all the levels above
9.641 MeV, any missed levels, and any direct four-body breakup; therefore, the threshold
for MT=91 may be lower than implied by the fourth level of 12 C. Note the value used for
QI.

3.3.3

Relationship Between File 3 and File 2

If File 2 (Resonance Parameters) contains resolved and/or unresolved parameters (LRP=1),
then the cross sections or self-shielding factors computed from these parameters in the resonance energy range for elastic scattering (MT=2), fission (MT=18), and radiative capture
(MT=102) must be combined with the cross sections given in File 3. The resonance contributions must also be included in any summation reactions that involve the three resonance
reactions (for example, MT=1, 3, or 5). The resonance energy range is defined in File 2.
Double-valued energy points will normally be given in File 3 at the upper and lower limits
of the unresolved and resolved resonance regions.
95

CHAPTER 3. FILE 3: REACTION CROSS SECTIONS
Some material evaluations for incident neutrons will not have resonance parameters but
will have a File 2 (LRP=0) that contains only the effective scattering radius. This quantity
is sometimes used to calculate the potential scattering cross section in self-shielding codes.
For these materials, the potential scattering cross section computed from File 2 must not be
added to the cross section given in File 3. The File 3 data for such materials comprise the
entire scattering cross section.
In certain derived libraries, the resonance cross sections have been reconstructed and
stored in File 3. Such files may have LRP=0 as described in the preceding paragraph.
Alternatively, they may have LRP=2 and include a full File 2 with complete resonance
parameters. In this case, resonance cross sections or self-shielding factors computed from
File 2 are not to be combined with the cross sections in File 3.

3.4

Procedures for Incident Neutrons

Cross section data for non-threshold reaction types must cover the energy range from a
lower limit of 10−5 eV to an upper limit of at least 20 MeV for all materials. For nonthreshold reactions, a cross section value must be given at 0.0253 eV. The limit on the
number of energy points (NP) to be used to represent a particular cross section is given in
Appendix G. The evaluator should not use more points than are necessary to represent the
cross section accurately.
The choice of data to be included in an evaluation depends on the intended application.
For neutron sublibraries, it is natural to define ”transport” evaluations and ”reaction” evaluations. The transport category can be further subdivided into ”low-energy transport” and
”high-energy transport.”
A reaction evaluation will contain File 1, File 2, File 3, and sometimes File 32 and/or
File 33. File 2 can contain resonance parameters. If radioactive products must be described,
Files 8, 9, 10, 39, and/or 40 may be present. File 3 may tabulate one or more reaction cross
sections. The total cross section is usually not well defined in reaction evaluations since
they may be incomplete. Examples of this class of evaluations include activation data and
dosimetry data.
A low-energy transport evaluation should be adequate for calculating neutron transport and simple transmutations for energies below about 6-10 MeV. Photon production and
covariance data should be included when possible. Typical evaluations will include Files 1,
2, 3, 4, 5, and sometimes Files 8, 9, 10, 12, 13, 14, 15, 31, 32, 33, 34, 35, 39, and/or 40.
Resonance parameters will usually be given so that self shielding can be computed. Chargedparticle spectra (MT=600-849) and neutron energy-angle correlation (MF=6) will often be
omitted. File 3 should include all reactions important in the target energy range, including the total (MT=1) and elastic scattering (MT=2). Other reactions commonly included
are inelastic scattering (MT=4,51-91), radiative capture (MT=102), fission (MT=18,1921,38), absorption (MT=103,104,105, etc.), and other neutron emitting reactions such as
MT=16,17,22,28, etc. Specific procedures for some specific reactions are given below. Examples of this class of evaluations include fission-product data and actinide data.
A high-energy transport evaluation should be adequate for calculating neutron
transport, transmutation, photon production, nuclear heating, radiation damage, gas pro96

CHAPTER 3. FILE 3: REACTION CROSS SECTIONS
duction, radioactivity, and charged-particle source terms for energies up to at least 20 MeV.
In some cases, the energy limit needs to be extended to 60 or even 150 MeV. These evaluations use Files 1, 2, 3, 4, 5, 6, 12, 13, 14, 15, and sometimes 8, 9, 10, 31, 32, 33, 34, 35,
39, and/or 40. Once again, File 3 should give cross sections for all reactions important in
the target energy range, including MT=1 and 2. This will normally include many of the
reactions mentioned above plus the series MT=600-849. At high energies, some reactions
may be combined using the ”complex reaction” identifier MT=5. File 6 will normally be
needed at high energies to represent energy-angle correlation for scattered neutrons and to
give particle and recoil energies for heating and damage calculations. Special attention to
energy balance is required. High-energy evaluations are important for materials used in fusion reactor designs, in shielding calculations, and in medical radiation-therapy equipment
(including the constituents of the human body).

3.4.1

Total Cross Section (MT=1)

The total is often the best-known cross section, and it is generally the most important cross
section of a shielding material. Considerable care should be exercised in evaluating this cross
section and in deciding how to represent it.
Cross section minima (potential windows) and cross section structure should be carefully
examined. Sufficient energy points must be used in describing the structure and minima
to reproduce the experimental data to the measured degree of accuracy. However, the
maximum number of points should not exceed the limits specified in Appendix G.
The set of points or energy mesh for the total cross section must be a union of all energy
meshes used for the partial cross sections. Within the above constraints, every attempt
should be made to minimize the number of points used. The total cross section must be the
sum of MT=2 (elastic) and MT=3 (nonelastic). If MT=3 is not given explicitly, then the
elastic cross section plus all implied components of the nonelastic cross section must sum to
the total cross section.
The fact that the total cross section is given at every energy point at which at least
one partial cross section is given allows the partial cross sections to be added together and
checked against the total for any possible errors. In certain cases, more points may be
necessary in the total cross section over a given energy range than are required to specify
the corresponding partial cross sections. For example, a constant elastic scattering cross
section and a 1/v radiative capture cross section could be exactly specified over a given
energy range by log-log interpolation (INT=5), but the sum of the two cross sections would
not be exactly linear on a log-log scale. As a general rule, the total cross section at any
energy should be calculated from the sum of the partial cross sections.

3.4.2

Elastic Scattering Cross Section (MT=2)

The elastic scattering cross section is generally not known to the same accuracy as the total
cross section. Frequently, the elastic scattering cross section is evaluated as the difference
between the total and the nonelastic cross section. This procedure can cause problems: the
resulting elastic scattering cross section shape may contain unreal structure. There may
be several causes. First, the nonelastic cross section, or any part thereof, is not generally
97

CHAPTER 3. FILE 3: REACTION CROSS SECTIONS
measured with the same energy resolution as the total cross section. When the somewhat
poorer resolution nonelastic data are subtracted from the total, the resolution effects appear
in the elastic cross section. Second, if the evaluated structure in the nonelastic cross section is
incorrect or improperly correlated with the structure in the total cross section (energy-scale
errors), an unrealistic structure is generated in the elastic scattering cross section.
The experimental elastic cross section is obtained by integrating measured angular distributions. These data may not cover the entire angular range or may contain contributions
from nonelastic neutrons. Such contamination is generally due to contributions from inelastic scattering to low-lying levels that were not resolved in the experiment. Care must be
taken in evaluating such results to obtain integrated cross sections. Similarly, experimental
angular distribution data can also cause problems when evaluating File 4.

3.4.3

Nonelastic Cross Section (MT=3)

The nonelastic cross section is not required unless any part of the photon production multiplicities given in File 12 uses MT=3. In this case, MT=3 is required in File 3. If MT=3
is given, then the set of points used to specify this cross section must be a union of the sets
used for its partials.

3.4.4

Inelastic Scattering Cross Sections (MT=4,51-91)

The total inelastic scattering cross section (MT=4) must be given if any partials are given;
that is, discrete level excitation cross sections (MT=51-90), or continuum inelastic scattering
(MT=91). The set of incident energy points used for the total inelastic cross section must
be a union of all the sets used for the partials.
Values should be assigned to the level excitation cross sections for as many levels as
possible and extended to as high an energy as possible. Any remaining inelastic scattering
should be treated as continuum. In particular, low-lying levels with significant direct interaction contributions (such as deformed nuclei with 0+ ground states) should be extended to
the upper limit of the file (at least 20 MeV) in competition with continuum scattering. The
secondary energy distribution for such neutrons resembles elastic scattering more than an
evaporation spectrum.
Level excitation cross sections must start with zero cross section at the threshold energy.
If the cross section for a particular level does not extend to the upper limit for the file (e.g.,
20 MeV), it must be double-valued at the highest energy point for which the cross section
is non-zero. The second cross section value at the point must be zero, and it should be
followed by another zero value at the upper limit. This will positively show that the cross
section has been truncated.
If LR=0, a section defined by a particular MT represents the (n,n’γ) reaction. The
angular distribution for the scattered neutron must be given in the corresponding section of
File 4 or 6. The associated photons should be given in a corresponding section of File 6 or
12 and 14, if possible. If the inelastic photons cannot be assigned to particular levels, they
can be represented using MT=4 in File 6, 12 or 13 and 14. When inelastic photons cannot
be separated from other nonelastic photons, they can be included under MT=3 in Files 13,
14 and 15.
98

CHAPTER 3. FILE 3: REACTION CROSS SECTIONS
A LR flag greater than zero indicates inelastic scattering to levels that de-excite by
breakup, particle emission, or pair production rather than by photon emission (see Section 0.5)1 . If LR=1, the identities, yields, and distributions for all particles and photons
can be given in File 6. If LR > 1, angular distributions for the neutron must be given
in File 4, and distributions are not available for the other emitted particles. In this case,
photon production is handled as described above for LR=0.
If a particular level decays in more than one way, then File 6 can be used or several
sections can be given in File 3 for that level. Consider the case in which an excited state
sometimes decays by emitting a proton, and sometimes by emitting an alpha particle. That
part of the reaction that represents (n,n’ α) would use LR=22, and the other part would
be given the next higher section number (MT) and would use LR=28 (n,n’p). The angular
distribution for the neutron would have to be given in two different MT numbers in File 4,
even though they represent the same neutron. The sections must be ordered by decreasing
values of QI (increasing excitation energy).

3.4.5

Fission (MT=18,19-21,38)

The total fission cross section is given in MT=18 for fissionable materials. Every attempt
should be made to break this cross section up into its various parts: first-chance fission (n,f),
MT=19; second-chance fission (n,n’f), MT=20; third-chance fission (n,2nf), MT=21; and
fourth-chance fission (n,3nf), MT=38. The data in MT=18 must be the sum of the data in
MT=19, 20, 21, and 38. The energy grid for MT=18 must be the union of the grids for all
the partials.
If resolved or unresolved resonance parameters are given in File 2, the fission cross section
computed from the parameters must be included in both MT=18 and MT=19.
The Q-value for MT=18, 19, 20, 21, and 38 is the energy released per fission minus the
neutrino energy. It should agree with the corresponding value given in MT=458 in File 1.
Secondary neutrons from fission are usually stated to be isotropic in the laboratory
system in File 4 and energy distributions are given in File 5. However, higher-chance fission
neutrons can have some degree of anisotropy and the data can be coded in File 6. The
complex rules associated with the partial fission reactions are described in Section 5.

3.4.6

Charged-Particle Emission to Discrete and Continuum Levels (MT=600-849)

The (n,p) reaction can be represented using a summation cross section (MT=103), or discrete levels, and a continuum (MT=600-648, and 649) in the same way that the (n,n’)
reaction is represented using MT=4, 51-90, and 91 (see Section 3.4.4). Similarly, (n,d) uses
MT=104 and 650-699, and so on for t, 3 He, and α emission. Of course, MT=600, 650, 700,
etc., represent the ground state and would not have corresponding sections in the photon
production files, unless the flag LR > 0 (such as in the 10 B(n,t)8 Be reaction).
1

LR=31 is still allowed, to uniquely define the γ-decay when using MF=3, and MF=12 (or 15) and
MF=4.

99

CHAPTER 3. FILE 3: REACTION CROSS SECTIONS

3.5

Procedures for Incident Charged Particles and
Photons

See Table 3 for sublibrary numbers for incident charged particles and photons. Procedures
for incident charged particles are generally the same as for neutrons, as given in Section 3.4.
The exceptions are noted below.

3.5.1

Total Cross Sections

The total cross section is undefined for incident charged particles. MT=1 should be used
for the photonuclear total cross section, while MT=501 is used for the total atomic photon
interaction cross section.

3.5.2

Elastic Scattering Cross Sections

As discussed in detail in Section 6.2.6, it is not possible to construct an integrated cross
section for charged-particle elastic scattering because of the Coulomb term. Therefore, the
cross section is either set to 1.0 or to a ”nuclear plus interference” value using a cutoff angle.
This value may in theory be 0.0, but for the purpose of the evaluation should be chosen as
a finite (small) number, e.g., 10−38 . The first and the last energy points used for MT=2 in
File 3 define the range of applicability of the cross section representation given in File 6.
The cross section need not cover the complete range from 10−5 eV to 20 MeV. MT=2 is
used for the elastic scattering cross section for all incident particles and photons (resonance
fluorescence). For photons, MT=502 and 504 are used for coherent and incoherent atomic
scattering, respectively.

3.5.3

Inelastic Scattering Cross Sections

The procedure for inelastic cross section for incident charged particles and photons is the
same as for neutrons. The following MT combinations should be used.
Incident
Particle
γ
n
p
d
t
3
He
α

MT’s for
Excited States
undefined
51-91
601-649
651-699
701-749
751-799
801-849

100

MT for
Total Inelastic
102
4
103
104
105
106
107

CHAPTER 3. FILE 3: REACTION CROSS SECTIONS

3.5.4

Stopping Power

The total charged-particle stopping power in eV·barns is given in MF=3, MT=500. This is
basically an atomic property representing the shielding of the nuclear charge by the electrons.
The same data should be included in all isotopic evaluations of an element, if applicable. It
is a ”total” stopping power in that most tabulations implicitly include large-angle coulomb
scattering which is also represented here in File 6. In practice, this contribution is probably
small enough to keep double counting from being a problem. At low particle energies, mixture effects are sometimes noticeable. They are not accounted for by this representation.

101

Chapter 4
FILE 4: ANGULAR
DISTRIBUTIONS OF SECONDARY
PARTICLES
4.1

General Description

File 4 is used to describe the angular distribution of emitted particles. It is used for reactions
with incident neutrons only, and should not be used for any other incident particle. Angular
distributions of emitted neutrons should be given for elastically scattered neutrons, and for
the neutrons resulting from discrete level excitation due to inelastic scattering. However,
angular distributions may also be given for particles resulting from (n,n’continuum ), (n,2n),
and other neutron emitting reactions. In these cases, the angular distributions will be
integrated over all final energies. File 4 may also contain angular distributions of emitted
charged particles for a reaction where only a single outgoing charged particle is possible
(MT=600 through 849, see Section 3.4.6). Emitted photon angular distributions are given
in File 14 when the particle angular distributions are given in File 4.
The use of File 6 to describe all emitted particle angular distributions is preferred when
charged particles are emitted, or when the emitted particle’s energy and angular distributions
are strongly correlated. In these cases, Files 4 and 14 should not be used.
In some cases, it may be possible to compute the angular distributions in the resolved
range from resonance parameters (see Section 2.4.18 for further discussion). In such cases,
the computed distributions may be preferable to the distributions from File 4 for deep penetration calculations. However, for many practical applications, the smoothed distributions
in File 4 are adequate.
Angular distributions for a specific reaction type (MT number) are given for a series of
incident energies, in order of increasing energy. The energy range covered should be the
same as that for the same reaction type in File 3. Angular distributions for several different
reaction types (MT’s) may be given in File 4 for each material, in ascending order of MT
number.
The angular distributions are expressed as normalized probability distributions, i.e.,

102

CHAPTER 4. FILE 4: ANGULAR DISTRIBUTIONS
Z

1

f (µ, E)dµ = 1

(4.1)

−1

where f (µ, E) dµ is the probability that a particle of incident energy E will be scattered into
the interval dµ about an angle whose cosine is µ. The units of f (µ, E) are (unit-cosine)−1 .
Since the angular distribution of scattered neutrons is generally assumed to have azimuthal
symmetry, the distribution may be represented as a Legendre polynomial series,
NL

f (µ, E) =
where:

X 2l + 1
2π
σ(µ, E) =
al (E) Pl (µ)
σs (E)
2
l=0

(4.2)

µ cosine of the scattered angle in either the laboratory or the center-of-mass
system
E energy of the incident particle in the laboratory system
σs (E) the scattering cross section, e.g., elastic scattering at energy E as given in
File 3 for the particular reaction type (MT)
l order of the Legendre polynomial
σ(µ, E) differential scattering cross section in units of barns per steradian
al the lth Legendre polynomial coefficient; it is implicitly understood that
a0 = 1.0 and is not given in the file.
The angular distributions may be given by one of two forms, and in either the centerof-mass (CM) or laboratory (LAB) coordinate systems. In the first form, the distributions
are given by tabulating the normalized probability distribution, f (µ, E), as a function of
incident energy. In the second form, the Legendre polynomial expansion coefficients, al (E),
are tabulated as a function of incident neutron energy.
Absolute differential cross sections are obtained by combining data from Files 3 and 4.
If tabulated distributions are given, the absolute differential cross section (in barns per
steradian) is obtained by
σs (E)
f (µ, E)
(4.3)
2π
where σs (E) is given in File 3 (for the same MT number) and f (µ, E) is given in File 4. If
the angular distributions are represented as Legendre polynomial coefficients, the absolute
differential cross sections are obtained by
σ(µ, E) =

NL

σ(µ, E) =

σs (E) X 2l + 1
al (E) Pl (µ)
2π l=0 2
103

(4.4)

CHAPTER 4. FILE 4: ANGULAR DISTRIBUTIONS
where σs (E) is given in File 3 (for the same MT number) and the coefficients al (E) are given
in File 4.
Transformation matrices to convert Legendre coefficients from CM to LAB coordinate
system and vice versa are no longer permitted in ENDF-6 formatted files. Expressions
defining elements of the transformation matrices can be found in papers by Zweifel and
Hurwitz1 and Amster2 .

4.2

Formats

File 4 is divided into sections, each containing data for a particular reaction type (MT
number) and ordered by increasing MT number. Each section always starts with a HEAD
record and ends with a SEND record. The following quantities are defined.
LTT Flag to specify the representation used and it may have the following values:
LTT=0, all angular distributions are isotropic
LTT=l, the data are given as Legendre expansion coefficients, al (E)
LTT=2, the data are given as tabulated probability distributions, f (µ, E)
LTT=3, low energy region is represented by as Legendre coefficients; higher
region is represented by tabulated data.
LI Flag to specify whether all the angular distributions are isotropic
LI=0, not all isotropic
LI=1, all isotropic
LCT Flag to specify the frame of reference used
LCT=l, the data are given in the LAB system
LCT=2, the data are given in the CM system
NE Number of incident energy points at which angular distributions are given
(See Appendix G for the limit on NE).
NL Highest order Legendre polynomial that is given at each energy (NL ≤ NM)
NM Maximum order Legendre polynomial that is required to describe the angular distributions in either the center-of-mass or the laboratory system. NM
should be an even number. See Appendix G for the limit on NM.
NP Number of angular points (cosines) used to give the tabulated probability
distributions for each energy (See Appendix G for the limit on NP).
Other commonly used variables are given in the Glossary (Appendix A).
The structure of a section depends on the values of LTT (i.e. the representation used:
al (E) Legendre polynomial expansion or f (µ, E) tabulated normalised probability distribution) but it always starts with a HEAD record of the form
1
2

P.F. Zweifel and H. Hurwitz, Jr., J. Appl. Phys. 25,1241 (1954).
H. Amster, J. Appl. Phys. 29, 623 (1958).

104

CHAPTER 4. FILE 4: ANGULAR DISTRIBUTIONS
[MAT, 4, MT/ ZA, AWR, 0, LTT, 0, 0]HEAD

4.2.1

Legendre Polynomial Coefficients (LTT=1, LI=0)

When LTT=1 (angular distributions given in terms of Legendre polynomial coefficients),
the structure of the section is:
[MAT, 4, MT/ ZA, AWR, 0, LTT, 0, 0]HEAD
(LTT=1)
[MAT, 4, MT/ 0.0, AWR, LI, LCT, 0, 0]CONT
(LI=0)
[MAT, 4, MT/ 0.0, 0.0, 0,
0, NR, NE/ Eint ]TAB2
[MAT, 4, MT/
T, E1 , LT,
0, NL, 0/ al (E1 )]LIST
[MAT, 4, MT/
T, E2 , LT,
0, NL, 0/ al (E2 )]LIST
------------------------------------------------------------------------------------------[MAT, 4, MT/
T,EN E , LT,
0, NL, 0/ al (EN E )]LIST
[MAT, 4, 0/ 0.0, 0.0, 0,
0, 0, 0]SEND
Note that T and LT refer to temperature (in K) and a test for temperature dependence,
respectively. These values are normally zero, however.

4.2.2

Tabulated Probability Distributions (LTT=2, LI=0)

If the angular distributions are given as tabulated probability distributions, LTT=2, the
structure of a section is:
[MAT, 4, MT/ ZA, AWR,
0, LTT, 0, 0]HEAD
(LTT=2)
[MAT, 4, MT/ 0.0, AWR, LI, LCT, 0, 0]CONT
(LI=0)
[MAT, 4, MT/ 0.0, 0.0,
0,
0, NR, NE/ Eint ] TAB2
[MAT, 4, MT/
T, E1 , LT,
0, NR, NP/ µint /f (µ, E1 )]TABl
[MAT, 4, MT/
T, E2 , LT,
0, NR, NP/ µint /f (µ, E2 )]TABl
------------------------------------------------------------------------------------------[MAT, 4, MT/
T, EN E , LT,
0, NR, NP/ µint /f (µ, EN E )]TAB1
[MAT, 4, 0/ 0.0, 0.0,
0,
0, 0, 0]SEND
Parameters T and LT are normally zero.

4.2.3

Purely Isotropic Angular Distributions (LTT=0, LI=1)

When all angular distributions for a given MT are assumed to be isotropic then the section
structure is:
[MAT, 4, MT/ ZA, AWR,
0, LTT, 0, 0]HEAD
(LTT=0)
[MAT, 4, MT/ 0.0, AWR, LI, LCT, 0, 0]CONT
(LI=1)
[MAT, 4, 0/ 0.0, 0.0,
0,
0, 0, 0]SEND

105

CHAPTER 4. FILE 4: ANGULAR DISTRIBUTIONS

4.2.4

Angular Distribution Over Two Energy Ranges (LTT=3,
LI=0)

If LTT=3, angular distributions are given as Legendre coefficients over the lower energy
range and as Probability Distributions over the higher energy range. The structure of a
subsection is:
[MAT, 4, MT/ ZA, AWR,
0, LTT, 0,
0]HEAD
(LTT=3)
[MAT, 4, MT/ 0.0, AWR, LI, LCT, 0, NM]CONT
(LI=0)
(Legendre coefficients)
[MAT, 4, MT/ 0.0, 0.0,
0,
0, NR, NE1/ Eint ]TAB2
[MAT, 4, MT/
T, E1 , LT,
0, NL,
0/ al (E1 )]LIST
------------------------------------------------------------------------------------------[MAT, 4, MT/
T,EN E1 , LT,
0, NL,
0/ al (EN E1 )]LIST
(Tabulated data)
[MAT, 4, MT/ 0.0, 0.0,
0,
0, NR, NE2/ Eint ]TAB2
[MAT, 4, MT/
T,EN E1 , LT,
0, NR, NP/ µint / f (µ, EN E1 ) ]TABl
------------------------------------------------------------------------------------------[MAT, 4, MT/
T,EN ET , LT,
0, NR, NP/ µint / f (µ, EN ET ) ]TABl
(NET = NE1+NE2-1)
[MAT, 4, 0/ 0.0, 0.0,
0,
0,
0,
0]SEND
Note that there is a double energy point at the boundary EN E1 , which appears as the
last point in the Legendre expansion tables and the first point in the tabulated distribution
tables.

4.3

Procedures

The angular distributions for two-body reactions should be given in the CM system
(LCT=2). It is recommended that other reactions (such as continuum inelastic, fission,
etc.) should be given in the LAB system. All angular distribution data should be given
at the minimum number of incident energy points that will accurately describe the energy
variation of the distributions. Legendre coefficients are preferred unless they cannot give an
adequate representation of the data.
When the data are represented as Legendre polynomial coefficients, certain procedures
should be followed. Enough Legendre coefficients should be used to accurately represent
the recommended angular distribution at a particular energy point, and to ensure that the
interpolated distribution is everywhere positive. The number of coefficients (NL) may vary
from energy point to energy point; in general, NL will increase with increasing incident
energy. A linear-linear interpolation scheme (INT=2) must be used to obtain coefficients at
intermediate energies. This is required to ensure that the interpolated distribution is positive
over the cosine interval from -1.0 to +1.0; it is also required because some coefficients may
be negative. In no case should NL exceed the limit NM, defined in Appendix G. If more
coefficients appear to be required to obtain a non-negative distribution, either a constrained
106

CHAPTER 4. FILE 4: ANGULAR DISTRIBUTIONS
Legendre polynomial fit to the data should be given, or the evaluator should switch to
tabular distributions at some energy (LTT=2 or LTT=3 representation). NL=1 is allowed
at low energies to specify an isotropic angular distribution.
When angular distributions are represented as tabular data, certain procedures should
be followed. Sufficient angular points (cosine values) should be given to accurately represent
the recommended distribution. The number of angular points may vary from distribution
to distribution. The cosine interval must be from -1.0 to +1.0. The log-linear interpolation
scheme (INT=4) for f (µ, E) vs. µ is recommended; the linear-linear (INT=2) interpolation
scheme for f (µ, E) vs. E is recommended.
Accurate angular distributions for the thermal energy range must be obtained using
File 7 or a detailed free-gas calculation. File 4 can only give distributions for stationary free
targets.
The ENDF-6 format rules do not allow transformation matrices for conversion between
CM and LAB coordinate systems. If needed, the conversion has to be done at the level of
the processing codes.

4.4

Procedures for Specific Reactions

4.4.1

Elastic Scattering (MT=2)

1. Legendre polynomial representation in the CM coordinate system is preferred for angular distributions of the elastic scattering reaction channel. However, if the number
of Legendre coefficients needed to adequately describe the distribution (making sure
it is non-negative at all angles) would exceed the NM limit (see Appendix G), the
evaluator should switch to tabular representation at some energy.
2. Care must be exercised in selecting an incident energy mesh for certain light-to-medium
mass materials. Here it is important to relate any known structure in the elastic
scattering cross section to the energy dependent variations in the angular distributions.
These two features of the cross sections cannot be analyzed independently of one
another. Remember, processing codes operate on MT=2 data that are given in Files 3
and 4. (Structure of the total cross section is not considered when generating energy
transfer arrays). It is better to maintain consistency in any structure effects between
File 3 and File 4 data than to introduce structure in one File and ignore it in the other.
3. Consistency must be maintained between angular distribution data given for elastic
and inelastic scattering. This applies not only to structural effects, but also to how
the distributions were obtained.
Frequently, the evaluated elastic scattering angular distributions are based on experimental results that, at times, contain contributions from inelastic scattering to
low-lying levels (which in turn may contain direct interaction effects). If inelastic
contributions have been subtracted from the experimental angular distributions, the
same have to be subtracted for the angle-integrated elastic scattering cross sections
and added to the inelastic cross sections in File 3 for consistency. This is particularly
107

CHAPTER 4. FILE 4: ANGULAR DISTRIBUTIONS
important when the inelastic contributions are due to direct interaction, since the angular distributions are not isotropic or symmetric about 90o , but they are generally
forward peaked.
4. Do not use an excessive number of incident energy points for the angular distributions.
The number used should be determined by the amount of variation in the angular
distributions but should not exceed the NE limit defined in Appendix G.
5. The energy range for which the angular distributions are given must correspond exactly
with the range given in File 3 for the same reaction channel (i.e. the same MT number).
6. In the case of neutrons, a relationship exists between the total cross section and the
differential cross section at forward angles (Wick’s limit or optical theorem):

◦

σ(0 ) ≥ σW =



kσT
4π
−28

σW = 10

2

m u
2h2 c2



AWR
1 + AWR

2

E σT2

where:
√
k = 2π 2m∗ E0 /h ; the wave number
h is Planck’s constant in units eV.s (see Appendix H),
m mass in atomic mass units,
m∗ = mu/c2 ; mass in absolute units,
u atomic mass unit (amu) in eV.s/c2 (see Appendix H),
c speed of light in vacuum in m.s−1 (see Appendix H),
E incident energy in the lab system in eV,
E0 incident energy in the center-of-mass system in eV,

2
AWR
E0 =
E
1 + AWR
AWR atomic mass ratio to the neutron,
σT

4.4.2

total cross section.

Discrete Channel Two-Body Reactions

1. Do not give angular distribution data for cumulative reactions (MT=4, 103, 104, 105,
106, 107) if discrete level data are present.

108

CHAPTER 4. FILE 4: ANGULAR DISTRIBUTIONS
2. Always give angular distribution data for single-particle emission (two-body) discrete
level reactions, if they are given in File 3; namely MT = 50 through 91, 600 through
649, 650 through 699, 700 through 749, 750 through 799 and 800 through 849.
3. Discrete channel (two body) angular distributions (e.g., MT = 2, 50 through 91, 600
through 649, 650 through 699, 700 through 749, 750 through 799 and 800 through
849) should be given as Legendre coefficients in the CM system, if possible.
4. The continuum reactions (MT=91, 649, 699, 749, 799, 849) should normally be given
in the LAB system.
5. Isotropic angular distributions should be used unless the degree of the isotropy exceeds 5%. If any level excitation cross sections contain significant direct interaction
contributions, angular distributions are very important.
6. Use the precautions outlined above when dealing with the level excitation cross sections
that contain a large amount of structure.
7. Do not overcomplicate the data files. Restrict the number of distributions to the
minimum required to accurately represent the data.

4.4.3

Other Particle-Producing Reactions

Neutron angular distribution data must be given for all other neutron producing reactions,
such as fission, (n,n′ α), or (n,2n) in File 4 or File 6. File 4 is only appropriate if the
distributions are fairly isotropic without strong pre-equilibrium components. The LAB
system should be used.
If angular data is needed for other more complex reactions, File 6 is usually more appropriate.

109

Chapter 5
File 5: ENERGY DISTRIBUTIONS
OF SECONDARY PARTICLES
5.1

General Description

File 5 describes the energy distributions of secondary particles expressed as normalized
probability distributions. It is designed for incident neutron reactions and spontaneous
fission only, and should not be used for any other incident particle. Data are given in
File 5 for all reaction types that produce secondary neutrons, unless the secondary neutron
energy distributions can be implicitly determined from the data given in File 3 and/or
File 4. For example, no data will be given in File 5 for elastic scattering (MT=2), since the
secondary energy distributions can be obtained from the angular distributions in File 4 and
the kinematic equations for a two-body interaction (see Appendix E). Similarly, no data
will be given for neutrons that result from the excitation of discrete inelastic levels (MT=51,
52, ..., 90).
Data should be given in File 5 for MT=91 (inelastic scattering to a continuum of levels),
MT=18 (prompt neutrons from fission), MT=16 (n,2n), MT=17 (n,3n), MT=455 (delayed
neutrons from fission), and certain other nonelastic reactions that produce secondary neutrons, unless they are given in File 6. The energy distribution for spontaneous fission is
given in File 5 (in sub-library 4).
Despite its important application in equilibrium reactor calculations, the ”total” fission
spectrum (prompt plus delayed) corresponding to the total number of neutrons per fission
tabulated in File 1 under MT 452, is not specified in ENDF, but must be reconstituted by
the processing code.
File 5 may also contain energy distributions of secondary charged particles for continuum
reactions where only a single outgoing charged particle is possible (MT=649, 699, etc.).
Continuum photon distributions should be described in File 15.
The use of File 6 to describe all particle energy distributions is preferred when several
charged particles are emitted or the particle energy and angular distribution are strongly
correlated. In these cases Files 5 and 15 should not be used.
Each section of the file gives the data for a particular reaction type (MT number). The
sections are then ordered by increasing MT number. The energy distributions p(E → E ′ )
110

CHAPTER 5. FILE 5: ENERGY DISTRIBUTIONS
are normalized so that
Z

0

′
Emax

p(E → E ′ )dE ′ = 1

(5.1)

′
where Emax
is the maximum possible secondary particle energy and its value depends on the
incoming particle energy E and the analytic representation of p(E → E ′ ). The secondary
particle energy E ′ is always expressed in the laboratory system.
The differential cross section is obtained from

dσ(E → E ′ )
= y σ(E) p(E → E ′ )
(5.2)
dE ′
where σ(E) is the cross section as given in File 3 for the same reaction type number (MT)
and y is the neutron multiplicity for this reaction (for some reactions y is implicit; e.g., y = 2
for the (n,2n) reaction, etc.).
The energy distributions p(E → E ′ ) can be broken down into partial energy distributions
fk (E → E ′ ), where each of the partial distributions can be described by a different analytic
representation;
p(E → E ′ ) =

NK
X
k=1

pk (E)fk (E → E ′ )

(5.3)

and at a particular incident neutron energy E,
NK
X

pk (E) = 1

(5.4)

k=1

where pk (E) is the fractional probability that the distribution fk (E → E ′ ) can be used at
energy E.
The partial energy distributions fk (E → E ′ ) are represented by various analytical formulations. Each formulation is called an energy distribution law and has an identification
number associated with it (LF number). The allowed energy distribution laws are given
below. 1

5.1.1

Secondary Energy Distribution Laws

The data are given in each section by specifying the number of partial energy distributions
that will be used. The same energy mesh should be used for each one. The partial energy
distributions may all use the same energy distribution law (LF number) or they may use
different laws.
1

Distribution laws for LF=2, 3, 4, 6, 8, and 10 are not presented because these laws are no longer
supported in ENDF-6 formats.

111

CHAPTER 5. FILE 5: ENERGY DISTRIBUTIONS
5.1.1.1

Arbitrary Tabulated Function (LF=1)
f (E → E ′ ) = g(E → E ′ )

For a set of incident energy points E, the distribution g(E → E ′ ) is tabulated as a function
of outgoing particle energy E ′ .
5.1.1.2
where:

General Evaporation Spectrum (LF=5)
f (E → E ′ ) = g (E ′ /θ(E))
θ(E) is tabulated as a function of incident neutron energy E,
g(x) is tabulated as a function of x, and x = E ′ /θ(E).

5.1.1.3

Simple Maxwellian Fission Spectrum (LF=7)
√
E ′ −E ′ /θ(E)
′
f (E → E ) =
e
I

where
I is the normalization constant, defined by

√
 p
π p
−(E−U )/θ
3/2
(E − U )/θ − (E − U )/θ e
I=θ
erf
2
θ is tabulated as a function of energy, E;

U is a constant introduced to define the proper upper limit for the final particle
energy such that 0 ≤ E ′ ≤ (E − U ).
5.1.1.4

Evaporation Spectrum (LF=9)
f (E → E ′ ) =

E ′ −E ′ /θ(E)
e
I

where
I is the normalization constant:



E−U
−(E−U )/θ
2
1+
I =θ 1−e
θ
θ is tabulated as a function of incident neutron energy, E;
U is a constant introduced to define the proper upper limit for the final particle
energy such that 0 ≤ E ′ ≤ (E − U ).
112

CHAPTER 5. FILE 5: ENERGY DISTRIBUTIONS
5.1.1.5

where

Energy-Dependent Watt Spectrum (LF=11)
′
√ 
e−E /a
′
bE ′
sinh
f (E → E ) =
I
I is the normalization constant:
r
r !
r !#
r
r
 "
1 πa3 b
E−U
ab
E−U
ab
ab
I =
exp
erf
−
+
+ erf
2
4
4
a
4
a
4

 
p
E−U
sinh b(E − U )
− a exp −
a

a and b are tabulated energy-dependent parameters;
U is a constant introduced to define the proper upper limit for the final particle
energy such that 0 ≤ E ′ ≤ (E − U ).
5.1.1.6

Energy-Dependent Fission Neutron Spectrum (Madland and Nix)
(LF=12)

1
[g(E ′ , EF (L)) + g(E ′ , EF (H))]
2





3
3
1
3/2
3/2
′
, u2 − γ
, u1
u2 E1 (u2 ) − u1 E1 (u1 ) + γ
g(E , EF ) = p
2
2
3 (EF TM )
√
p 2
′
E − EF /TM
u1 =
√
p 2
E ′ + EF /TM
u2 =

f (E → E ′ ) =

where:

EF (X) are constant, which represent the average kinetic energy per nucleon of the
fission fragment; arguments L and H refer to the average light fragment
(given by the parameter EFL in the file) and the average heavy fragment
(given by the parameter EFH in the file), respectively.
TM parameter tabulated as a function of incident neutron energy,
E1 (x) is the exponential integral,
γ(a, x) is the incomplete gamma function. The integral of this spectrum between
zero and infinity is one. The value of the integral for a finite integration
range is given in Section 5.4.10.

113

CHAPTER 5. FILE 5: ENERGY DISTRIBUTIONS

5.2

Formats

Each section of File 5 contains the data for a particular reaction type (MT number), starts
with a HEAD record, and ends with a SEND record. Each subsection contains the data for
one partial energy distribution. The structure of a subsection depends on the value of LF
(the energy distribution law). The following quantities are defined.
NK Number of partial energy distributions. There will be one subsection for each
partial distribution.
U Constant that defines the upper energy limit for the secondary particle so
that 0 ≤ E′ ≤ E - U (given in the LAB coordinate system).
θ

Effective temperature used to describe the secondary energy distribution for
LF = 5, 7, or 9.

LF Flag specifying the energy distribution law used for a particular subsection
(partial energy distribution). (The definitions for LF are given in Section 5.1).
pk (Ei ) Fractional part of the particular cross section which can be described by the
k th partial energy distribution at the i-th incident energy point, subject to
the condition
NK
X
pk (Ei ) = 1.0
k=1

fk (E → E ′ ) k th partial energy distribution. The definition depends on the value of LF.
NR Number of interpolation ranges.
NP Number of incident energy points at which pk (E) is given.
a,b Parameters used in the energy dependent Watt spectrum, LF = 11.
EFL,EFH Constants EF (L) and EF (H) used in the energy-dependent fission neutron
spectrum (Madland and Nix), LF = 12.
TM Maximum temperature parameter, TM (E), of the energy-dependent fission
neutron spectrum (Madland and Nix), LF =12.
NE Number of incident energy points at which a tabulated distribution is given;
For the maximum limit on NE see Appendix G.
NF Number of secondary energy points in a tabulation; For the maximum limit
on NF see Appendix G.

114

CHAPTER 5. FILE 5: ENERGY DISTRIBUTIONS
The structure of a section has the following form:
[MAT, 5, MT/ ZA, AWR, 0, 0, NK, 0]HEAD


--------------------------------------------
[MAT, 5, MT/ 0.0, 0.0, 0, 0, 0, 0]SEND
The structure of a subsection depends on the value of LF. The formats for the various values
of LF are given below.

5.2.1

Arbitrary Tabulated Function (LF=1)

The structure of a section has the following form:
[MAT, 5, MT/ 0.0, 0.0,
0,
LF,
NR,
NP/ Eint /p(E) ]TAB1
[MAT, 5, MT/ 0.0, 0.0,
0,
0,
NR,
NE/ Eint ]TAB2
[MAT, 5, MT/ 0.0, E1 ,
0,
0,
NR,
NF/ E’int /
′
′
′
E1
0.0, E2 , g(E1 → E2 ), E3′ , g(E1 → E3′ )
--------------------------------------′
′
′
--- EN
g(E1 → EN
EN
F −1 ,
F −1 ),
F , 0.0]TAB1
[MAT, 5, MT/ 0.0, E2 ,
0,
0,
NR,
NF/ E’int /
E1′
0.0, E2′ , g(E2 → E2′ ), E3′ , g(E2 → E3′ )
--------------------------------------′
′
′
g(E2 → EN
EN
--- EN
F −1 ,
F −1 ),
F , 0.0]TAB1
[MAT, 5, MT/ 0.0,EN E ,
0,
0,
NR,
NF/ E’int /
′
′
′
E1
0.0, E2 , g(EN E → E2 ), E3′ , g(EN E → E3′ )
--------------------------------------′
′
′
--- EN
g(E2 → EN
EN
F −1 ,
F −1 ),
F , 0.0]TAB1

LF=1

Note that the incident energy mesh for pk (E) does not have to be the same as the E mesh
used to specify the energy distributions. The interpolation scheme used between incident
energy points E, and between secondary energy points E ′ , should be linear-linear.

5.2.2

General Evaporation Spectrum (LF=5)

The structure of a section has the following form:
[MAT, 5, MT/
U, 0.0,
[MAT, 5, MT/ 0.0, 0.0,
[MAT, 5, MT/ 0.0, 0.0,

0, LF, NR, NP/ Eint / p(E)]TAB1 (LF=5)
0, 0, NR, NE/ Eint / θ(E)]TAB1
0, 0, NR, NF/ xint / g(x)]TAB1 x = E ′ /θ(E)

115

CHAPTER 5. FILE 5: ENERGY DISTRIBUTIONS

5.2.3

Simple Maxwellian Fission Spectrum (LF=7)

The structure of a section has the following form:
[MAT, 5, MT/
U, 0.0,
[MAT, 5, MT/ 0.0, 0.0,

5.2.4

0, LF, NR, NP/ Eint / p(E)]TAB1 (LF=7)
0, 0, NR, NE/ Eint / θ(E)]TAB1

Evaporation Spectrum (LF=9)

The structure of a section has the following form:
[MAT, 5, MT/
U, 0.0,
[MAT, 5, MT/ 0.0, 0.0,

5.2.5

0, LF, NR, NP/ Eint / p(E)]TAB1 (LF=9)
0, 0, NR, NE/ Eint / θ(E)]TAB1

Energy-Dependent Watt Spectrum (LF=11)

The structure of a section has the following form:
[MAT, 5, MT/
U, 0.0,
[MAT, 5, MT/ 0.0, 0.0,
[MAT, 5, MT/ 0.0, 0.0,

5.2.6

0, LF, NR, NP / Eint / p(E)]TAB1 (LF=11)
0, 0, NR, NE / Eint / a(E)]TAB1
0, 0, NR, NE / Eint / b(E)]TAB1

Energy-Dependent Fission Neutron Spectrum
(Madland and Nix) (LF=12)

The structure of a section has the following form:
[MAT, 5, MT/ 0.0, 0.0,
[MAT, 5, MT/ EFL, EFH,

5.3

0, LF, NR, NP/ Eint / p(E) ]TAB1 (LF=12)
0, 0, NR, NE/ Eint / TM (E)]TAB1

Procedures

As many as three different energy meshes may be required to describe the data in a subsection
(one partial distribution). These are the incident energy mesh for pk (E), the incident energy
mesh at which the secondary neutrons are given, fk (E → E ′ ), and the secondary energy
mesh for fk (E → E ′ ). It is recommended that a linear-linear or a linear-log interpolation
scheme be used for the first two energy meshes, and a linear-linear interpolation for the last
energy mesh.
Double energy points must be given in the incident energy mesh whenever there is a
discontinuity in any of the pk (E)’s (this situation occurs fairly frequently). This energy
mesh must also include threshold energy values for all reactions being described by the
pk (E)’s. Zero values for pk must be given for energies below the threshold (if applicable).
Two nuclear temperatures may be given for the (n,2n) reaction. Each temperature θ
may be given as a function of incident neutron energy. In this case p1 (E) = p2 (E) = 0.5. A
similar procedure may be followed for the (n,3n) and other reactions.
A constant, U , is given for certain distribution laws (LF=5, 7, 9, or 11). The constant,
U , is provided to define the proper upper limit for the secondary energy distribution so that
116

CHAPTER 5. FILE 5: ENERGY DISTRIBUTIONS
0 ≤ E ′ ≤ E − U . The value of U depends on how the data are represented for a particular
reaction type. Consider U for inelastic scattering.
Case A: The total inelastic scattering cross section is described as a continuum. U is
the threshold energy for exciting the lowest level in the residual nucleus.
Case B: For the energy range considered, the first three levels are described explicitly
(either in File 3, MT = 51, 52, and 53, or in File 5), and the rest of the
inelastic cross section is treated as a continuum. U is the threshold energy
(known or estimated) for the fourth level in the residual nucleus.
If the reaction being described is fission, then U should be a large negative value
(U = −20.0 × 106 eV to −30 × 106 eV). In this case neutrons can be born with energies
much larger than the incident neutron energy. It is common practice to describe the inelastic cross section as the sum of excitation cross sections (for discrete levels) for neutron
energies up to the point where level positions are no longer known. At this energy point, the
total inelastic cross section is treated as a continuum. This practice can lead to erroneous
secondary energy distributions for incident neutron energies just above the cutoff energy. It
is recommended that the level excitation cross sections for the first several levels (e.g., 4 or
5 levels) be estimated for several MeV above the cutoff energy. The continuum portion of
the inelastic cross section will be zero at the cutoff energy, and it will not become the total
inelastic cross section until several MeV above the cutoff energy.
It is recommended that the cross sections for excitation of discrete inelastic levels be
described in File 3 (MT=51, 52, ..., etc.). The angular distributions for the neutrons resulting from these levels should be given in File 4 (the same MT numbers). The secondary
energy distributions for these neutrons can be obtained analytically from the data in Files 3
and 4. This procedure is the only way in which the energy distributions can be given for
these neutrons. For inelastic scattering, the only data required in Files 5 are for MT=91
(continuum part).

5.4
5.4.1

Additional Procedures
General Comments

1. Do not give File 5 data for the discrete level excitation given in File 3 as MT=51, 52,
..., 90. If MT=91 is given in File 3, a section for MT=91 must be given in File 5 or
File 6. A section must also be given in File 5 or File 6 for all other neutron-producing
reactions. Continuum energy distributions for emitted protons, deuterons etc., may
be given in MT=649 etc., and for photons, in File 6 or File 15. When more than one
particle type is emitted, File 6 should be used to assure energy conservation.
2. Care must be used in selecting the distribution law number (LF) to represent the data.
As a rule, use the simplest law that will accurately represent the data.

117

CHAPTER 5. FILE 5: ENERGY DISTRIBUTIONS
3. A section in File 5 must cover the same incident energy range as was used for the same
MT number in File 3. The sum of the probabilities for all laws used must be equal to
unity for all incident energy points.
4. If the incident neutron energy exceeds several MeV, pre-equilibrium neutron emission
can be important, as illustrated from high-resolution neutron and proton spectra measurements and analysis of pulsed sphere experiments. In these cases either tabulated
spectra or ”mocked-up” levels can be constructed to supplement or replace simple
evaporation spectra.
5. Prompt-neutron fission spectra are given under MT=18, 19, 20, 21, and 38. Delayedneutron fission spectra are given under MT=455.
6. The energy distribution of prompt-neutrons from spontaneous fission is given in File 5
for MT=18, but in the decay data sublibrary (NSUB=4). It is used with νp from
File 1 (MT=456) to determine the prompt-neutron spontaneous-fission spectrum.
The delayed-neutron spontaneous-fission spectrum is determined from νd from File 1
(MT=455) and the delayed-neutron energy-distribution in File 5 MT=455. Note that
for the specification of spontaneous-fission spectra no cross sections in File 3 are given.

5.4.2

Tabulated Distributions (LF=1)

Use only tabulated distributions to represent complicated energy distributions. Use the minimum number of incident energy points and secondary neutron energy points to accurately
represent the data. The integral over secondary neutron energies for each incident energy
point must be unity to within four significant figures. All interpolation schemes must be
with linear-linear or linear-log (INT=1,2, or 3) to preserve probabilities upon interpolation.
All secondary energy distributions must start and end with zero values for the distribution
function g(E → E ′ ).

5.4.3

Maxwellian Spectrum (LF=7)

A linear-linear interpolation scheme is preferred for specifying the nuclear temperature as a
function of energy.

5.4.4

Evaporation Spectrum (LF=9)

An evaporation spectrum is preferred for most reactions. Care must be taken in describing
the nuclear temperature near the threshold of a reaction. Nuclear temperatures that are too
large can violate conservation of energy.

5.4.5

Watt Spectrum (LF=11)

A linear-linear interpolation scheme is preferred for specifying the parameters a and b as a
function of energy.

118

CHAPTER 5. FILE 5: ENERGY DISTRIBUTIONS

5.4.6

Madland-Nix Spectrum (LF=12)

A log-log interpolation scheme may be used for specifying the parameter TM as a function
of incident neutron energy.

5.4.7

Selection of the Integration Constant, U

1. When LF = 5, 7, 9, or 11 is used, an integration constant U is required. This constant
′
is used in defining the upper energy limit of secondary neutrons; i.e., Emax
= En − U ,
where En is the incident neutron energy. U is a constant for the complete energy range
covered by a subsection in File 5 and is given in the LAB system.
2. U is negative for fission reactions. The preferred value is -20 MeV.
3. In practice, U can be taken to be the absolute value of Q for the lowest level (known
or estimated) that can be excited by the particular reaction within the incident energy
range covered by the subsection. U is actually a function of the incident neutron
energy, but it can be shown that it is always greater than the absolute value of Q and
less than the threshold energy of the reaction. At large AWR, since Eth and |Q| are
approximately equal, either could be used but the absolute value of Q is preferred. At
small AWR, using |Q| for U is the best approximation and must be used.
4. The following four cases commonly occur in data files; procedures are given for obtaining U values.

Case A: The complete reaction is treated as a continuum.
U = −Q, where Q is the reaction Q-value.
Case B: The reaction is described by excitation of three levels (in File 3 as MT = 51,
52, 53) and a continuum part where Q4 is the known or estimated Q-value
for the fourth level:
U = −Q4 .
Case C: The reaction is described by excitation of three levels (in File 3 as MT=51,
52, and 53) and a continuum part which extends below the threshold for
MT=51. If, for example, the reaction is a 3-body breakup reaction, use
U = −Q ,
where Q is the energy required for 3-body breakup.
Case D: The reaction is described by excitation of the first three levels (in File 3 as
MT=51, 52, 53) for neutron energies from the level thresholds up to 20 MeV,
excitation of the next five levels (in File 3 as MT=54, ..., 58) from their
thresholds up to 8 MeV, and by a continuum part that starts at 5 MeV.
In this case two subsections should be used, one to describe the energy range
from 5 to 8 MeV and another to describe the energy region from 8 to 20 MeV.
In the first subsection (5 - 8 MeV),
119

CHAPTER 5. FILE 5: ENERGY DISTRIBUTIONS
U = −Q9 ,

and the second (8 - 20 MeV),
U = −Q4 .

5.4.8

Multiple Nuclear Temperatures

Certain reactions, such as (n,2n), may require specification of more than one nuclear temperature. θ(E) should be given for each neutron in the exit channels; this is done by using
more than one subsection for a reaction. The U value is the same for all subsections. The
upper energy limit is determined by the threshold energy and not by level densities in the
residual nuclei.

5.4.9

Average Energy for a Distribution

The average energy of a secondary neutron distribution must be less than the available
energy for the reaction:
1 + AWR
Q,
AWR
where Eavail is greater than the neutron multiplicity times the average energy of all the
emitted neutrons, νE ′ , where ν is the multiplicity. The mean energy E ′ should be calculated
from the distribution at each value of E. This mean is analytic in the four cases given below:
Eavail = E +

LF=7 E ′ = 32 θ −

θ5/2
I

LF=9 E ′ = 2θ −

θ3
I

L=11





E−U 3/2 −(E−U )/θ
e
θ



E−U 2 −(E−U )/θ
e
θ

q i
q

 q a3 b 3 ab 
 h q Er q ab 
Er
ab
π
a exp ab
+
erf
+
−
+
erf
4
4
2
4
a
4
4
4
q
i
hq
√


√
√
Er
− 3a2 ab exp − Ear
cosh bEr − ab
sinh bEr
a
4


  Er ab 

√
√
√
− 2a2 exp − Ear
+ 4 sinh bEr − bEr cosh bEr
a
where Er = E − U
E′

=

1
2I

LF=12 E ′ = 12 [EF (L) + EF (H)] + 43 TM
Parameter U is described in Section 5.3. The analytic functions for I are given in Section 5.1
for LF=7, 9, 11. For LF=12, Section 5.4.10 gives the method for obtaining the integral of
the distribution function.

120

CHAPTER 5. FILE 5: ENERGY DISTRIBUTIONS

5.4.10

Additional Procedures for Energy-Dependent
Fission Neutron Spectrum (Madland and Nix)

To define√the integral over a finite energy range [a, b], set:
α = √TM
β = EF
√
2
A = ( a + β) /α2
√
2
B=
b + β /α2
√
2
A′ = ( a − β) /α2
√
2
B′ =
b − β /α2
The integral is given by one of the following three expressions depending on the region of
integration in which a and b lie.
5.4.10.1

Region I (a > EF , b > EF )

p
3 ET TM

Z

b

g(E ′ , EF )dE ′ =





2 2 5/2 1
2 2 5/2 1
2
2
=
α B − αβB E1 (B) −
α A − αβA E1 (A)
5
2
5
2





2 2 ′5/2 1
2 2 ′5/2 1
′
′
′2
′2
−
E1 (B ) −
α B
+ αβB
α A
+ αβA E1 (A )
5
2
5
2

 3  
  3 
√
√
2
2
+ α B − 2αβ B γ
, B − α A − 2αβ A γ
,A
2
2
 




√ 
√ 
3 ′
3 ′
2 ′
2 ′
′
′
, B − α A + 2αβ A γ
,A
− α B + 2αβ B γ
2
2
 







3 2
5
5 ′
5
5 ′
− α γ
,B − γ
,A − γ
,B − γ
,A
5
2
2
2
2
h
i
3
′
′
− αβ e−B (1 + B) − e−A (1 + A) + e−B (1 + B ′ ) − e−A (1 + A′ )
5
a

121

CHAPTER 5. FILE 5: ENERGY DISTRIBUTIONS
5.4.10.2

Region II (a < EF , b < EF )

p
3 ET TM

5.4.10.3

Z

b

g(E ′ , EF )dE ′ =
a





2 2 5/2 1
2 2 5/2 1
2
2
=
α B − αβB E1 (B) −
α A − αβA E1 (A)
5
2
5
2





2 2 ′5/2 1
2 2 ′5/2 1
′
′
′2
′2
−
E1 (B ) −
α B
+ αβB
α A
+ αβA E1 (A )
5
2
5
2



 


√
√ 
3
3
2
2
+ α B − 2αβ B γ
, B − α A − 2αβ A γ
,A
2
2
 




√ 
√ 
3
3
2
′
′
′
2 ′
, B − α A − 2αβ A′ γ
,A
− α B − 2αβ B ′ γ
2
2
 







3 2
5
5 ′
5
5 ′
− α γ
,B − γ
,A − γ
,B − γ
,A
5
2
2
2
2
h
i
3
−B
−A
−B ′
′
−A′
′
− αβ e (1 + B) − e (1 + A) + e (1 + B ) + e (1 + A )
5

Region III (a < EF , b > EF )

p
3 ET TM

Z

b

g(E ′ , EF )dE ′ =
a





2 2 5/2 1
2 2 5/2 1
2
2
=
α B − αβB E1 (B) −
α A − αβA E1 (A)
5
2
5
2





2 2 ′5/2 1
2 2 ′5/2 1
′
′
′2
′2
−
E1 (B ) −
α B
+ αβB
α A
+ αβA E1 (A )
5
2
5
2



 


√
√ 
3
3
2
2
+ α B − 2αβ B γ
, B − α A − 2αβ A γ
,A
2
2
 




√ 
√ 
3
3
2
′
′
′
2 ′
, B − α A + 2αβ A′ γ
,A
− α B + 2αβ B ′ γ
2
2
 







3 2
5
5 ′
5
5 ′
− α γ
,B − γ
,A − γ
,B + γ
,A
5
2
2
2
2
h
i
3
′
′
− αβ e−B (1 + B) − e−A (1 + A) + e−B (1 + B ′ ) + e−A (1 + A′ ) − 2
5

The expression for Region III would be used to calculate a normalization integral I for
the finite integration constant U , if a physical basis existed by which U could be well
determined.

122

Chapter 6
File 6: PRODUCT
ENERGY-ANGLE
DISTRIBUTIONS
6.1

General Description

This file is provided to represent the distribution of reaction products (i.e., neutrons, photons, charged particles, and residual nuclei) in energy and angle. It works together with
File 3, which contains the reaction cross sections, and replaces the combination of Files 4
and 5. Radioactive products are identified in File 8. The use of File 6 is recommended
when the energy and angular distributions of the emitted particles must be coupled, when
it is important to give a concurrent description of neutron scattering and particle emission,
when so many reaction channels are open that it is difficult to provide separate reactions,
or when accurate charged-particle or residual-nucleus distributions are required for particle
transport, heat deposition, or radiation damage calculations.
For the purposes of this file, any reaction is defined by giving the production cross section
for each reaction product in barns/steradian assuming azimuthal symmetry:
σi (µ, E, E ′ ) = σ(E) yi (E) fi (µ, E, E ′ )/2π

(6.1)

where:
i denotes one particular product,
E is the incident energy,
E ′ is the energy of the product emitted with cosine µ,
σ(E) is the interaction cross section (File 3),
yi is the product yield or multiplicity, and
fi is the normalized distribution with units (eV.unit-cosine)−1 where
Z
Z
′
dE
dµ fi (µ, E, E ′ ) = 1
123

(6.2)

CHAPTER 6. FILE 6: PRODUCT ENERGY-ANGLE DISTRIBUTIONS
This representation ignores most correlations between products and most sequential reactions; that is, the distributions given here are those, which would be seen by an observer
outside of a ”black box” looking at one particle at a time. The process being described
may be a combination of several different reactions, and the product distributions may be
described using several different representations.

6.2

Formats

The following quantities are defined for all representations:
ZA, AWR Standard material charge and mass parameters.
LCT Reference system for secondary energy and angle (incident energy is always
given in the LAB system).
LCT=1, laboratory (LAB) coordinates used for both;
LCT=2, center-of-mass (CM) system used for both;
LCT=3, center-of-mass system for both angle and energy of light particles
(A≤ 4), laboratory system for heavy recoils (A>4).
NK Number of subsections in this section (MT). Each subsection describes one
reaction product. There can be more than one subsection for a given particle
or residual nucleus (see LIP). For the limit on NK see Appendix G.
ZAP Product identifier 1000 ∗ Z + A with Z = 0 for photons and A = 0 for
electrons and positrons. A section with A = 0 can also be used to represent
the average recoil energy or spectrum for an elemental target (see text).
AWP Product mass in neutron units. When ZAP=0, this field can contain the
energy of a primary photon. In that case, this section will contain an angular
distribution (LAW=2) for the primary photon.
LIP Product modifier flag. Its main use is to identify the isomeric state of a
product nucleus. In this case, LIP=0 for the ground state, LIP=1 for the
first isomeric state, etc. These values should be consistent with LISO in
File 8, MT=457.
In some cases, it may be useful to use LIP to, distinguish between different
subsections with the same value of ZAP for light particles. For example,
LIP=0 could be the first neutron out for a sequential reaction, LIP=1 could
be the second neutron, and so on. Other possible uses might be to indicate
which compound system emitted the particles, or to distinguish between the
neutron from the (n,np) channel and that from the (n,pn) channel. The
exact meaning assigned to LIP should be explained in the File 1, MT=451
comments.
LAW Flag to distinguish between different representations of the distribution function, fi :
124

CHAPTER 6. FILE 6: PRODUCT ENERGY-ANGLE DISTRIBUTIONS
LAW=0,
LAW=1,
LAW=2,
LAW=3,
LAW=4,
LAW=5,
LAW=6,
LAW=7,

unknown distribution;
continuum energy-angle distribution;
two-body reaction angular distribution;
isotropic two-body distribution;
recoil distribution of a two-body reaction;
charged-particle elastic scattering;
n-body phase-space distribution; and
laboratory angle-energy law.

NR, NP, Eint Standard TAB1 interpolation parameters.
A section in File 6 has the following form:
[MAT, 6, MT/ ZA, AWR,
0, LCT, NK,
0]HEAD
[MAT, 6, MT/ ZAP, AWP, LIP, LAW, NR, NP/ Eint / yi (E)]TAB1

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

---------------------------[MAT, 6, MT/ 0.0, 0.0,
0,
0,
0,
0]SEND
File 6 should have a subsection for every product of the reaction or sum of reactions being
described except for MT = 3, 4, 103-107 when they are being used to represent lumped
photons. An exception to this ordering is made when capture primary photons are being
described. Then the ordering is (1) angular distribution of primary photon to ground state
(LAW=2), (2) corresponding recoil (LAW=4), (3) angular distribuiton of primary photon
to first excited state, (4) corresponding recoil, (5) energy distribution of cascade photons
from first excited state (LAW=1 delta function), and so on, until all primary photons have
been described. The subsections are arranged in the following order:
1. particles (n, p, d, etc.) in order of ZAP and LIP,
2. residual nuclei and isomers in order of ZAP and LIP,
3. photons, and
4. electrons.
The contents of the subsection for each LAW are described below.

6.2.1

Unknown Distribution (LAW=0)

This law simply identifies a product without specifying a distribution. It can be used to
give production yields for particles, isomers, radioactive nuclei, or other interesting nuclei
in materials that are not important for particle transport, heating, or radiation damage
calculations. No law-dependent structure is given.

125

CHAPTER 6. FILE 6: PRODUCT ENERGY-ANGLE DISTRIBUTIONS

6.2.2

Continuum Energy-Angle Distributions (LAW=1)

This law is used to describe particles emitted in multi-body reactions or combinations of
several reactions, such as scattering through a range of levels or reactions at high energies
where many channels are normally open. For isotropic reactions, it is very similar to File 5,
LF=1 except for a special option to represent sharp peaks as ”delta functions” and the use
of LIST instead of TAB1 record. The following quantities are defined for LAW=1:
LANG Indicator which selects the angular representation to be used; if
LANG=1, Legendre coefficients are used,
LANG=2, Kalbach-Mann systematics are used,
LANG=11-15, a tabulated angular distribution is given using NA/2 cosines
and the interpolation scheme specified by LANG-10 (for example, LANG=12
selects linear-linear interpolation).
LEP Interpolation scheme for secondary energy;
LEP=1 for histogram,
LEP=2 for linear-linear, etc.
NR, NE, Eint Standard TAB2 interpolation parameters.
INT=1 is allowed (the upper limit is implied by File 3),
INT=12-15 is allowed for corresponding-point interpolation,
INT=21-25 is allowed for unit base interpolation.
NW Total number of words in the LIST record; NW = NEP (NA+2).
NEP Number of secondary energy points in the distribution.
ND Number of discrete energies given.
The first ND≥0 entries in the list of NEP energies are discrete, and the
remaining (NEP-ND)≥ 0 entries are to be used with LEP to describe a
continuous distribution. Discrete primary photons should be flagged with
negative energies.
NA Number of angular parameters.
Use NA=0 for isotropic distributions (note that all options are identical if
NA=0).
Use NA=1 or 2 with LANG=2 (Kalbach-Mann).
The structure of a subsection is:
[MAT, 6, MT/ 0.0,
0.0, LANG, LEP,
NR,
NE/ Eint ]TAB2
[MAT, 6, MT/ 0.0,
E1,
ND,
NA,
NW, NEP/
E1′ ,
b0 (E1 , E1′ ),
b1 (E1 , E1′ ), -------- bNA (E1 , E1′ ),
E2′ ,
b0 (E1 , E2′ ),
b1 (E1 , E2′ ), -------- bNA (E1 , E2′ ),
-------------------------------------------′
′
′
′
EN
b1 (E1 , ENEP
), ---- bNA (E1 , ENEP
)]LIST
EP , b0 (E1 , ENEP ),

126

CHAPTER 6. FILE 6: PRODUCT ENERGY-ANGLE DISTRIBUTIONS
where the contents of the bi depend on LANG.
The angular part of fi can be represented in several different ways (denoted by LANG).
6.2.2.1

Legendre Coefficients Representation (LANG=1)

Legendre coefficients are used as follows:
′

fl (µ, E, E ) =

NA
X
2l + 1
l=0

2

fl (E, E ′ ) Pl (µ)

(6.3)

where NA is the number of angular parameters, and the other parameters have their previous
meanings. Note that these coefficients are not normalized like those for discrete two-body
scattering (LAW=2); by definition, f0 (E, E ′ ) gives the total probability of scattering from
E to E ′ integrated over all angles. This is equivalent to the function g(E, E ′ ) that would
normally be given in File 5. The Legendre coefficients are stored with f0 in b0 , f1 in b1 , etc.
6.2.2.2

Kalbach-Mann Systematics Representation (LANG=2)

The angular distributions are represented by using the Kalbach-Mann systematics [Ref.1]
in the extended form developed by Kalbach [Ref.2], hereinafter referred to as KA88. The
distribution is given in terms of the parameters r and a, which are described below. If
NA=1, the parameter r is given and a is calculated. If NA=2, then both parameters r and
a are given explicitly.
The formulation addresses reactions of the form
A + a → C → B + b,
where:
A is the target,
a is the incident projectile,
C is the compound nucleus,
b is the emitted particle,
B is the residual nucleus.
The following quantities are defined:
Ea energy of the incident projectile a in the laboratory system
ǫa entrance channel energy, the kinetic energy of the incident projectile a and
the target particle A in the center-of-mass system, defined by:
ε a = Ea ×
127

AWRA
AWRA + AWRa

CHAPTER 6. FILE 6: PRODUCT ENERGY-ANGLE DISTRIBUTIONS
Eb energy of the emitted particle in the laboratory system
ǫb emission channel energy, the kinetic energy of the emission particle b and the
residual nucleus B in the center-of-mass system, defined by:
εb = Eb ×

AWRB
AWRB + AWRb

µb cosine of the scattering angle of the emitted particle b in the center-of-mass
system
It is required that LCT=2 with LANG=2.
The KA88 distribution is represented by:


a(Ea , Eb )f0 (Ea , Eb )
f (µb , Ea , Eb ) =
cosh (a(Ea , Eb )µb ) + r(Ea , Eb ) sinh (a(Ea , Eb )µb )
2 sinh (a(Ea , Eb ))
(6.4)
where r(Ea , Eb ) is the pre-compound fraction as given by the evaluator and a(Ea , Eb ) is a
simple parameterized function that depends mostly on the center-of-mass emission energy
Eb , but also depends slightly on particle type and the incident energy at higher values of
Ea .
The center-of-mass energies and angles Eb and µb are transformed into the laboratory
system using the expressions:

Eb,lab
µb,lab

√
√
AWRa AWRb p
AWRa AWRb
= Eb,cm +
Ea,lab + 2
Ea,lab Eb,cm µb,cm
(AWRA + AWRa )2
AWRA + AWRa
s
s
√
√
AWRa AWRb Ea,lab
Eb,cm
µb,cm +
(6.5)
=
Eb,lab
AWRA + AWRa Eb,lab

The pre-compound fraction r, where r ranges from 0.0 to 1.0, is usually computed by a
model code, although it can be chosen to fit experimental data.
The formula1 for calculating the slope value a(Ea , Eb ) is:
a(Ea , Eb ) = C1 X1 + C2 X13 + C3 Ma mb X34
where:
ea = ǫa + Sa
R1 = minimum(ea , Et1 )
X1 = R1 eb /ea
The parameter values for light
1
2

eb = ǫb + Sb
R3 = minimum(ea , Et3 )
X3 = R3 eb /ea
particle induced reactions as given in KA882 are:

Equation 10 of Ref. 2.
Table V of ref. 2.

128

CHAPTER 6. FILE 6: PRODUCT ENERGY-ANGLE DISTRIBUTIONS
C2 = 1.8×10−6 /(MeV)3

C1 = 0.04/MeV
C3 = 6.7×10−7 /(MeV)4
Et1 = 130 MeV
Mn = 1
Md = 1
mn = 1/2
md = 1
m3He = 1

Et3 = 41 MeV
Mp = 1
Mα = 0
mp = 1
mt = 1
mα = 2

Sa and Sb are the separation energies for the incident and emitted particles, respectively,
neglecting pairing and other effects for the reaction A + a → C → B + b. The formulae for
the separation energies3 in MeV are:

Sa


(NC − ZC )2 (NA − ZA )2
−
= 15.68 [AC − AA ] − 28.07
AC
AA
"
#
i
h
(NC − ZC )2 (NA − ZA )2
2/3
2/3
+ 33.22
− 18.56 AC − AA
−
4/3
4/3
AC
AA
"
#

 2
ZC2
ZA2
ZA2
ZC
− Ia
−
− 0.717
− 1/3 + 1.211
1/3
AC
AA
AC
AA


and

Sb = 15.68 [AC
h
2/3
− 18.56 AC
− 0.717

"

ZC2

1/3

AC


(NC − ZC )2 (NB − ZB )2
−
− AB ] − 28.07
AC
AB
"
#
i
2
2
(N
−
Z
)
(N
−
Z
)
C
C
B
B
2/3
− AB + 33.22
−
4/3
4/3
AC
AB
#

 2
ZB2
ZB2
ZC
− Ib
−
− 1/3 + 1.211
AC
AB
AB


where:
A, B, C subscripts refer to the target nucleus, the residual nucleus, and the compound
nucleus, as before,
N, Z, A are the neutron, proton, and mass numbers of the nuclei,
Ia , Ib are the energies required to separate the incident and emitted particles into
their constituent nucleons (see Appendix H for values used for given particles).
3

Equation 4 of Ref. 2.

129

CHAPTER 6. FILE 6: PRODUCT ENERGY-ANGLE DISTRIBUTIONS
The parameter f0 (Ea , Eb ) has the same meaning as f0 in equation (6.3); that is, the
total emission probability for this Ea and Eb . The number of angular parameters (NA) for
LANG=2 can be:
NA=1 in which case f0 and r are stored in the positions b0 and b1 , respectively, or
NA=2 where f0 , r and a are stored in the positions of b0 , b1 and b2 , respectively.
This formulation uses a single-particle-emission concept; it is assumed that each and
every secondary particle is emitted from the original compound nucleus C. When the incident
projectile a, and the emitted particle b, are the same, Sa = Sb , regardless of the reaction.
For incident projectile z, if neutrons emitted from the compound nucleus C are detected,
there will be one and only one Sb appropriate for all reactions, for example, (z,nα), (z,n3α),
(z,2nα), (z,np), (z,2n2α), and (z,nt2α). Furthermore, if the incident projectile is a neutron
(z=n in previous examples), then Sa = Sb in all cases; even for neutrons emitted in neutroninduced reactions, Sa and Sb will be identical.
6.2.2.3

Tabulated Function Representation (LANG=11-15)

For LANG=11-15, a tabulated function is given for f (µ) using the interpolation scheme
defined by (LANG - 10). For example, if LANG=12, use linear-linear interpolation (do not
use log interpolation with the cosine). The cosine grid of NA/2 values, µi , must span the
entire angular range open to the particle for E and E ′ , and the integral of f (µ) over all
angles must give the total emission probability for this E and E ′ (that is, it must equal f0 ,
as defined above). The value of f below µ1 or above µNA/2 is zero.
The tabulation is stored in the angular parameters as follows:
b0 = f0 ,
b1 = µ 1 ,
b2 = 0.5 f1 (µ1 )/f0 ,
b3 = µ 2 ,
...
...
bNA = 0.5 fNA/2 (µNA/2 )/f0 .
The preferred values for NA are 4, 10, 16, 22, etc.
In order to provide a good representation of sharp peaks, LAW=1 allows for a superposition of a continuum and a set of delta functions. These discrete lines could be used to
represent particle excitations in the CM frame because the method of corresponding points
can be used to supply the correct energy dependence. However, the use of LAW=2 together with MT=50-90, 600-650, etc., is preferred. This option is also useful when photon
production is given in File 6.
130

CHAPTER 6. FILE 6: PRODUCT ENERGY-ANGLE DISTRIBUTIONS

6.2.3

Discrete Two-Body Scattering (LAW=2)

This law is used to describe the distribution in energy and angle of particles described
by two-body kinematics. It is very similar to File 4, except its use in File 6 allows the
concurrent description of the emission of positrons, electrons, photons, neutrons, charged
particles, residual nuclei, and isomers. Since the energy of a particle emitted with a particular
scattering cosine µ is determined by kinematics, it is only necessary to give:

pi (µ, E) =

Z

dE ′ fi (µ, E, E ′ )
NL

1 X 2l + 1
=
+
al (E) Pl (µ)
2 l+1 2

(6.6)

,
where the Pl are the Legendre polynomials with the maximum order NL. Note that the
angular distribution pi is normalized. The following quantities are defined for LAW=2:
LANG flag that indicates the representation:
LANG=0, Legendre expansion;
LANG=12, tabulation with pi (µ) linear in µ;
LANG=14, tabulation with log(pi ) linear in µ.
NR, NE, Eint standard TAB2 interpolation parameters.
NL for LANG=0, NL is the highest Legendre order used;
for LANG>0, NL is the number of cosines tabulated.
NW number of parameters given in the LIST record:
for LANG=0, NW=NL;
for LANG>0, NW=2*NL.
Al for LANG=0, the Legendre coefficients,
for LANG>0, the, (µ, pi ) pairs for the tabulated angular distribution

The format for a subsection with LAW=2 is:
[MAT, 6, MT/ 0.0, 0.0,
0, 0, NR, NE/ Eint ]TAB2
[MAT, 6, MT/ 0.0, E1 ,LANG, 0, NW, NL/ Al (E)]LIST
--------------------------
--------------------------Note that the Legendre expansion option LANG=0 is very similar to File 4, LTT=1. The
tabulated option LANG¿0 is similar to File 4, LTT=2, except that a LIST record is used
instead of TAB1. The kinematical equations require AWR and AWP from File 6 and QI
from File 3.
131

CHAPTER 6. FILE 6: PRODUCT ENERGY-ANGLE DISTRIBUTIONS
LAW=2 can be used in sections with MT=50-90, 600-648, 650-698, etc., only, and the
center-of-mass system must be used (LCT=2). In addition, LAW=2 can be used to give the
angular distributions for primary photons in sections with MT=102.

6.2.4

Isotropic Discrete Emission (LAW=3)

This law serves the same purpose as LAW=2, but the angular distribution is assumed to be
isotropic in the CM system for all incident energies. No LAW-dependent structure is given.
This option is similar to LI=1 in File 4. The energy of the emitted particle is completely
determined by AWR and AWP in this section and QI from File 3.

6.2.5

Discrete Two-Body Recoils (LAW=4)

If the recoil nucleus of a two-body reaction (e.g., (n,n’), (p,n), etc.) described using LAW=2
or 3 does not break up, its energy and angular distribution can be determined from the
kinematics. No LAW-dependent structure is given. If isomer production is possible, multiple
subsections with LAW=4 can be given to define the energy-dependent branching ratio for
the production of each excited nucleus. Finally, LAW=4 may be used to describe the recoil
nucleus after radiative capture (MT=102), with the understanding that photon momentum
at low energies must be treated approximately, or when detailed angular distributions are
given for primary photons.

6.2.6

Charged-Particle Elastic Scattering (LAW=5)

Elastic scattering of charged particles includes components from Coulomb scattering, nuclear
scattering, and the interference between them. The Coulomb scattering is represented by
the Rutherford formula and electronic screening is ignored. The following parameters are
defined.
σcd (µ, E) differential Coulomb scattering cross section the center-of-mass system for
distinguishable particles (barns/sr)
σci (µ, E) cross section the center-of-mass system for identical particles (barns/sr)
E energy of the incident particle in the laboratory system (eV)
µ cosine of the scattering angle in the center-of-mass system
m1 incident particle mass (amu)
Z1 , Z2 charge numbers of the incident particle and target, respectively
s spin (identical particles only, s = 0, 21 , 1, 32 , etc.)
A target/projectile mass ratio
k particle wave number (barns−1/2 )
132

CHAPTER 6. FILE 6: PRODUCT ENERGY-ANGLE DISTRIBUTIONS
η dimensionless Coulomb parameter
u, c, h̄, α fundamental constants (see Appendix H).
The cross sections can then be written:
η2
k 2 (1 − µ)2



2η 2
1+µ
1 + µ2 (−1)2s
σci (µ, E) = 2
+
cos η ln
k (1 − µ2 ) 1 − µ2
2s + 1
1−µ
r
2 u
A
where
k =
m1 E × 10−14
1 + A h̄2 c2
r
α 2 u m1
η = Z 1 Z2
2 E
σcd (µ, E) =

(6.7)
(6.8)
(6.9)
(6.10)

Note that A = 1 and Z1 = Z2 for identical particles.
The net elastic scattering cross section for distinguishable particles may be written as:
(
)
 NL

2η
1 − µ X 2l + 1
σed (µ, E) = σcd (µ, E) −
Re exp iη ln
al (E)Pl (µ)
1−µ
2
2
l=0
+

2NL
X
2l + 1
l=0

2

bl (E)Pl (µ)

(6.11)

and the cross section for identical particles is:
σei (µ, E) = σci (µ, E)

( NL 
)

 
X
2l
+
1
2η
(1 + µ) exp iη ln 1−µ
2 

Re
al (E)Pl (µ)
−
+(−1)l (1 − µ) exp iη ln 1+µ
1 − µ2
2
2
l=0
+

NL
X
4l + 1
l=0

2

bl (E)P2l (µ)

(6.12)

where the al are complex coefficients for expanding the trace of the nuclear scattering amplitude matrix and the bl are real coefficients for expanding the nuclear scattering cross section.
The value of NL represents the highest partial wave contributing to nuclear scattering. Note
that σei (−µ, E) = σei (µ, E).
The three terms in Equations (6.11) and (6.12) are Coulomb, interference, and nuclear
scattering, respectively. Since an integrated cross section is not defined for this representation, a value of 1.0 is used in File 3.
When only experimental data are available, it is convenient to remove the infinity due
to σC by subtraction and to remove the remaining infinity in the interference term by
multiplication, thereby obtaining the residual cross sections:
133

CHAPTER 6. FILE 6: PRODUCT ENERGY-ANGLE DISTRIBUTIONS

and



σRd (µ, E) = (1 − µ) σed (µ, E) − σcd (µ, E)



σRi (µ, E) = (1 − µ2 ) σei (µ, E) − σci (µ, E)

(6.13)
(6.14)

Then σR can be given as a Legendre polynomial expansion in the forms:
NL
X
2l + 1

σRd (µ, E) =

2

l=0

and
σRi (µ, E) =

NL
X
4l + 1
l=0

2

cld (E)Pl (µ)

(6.15)

cli (E)P2l (µ)

(6.16)

A cross section value of 1.0 is used in File 3.
Because the interference term oscillates as µ goes to 1, the limit of the Legendre representation of the residual cross section at small angles may not be well defined. However,
if the coefficients are chosen properly, the effect of this region will be small because the
Coulomb term is large.
It is also possible to represent experimental data using the ”nuclear plus interference”
cross section and angular distribution in the CM system defined by:
σN I (µ, E) =

µ
Zmax

µmin



σe (µ, E) − σc (µ, E) dµ

(6.17)

and
PN I (µ, E) =

σe (µ, E) − σc (µ, E)
,
σN I (E)
0
otherwise,

µmin ≤ µ ≤ µmax

(6.18)

where µmin = −1 for different particles and 0 for identical particles. The maximum cosine
should be as close to 1.0 as possible, especially at high energies where Coulomb scattering is
less important. The Coulomb cross section σc (µ, E) is to be computed using equations (6.7)
or (6.8) for different or identical particles, respectively. The angular distribution pN I is given
in File 6 as a tabulated function of µ, and σN I (E) in barns is given in File 3. The following
quantities are defined for LAW=5:
SPI Spin of the particle. Used for identical particles (SPI=0, 1/2, 1, etc.).
LIDP Indicates that the particles are identical when LIDP=1; otherwise, LIDP=0.
LTP Indicates the representation:
LTP=1 nuclear amplitude expansion, equations (6.11) and (6.12);
134

CHAPTER 6. FILE 6: PRODUCT ENERGY-ANGLE DISTRIBUTIONS
LTP=2 residual cross section expansion as Legendre coefficients, equations (6.13) through (6.16);
LTP=12 nuclear plus interference distribution with pN I linear in µ,
equations (6.17) and (6.18);
LTP=14 tabulation with ln(PN I ) linear in µ, equations (6.17)
and (6.18).
LTP=15 tabulation with PN I linear in µ, equations (6.17) and (6.18).
NR, NE, Eint Standard TAB2 parameters.
NL For LTP≤2, NL is the highest Legendre order of nuclear partial waves used;
For LTP>2, NL is the number of cosines tabulated.
NW Number of parameters given in the LIST record:
for LTP=1 and LIDP=0, NW=4*NL+3;
for LTP=1 and LIDP=1, NW=3*NL+3;
for LTP=2, NW=NL+1; and for LTP>2, NW=2*NL.
Ai (E) Coefficients (ai , bi , or ci as described below) in barns/sr or (µ, p) pairs, where
p is dimensionless.
A subsection for LAW=5 has the following form:
[MAT, 6, MT/ SPI, 0.0, LIDP, 0, NR, NE/ Eint ]TAB2
[MAT, 6, MT/ 0.0, E1 , LTP, 0, NW, NL/ A1 (E1 )]LIST
------------------------------
------------------------------The coefficients witin the LIST array Ai are organized as follows:
LTP=1 and LIDP=0,
b0 , b1 , ...b2N L , Ra0 , Ia0 , Ra1 , Ia1 , ...IaN L ;
LTP=1 and LIDP=1,
b0 , b1 , ...b2N L , Ra0 , Ia0 , Ra1 , Ia1 , ...IaN L ;
LTP=2,
c0 , c1 , ...

cN L

LTP>2,
µ1 , pN I (µ1 ), ...

µN L , pN I (µN L ).

135

CHAPTER 6. FILE 6: PRODUCT ENERGY-ANGLE DISTRIBUTIONS

6.2.7

N-Body Phase-Space Distributions (LAW=6)

In the absence of detailed information, it is often useful to use n-body phase-space distributions for the particles emitted from neutron and charged-particle reactions. These
distributions conserve energy and momentum, and they provide reasonable kinematic limits
for secondary energy and angle in the LAB system.
The phase-space distribution for particle i in the CM system is

where

√
PiCM (µ, E, E ′ ) = Cn E ′ (Eimax − E ′ )(3n/2)−4

(6.19)

Eimax is the maximum possible center-of-mass energy for particle i,
µ, E ′ are cosine of the scattering angle and outgoing particle energy in the CM
system, and
Cn are normalization constants:
4

C3 =
C4
C5

π(Eimax )2
105
=
32(Eimax )7/2
256
=
14π(Eimax )5

In the laboratory system, the distributions become
Pilab (µ, E, E ′ )

= Cn

√

E′

h

Eimax


i(3n/2)−4
√
∗
′
∗
′
− E + E − 2µ E E

(6.20)

where µ and E ′ are in the laboratory system and E ∗ is given by:
E∗ = E

Aincident
AWR + Aexit

and Aincident and Aexit are the mass ratios to the neutron mass of the incident and exit
particles, respectively. In the general case, the range of both E ′ and µ is limited by the
condition that the quantity in square brackets remains non-negative.
The value of Eimax is a fraction of the energy available in CM,
Eimax =

M − mi
Ea
M

where M is the total mass of the n particles being treated by this law. Note that M may
be less than the total mass of the products for reactions such as:
α + 9 Be → n + 3α
where the neutron can be treated as a two-body event and the alphas by a 3-body phasespace law. The parameter M =APSX is provided so that Eimax can be determined without
having to process the other subsections of this section.
136

CHAPTER 6. FILE 6: PRODUCT ENERGY-ANGLE DISTRIBUTIONS
The energy available in CM for one-step reactions is
mT
Ea =
E+Q
mp + mT
where
mT

is the target mass,

mp is the projectile mass,
E is the energy in the LAB system, and
Q is the reaction QI value from File 3.
For two-step reactions such as the one discussed above, Ea is just the recoil energy from the
first step. The following quantities are defined for LAW=6:
APSX total mass in neutron units of the n particles being treated by the law.
NPSX number of particles distributed according to the phase-space law.
Only a CONT record is given
[MAT, 6, MT/ APSX, 0.0,

6.2.8

0,

0,

0, NPSX]CONT

Laboratory Angle-Energy Law (LAW=7)

The continuum energy-angle representation (LAW=1) is good for nuclear model code results
and for experimental data that have been converted to Legendre coefficients. However, since
experiments normally give spectra at various fixed angles, some evaluators may prefer to
enter data sorted according to (E, µ, E ′ ), rather than the LAW=1 ordering (E, E ′ , µ). The
following quantities are defined for LAW=7:
NR, NE, Eint
NRM, NMU, µint
NRP, NEP, E’int

normal TAB2 interpolation parameters for incident energy, E.
normal TAB2 interpolation parameters for emission cosine, µ.
normal TAB1 interpolation parameters for secondary energy, E ′ .

The structure of a subsection is:
[MAT, 6, MT/ 0.0,
0.0,
0, 0,
NR, NE/Eint ]TAB2
[MAT, 6, MT/ 0.0,
E1 ,
0, 0, NRM, NMU/µint ]TAB2
[MAT, 6, MT/ 0.0,
µ1 ,
0, 0, NRP, NEP/E′int /
′
′
′
E1 ,f (µ1 , E1 , E1 ), E2 ,f (µ1 , E1 , E2′ ), ----′
′
------------------------- EN
EP ,f (µ1 , E1 , EN EP )]TAB1
-----------------------------------
------------------------------------
-----------------------------------The distribution f(µ, E, E ′ ) is defined by equation (6.1). Emission cosine and secondary
energy must be given in the laboratory system for LAW=7. Also, both variables must cover
the entire angle-energy range open to the emitted particle.
137

CHAPTER 6. FILE 6: PRODUCT ENERGY-ANGLE DISTRIBUTIONS

6.3

Procedures

File 6 and incident charged particles are new for ENDF-6, and it will take time for detailed
procedures to evolve. The following comments are intended to clarify some features of the
format.

6.3.1

Elastic Scattering

According to ENDF-6 format the neutron elastic scattering is represented by giving a cross
section in File 3, MT=2 (with resonance contributions in File 2) and angular distributions
in File 4, MT=2. This representation is compatible with previous versions of the ENDF
format.
Charged-particle elastic cross section is infinite due to the Coulomb contribution. For
formal reasons MT=2 in File 3 must be present. With LPT=1 and LPT=2 representations
the elastic cross section in File 3 is arbitrarily set to 1.0 at all energies. With LPT=12 and
LPT=14 representation, MT=2 in File 3 contains σN I (E) in barns. The elastic angular
distributions in File 6 use LAW=5.
Whenever possible, the nuclear amplitude expansion should be used with LAW=5. Note
that the ai and bi coefficients are not independent, being related by their mutual dependence
on the nuclear scattering amplitudes, which are themselves constrained by unitarity and
various conservation conditions. Thus, any attempt to fit data directly with expressions
(6.13) or (6.14) would under-determine the ai ’s and bi ’s, giving spurious values for them. The
only feasible procedure is to fit the experimental data in terms of a direct parametrization
of the nuclear scattering amplitudes (phase shifts, etc.) and extract the ai and bi coefficients
from them.
The second representation (LTP=2) can be used when an approximate direct fit to the
experimental data is desired. The simple pole approximation for the Coulomb amplitude implied by this representation becomes increasingly poor at lower energies and smaller angles.
Since the deficiencies of the approximation are masked by the dominance of the Rutherford
cross section in the same region, however, one could expect a reasonable representation of
the net scattering cross section at all energies and angles, provided that the coefficients
C1 are determined by fitting data excluding the angular region where the Rutherford cross
section is dominant.
Tabulated distributions (LTP=12 or 14) are also useful for direct fits to experimental
data. In this case, the choice of the cutoff cosine is used to indicate the angular region where
Rutherford scattering is dominant.
Figures 6.1 and 6.2 illustrate a typical cross section computed with amplitudes and the
corresponding residual cross-section representation.

6.3.2

Photons

Emitted photons are described using a subsection with ZAP=0. The spectrum is obtained
as a sum of discrete photons (delta functions) and a continuum distribution packed into one
LIST record. The discrete photons (if any) are given first. They are tabulated in order of
decreasing energy, and their energy range may overlap the continuum. The continuum (if
138

CHAPTER 6. FILE 6: PRODUCT ENERGY-ANGLE DISTRIBUTIONS

dσ(E)/dΩ (mb/sr)

500
3

400

He(p,el) - 10 MeV

300

200

100

0

0

30

60
90
120
c.m. Scattering Angle (θ )

150

180

Figure 6.1: Example of Charged-Particle Elastic Scattering Cross Section

dσ(E)/dΩ (mb/sr)

0.4
3

0.3

He(p,el) - 10 MeV

0.2

0.1

0.0

-0.1

0

30

60
90
120
c.m. Scattering Angle (θ )

150

180

Figure 6.2: Example of Residual Cross Section for Elastic Scattering

139

CHAPTER 6. FILE 6: PRODUCT ENERGY-ANGLE DISTRIBUTIONS
any) is given next, and the energies must be in increasing order. Corresponding-point or
unit-base interpolation is applied separately to the discrete and continuum segments of the
record. A separate angular distribution can be attached to each discrete photon or to each
energy of a distribution, but the isotropic form (NA=0) is usually adequate.
For a two-body discrete-level reaction, all the discrete photons produced by cascades
from the given level should be included under the same reaction (MT) so that the reaction
explicitly conserves energy. This scheme also gives simple energy-independent yields and
simple spectra. If the level structure is not known well enough to separate the contributions
to the intensity of a particular photon by reaction, the photons can be lumped together in
a summation MT with the restriction that energy be conserved for the sum of all reactions.
As an example, consider the typical level structure for the reaction A(i,p)R shown in
Figure 6.3. Assume that the secondary protons are described by discrete levels in MT=600603 and a continuum in MT=649. As many discrete photons as possible should be given with
their associated direct level. Thus, the production of photons arising from direct excitation
of the first level should be given in MT=601 (the yield will be 1.0 in the absence of internal
conversion).

649

603
γ2
602
γ1

γ3
601

Figure 6.3: Typical Level Structure for Proton-Induced Photon Production
Photons (γ1 , γ2 , and γ3 ) should be given in MT=602 using the simple constant yields
computed from the branching factors and conversion ratios. This process should be continued until the knowledge of the cascades begins to get fuzzy. All the remaining production
of γ1 , γ2 , γ3 , and all photons associated with higher levels (603 and 649 in this case), are
then given in the redundant MT=103 using energy-dependent yields and a combination of
discrete bins and a continuous distribution.
Photons produced during multi-body reactions should also be tabulated under the reaction MT number so that each reaction independently conserves energy when possible. If
necessary, the photons can be lumped together under the redundant MT=3 as long as energy
140

CHAPTER 6. FILE 6: PRODUCT ENERGY-ANGLE DISTRIBUTIONS
is conserved for the sum of all reactions.
A special represention of primary capture photons for light isotopes is allowed using
MT=102. The first subsection gives the angular distribution in the center of mass for the
primary photon to the ground state using LAW=2. The next subsection gives the two-body
recoil for this event using LAW=4. This is followed by a subsection giving the angular
distribution for the primary photon to the first excited state and a subsection giving the
corresponding recoil. The next subsection gives the energy spectrum of the gamma cascade
from de-exciting this level using LAW=1, NA=0, and ND=NEP discrete photon lines. This
pattern is continued until all the primary photons have been described. In each section
with LAW=2, ZA=0 and the AWP field contains the energy of the primary photon. The
laboratory energies and cosines for the primary gamma and its corresponding recoil particle
must be computed using relativistic kinematics as described in Appendix E.

6.3.3

Particles

Isotropic or low-order distributions are often sufficient for the charged particles emitted
in continuum reactions because of their short ranges. The angular distributions of emitted
neutrons may be needed in more detail because of their importance in shielding calculations.
Note that the angular distributions of identical particles must be symmetric in the CM
system. This is true whether the identical particles are in the entrance channel or the exit
channel. Symmetry is enforced by setting all odd Legendre components to zero, or by making
pi (µ) = pi (−µ).

6.3.4

Neutron Emission

It is important to represent the spectrum of emitted neutrons as realistically as possible
due to their importance for shielding, activation, and fission. Small emission probabilities
for low-energy neutrons may acquire increased importance due to the large cross sections
at low energies. However, many modern evaluations are done with nuclear model codes
that represent emission in energy bins (that is, histograms). Direct use of such calculations
would severely distort the effects of emitted neutrons (although the representation would
be reasonable for emitted charged particles). In such cases, the evaluator should fit a realistic evaporation shape to his low-energy neutron emission and use this shape to generate
additional points for the energy distribution.

6.3.5

Recoil Distributions

The energy distribution of the recoil nucleus is needed to compute radiation damage and
should be provided for structural materials whenever possible. Nuclear heating depends
on the average recoil energy, and an average or full distribution should be provided for all
isotopes that are used in reasonable concentrations in the common applications. All recoil
information can be omitted for minor isotopes that only affect activation. Recoil angular
distributions are rarely needed. Particle, photon, and recoil distributions taken together
should conserve energy.

141

CHAPTER 6. FILE 6: PRODUCT ENERGY-ANGLE DISTRIBUTIONS
To enter only recoil average energy, use NEP=1 and ND=1; the recoil spectrum becomes
a delta function at the average energy. Average energy must be entered even though it can
be computed from the other distributions.
At high energies, it becomes difficult to represent the recoil distribution in the center-ofmass frame, even when that frame remains appropriate for the distribution of light particles
(A≤ 4). In such cases, use the LCT=3 option4 , giving the light particles in the CM frame
(normally with Kalbach systematics), and the heavy recoil energy spectrum as isotropic in
the laboratory frame. Such isotropic distributions are normally adequate for representing
energy deposition and damage.

6.3.6

Elements as Targets

Targets which are elements can be represented by using a ZA with A=0 as usual. An attempt
should be made to tabulate every product of a reaction on that element.
As an example, nat Fe(n,2n) will produce 53 Fe and 54 Fe in addition to more 56 Fe and 57 Fe.
The product yields in File 6 can be converted to production or activation cross sections for
each of these species without having recourse to isotopic evaluations. However, it would
be difficult to give a recoil spectrum for each of these nuclei in full detail. Therefore, the
evaluator is allowed to give a single total recoil spectrum with ZAP=ZA, where A=0. The
yield should be 1.0 and AWP should be an appropriate average recoil mass.

6.3.7

CM versus LAB Coordinate System

Some energy-angle distributions show relatively sharp features (from levels) superimposed
on a smoother continuum (see Figure 6.4). If such a distribution is given in the CM system,
the position of these peaks in E ′ is independent of the scattering angle as shown in Equation (C.5). This helps assure that the angular distribution given for each E and E ′ will be
fairly simple. Furthermore, the E ′ for a given sharp peak is a linear function of incident
energy E, thus the corresponding-point or unit-base interpolation schemes can be set up to
follow the peak exactly. Sharp lines can be represented as delta functions as described for
photons, or a more realistic width and shape can be given in tabulated form.
Therefore, the CM system should be used for representing secondary energy and scattering angle whenever relatively sharp features are found, even for reactions with three or more
particles in the final state. The transformation to LAB coordinates is made by doing vector
sums of the emitted particle CM velocities and the LAB velocity of the center of mass of
the initial colliding system.
Experimental data are usually provided at fixed angles in the LAB system. It may often
be difficult to convert the data to constant energies in the CM system as recommended
by this format. However, transport calculations require data for the full range of angles
and energies, and full ranges are required to get accurate values of the integrated cross
sections from the experimental distributions. The most accurate way to do this process of
interpolation and extrapolation is probably to model the distribution in the CM and adjust
4

LCT=3 is only appropriate for LAW=1 .

142

CHAPTER 6. FILE 6: PRODUCT ENERGY-ANGLE DISTRIBUTIONS

Figure 6.4: Typical Energy Spectrum Showing Levels on a Continuum
it to represent the LAB data. The numbers recommended for this format automatically
arise from this process.

6.3.8

Phase Space

Comparison of experimental data with a phase-space prediction will often show overall
qualitative agreement except for several broad or narrow peaks. It is desirable to represent
those peaks using LAW=2 or 3. The remainder may be small enough to represent reasonably
well with one of the phase-space laws.
In the absence of complete experimental data, it is recommended that the evaluator
supply a phase-space distribution. This assures that energy will be conserved and gives
reasonable kinematic limits on energy and angle in the laboratory system. Later comparisons
between the evaluation and data may indicate possible improvements in the evaluation.

References for Chapter 6
1. C. Kalbach and F. M. Mann, Phenomenology of continuous angular distributions. I.
Systematics and parametrization, Phys. Rev.C 23 (1981) 112
2. C. Kalbach, Systematics of Continuum Angular Distributions: Extensions to Higher
Energies, Phys. Rev. C 37 (1988) 2350

143

Chapter 7
File 7: THERMAL NEUTRON
SCATTERING LAW DATA
7.1

General Description

File 7 contains neutron scattering data for the thermal neutron energy range (E < 5 eV)
for moderating materials. Sections are provided for elastic (MT=2) and inelastic (MT=4)
scattering. In the ENDF-6 formats, File 7 is complete in itself, and Files 3 and 4 are no
longer required to obtain the total scattering cross section in the thermal energy range.

7.2

Coherent Elastic Scattering

The coherent elastic scattering from a powdered crystalline material may be represented as
follows:

where:

Ei 
-------------------------[MAT, 7, 0/0.0, 0.0,
0, 0, 0, 0]
1

HEAD
LTHR=1
Eint / S(E,T0 ) ] TAB1
S(Ei ,T1 ) ] LIST

SEND

As an example, the HEXSCAT code [Ref. 1] can be used for hexagonal crystal lattices.

145

CHAPTER 7. FILE 7: THERMAL NEUTRON SCATTERING LAW DATA

7.2.2

Procedures for Coherent Elastic Scattering

The coherent elastic scattering cross section is easily computed from S(E, T ) by reconstructing an appropriate energy grid and dividing S by E at each point on the grid. A discontinuity
should be supplied at each Ei , and log-log interpolation should be used between Bragg edges.
The cross section is zero below the first Bragg edge.
The function S(E, T ) should be defined up to 5 eV. When the Bragg edges get very close
to each other (above 1 eV), the ”stair steps” are small. It is permissible to group edges
together in this region in order to reduce the number of steps given while still preserving
the average value of the cross section. Either discrete angle or Legendre representations of
the angular dependence of coherent elastic scattering can be constructed. It is necessary to
recover the values of si (T ) from S(E, T ) by subtraction.

7.3

Incoherent Elastic Scattering

Elastic scattering can be treated in the incoherent approximation for partially ordered systems such as ZrHx and polyethylene. The differential cross section is given by:
d2 σ
σb −2EW ′ (T )(1−µ)
e
(E → E ′ , µ, T ) =
δ(E − E ′ )
′
dE dΩ
4π

(7.4)

where:
σb is the characteristic bound cross section (barns),
W ′ is the DebyeWaller integral divided by the atomic mass (eV−1 ),
and all the other symbols have their previous meanings. The integrated cross section is
easily obtained:

′
σb 1 − e−4EW
σ(E) =
(7.5)
2
2EW ′
Note that the limit of σ for small E is σb .

7.3.1

Format for Incoherent Elastic Scattering

The parameters for incoherent elastic scattering are also given in a section of File 7 with
MT=2, because coherent and incoherent representations never occur together for a material.
The following quantities are defined:
ZA, AWR Standard charge and mass parameters.
LTHR Flag indicating which type of thermal data is being represented.
LTHR=2 for incoherent elastic scattering.
NP Number of temperatures.
SB characteristic bound cross section (barns)
146

CHAPTER 7. FILE 7: THERMAL NEUTRON SCATTERING LAW DATA
W ′ (T ) Debye-Waller integral divided by the atomic mass (eV−1 ) as a function of
temperature (K).
The structure of a section is
[MAT, 7, 2/ ZA, AWR, LTHR, 0, 0, 0]HEAD
(LTHR=2)
[MAT, 7, 2/ SB, 0.0,
0, 0, NR, NP/ Tint / W ′ (T ) ]TAB1
[MAT, 7, 0/ 0.0, 0.0,
0, 0, 0, 0] SEND

7.3.2

Procedures for Incoherent Elastic Scattering

This formalism can be used for energies up to 5 eV.
For some moderator materials containing more than one kind of atom, the incoherent
elastic cross section is computed as the sum of contributions from two different materials.
As an example, H in ZrHx is given in MAT 0007, and Zr in ZrHx is given in MAT 0058.

7.4

Incoherent Inelastic Scattering

Inelastic scattering is represented by the thermal neutron scattering law, S(α, β, T ), and is
defined for a moderating molecule or crystal by:
NS

X Mn σbn
d2 σ
′
(E
→
E
,
µ,
T
)
=
dΩ dE ′
4πkT
n=0

r

E ′ −β/2
e
Sn (α, β, T )
E

(7.6)

where (NS+1) types of atoms occur in the molecule or unit cell (i.e., for H2 O, NS=1) and
Mn Number of atoms of n-type in the molecule or unit cell
T Moderator temperature (K)
E Incident neutron energy (eV)
E ′ Secondary neutron energy (eV)
β Energy transfer, β = (E ′ − E)/kT
i
h
√
α Momentum transfer, α = E ′ + E − 2µ EE ′ /A0 kT

An Mass of the nth type atom

A0 Mass of the principal scattering atom in the molecule
σf n Free atom scattering cross section of the nth type atom
σbn = σf n

147



An + 1
An

2

(7.7)

CHAPTER 7. FILE 7: THERMAL NEUTRON SCATTERING LAW DATA
k Boltzmann’s constant (see Appendix H)
µ Cosine of the scattering angle (in the lab system)
The data in File 7 for any particular material contain only the scattering law for the
principal scatterer, S(α, β, T ), i.e., the 0th atom in the molecule. These data are given
as an arbitrary tabulated function. The scattering properties for the other atom types
(n=1,2,...,NS) are represented by analytical functions. Note that the scattering properties of all atoms in the molecule may be represented by analytical functions. In this case
there is no principal scattering atom.
In some cases, the scattering properties of other atom types in a molecule or crystal may
be described by giving S0 (α, β, T ) in another material. As an example, H in ZrHx and Zr in
ZrHx are given in separate MATs.
For high incident energies, α and/or β values may be required that are outside the ranges
tabulated for S(α, β). In these cases, the short-collision-time (SCT) approximation should
be used as follows:
h
i
2
T
|β|
exp − (α−|β|)
−
4αTeff (T )
2
q
S SCT (α, β, T ) =
(7.8)
Teff (T )
4πα T
where Tef f (T ) is the effective temperature, and the other symbols have their previous meanings.
The constants required for the scattering law data and the analytic representations for the
non-principal scattering atoms are given in an array, B(N), N=1,2,...,NI, where NI=6(NS+1).
Six constants are required for each atom type (one 80-character record). The first six
elements pertain to the principal scattering atom, n=0.
The elements of the array B(N) are defined as:
B(1) M0 f0 , the total free atom cross section for the principal scattering atom.
If B(1) = 0.0, there is no principal scattering atom and the scattering properties for this material are completely described by the analytic functions for
each atom type in this material.
B(2) ǫ, the value of E/kT above which the static model of elastic scattering is
adequate (total scattering properties may be obtained from MT=2 as given
in File 2 or 3 and File 4 of the appropriate materials).
B(3) A0 , the ratio of the mass of the atom to that of the neutron that was used
to compute α.
√
E ′ + E − 2µ EE ′
α=
A0 kT
B(4) Emax , the upper energy limit for the constant σf 0 (upper energy limit in
which S0 (α, β, T ) may be used).
B(5) not used.
148

CHAPTER 7. FILE 7: THERMAL NEUTRON SCATTERING LAW DATA
B(6) M0 , the number of principal scattering atoms in the material. (For example,
M = 2 for H in H2 O).
The next six constants specify the analytic functions that describe the scattering properties of the first non-principal scattering atom, (n=1); i.e., for H2 0, this atom would be
oxygen if the principal atom were hydrogen.
B(7) a1 , a
a1 =
a1 =
a1 =

test indicating the type of analytic function used for this atom type.
0.0, use the atom in SCT approximation only (see below).
1.0, use a free gas scattering law.
2.0, use a diffusive motion scattering law.

B(8) M1 σf 1 , the total free atom cross section for this atom type.
B(9)

A1 , effective mass for this atom type.

B(10) not used.
B(11) not used.
B(12) M1 , the number of atoms of this type in the molecule or unit cell.
The next six constants, B(13) through B(18), are used to describe the second nonprincipal scattering atom (n=2), if required. The constants are defined in the same way as
for n=1; e.g., B(13) is the same type of constant as B(7).
A mixed S(α, β) method has sometimes been used. Using BeO as an example, the
S(α, β) for Be in BeO is combined with that for O in BeO and adjusted to the Be free atom
cross section and mass as a reference. The mixed S(α, β) is used for the principal atom in
equation (7.6) as if NS were zero. However, all of the NS+1 atoms are used in the SCT
contribution to the cross section.
The scattering law is given by S(α, β, T ) for a series of β values. For each β value, the
function versus β is given for a series of temperatures. Thus, the looping order is actually
first β, then T , then α. S(α, β) is normally a symmetric function of β and only positive
values are given. For ortho and parahydrogen and deuterium, this is no longer true. Both
negative and positive values must be given in increasing value of β and the flag LASYM is
set to one.
In certain cases, a more accurate temperature representation may be obtained by replacing the value of the actual temperature, T , that is used in the definition of α and β with
a constant, T0 (T0 = 0.0253 eV or the equivalent depending on the units of Boltzmann’s
constant). A flag (LAT) is given for each material to indicate which temperature has been
used in generating the S(α, β) data.
For down-scattering events with large energy losses and for low temperatures, β can be
large and negative. The main contribution to the cross section comes from the region near
α+β = 0. Computer precision can become a real problem in these cases. As an example, for
water at room temperature, calculations using equation (7.6) for incident neutrons at 4 eV
149

CHAPTER 7. FILE 7: THERMAL NEUTRON SCATTERING LAW DATA
require working with products like e80 × 10−34 . For liquid hydrogen at 20 Kelvin and for
1 eV transfers, the products can be e300 × 10−130 . These very large and small numbers are
difficult to handle on most computers, especially 32-bit machines. The LLN flag is provided
for such cases: the evaluator simply stores ln S instead of S and changes the interpolation
scheme accordingly (that is, the normal log-log law changes to log-lin). Values of S = 0.0
like those found in the existing ENDF/B-III thermal files really stand for some very small
number less than 10−32 and should be changed to some large negative value, such as -999.

7.4.1

Formats for Incoherent Inelastic Scattering

The parameters for incoherent inelastic scattering are given in a section of File 7 with MT=4.
The following quantities are defined:
LAT Flag indicating which temperature has been used to compute α and β
LAT=0, the actual temperature has been used.
LAT=1, the constant T0 = 0.0253 eV has been used.
LASYM Flag indicating whether an asymmetric S(α, β) is given
LASYM=0, S is symmetric.
LASYM=1, S is asymmetric
LLN Flag indicating the form of S(α, β) stored in the file
LLN=0, S is stored directly.
LLN=1, ln(S) is stored.
NS Number of non-principal scattering atom types. For most moderating materials there will be (NS+1) types of atoms in the molecule (NS≤ 3).
NI Total number of items in the B(N ) list. NI=6(NS+1).
B(N ) List of constants. Definitions are given above (Section 7.4).
NR Number of interpolation ranges for a particular parameter, either β or α .
LT Temperature dependence flag. The data for the first temperature are given
in a TAB1 record, and the data for the LT subsequent temperatures are given
in LIST records using the same α grid as for the first temperature.
LI Interpolation law to be used between this and the previous temperature.
Values of LI are the same as those specified for INT in a standard TAB1
interpolation table.
NT Total number of temperatures given. Note that NT = LT+1.
Tef f 0 Table of effective temperatures (K) for the shortcollision-time approximation
given as a function of moderator temperature T (K) for the principal atom.

150

CHAPTER 7. FILE 7: THERMAL NEUTRON SCATTERING LAW DATA
Tef f 1 ,Tef f 2 , Tef f 3 Table for effective temperatures for the first, second, and third nonprincipal atom. Given if a1 = 0.0 only.
NB Total number of β values given.
NP Number of α values given for each value of β for the first temperature described, NP is the number of pairs, α and S(α, β), given.
βint , αint Interpolation schemes used.
The structure of a section is
[MAT, 7, 4 / ZA, AWR,
0, LAT, LASYM, 0]HEAD
[MAT, 7, 4 / 0.0, 0.0, LLN,
0,
NI, NS/B(N) ] LIST
[MAT, 7, 4 / 0.0, 0.0,
0,
0,
NR, NB/βint ] TAB2
[MAT, 7, 4 / T0 , β1 , LT,
0,
NR, NP/ αint / S(α, β1 ,T0 ) ] TAB1
[MAT, 7, 4 / T1 , β1 , LI,
0,
NP, 0/ S(α, β1 , T1 ) ] LIST
-----------------------------
-----------------------------[MAT, 7, 4 / T0 , β 2 , LT, 0, NR, NP/ αint / S(α,β 2 ,T0 ) ] TAB1
-----------------------------
-----------------------------[MAT, 7, 4 / 0.0, 0.0, 0, 0, NR, NT/ Tint / Tef f 0 (T) ] TAB1
-----------------------------
-----------------------------[MAT, 7, 0 / 0.0, 0.0, 0, 0, 0, 0] SEND
If the scattering law data are completely specified by analytic functions (no principal scattering atom type, as indicated by B(1)=0), tabulated values of S(α, β) are omitted and the
TAB2 and TAB1 records are not given.

7.4.2

Procedures for Incoherent Inelastic Scattering

The data in MF=7, MT=4 should be sufficient to describe incoherent inelastic scattering
for incident neutron energies up to 5 eV. The tabulated S(α, β) function should be useful to
energies as high as possible in order to minimize the discontinuities that occur when changing
to the short-collisiontime approximation. The β mesh for S(α, β) should be selected in such
a manner as to accurately represent the scattering properties of the material with a minimum
of β points. The α mesh at which S(α, β) is given should be the same for each β value and
for each temperature.

151

CHAPTER 7. FILE 7: THERMAL NEUTRON SCATTERING LAW DATA
Experience has shown that temperature interpolation of S(α, β) is unreliable. It is recommended that cross sections be computed for the given moderator temperatures only. Data
for other temperatures should be obtained by interpolation between the cross sections.

References for Chapter 7
1. Y.D. Naliboff and J.V. Koppel, HEXSCAT:Coherent Scattering of Neutrons by Hexagonal Lattices, General Atomic report GA6026 (1964).

152

Chapter 8
File 8: RADIOACTIVE DECAY
AND FISSION PRODUCT YIELD
DATA
8.1

General Description

Information concerning the decay of the reaction products (any MT) is given in this file. In
addition, fission product yield data (MT=454 and 459) for fissionable materials (see Section 8.3) and spontaneous radioactive decay data (MT=457) for the nucleus (see Section 8.4)
are included. See descriptions of File 9 and File 10 for information on isomeric state production from the various reactions. Since a reaction may result in more than one unstable end
product, data for the most important product should be entered, while others are allowed.

8.2

Radioactive Nuclide Production

For any isotope, sections for reactions defined by the MT number may be given, which specify
that the end product from the interaction of any incident particle or photon is radioactive.
The end-products of the reaction are identified by the ZAP (ZA for the product), and noting
how these end products decay. A section will contain only minimal information about the
chain that follows each reaction. One or more isomeric states of the target or the radioactive
end product isotope will be described.
The following quantities are defined:
ZA Designation of the original nuclide (ZA = 1000Z + A).
ZAP Designation of the nuclide produced in the reaction (ZAP = 1000Z + A).
NS Total number of states (LFS) of the radioactive reaction product for which
decay data are given.
LMF File number (3, 6, 9, or 10) in which the multiplicity or cross section for this
MT number will be found.
153

CHAPTER 8. FILE 8: DECAY AND FISSION PRODUCT YIELDS
LIS State number (including ground and all levels) of the target (ZA).
LISO Isomeric state number of the target.
LFS Level number (including ground and all levels) of the state of ZAP formed
by the neutron interaction (to be given in ascending order).
ELFS Excitation energy of the state of ZAP produced in the interaction (in eV
above ground state).
NO Flag denoting where the decay information is to be given for an important
radioactive end product.
NO = 0, complete decay chain given under this MT.
NO = 1, decay chain given in MT = 457 in the decay data file.
ND Number of branches into which the nuclide ZAP decays.
HL half-life of the nuclide ZAP in seconds.
ZAN Z and mass identifier of the next nuclide produced along the chain.
BR Branching ratio for the production of that particular ZAN and level.
END Endpoint energy of the particle or quantum emitted (this does not include
the gamma energy, following beta decay, for example).
CT Chain terminator that gives minimal information about the formation and
decay of ZAN. The hundredths digit of CT designates the excited level in
which ZAN is formed.
• 1.0 ≤ CT < 2.0 indicates that the chain terminates with ZAN, possibly
after one or more gamma decays.
• CT ≥ 2.0 indicates that ZAN is unstable and decays further to other
nuclides. For example, consider the nuclide (ZAP) formed via a neutron
reaction (MT number) in a final state (LFS number); ZAP then decays
to a level in ZAN; the level number is part of the CT indicator and
includes non-isomeric states in the count.
The following examples may help explain the use of CT:
CT = 1.00, ZAN was formed in the ground state which is stable.
CT = 1.06, ZAN was formed in the sixth excited state; the sixth state
decayed to the ground state which is stable.
CT = 2.00, ZAN was formed in the ground state which is unstable.
(No delayed gammas are associated with the formation and decay of
this particular ZAN). The next decay in the chain is specified under the
RTYP.

154

CHAPTER 8. FILE 8: DECAY AND FISSION PRODUCT YIELDS
CT = 2.11, ZAN was formed in the 11th excited state but the chain does
not terminate with that ZAN. The next decay in the chain is specified
under the RTYP.
It is readily apparent from the above that CT = ”1.” indicates that the chain
terminates with that particular ZAN and CT = ”2.” means that one or more
decays are involved before stability is reached. Note, however, that stability
can be reached instantaneously upon occasion with the emission of one or
more light particles.
RTYP Mode of decay using the same definitions specified in MT=457 (see Section 8.4).
As an example, consider MT=102. Then RTYP = 1.44 would be interpreted
as follows:
The first two columns of the RTYP (1.) indicates β − decay of ZAP; the third
and fourth columns (44) indicate that the nucleus ZAN (formed in the β −
decay) then immediately emits two α particles.
This example is represented by the following reaction:
n + 7 Li → γ + 8 Li (ground state)
β−
ց
2.94 2+ ,0
8
Be
ց

(MT=102)

2α

For this example:
ZA(7 Li) = 3007
ZAP(8 Li) = 3008
ZAN(8 Be) = 4008

LIS(7 Li) = 0
LISO(7 Li) = 0
LFS(8 Li) = 0
CT(8 Be) = 2.01

Since Be has a half-life of the order of compound-nucleus formation times,
decay data for MT=457 are not required, and the complete chain can easily
be represented and read from the information given here.

8.2.1

Formats

The structure of each section always starts with a HEAD record and ends with a SEND
record. Subsections contain data for a particular final state of the reaction product (LFS).
The number of subsections NS is given on the HEAD record for the section. The subsections are ordered by increasing value of LFS. The structure of a section is:

155

CHAPTER 8. FILE 8: DECAY AND FISSION PRODUCT YIELDS
[MAT, 8, MT/ ZA, AWR, LIS, LISO, NS, NO] HEAD


---------------------------------
[MAT, 8, 0/ 0.0, 0.0, 0, 0, 0, 0] SEND
For NO=0 the structure of the subsection is:
[MAT, 8, MT/ ZAP, ELFS, LMF, LFS, 6*ND, 0/
HL1 , RTYP1 , ZAN1 , BR1 , END1 , CT1 ,
HL2 , RTYP2 , ZAN2 , BR2 , END2 , CT2 ,
--------------------------------HLN D ,RTYPN D ,ZANN D ,BRN D , ENDN D , CTN D ] LIST
If NO=1, then the reaction gives rise to a significant product which is radioactive, and
the evaluator wishes only to identify the radioactive product. The evaluator must supply
MF=8, MT=457 data elsewhere to describe the decay of the product. It is understood that
the cross section for producing this radioactive product can be determined from the data in
File 3, 6, 9, or 10 depending upon the value of LMF.
For NO=1, the structure of the subsection is:
[MAT, 8, MT/ ZAP, ELFS, LMF, LFS, 0, 0] CONT

8.2.2

Procedures

1. Data should be given for all unstable states of the reaction product nucleus for which
cross sections are given in File 3 or File 10 or multiplicities in File 6 or File 9. No
information of this type is allowed in evaluations for mixtures of elements, molecules,
or elements with more than one naturally occurring isotope.
2. In order to provide more general usefulness as these files are being constructed, the
following procedures are mandatory. For each reaction type (MT), File 6 yields, File 9
multiplicities, or File 10 cross sections must be provided, except when LMF=3.
3. If the ENDF file also contains a complete evaluation of the neutron cross sections
for the reaction product nucleus (ZAP, LIS), then the radioactive decay data for the
evaluation of (ZAP, LIS) found in MF=8, MT=457 must be consistent with the decay
data in this section.
4. The method for calculating the nuclide production cross section is determined by the
choice of LMF:
LMF=3 implies that the production cross section is taken directly from the corresponding sections in File 3.
LMF=6 implies that the production cross section is the product of the cross section
in File 3 and the yield in File 6.
156

CHAPTER 8. FILE 8: DECAY AND FISSION PRODUCT YIELDS
LMF=9 implies that the production cross section is the product of the cross section
in File 3 and the multiplicity in File 9.
LMF=10 implies that the production cross section is given explicitly in File 10 (in
barns).

8.3

Fission Product Yield Data
(MT=454, MT=459)

MT numbers 454 and 459 specify the energy-dependent fission product yield data for each
incident particle or photon. These MT numbers can also be used to identify yields for spontaneous fission. A complete set of fission product yield data is given for a particular incident
particle energy. Data sets should be given at sufficient incident energies to completely specify
yield data for the energy range given for the fission cross section (as determined from Files 2
or 3). These data are given by specifying fission product identifiers and fission product
yields.
MT=454 is used for independent yields (YI), and MT=459 is used for cumulative yields
(YC). The formats for MT=454 and MT=459 are identical. Independent yields (YI) are
direct yields per fission prior to delayed neutron, beta, etc., decay. The sum of all independent yields is 2.0 for any particular incident particle energy, when light charged particles are
ignored (e.g. 1 H, 2 H, 3 H, 3 He and 4 He). Cumulative yields (YC) are specified for the same
set of fission products. These account for all decay branches, including delayed neutrons.
The fission products are specified by giving an excited state designation (FPS) and a
(charge, mass) identifier (ZAFP). Thus, fission product nuclides are given, not mass chains.
More than one (Z,A) may be used to represent the yields for a particular mass chain.
The following quantities are defined
NFP Number of fission product nuclide states to be specified at each incident
energy point (this is actually the number of sets of fission product identifiers
fission product yields). (NFP≤ 2500).
ZAFP (Z,A) identifier for a particular fission product. (ZAFP = (1000Z + A)).
FPS State designator (floating-point number) for the fission product nuclide (FPS
= 0.0 means the ground state, FPS=1.0 means the first excited state, etc.)
YI (MT=454), independent yield for a particular fission product prior to particle
decay.
DYI (MT=454) 1σ uncertainty in YI.
YC (MT=459) cumulative yield.
DYC (MT=459) 1σ uncertainty in YC.

157

CHAPTER 8. FILE 8: DECAY AND FISSION PRODUCT YIELDS
Cn (Ei ) Array of yield data for the ith energy point. This array contains NFP sets
of four parameters in the order ZAFP, FPS, YI, and DYI in MT=454 and
ZAFP, FPS, YC, and DYC in MT=459.
NN Number of items in the Cn (Ei ) array, equal to 4*NFP.
Ei Incident particle energy of the ith point (eV).
LE Test to determine whether energy-dependent fission product yields given:
LE=0 implies no energy-dependence (only one set of fission product
yield data given);
LE>0 indicates that (LE+1) sets of fission product yield data are
given at (LE+1) incident particle energies.
Ii Interpolation scheme (see paragraph on Two-dimensional Interpolation
Schemes in Section 0.5.2) to be used between the Ei−1 and Ei energy points.

8.3.1

Formats

The structure of a section always starts with a HEAD record and ends with a SEND record.
Sets of fission product yield data are given for one or more incident energies. The sets are
ordered by increasing incident energy. For a particular energy the data are presented by
giving four parameters (ZAFP, FPS, YI, and DYI in MT=454 and ZAFP, FPS, YC, and
DYC in MT 459) for each fission product state. The data are first ordered by increasing
values of ZAFP. If more than one yield is given for the same (Z,A) the data are ordered by
increasing value of the state designator (FPS). The structure for a section is:
[MAT, 8, MT/ ZA, AWR, LE+1, 0, 0,
0]HEAD
[MAT, 8, MT/ E1 , 0.0,
LE, 0, NN, NFP/ Cn (E1 ) ]LIST
[MAT, 8, MT/ E2 , 0.0,
I, 0, NN, NFP/ Cn (E2 ) ]LIST
[MAT, 8, MT/ E3 , 0.0,
I, 0, NN, NFP/ Cn (E3 ) ]LIST
------------------------[MAT, 8, 0 /0.0, 0.0,
0, 0, 0,
0]SEND
where MT=454 for independent yield data, and MT=459 for cumulative yield data. There
are (LE+1) LIST records.

8.3.2

Procedures

The data sets for fission product yields should be given over the same energy range as that
used in Files 2 and/or File 3 for the fission cross section. The yields are given as a fractional
value at each energy, and the independent yields (ignoring light charged particle yields) will
sum to 2.0.
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CHAPTER 8. FILE 8: DECAY AND FISSION PRODUCT YIELDS
This format provides for the yields (YI or YC) to each excited state (FPS) of the nuclide
designated by ZAFP, and hence accommodates the many metastable fission products having
direct fission yields. Data may be given for one or more fission product nuclide states to
represent the yield for a particular mass chain. If independent yield data are given for more
than one nuclide, the yield for the lowest Z (charge) nuclide state for a particular mass chain
should be the same as the cumulative yield in MT=459, and all other independent yields
for this same chain should be direct yields.
The direct fission product yields are those prior to delayed neutron emission; for this
reason, the summation of independent yields over the nuclides in each mass chain does not
necessarily equal the isobaric chain yield.
The cumulative yields in MT=459 for each nuclide (ZAFP, FPS) should be consistent
with the decay branching fractions in MT=457 and the independent yields in MT=454. It
should be noted that the main use of cumulative yields is to carry out calculations without
solving the full decay equations. Several extremely long-lived fission products exist that have
non-isobaric decay modes (e.g. alpha decay). If these are included in the calculation of their
daughter products cumulative yields, the yields will be larger that seen in practice. Some
evaluators omit nuclides with long half-lives from the cumulative yield of their daughterproducts1 The users should be careful about the interpretation of cumulative yield data.
Yields for the same fission product nuclides should be given at each energy point. This
will facilitate interpolation of yield data between incident energy points. Also, a linear-linear
interpolation scheme should be used.

8.4

Radioactive Decay Data (MT=457)

The spontaneous radioactive decay data are given in Section 457. This section is restricted to
single nuclides in their ground state or an isomeric state. (An isomeric state is a ”long-lived”
excited state of the nucleus.) The main purpose of MT=457 is to describe the energy spectra
resulting from radioactive decay and give average parameters useful for applications such as
decay heat, waste disposal, depletion and buildup studies, shielding, and fuel integrity. The
information in this section can be divided into three parts:
a.) General information about the material
ZA Designation of the original (radioactive) nuclide (ZA=1000Z + A).
AWR Ratio of the LIS state nuclide mass to that of neutron.
LIS State of the original nuclide (LIS=0, ground state, LIS=1, first excited state,
etc.)
LISO Isomeric state number for the original nuclide (LISO=0, ground state;
LISO=1, first isomeric state; etc.)
NST Nucleus stability flag (NST=0, radioactive; NST=1, stable)
1

The JEFF-3.1 evaluator oxcluded nuclides with half-lives greater than 1013 seconds (0.32 million years).

159

CHAPTER 8. FILE 8: DECAY AND FISSION PRODUCT YIELDS
T1/2 half-life of the original nuclide (seconds).
NC Total number of decay energies (eV) given (NC = 3 or 17).
E x Average decay energy (eV) of radiation x, e.g., for decay heat applications.
The average energies must be given in an order specified in Section 8.4.2
Unknown average radiation energies are indicated by a value of -1.0.
SPI Spin of the nuclide in its LIS state.
(SPI=-77.777 implies spin unknown)
PAR Parity of the nuclide in its LIS state (±1.0).

b.) Decay mode information for each mode of decay
NDK Total number of decay modes given.
RTYP Mode of decay of the nuclide in its LIS state.
Decay modes are defined as follows:
RTYP

Decay Mode

0. γ

γ-ray (not used in MT457)

1. β

Beta decay

2. e.c., β + Electron capture and/or positron emission
3. IT

Isomeric transition (will in general be present only when the
state being considered is an isomeric state)

4. α

Alpha decay

5. n

Neutron emission (not delayed neutron decay, see below)

6. SF

Spontaneous fission

7. p

Proton emission

10.

Unknown origin

Multiple particle decay is also allowed using any combination of the above
RTYP variables as illustrated in the following examples:
RTYP

Decay Mode

1.5 β − ,n

Beta decay followed by neutron emission (delayed neutron decay)

1.4 β − ,α

Beta decay followed by alpha emission (e.g.

2.4 β + ,α

Positron decay followed by alpha emission.
160

16

N decay)

CHAPTER 8. FILE 8: DECAY AND FISSION PRODUCT YIELDS
RFS Isomeric state flag for daughter nuclide.
RFS=0.0, ground state;
RFS=1.0, first isomeric state, etc.
Q Total decay energy (eV) available in the corresponding decay process. This
is not necessarily the same as the maximum energy of the emitted radiation.
In the case of an isomeric transition Q will be the energy of the isomeric
state. For both β + and β − , Q equals the energy corresponding to the mass
difference between the initial and final atoms).
BR Fraction of the decay of the nuclide in its LIS state which proceeds by the
corresponding decay mode, e.g., if only β − occurs and no isomeric states in
the daughter nucleus are excited then BR=1.0 for β − decay.

c.) Resulting radiation spectra
NSP Total number of radiation types (STYP) for which spectral information is
given (NSP may be zero).
STYP Decay radiation type
Decay radiations are defined as follows:
STYP

Radiation type

0. γ

Gamma rays

1. β −

Beta rays

2. e.c. β + Electron capture and/or positron emission
4. α

Alpha particles

5. n

Neutrons

6. SF

Spontaneous fission fragments

7. p

Protons

−

8. e

”Discrete electrons”

9. x

X-rays and annihilation radiation (photons not arising as transitions between nuclear states)

ER discrete energy (eV) of radiation produced(Eγ , Eβ , Ee.c. , etc.)
RI intensity of discrete radiation produced (relative units).
RP spectrum of the continuum
component of the radiation in units of probabilR
ity/eV such that RP(E)dE = 1.

TYPE Type of transition for beta and electron capture.
Types Defined:
161

CHAPTER 8. FILE 8: DECAY AND FISSION PRODUCT YIELDS
TYPE Spectrum Definition
0.0 Not required for STYP
1.0 Allowed, non-unique
2.0 First-forbidden unique
3.0 Second-forbidden unique
RICC Total internal conversion coefficient (STYP=0.0 only)
RICK K-shell internal conversion coefficient (STYP=0.0 only)
RICL L-shell internal conversion coefficient (STYP=0.0 only)
RIS Internal pair formation coefficient (STYP=0.0)
STYP=2.0, positron intensity,
STYP=0.0 otherwise.
LCON

2

Continuum spectrum flag
LCON=0, no continuous spectrum given
LCON=1, only continuous spectrum given
LCON=2, both discrete and continuum spectra.

NT Number of entries given for each discrete energy (ER).
FC Continuum spectrum normalization factor (absolute intensity/relative intensity).
FD Discrete spectrum normalization factor (absolute intensity/relative intensity).
NER Total number of tabulated discrete energies for a given spectral type (STYP).
ERAV

3

Average decay energy of radiation produced.

NR Number of interpolation ranges for the continuum spectrum.
NP Number of points at which the distribution is given.
Eint Interpolation scheme for the continuum spectrum.
NK Number of partial energy distributions when LCON = 5 is used.
∆ Uncertainty in any quantity.
2

RTYP=6.0 , STYP=5.0, spontaneous fission neutron spectra: FC=ν total for LCON=1; FC=νp , FD=νd
for LCON=2.
3
For STYP=2, this is the average positron energy; for STYP=4, this includes energy of recoil nucleus.

162

CHAPTER 8. FILE 8: DECAY AND FISSION PRODUCT YIELDS
LCOV Flag indicating whether covariance data are given for continuum spectrum
data. (LCON=1 or 2).
LCOV=0, no covariance data given.
LCOV=1, covariance data given.
LB Flag indicating the meaning of the numbers given in the array {Ek ,Fk }. (Only
LB=2 presently allowed, See chapter 33).
NPP Number of pairs of numbers in the {Ek ,Fk } array.
The array {Ek ,Fk } contains pairs of numbers, referred to as an Ek table. In each Ek
table the first member of a pair is an energy, Ek , the second member of the pair, Fk , is a
number associated with the energy interval between the two entries Ek and Ek+1 .
The Ek table must cover the complete range of secondary particle energies. Some of the
Fk ’s may be zero, as must be the case below threshold for a threshold reaction, and the last
value of Fk in an Ek table must be zero or blank since it is not defined.
The Fk values in the Ek table for the allowed LB=2 representation are fractional components of the covariance matrix, fully correlated over all Ek intervals:
X
′
Sik Sjk Fk Fk′ Xi Xj
Cov(Xi , Xj ) =
k,k′

where Sik = 1 when the energy Ei is in the interval Ek to Ek+1 of the Ek table,
0 otherwise.
Here Xi is the normalized spectral intensity at decay particle emitted kinetic energy
range Ei obtained from the {E,RP} TAB1 record indicated.

8.4.1

Formats

The structure of this section always starts with a HEAD record and ends with a SEND
record. For a radioactive nucleus (NST=0), this section is divided into subsections as follows4 :
[MAT, 8,457/ ZA,
AWR, LIS, LISO,
NST, NSP]HEAD
(NST=0)
[MAT, 8,457/ T1/2 , ∆T1/2 ,
0,
0, 2*NC,
0/(Ex ,∆Ex ) / LIST
[MAT, 8,457/ SPI,
PAR,
0,
0, 6*NDK, NDK/
RTYP1 , RFS1 , Q1 , ∆Q1 , BR1 , ∆BR1 ,
--------------------------------RTYPN DK , RFSN DK , QN DK , ∆QN DK , BRN DK ,∆BRN DK ] LIST


--------------------------------
[MAT, 8, 0/ 0.0, 0.0, 0, 0, 0, 0] SEND
4

Data in each LIST record must be given in order specified in Section 8.4.2

163

CHAPTER 8. FILE 8: DECAY AND FISSION PRODUCT YIELDS
The structure of a subsection is:
[MAT, 8,457/ 0.0, STYP, LCON, 0, 6, NER/
FD, ∆FD, ERAV , ∆ERAV , FC, ∆FC] LIST
[MAT, 8,457/ ER1 , ∆ER1 , 0, 0, NT, 0/
RTYP1 , TYPE1 , RI1 , ∆RI1 , RIS1 , ∆RIS1 ,
RICC1 ,∆RICC1 , RICK1 ,∆RICK1 , RICL1 ,∆RICL1 ] LIST
-------------------------------------ERN ER , ∆ERN ER , 0, 0, NT, 0/
RTYPN ER ,TYPEN ER , RIN ER , RIN ER , ------------] LIST
(omit these LIST records if LCON=1)
[MAT, 8,457/ RTYP, 0.0, 0, LCOV, NR, NP/ Eint / RP(E) ] TAB1
(omit if LCON=0)
[MAT, 8,457/ 0.0, 0.0, 0, LB, 2*NPP, NPP/ (Ek ,Fk ) ] LIST
(omit if LCOV=0 or LCON=0)
For a stable nucleus (NST=1), this section is divided into subsections as follows:
[MAT, 8,457/ ZA,
[MAT, 8,457/ 0.0,
0.0,
[MAT, 8,457/ SPI,
0.0,

8.4.2

AWR,
0.0,
0.0,
PAR,
0.0,

LIS,
0,
0.0,
0,
0.0,

LISO,
0,
0.0,
0,
0.0,

NST,
6,
0.0,
6,
0.0,

0]HEAD
0/
0.0] LIST
0]
0.0] LIST

(NST=1)

Procedures

1. The initial state of the parent nucleus is designated by LISO, which equals 0 for the
ground state and equals n for the nth isomeric state. Only isomeric states are included
in the count of LISO. (In other files isomeric and nonisomeric states may be included
in the count of levels.)
2. The average decay energy E x for decay heat application is given for three general radiation types, E LP (for light particles), E EM (for electromagnetic radiation), and E HP
(for heavy particles), followed by the individual components. The sum of these three
general quantities is the total average (neutrino energies excluded) energy available
per decay to the decay heat problem. The three quantities are more precisely defined
as:
E LP = E β − + E β + + E e− + . . .
E EM = E γ + E x−ray + E annih.rad. + . . .
E HP = E α + E SF + E p + E n + . . .
where E LP means the average energy of all ”electron-related” radiation such as β − , β +
conversion-electrons, Auger, etc. The quantity E EM means the average energy of all
”electromagnetic” radiations such as gamma-rays, x-rays, and annihilation radiation.
The quantity E HP is the average energy of all heavy charged particles and neutrons,
and also includes the recoil energy; but the alpha energy alone can be separated out
164

CHAPTER 8. FILE 8: DECAY AND FISSION PRODUCT YIELDS
by the usual MR /(MR + Mα ) factor, where MR and Mα are the recoil nucleus and
alpha masses, respectively.
The average decay energies E x must be given in the following order:
E LP

Average energy of all light particles.

E EM

Average energy of all electromagnetic radiation.

E HP

Average energy of all heavy particles.

E β−

Average β − energy.

E β+

Average β + energy.

E Ae−

Average Auger-electron energy.

E ce−

Average conversion-electron energy.

Eγ

Average gamma-ray energy.

E x−ray

Average X-ray energy.

E InB

Average internal Bremsstrahlung energy.

E annih.rad.

Average annihilation energy.

Eα

Average α energy.

E recoil

Average recoil energy.

E SF

Average SF energy.

En

Average prompt and/or delayed neutron energy.

Ep

Average proton energy.

Eν

Average neutrino or antineutrino energy.

3. The symbol RTYP indicates the mode of decay as determined by the initial event.
A nucleus undergoing beta decay to an excited state of the daughter nucleus, which
subsequently decays by gamma emission, is in the beta decay mode. RTYP = 0.0 is
not allowed in MT = 457 (although used under Section 8.2).
An isomeric state of the daughter nuclide resulting from the decay of parent nuclides
is designated by RFS following the procedures used for LISO. Q represents the total
energy available in the decay process and is equal to the energy difference available
between the initial and final states (both of which may be isomeric). The branching
ratio BR for each decay mode is given as a fraction and the sum over all decay modes
must equal unity.
Multiple particle emission is also allowed by using any combination of the RTYP
variables. This will account for particle emission from nuclear states excited in the
decay of the parent (”delayed-particle” emission) whose half-lives are too short to
warrant separate entry in the file. It also allows users and processing codes to identify
the various intermediate states, without having to examine all the spectrum listings to
determine radiation types. The multiple-particle RTYP should be constructed in the
order in which the particles are emitted. (e.g., RTYP = 1.5 indicates decay followed
by neutron emission).
165

CHAPTER 8. FILE 8: DECAY AND FISSION PRODUCT YIELDS
4. The source of radiation should be specified for each spectral line or continuous spectra. The source of radiation is a floating-point integer corresponding to the RTYP
definitions. If the source of radiation is not known RTYP = 10. should be used.
5. The energy spectra should be specified if they are known and identified by STYP.
Gamma spectra are described using STYP = 0.0. Relative intensities and errors in
the relative intensity should be specified. Absolute normalization is made through
multiplication by FC and FD. If absolute discrete spectra are given, FD must equal
unity. The radiation intensity should total the contributions from all decays leading
to radiation within a particular decay type, STYP, having energy Er ± ∆Er .
(a) For gamma ray emission (STYP=0.0), no other information is required if X-ray,
Auger electron, conversion electron, and pair formation intensities have not been
calculated for these transitions. In this case NT=6.
The amount of additional information depends upon the detail in which quantities
were obtained for inclusion in STYP=8. or 9. spectra, and the number of decay
modes. (This detail will also be reflected in the uncertainties assigned in STYP=8.
or 9. spectra.) If only the total conversion electron emission is calculated, RICC
and ∆RICC should be included and NT is specified as 8. If contributors from
the individual K, L, and M shells are calculated, the K and L shell conversion
coefficients should be included and NT = 12. In the rare case (i.e., 167 N), where
internal pair formation is needed, the internal pair formation coefficient should
be included as the quantities RIS and ∆RIS.
(b) For electron capture (STYP=2.), the quantity RIS is 0.0 provided Ee.c. ≤ 1.022
MeV. If positron emission is energetically possible, RIS and ∆RIS must be specified (as Iβ+ and ∆Iβ+ ).
(c) The spectra should be ordered in increasing values of STYP, and discrete spectral
data should be specified before continuous spectra.
(d) For STYP=5. (spontaneous fission neutrons), LCON=0, NER=0, and EAV and
∆EAV should be given.
(e) For STYP=6. (spontaneous fission fragments) LCON=0, NER=0, and E SF and
∆E SF should be given.
6. The specification of data uncertainties is an important quantity which is difficult to
represent in a simple way. Although a one sigma variance is desired, a number should
be entered that at least indicates qualitatively how well the parameter is known.
For STYP=8. and 9., ∆E will reflect the detail in which these values were derived.
For example, if only the total conversion electron emission has been calculated, ∆E
would be the spread between K-conversion and M-conversion electron energies. If a
very detailed calculation has been made, ∆E would reflect the uncertainties in the
electron binding energy and the transition energy.
7. The spontaneous fission spectrum is specified in File 5 of sub-library 4 (no incoming
projectile).
166

CHAPTER 8. FILE 8: DECAY AND FISSION PRODUCT YIELDS
8. Every effort should be made to determine the spin and parity of the original nucleus,
either by experimental evidence or by strong theoretical arguments. If the spin cannot
be determined, it should be reported as 77.777; if the parity cannot be determined it
should be reported as zero.
9. Because the continuum spectrum is normalized, the absolute covariance matrix of a
multi-component normalized spectrum processed from this file must have zero for the
sum of each row and column. (Processing codes should perform this check).
Since the covariance form for radioactive product spectra is confined to LB=2, meeting
this test is equivalent to the following condition on the Fk of the Ek covariance table:
X
Fk yk = 0,
(8.1)
k

where

Z

yk =

Ek+1

RP(E)dE

(8.2)

Ek

and yk is P
the energy spectrum on the uncertainty evaluation grid, subject to the
condition
yk = 1. If the initial Fk do not meet this condition, the corrected values
Fk′ are given by:
X
Fk yk
Fk′ = Fk −
k

Note that unlike the case for File 33, some of the Fk′ must be negative. Also, the
processed multigroup correlation matrix will show some off-diagonal components that
are -1 as well as others that are +1.
When a processing code constructs the absolute covariance Vmn on the user’s energy
grid Em , the simplest relations to use are
Vmn = σm σn
Z En+1
where σn =
F (E) RP(E) dE
En

and the integral is easy because F (E) is piece-wise continuous on the Ek grid. By
this construction we are assured that the null sum condition will be retained for the
covariance matrix of the processed multigroup spectrum.

167

Chapter 9
File 9: MULTIPLICITIES FOR
PRODUCTION OF RADIOACTIVE
NUCLIDES
9.1

General Description

Neutron cross sections for the excitation of metastable states (i.e. the activation cross
sections) can be reconstructed from the cross sections in File 3 and the multiplicities in
File 9. The multiplicity represents that fraction of the cross section in File 3 that produces
the LFS state in the daughter nucleus. The multiplicities are given as a function of energy,
E, where E is the incident neutron energy (in eV) in the laboratory system. They are given
as energy-multiplicity pairs. An interpolation scheme must be given to specify the energy
variation of the data for incident energies between a given energy point and the next higher
energy point. File 9 is divided into sections, each section containing data for a particular
reaction type (MT number). The sections are ordered by increasing MT number. Within
a section for a given MT are subsections for different final states of the daughter product
(LFS). File 9 is only allowed for evaluations that represent data for single nuclides.

9.2

Formats

File 9 is made up of sections where each section gives the multiplicity for a particular reaction
type (MT number). Each section always starts with a HEAD record and ends with a SEND
record. For File 9, the following quantities are defined:
LIS Indicator to specify the level number of the target.
LFS Indicator to specify the level number of the nuclide (ZAP) (as defined in
File 8) produced in the reaction (MT number).
LFS = 0: the final state is the ground state.
LFS = 1: the final state is the first excited state.
168

CHAPTER 9. FILE 9: MULTIPLICITIES OF RADIOACTIVE PRODUCTS
LFS = 2: the final state is the second excited state.
——————LFS = 98: an unspecified range of final states.
QM Mass-difference Q value (eV); see Section 3.3.2.
QI Reaction Q value (eV) for the state described by the subsection.
For isomeric, states QI is defined as QM minus the residual excitation energy
of the isomer.
For the ground state, QI=QM for reactions with no intermediate states in
the residual nucleus.
(See Section 3.3.2)
IZAP 1000 × Z + A for the product nucleus
NS Number of final states for each MT for which multiplicities are given.
NR Number of energy ranges. A different interpolation scheme may be given for
each range. For the limit on the maximum value of NR see Appendix G.
NP Total number of energy points used to specify the data. For the limit on the
maximum value of NP see Appendix G.
Eint Interpolation scheme for each energy range. (For details, see Section 0.5.2)
Y (E) Multiplicity for a particular reaction type at incident energy E (eV). Data
are given for energy-multiplicity pairs.
The structure of a section is:
[MAT, 9, MT/ ZA, AWR, LIS, 0, NS, 0]HEAD

[MAT, 9, 0/ 0.0, 0.0,
0, 0, 0, 0]SEND
The structure of a subsection is:
[MAT, 9, MT/ QM, QI, IZAP, LFS, NR, NP/ Eint / Y (E)]TAB1

9.3

Procedures

Multiplicities must be given in File 9 for those reactions described in File 8 which have
LMF=9 in the LIST record of the subsection for that particular MT number and value of
LFS. The multiplicities in File 9 describe the fraction of the cross section that produces the
LFS state in the daughter nucleus. For a reaction represented by resonance parameters in
File 2, File 10 cannot be used; only multiplicities in File 9 are allowed.
169

CHAPTER 9. FILE 9: MULTIPLICITIES OF RADIOACTIVE PRODUCTS
The data in File 9 must cover the entire energy range for each reaction in File 3 from
threshold to at least 20 MeV. That is, multiplicities cannot be used over a portion of the
incident neutron energy range with cross sections covering another portion. For reactions
with negative Q-values , the first energy point should be at the threshold given in File 3. If
a subsection QI is not equal to the QI in File 3, the multiplicity should be given as zero up
to the energy point corresponding to the threshold of the subsection.
The set of points or energy mesh used for the total cross section in File 3 must include
the union of all energy meshes in File 9 for each MT number. Although a large number
of incident energy points are allowed for the total cross section (see Appendix G), every
attempt should be made to minimize the number of points in File 9.
The multiplicities in File 9 should be equal to or less than unity since the cross sections to
be generated must be equal to or less than the cross sections in File 3 for each MT number.
In summary, the proper procedure would be to defer entering data into File 8 for a given
MT until the File 9 multiplicities (or File l0 cross sections) are added to the evaluations.
That is, every MT number (except MT=454, 457, or 459) in File 8 with LMF=9 as an
indicator in the LIST record of the subsection for that particular MT and LFS value must
have the corresponding multiplicities in File 9.

170

Chapter 10
File 10: CROSS SECTIONS FOR
PRODUCTION OF RADIOACTIVE
NUCLIDES
10.1

General Description

Neutron activation cross sections [such as the (n,p) and (n,2n) cross sections] and cross
sections for a particular state of a radioactive target are given in File 10. These cross
sections are given as a function of energy E, where E is the incident particle or photon
energy (in eV) in the laboratory system. They are given as energy-cross-section pairs. An
interpolation scheme must specify the energy variation of the data for energies between a
given energy point and the next higher energy point.
File 10 is divided into sections, each section containing the data for a particular reaction
type (MT number). The sections are ordered by increasing MT number. Within a section for
a given MT are subsections for different final states (LFS) of the daughter product nucleus.
File 10 is allowed only for evaluations that represent the data for single isotopes.

10.2

Formats

File 10 is made up of sections where each section gives the cross section for a particular
reaction type (MT number). Each section always starts with a HEAD record and ends with
a SEND record. For File 10, the following quantities are defined:
LIS Indicator to specify the level number of the target.
LFS Indicator to specify the level number of the nuclide (IZAP) produced in the
reaction (MT) number.
LFS=0 the final state is the ground state.
LFS=1 the final state is the first excited state.
LFS=2 the final state is the second excited state.
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CHAPTER 10. FILE 10: PRODUCTION CROSS SECTIONS FOR RADIONUCLIDES
——LFS=98 an unspecified range of final states.
QM Mass-difference Q value (eV); see Section 3.3.2.
QI Reaction Q value (eV) for the state described by the subsection.
For isomeric, states QI is defined as QM minus the residual excitation energy
of the isomer.
For the ground state, QI=QM for reactions with no intermediate states in
the residual nucleus.
(See Section 3.3.2)
IZAP 1000 × Z + A for the product nucleus
NS Number of final states for each MT for which cross sections are given.
NR Number of energy ranges. A different interpolation scheme may be given for
each range. For the limit on the maximum value of NR see Appendix G.
NP Total number of energy points used to specify the data. For the limit on the
maximum value of NP see Appendix G.
Eint Interpolation scheme for each energy range. (For details, see Section 0.5.2)
σ(E) Cross section in barns for a particular reaction type at incident energy E
(eV). Data are given for energy, cross-section pairs.
The structure of a section is:
[MAT, 10, MT/ ZA, AWR, LIS, 0, NS, 0]HEAD

[MAT, 10, 0/ 0.0, 0.0,
0, 0, 0, 0]SEND
The structure of a subsection is:
[MAT, 10, MT/ QM, QI, IZAP, LFS, NR, NP/ Eint / σ(E)]TAB1

10.3

Procedures

Isomer production cross sections must be given in File 10 for those reactions described in
File 8 which have LMF=10 in the LIST record of the subsection for that particular MT
number and value of LFS. The data in File 10 are the cross sections for the production of a
final state (LFS) of the daughter product nucleus. For a reaction represented by resonance
parameters in File 2, File 10 cannot be used; only multiplicities in File 6 or File 9 are
allowed.
172

CHAPTER 10. FILE 10: PRODUCTION CROSS SECTIONS FOR RADIONUCLIDES
The data in File 10 must cover the entire energy range for each reaction from the threshold
of the subsection in File 10 up to at least 20 MeV. That is, cross sections cannot be used over
a portion of the incident neutron energy range with multiplicities covering another portion.
For reactions with negative Q-values, the first energy point should be at the threshold of
the subsection in File 10 and the cross section at this point must be zero.
The set of points or energy mesh used for the total cross section in File 3 must be the
union of all energy meshes in File 10 for each MT number. The maximum number of points
NT for the total cross section is given in Appendix G. However, every attempt should be
made to minimize the number of points in File 10.
Using the 93 Nb(n,2n)92 Nb cross section as an example, only the cross section for the
production of the 10.16-day isomer in 92 Nb would appear under MT=16 with LIS=0 and
LFS=1 in File 10. The sum of all partial cross sections for the (n,2n) reaction would still
be found in File 3 under MT=16 [note that this is the only (n,2n) cross section required
for neutron transport calculations]. It should be noted, however, in this particular case,
that the evaluator would have the choice of using energy-dependent multiplicities in File 9
instead of cross sections in File 10.
The cross sections that appear in File 10 are redundant; that is, they should not be
included in the check sum for the total cross section. The cross sections in File 10 must be
equal to or less than the cross sections for that MT number that appears in File 3.
In summary, the proper procedure is to defer entering data into File 8 for a given MT
until the File 10 cross sections (or File 9 multiplicities) are added to the evaluations. That is,
every MT number (except MT=454, 457, or 459) with LMF=10 as an indicator in the LIST
record of the subsection for that particular MT and LFS value must have the corresponding
cross sections in File 10.

173

Chapter 11
File 11: GENERAL COMMENTS
ON PHOTON PRODUCTION
11.1

General Description

Photon production data not represented in File 6 may be presented in four distinct files as
follows:
File Description
12 Multiplicities and transition probability arrays
13 Photon production cross sections
14 Photon angular distributions
15 Continuous photon energy spectra
With the exception of File 12, all the files are closely analogous to the corresponding
neutron data files with the same number (modulo 10). The purpose of File 12 is to provide additional methods for representing the energy dependence of photon production cross
sections. The allowed reaction type (MT) numbers are the same as those assigned for neutron reactions, Files 1 through 7. However, they may have somewhat different meanings for
photon production that require additional explanation in some cases:
1. MT=3 should be used in File 12 through 15 to represent composite cross sections, that
is, photon production cross sections from more than one reaction type that have been
lumped together.
2. There is no apparent reason to have redundant or derived data for the photon production files, as is the case for the neutron files, i.e., MT=3, 4, etc. Therefore, to avoid
confusion, the join of all sections of Files 6, 12 and 13 should represent the photon
production, with each section being disjoint from all others.
3. Using Figure 11.1 as a guide, let us consider how one might represent inelastic γ-ray
production. The differential cross section for producing γ ray of energy Eγ resulting
from the excitation of the mth
0 level of the residual nucleus and the subsequent transition between two definite levels (j → i), which need not be adjacent, including the
174

CHAPTER 11. FILE 11: GENERAL COMMENTS ON PHOTON PRODUCTION
effects of cascading from the m0 − j levels higher than j, is given by:
m0 −j
X
dσ
Rm0 jα ,
(Eγ , E, m0 , i, j) = δ {Eγ − (εj − εi )} Aj,i σm0 (E)
dEγ
α=1

(11.1)

where:
σm0 Cross section for exciting the mth
0 level with incident particle of energy
E, taken from File 3 for the MT corresponding to the mth
0 level,
δ {Eγ − (εj − εi )} Delta function defining the discrete gamma-ray of energy Eγ that
results from the transition from level j to level i,
Aij Probability that a gamma ray of energy of Eγ is emitted in the transition from level j to i, taken as the gamma-ray branching ratio,
Rm0 jα Probability that the nucleus initially excited at level m0 will de-excite
to level j in α transitions, where α ranges from 1 to m0 − j,
Rm0 jα =

m
1 −1
X

m
0 −1
X

m1 =α+(j−1) m2 =α+(j−2)

···

mα−2 −1

X

j
α
X
Y

mα−1 =j+1 mα

Tml−1 ml

l=1

Tkj probability of the residual nucleus having a transition to the lth level
given that it was in the excited state corresponding to the k th level, i.e.,
the branching ratio for a gamma ray transition from the k → l level.
In general, Rm0 ja is the sum of the products of a transition probabilities (branching
ratios) leading from level m0 through intermediate levels to level j. In the example
shown for initial excitation of level m0 = 5 and interest being in the resulting γ-ray
due to transition between levels 2 and 1:
Rm0 j = Tm0 j = T52
Rm0 j2 = Tm0 m1 T m1 j + Tm0 m2 Tm2 j = T52 .T42 + T53 .T32 = 0 + 0 = 0
The relevant quantities for this example are [Eγ = Eγ2 = (ε2 − ε1 )], and
Rm0 j3 = Tm0 m1 Tm1 m2 Tm2 j = T54 .T43 .T32
If m0 and j are separated by many levels, the scheme becomes very involved.
We are at once beset by the problem that no clear choice of ENDF representation in
terms of section number is possible. The data may naturally be identified with both
th
the mth
level. To avoid this problem, we can sum equation (11.1)
0 level and the j
over m0 :
N
X
dσ
dσ1
(Eγ , E, i, j) =
(Eγ , E, m0 , i, j)
(11.2)
dEγ
dEγ
m =j
0

175

CHAPTER 11. FILE 11: GENERAL COMMENTS ON PHOTON PRODUCTION

Continuum
Incident Neutron Energy
E

m0
γ5a

5

ǫ5

4

ǫ4

3

ǫ3

2

ǫ2

1

ǫ1

0

Ground State

γ5b

m1
γ4a

γ4b

m2 j
γ3a

γ3b

m3 i

γ2
γ1

Figure 11.1: Schematic diagram of a level scheme to illustrate gamma production.
where N is the highest level that can be excited by a neutron of incident energy E [i.e.,
εN ≤ E.AWR/(AWR + 1)]. This gives a de-excitation cross section that can single out
a definite γ-ray transition and has the advantage when experimental data are to be
represented. The de-excitation cross section is identified with the j th level.
Alternatively, we can sum equation (11.1) over i and j:
j−1

m

0 X
X
dσ2
dσ
(Eγ , E, m0 ) =
(Eγ , E, m0 , i, j) .
dEγ
dEγ
j=1 i=0

(11.3)

This gives an excitation cross section that can single out a definite excited state and
has the advantage when calculated data are to be represented. The excitation cross
section is identified with the mth
0 level. If equation (11.2) is summed over i and j, or
if equation (11.3) is summed over m0 then:
dσ
(Eγ , E) =
dEγ
≡

N
X
dσ2
(Eγ , E, m0 )
dE
γ
m =1
0

j=l
N X
X
j=1 i=0

dσ1
(Eγ , E, i, j)
dEγ

(11.4)

This gives a cross section for all possible excitations and transitions and thus corresponds to the total inelastic neutron cross section for discrete levels. It is recommended
176

CHAPTER 11. FILE 11: GENERAL COMMENTS ON PHOTON PRODUCTION
that MT=4 be used for the data represented by equation (11.4), as well as for the continuum. If, however, it is expedient or useful to use MT=51 through 91, then one must
use either the de excitation cross sections of equation (11.2) or the excitation cross
sections of equation (11.3), but not both. A restriction is imposed if the transition
probability array option is used and if the entire neutron energy range is not covered
by the known transition probabilities. Then, for MT=51 through 90 in File 12 to be
used for the remaining neutron energy range, a representation by excitation multiplicities must be used. The integrated cross sections of File 13 are obtained by integrating
equations (11.1) through (11.4) over Eγ .
4. The remarks in Item 3 apply for discrete rays from (n,pγ), (n,dγ), (n,tγ), (n,3 Heγ),
(n,αγ) reactions, and the use of MT=103, 101, 105, 106, and 107 is recommended for
these cases.

177

Chapter 12
File 12: PHOTON PRODUCTION
MULTIPLICITIES AND
TRANSITION PROBABILITY
ARRAYS
12.1

General Description

File 12 can be used to represent the neutron energy dependence of photon production cross
sections or delayed photon source functions by means of either multiplicities or transition
probability arrays. Both methods rely upon processing codes that use either neutron cross
sections from File 2 and/or File 3 to generate absolute photon production cross sections or
time constants from File 1 (MT=460) to generate delayed photon source functions.
Multiplicities can be used to represent the cross sections of discrete photons and/or the
integrated cross sections of continuous photon spectra. The MT numbers in File 12 designate
the particular neutron cross sections (File 2 and/or File 3) to which the multiplicities are
referred. The use of multiplicities is the recommended method of presenting (n,γ) capture γray cross sections, provided, of course, that the (n,γ) cross section is adequately represented
in File 2 and/or File 3.
For well-established level decay schemes, the use of transition probability arrays offers
a concise method for presenting (n,xγ) information. With this method, the actual decay
scheme of the residual nucleus for a particular reaction (defined by MT number) is entered in
File 12. This information can then be used by a processing code together with discrete level
excitation cross sections from File 3 to calculate discrete γ-ray production cross sections.
This option cannot be used to represent the integrals of continuous photon spectra.

12.2

Formats

Each section of File 12 gives information for a particular reaction type (MT number), either
as multiplicities (LO=1) or as transition probability arrays (LO=2). Each section always
starts with a HEAD record and ends with a SEND record.
178

CHAPTER 12. FILE 12: PHOTON PRODUCTION YIELD DATA

12.2.1

Option 1: Multiplicities (LO=1)

The neutron energy dependence of photon production cross section is represented by tabulating a set of neutron energy and multiplicity pairs {E, yk (E)} for each discrete photon and
for the photon energy continuum1 . The subscript k designates a particular discrete photon
or a photon continuum, and the total number of such sets is represented by NK.
The multiplicity or yield yk (E) is defined by:
σkγ
yk (E) =
σ(E)

(photons)

(12.1)

where E designates the neutron energy and σ(E) is the neutron cross section in File 2 and/or
File 3 to which the multiplicity is referred (by the MT number). For discrete photons, σkγ (E)
is the photon production cross section for the discrete photon designated by k. For photon
continua, σkγ (E) is the cross section for the photon continuum integrated over photon energy.
In the continuum case,
R dσkγ
(Eγ ← E) dEγ
σkγ (E)
dEγ
yk (E) =
=
σ(E)
σ(E)
R
Z Eγmax
σ(E) yk (Eγ ← E) dEγ
yk (Eγ ← E) dEγ
=
=
σ(E)
0

(12.2)

where Eγ designates photon energy (eV), the term:
dσkγ
(Eγ ← E)
Eγ
is the absolute photon energy distribution in barns/eV, and yk (Eγ ← E) is the relative
energy distribution in photons/eV. The quantity yk (Eγ ← E) can be broken down further
as:
yk (Eγ ← E) = yk (E)fk (Eγ ← E),
which results in the requirements that
Z Eγmax
0

fk (Eγ ← E) dEγ = 1.

Any time a continuum representation is used for a given MT number in either File 12 or
13, then the normalized energy distribution fk (Eγ ← E) must be given in File 15 under the
same MT number.
As a check quantity, the total yield from the NK contributions:
Y (E) =

NK
X

yk (E)

(photons)

k=1

1

There should be no more than one energy continuum for each MT number used. If the decomposition
of a continuum into several parts is desired, this can be accomplished in File 15.

179

CHAPTER 12. FILE 12: PHOTON PRODUCTION YIELD DATA
is also tabulated for each MT number if NK>1.
The structure of a section for LO=1 is2 :
[MAT, 12, MT/ ZA, AWR, LO, 0, NK, 0]HEAD
(LO=1)
[MAT, 12, MT/ 0.0, 0.0, 0, 0, NR, NP/Eint / Y (E)] TAB1


......
......

[MAT, 12, 0/ 0.0, 0.0, 0, 0, 0, 0] SEND
and the structure of each subsection is

(if NK>1)

[MAT, 12, MT/ Egk , ESk , LP, LF, NR, NP/ Eint / yk (E)] TAB1
where:
NK number of discrete photons including the continuum.
ESk energy of the level from which the photon originates. If the level is unknown
or if a continuous photon spectrum is produced, then ESk ≡ 0.0 should be
used.
EGk photon energy for LP=0 or 1 or Binding Energy for LP=2. For a continuous
photon energy distribution, EGk ≡ 0.0 should be used.
LP indicator of whether or not the particular photon is a primary:
LP=0 origin of photons is not designated or not known, and the
photon energy is EGk ;
LP=1 for non-primary photons where the photon energy is again
simply EGk ;
LP=2 for primary photons where the photon energy EG’k is given
by
AWR
EG′k = EGk +
En .
AWR + 1
LF the photon energy distribution law number, which presently has only two
values defined:
LF=1, a normalized tabulated function (in File 15), and
LF=2, a discrete photon energy.
2

If the total number of discrete photons and photon continua is one (NK=1), the TAB1 record is omitted.

180

CHAPTER 12. FILE 12: PHOTON PRODUCTION YIELD DATA

12.2.2

Option 2: Transition Probability Arrays (LO=2)

With this option, the only data required are the level energies, de-excitation transition
probabilities, and (where necessary) conditional photon emission probabilities. Given this
information, the photon energies and their multiplicities can readily be calculated. Photon
production cross sections can then be computed for any given level from the excitation
cross sections in File 3, along with the transition probability array. Similarly, multiplicities
and photon production cross sections can be constructed for the total cascade. For any
given level, the transition and photon emission probability data given in the section are for
photons originating at that level only; any further cascading is determined from the data
for the lower levels. The following quantities are defined:
LG LG=1, simple case (all transitions are γ emission).
LG=2, complex case (internal conversion or other competing processes occur).
NS Number of levels below the present one, including the ground state. (The
present level is also uniquely defined by the MT number and by its energy
level).
NT Number of transitions for which data are given in a list to follow (i.e., number
of non-zero transition probabilities), NT≤ NS.
ESi Energy of the ith level, i=0,1,2... NS.
ES ≡ 0.0 implies the ground state.
TPi TPN S,i , probability of a direct transition from level NS to level i, i=0,1,2...
(NS1).
GPi GPN S,i , the probability that, given a transition from level NS to level i, the
transition is a photon transition (i.e., the conditional probability of photon
emission).
Bi Array of NT doublets or triplets depending on the LG value.
Note that each level can be identified by its NS number. Then the energy of a photon
from a transition to level i is given by Eγ = ESN S − ESi , and its multiplicity is given by
y(Eγ ← E) = (TPi )(GPi ). It is implicitly assumed that the transition probability array is
independent of incident neutron energy. The structure of a section for LO=2 is:
[MAT, 12, MT/
ZA, AWR, LO, LG,
NS,
0]HEAD
[MAT, 12, MT/ ESN S , 0.0, LP, 0, (LG+1)*NT, NT/B]LIST
[MAT, 12, 0/ 0.0, 0.0, 0, 0,
0,
0]SEND

(LO=2)

If LG=1, the array Bi consists of NT doublets {ESi ,TPi };
if LG=2, it consists of NT triplets {ESi ,TPi ,GPi }.
Here the subscript i is a running index over the levels below the level for which the transition
probability array is being given (i.e., below level NS). The doublets or triplets are given in
decreasing magnitude of energy ESi .
181

CHAPTER 12. FILE 12: PHOTON PRODUCTION YIELD DATA

12.3

Procedures

1. Under Option 1, the subsections are given in decreasing magnitude of EGk .
2. Under Option 1, the convention is that the subsection for the continuum photons, if
present, is last. In this case, the last value of EGk (EGN K ) is set equal to 0.0, and
logical consistency with Procedure 1 is maintained.
3. Under Option 1, the values of EGk should be consistent to within four significant
figures with the corresponding EGk values for the File 14 photon angular distributions.
This allows processing and ”physics” checking codes to match photon yields with the
corresponding angular distributions.
4. Under Option 1, ESk is the energy of the level from which the photon originates. If
ESk is unknown or not meaningful (as for the continuous photon spectrum), the value
0.0 should be entered.
5. If capture and fission resonance parameters are given in File 2, photon production for
these reactions should be given by using Option 1 of File 12, instead of using photon
production cross sections in File 13. This is due to the voluminous data required to
represent the resonance structure in File 13 and the difficulty of calculating multigroup
photon production matrices from such data.
6. Under Option 1, the total yield table, Y (E), should span exactly the same energy
range as the combined energy ranges of all the yk (E). Within that range,
Y (E) =

NK
X

yk (E)

k=1

should hold within four significant figures.
7. The excitation cross sections for all the levels appearing in the transition probability
arrays must, of course, be given in File 3.
8. The join of all sections, regardless of the option used, should represent the photon
production data, with no redundancy. For example, MT=4 cannot include any photons
given elsewhere under MT=51 through 91. Likewise, there can be no redundancy
between Files 12 and 13.
9. If only one energy distribution is given under Option 1 (NK=1), the TAB1 record for
the Y(E) table is deleted to avoid repetitive entries.
10. Data should not be given in File 12 for reaction types that do not appear in Files 2
and/or 3.
11. Under Option 2, the level energies, ESi , in the transition probability arrays are given
in decreasing magnitude.
182

CHAPTER 12. FILE 12: PHOTON PRODUCTION YIELD DATA
12. The MT numbers for which transition probability data are given should be for consecutive levels, beginning at the first level, with no embedded levels omitted.
13. The energies of photons arising from level transitions should be consistent within four
significant figures with the corresponding EGk values in File 14. Therefore, care must
be taken to specify level energies to the appropriate number of significant figures.
14. Under Option 2, the sum of the transition probabilities (TPi ) over i should equal
1.0000 (that is, should be unity to within five significant figures).
15. The limit on the number of energy points for the tabulations of Y (E) or yk (E) is
given in Appendix G. This is an upper limit that will rarely be approached in practice
because yields are normally smoothly varying functions of incident neutron energy.
16. The limit on the number of interpolation regions is also given in Appendix G.
17. Tabulations of non-threshold data should normally cover at least the energy range
10−5 eV ≤ E ≤ 2 × 107 eV, where practical. Threshold data should be given from
threshold energy up to at least 2 × 107 eV, where practical.
18. Transition Probability Arrays for (n,n’γ) photons:
(a) The use of transition probability arrays (File 12, LO=2) is a convenient way
to represent a portion of the γ-rays produced by deexcitation of discrete levels
populated by (n,n’γ) and other reactions.
(b) Several conditions must be met before this representation can be used. Level
excitation cross sections (e.g. given in File 3 for neutrons as MT=51,...) must be
given from threshold energies up to the same maximum energy (no exceptions).
Decay properties of all levels must be known. The information given in File 12
must be consistent with data given in File 3.
(c) Usually, not all the conditions can be met. Part of the problem is the recommendation that level excitation cross sections for the first few levels be given for
neutron energies up to at least 20 MeV. This might be difficult to achieve, unless
one relies on nuclear model calculations.

183

Chapter 13
File 13: PHOTON PRODUCTION
CROSS SECTIONS
13.1

General Description

The purpose of File 13 is the same as that of File 12, namely, it can be used to represent
the neutron and photon energy dependence of photon production cross sections. In File 13,
however, absolute cross sections in barns are tabulated, and there is no need to refer to the
cross sections in File 3.

13.2

Formats

As in File 12, each section in File 13 gives information for a particular reaction type (MT
number). Each section always starts with a HEAD record and ends with a SEND record.
The representation of the energy dependence of the cross sections is accomplished by
tabulating a set of neutron energy-cross section pairs {E, σkγ (E)} for each discrete photon
and for the photon energy continuum. The subscript k designates a particular discrete
photon or the photon continuum, and the total number of such sets in NK. For discrete
photons, σkγ (E) is the photon production cross section (barns) for the photon designated by
k. For the photon continuum, σkγ (E) is the cross section, integrated over photon energy, for
the photon continuum1 designated by k. In the continuum case,
Z Eγmax
dσkγ
γ
(Eγ ← E) dEγ
(barns)
(13.1)
σk (E) =
dEγ
0
where Eγ designates photon energy (eV), and dσkγ /dEγ (Eγ ← E) is the absolute photon
energy distribution in barns/eV. The energy distribution can be further broken down as:
dσkγ
(Eγ ← E) = σkγ (E) fk (Eγ ← E)
dE γ
1

(13.2)

There should be no more than one energy continuum for each MT number used. If the decomposition
of a continuum into several parts is desired, this can be accomplished in File 15.

184

CHAPTER 13. FILE 13: PHOTON PRODUCTION CROSS SECTIONS
which obviously requires that:
Z

Eγmax

0

fk (Eγ ← E) dEγ = 1.

Any time a continuum representation is used for a given MT number in File 13, the normalized energy distribution, fk (Eγ ← E), must be given in File 15 under the same MT
number.
As a check quantity, the total photon production cross section,
γ
σtot
(E)

=

NK
X

σkγ (E)

(barns),

(13.3)

k=1

is also tabulated for each MT number, unless only one subsection is present (i.e., NK=1).
The following quantities are defined.
NK Number of discrete photons including the continuum.
ESk Energy of the level from which the photon originates. If the level is unknown
or if a continuous photon spectrum is produced, then ESk ≡ 0.0 should be
used.
EGk Photon energy for LP=0 or 1 or Binding Energy for LP=2. For a continuous
photon energy distribution, EGk =0.0 should be used.
LP Indicates whether or not the particular photon is a primary:
LP=0 origin of photons is not designated or not known, and the
photon energy is EGk ;
LP=1 for non-primary photons where the photon energy is again
simply EGk ;
LP=2 for primary photons where the photon energy EG’′k is given
by:
AWR
EG′k = EGk +
En
AWR + 1
LF Photon energy distribution law number, which presently has only two values
defined:
LF=1, a normalized tabulated function (in File 15), and
LF=2, a discrete photon energy.
The structure of a section in File 13 is2 :
2

If the total number of discrete photons and photon continua is one (NK=1), this TAB1 record is omitted.

185

CHAPTER 13. FILE 13: PHOTON PRODUCTION CROSS SECTIONS
[MAT, 13, MT/ ZA, AWR, 0, 0, NK, 0]HEAD
γ
[MAT, 13, MT/ 0.0, 0.0, 0, 0, NR, NP/ Eint / σtot
(E)]TAB1


------------------------
[MAT, 13, 0/ 0.0, 0.0, 0, 0, 0, 0] SEND

(NK>1)

and the structure of each subsection is:
[MAT, 13, MT/ EGk , ESk , LP, LF, NR, NP/ Eint / σkγ (E)] TAB1,

13.3

Procedures

1. The subsections are given in decreasing magnitude of EGk .
2. The convention is that the subsection for the continuum photons, if present, is last.
In this case, EGN K ≡ 0.0.
3. The values of EGk should be consistent to within four significant figures with the
corresponding EGk values in File 14.
4. ESk is the energy of the level from which the photon originates, if known. Otherwise
ESk ≡ 0.0.
5. If capture and fission resonance parameters are given in File 2, the corresponding
photon production should be given by using Option 1 of File 12, instead of using
photon production cross sections.
γ
6. The total photon production cross section table, σtot
(E) should span exactly the same
energy range as the combined energy range of all the σkγ (E). Within that range,

γ
σtot
(E)

=

NK
X

σkγ (E)

k=1

should hold within four significant figures. If only one energy distribution is given,
γ
either discrete or continuous (NK=1), the TAB1 record for the σtot
(E) is deleted.
7. The join of all sections in File 12 and 13 combined should represent the photon production data with no redundancy. For example, MT=4 cannot include any photons
given elsewhere under MT=51 through 91.
8. The limit on the number of energy points in a tabulation for any photon production
subsection is given in Appendix G. This is an upper limit; in practice, the minimum
number of points possible should be used. If there is extensive structure, the use of
File 12 should be seriously considered, because yields are normally much smoother
functions of incident neutron energy than cross sections.
186

CHAPTER 13. FILE 13: PHOTON PRODUCTION CROSS SECTIONS
9. The limit on the number of interpolation regions is also given in Appendix G.
10. Tabulations of non-threshold data should normally cover at least the energy range
10−5 eV ≤ E ≤ 2 × 107 eV, where practical. Threshold data should be given from
threshold energy up to at least 2 × 107 eV, where practical.

13.4

Preferred Representations

1. The recommended representation for (n,n’γ) reactions is photon production cross section (File 13) using MT=4. All discrete and continuum γ rays are given in a series of
subsections.
2. Photon production cross sections resemble the frequently measured or reported results.
3. The use of MT=4 eliminates confusion about whether the data represent an excitation
or de-excitation cross section (see File 11).
4. If for any reason MT=51,52 ... is used, it is understood that these data represent
de-excitation and not excitation cross sections (see 3 above). MT=51, 52, ... in File 3,
of course, means excitation cross sections.
5. Combined use of MT=4 and MT=51, 52, ... is not allowed.
6. Above a certain energy point probably it will not be possible to separate the various
components of the total γ production cross section. When this happens, it is preferable
to represent the data as MT=3.
7. Data for all other reactions should be given as photon production cross sections
(File 13) using the appropriate MT numbers. The same general rules outlined above
should be used.

187

Chapter 14
File 14: PHOTON ANGULAR
DISTRIBUTIONS
14.1

General Description

The purpose of File 14 is to provide a means for representing the angular distributions of
secondary photons produced in neutron interactions. Angular distributions should be given
for each discrete photon and photon continuum appearing in Files 12 and 13, even if the
distributions are isotropic.
The structure of File 14 is, with the exception of isotropic flag (LI), closely analogous to
that of File 4. Angular distributions for a specific reaction type (MT number) are given for
a series of incident neutron energies in order of increasing neutron energy. The energy range
covered should be the same as that for the data given under the corresponding reaction type
in File 12 or File 13. The data are given in ascending order of MT number.
The angular distributions are expressed as normalized probability distributions, that is:
Z 1
pk (µ, E) dµ = 1 ,
−1

where pk (µ, E) is the probability that an incident neutron of energy E will result in a
particular discrete photon or photon energy continuum (specified by k and MT number)
being emitted into unit cosine about an angle whose cosine is µ. Because the photon angular
distribution is assumed to have azimuthal symmetry, the distribution may be represented
as a Legendre series expansion,
2π dσkγ
(Ω, E)
σkγ (E) dΩ
NL
X
2l + 1 k
=
al (E)Pl (µ)
2
l=0

pk (µ, E) =

where

188

(14.1)
(14.2)

CHAPTER 14. FILE 14: PHOTON ANGULAR DISTRIBUTIONS
µ cosine of the reaction angle in the laboratory system,
E energy of the incident neutron in the laboratory system,
σkγ (E) photon production cross section for the discrete photon or photon continuum
specified by k, as given in either File 13 or in File 2, 3, and 12 combined,
l order of the Legendre polynomial,
dσkγ /dΩ differential photon production cross section in barns/steradian,
akl (E) the lth Legendre coefficient associated with the discrete photon or photon
continuum specified by k. It is implicitly assumed that ak0 (E) ≡ 1.0.
Z 1
k
pk (µ, E) Pl (µ)dµ
al (E) =
−1

Angular distributions may be given in File 14 by tabulating as a function of incident
neutron energy either the normalized probability distribution function, pk (µ, E), or the
Legendre polynomial expansion coefficients, akl (E). Provision is made in the format for
simple flags to denote isotropic angular distributions, either for a block of individual photons
within a reaction type or for all photons within a reaction type taken as a group.
If File 14 is used to describe the angular distributions of continuum spectra, separability of the photon energy and angular distributions is implied. If this is not an adequate
representation, File 6 must be used instead.

14.2

Formats

As usual, sections are ordered by increasing reaction type (MT) numbers. The following
definitions are required:
LI LI=0, distribution is not isotropic for all photons from this reaction type,
but may be for some photons.
LI=1, distribution is isotropic for all photons from this reaction type.
LTT LTT=1, data are given as Legendre coefficients, where ak0 (E) ≡ 1.0 is understood.
LTT=2, data are given in tabular form.
NK Number of discrete photons including the continuum (must equal the value
given in File 12 or 13).
NI Number of isotropic photon angular distributions given in a section (MT
number) for which LI = 0, i.e., a section with at least one anisotropic distribution.
NE Number of neutron energy points given in a TAB2 record.
189

CHAPTER 14. FILE 14: PHOTON ANGULAR DISTRIBUTIONS
NLi Highest value of l required at each neutron energy Ei .
ESk Energy of the level from which the photon originates.
If the level is unknown or if a continuous photon spectrum is produced, then
ESk =0.0 should be used.
EGk Photon energy as given in File 12 or 13.
For a continuous photon energy distribution, EGk ,=0.0 should be used.

14.2.1

Isotropic Distribution (LI=1)

If LI=1, then all photons for the reaction type (MT) in question are assumed to be isotropic.
This is a flag that the processing code can sense, and thus needless isotropic distribution
data are not entered in the file. In this case, the section is composed of a HEAD card and
a SEND card, as follows:
[MAT, 14, MT/ ZA, AWR, LI, 0, NK, 0]HEAD
[MAT, 14, 0 / 0.0, 0.0, 0, 0, 0, 0]SEND

(LI=1)

If LI=0, there are two possible structures for a section, depending upon the value of LTT.

14.2.2

Anisotropic Distribution with Legendre Coefficient Representation (LI=0, LTT=1)

The structure of a section with LI=0 and LTT=1 is:
[MAT, 14, MT/ ZA, AWR, LI, LTT, NK, NI]HEAD


-----------------------
[MAT, 14, 0/ 0.0, 0.0, 0, 0, 0, 0] SEND

(LI=0, LTT=1)

The structure of each record in the first block of NI subsections (for the NI isotropic photons)
is:
[MAT, 14, MT/ EGk , ESk , 0, 0, 0, 0] CONT
There is just one CONT record for each isotropic photon. (The set of CONT records is
empty if NI=0). The subsections are ordered in decreasing magnitude of EGk (photon
energy), and the continuum, if present and isotropic, appears last, with EGk =0.0.
This block of NI subsections is then followed by a block of NK−NI subsections for
the anisotropic photons in decreasing magnitude of EGk . The continuum, if present and
anisotropic, appears last, with EGk =0.0. The structure for the last NK−NI subsections is:

190

CHAPTER 14. FILE 14: PHOTON ANGULAR DISTRIBUTIONS
[MAT, 14, MT/ EGk , ESk , 0, 0, NR, NE/ Eint ] TAB2
[MAT, 14, MT/ 0.0, E1 , 0, 0, NL1 , 0/ al k (E1 )] LIST
[MAT, 14, MT/ 0.0, E2 , 0, 0, NL2 , 0/ al k (E2 )] LIST
------------------------------------[MAT, 14, MT/ 0.0, EN E , 0, 0, NLN E , 0/ akl (EN E )] LIST
Note that lists of the akl (E) start at l = 1 because ak0 (E) ≡ 1.0 is implicitly assumed.

14.2.3

Anisotropic Distribution with Tabulated Angular Distributions (LI=0, LTT=2)

The structure of a section for LI=0 and LTT=2 is:
[MAT, 14, MT/ ZA, AWR, LI, LTT, NK, NI] HEAD


----------------------------
[MAT, 14, 0/ 0.0, 0.0, 0, 0, 0, 0] SEND

(LI=0, LTT=2)

The structure of the first block of NI subsection (where NI may be zero) is the same as for
the case of a Legendre representation; i.e., it consists of one CONT record for each of the NI
isotropic photons in decreasing magnitude of EGk . The continuum, if present and isotropic,
appears last, with EGk ≡ 0.0. The structure of the first NI subsections is:
[MAT, 14, MT/ EGk , ESk , 0, 0, 0, 0] CONT
This block of NI subsections is then followed by a block of NK−NI subsections for the
anisotropic photons, again in decreasing magnitude of EGk , with the continuum, if present
and anisotropic, appearing last, with EGk ≡ 0.0. The structure of the last NK−NI subsections is:
[MAT, 14, MT/ EGk , Esk , 0, 0, NR, NE/ Eint ] TAB2
[MAT, 14, MT/ 0.0, E1 , 0, 0, NR, NP/ µint / pk (µ, E1 )] TAB1
[MAT, 14, MT/ 0.0, E2 , 0, 0, NR, NP/ µint / pk (µ, E2 )] TAB1
-------------------------------------------------[MAT, 14, MT/ 0.0, EN E , 0, 0, NR, NP/ µint / pk (µ, EN E )] TAB1

14.3

Procedures

1. The subsections are given in decreasing magnitude of EGk within each of the isotropic
and anisotropic blocks.
2. The convention is that the subsection for the continuous photon spectrum, if present,
appears last in its block. In this case, EGN K ≡ 0.0.

191

CHAPTER 14. FILE 14: PHOTON ANGULAR DISTRIBUTIONS
3. The values of EGk should be consistent within four significant figures with the corresponding EGk values in File 12 or 13. File 12, Option 2 (transition probability arrays),
the values of EGk are implicitly determined by the level energies.
4. ESk is the energy of the level from which the photon originates, if known. Otherwise,
ESk = 0.0 (as is always the case for the continuum).
5. Data should not appear in File 14 for photons that do not have production data given
in File 12 or 13. Conversely, for every photon appearing in File 12 or 13 an angular
distribution must be given in File 14. The neutron energy range for which the angular
distributions are given should be the same as that for which the photon production
data are given in File 12 or 13.
6. For LTT=1 (Legendre coefficients), the value of NL should be the minimum number
of coefficients that will reproduce the angular distribution with sufficient accuracy and
be positive everywhere. In all cases, NL should be an even number, ≤ 20.
7. The TAB1 records for the pk (µ, Ei ) within a subsection are given in increasing order
of neutron energy, Ei .
8. The tabulated probability functions, pk (µ, Ei ), should be normalized within four significant figures (to unity).
9. The interpolation scheme for pk (µ, E) with respect to E must be linearlinear or loglinear (INT=2 or 4) to preserve normality of the interpolated distributions. It is
recommended that the interpolation in µ be linear-linear (INT=2).
10. For LI=1 (isotropic distribution), the parameter NK is the number of photons in that
section and should be consistent with the NK values in Files 12 and 13.
11. The minimum amount of data should be used that will accurately represent the angular
distribution as a function of both µ and E.
12. If all photons for a reaction type (MT number) are isotropic, the LI=1 flag should be
used. The use of LI=0 and NI=NK is strongly discouraged. Likewise, isotropic distributions should not be entered explicitly as a tabulation or as a Legendre expansion
with akl (E) ≡ 0, l ≥ 1.
13. Angular distributions for photons must be given for all discrete and continuum photons. This can be done by specifying the data explicitly (by giving distributions)
or implicitly by using a flag meaning that all photons for a particular reaction (MT
number) are isotropic. Isotropic angular distributions should be specified unless the
anisotropy is ≥ 20%.

192

Chapter 15
File 15: CONTINUOUS PHOTON
ENERGY SPECTRA
15.1

General Description

File 15 provides a means for representing continuous energy distributions of secondary photons, expressed as normalized probability distributions. The energy distribution of each
photon continuum occurring in Files 12 and 13 should be specified in File 15 over the same
neutron energy range used in Files 12 and 13. Each section of File 15 gives the data for
a particular reaction type (MT number) and the sections are ordered by increasing MT
number. The energy distributions, f (Eγ ← E), are in units of eV−1 and are normalized so
that:
Z Eγmax
f (Eγ ← E) dEγ = 1 ,
0

Eγmax

where
is the maximum possible secondary photon energy and its value depends on the
incoming neutron energy as well as the particular nuclei involved.1 The energy distributions
f (Eγ ← E) can be broken down into the weighted sum of several different normalized
distributions in the following manner:
f (Eγ ← E) =

NC
X
j=1

pj (E) gj (Eγ ← E)

(eV)−1

(15.1)

where:
NC the number of partial distributions used to represent f (Eγ ← E),
gj (Eγ ← E) the jth normalized partial distribution in the units eV−1 , and
1

Note that the subscript k used in describing Files 12 and 13 has been dropped from f (Eγ ← E). This
is done because only one energy continuum is allowed for each MT number, and the subscript k has no
meaning in File 15. It is, in fact, the NKth subsection in File 12 or 13 that contains the production data for
the continuum.

193

CHAPTER 15. FILE 15: CONTINUOUS PHOTON ENERGY SPECTRA
pj (E) the probability or weight given to the jth partial distribution, gj (Eγ ← E).
The following condition is imposed.
Z Eγmax
0

gj (Eγ ← E) dEγ = 1 .

Thus,
NC
X

pj (E) = 1 .

j=1

The absolute energy distribution cross section, σ γ (Eγ ← E), can be constructed from the
expression:
σ γ (Eγ ← E) = σ γ (E) f (Eγ ← E) (b/eV),
where σ γ (E) is the integrated cross section for the continuum given either directly in File 13
or through the combination of Files 2, 3, and 12.
The system used to represent continuous photon energy distributions in File 15 is similar
to that used in File 5. At present, however, there is only one continuous distribution law
activated for File 15, i.e.,
gj (Eγ ← E) = g (Eγ ← E) ,

where g(Eγ ← E) represents an arbitrary tabulated function. In the future, new laws (for
example, the fission gamma-ray spectrum) may be added.

15.2

Formats

The structure of a section is:
[MAT, 15, MT/ ZA, AWR, 0, 0, NC, 0] HEAD


-------------------------------
[MAT, 15, 0/ 0.0, 0.0, 0, 0, 0, 0] SEND
For LF=1, the structure of a subsection is:
[MAT,
[MAT,
[MAT,
[MAT,

15, MT/ 0.0, 0.0, 0, LF, NR, NP/ Eint / pj (E)] TAB1
(LF=1)
15, MT/ 0.0, 0.0, 0, 0, NR, NE/ Eint ] TAB2
15, MT/ 0.0, E1 , 0, 0, NR, NP/ Eγint / g(Eγ ← E1 )] TAB1
15, MT/ 0.0, E2 , 0, 0, NR, NP/ Eγint / g(Eγ ← E2 )] TAB1
----------------------------[MAT, 15, MT/ 0.0, EN E , 0, 0, NR, NP/ Eγint / g(Eγ ← EN E )] TAB1

Only one distribution law is presently available (tabulated secondary photon energy
distribution). Therefore, formats for other laws remain to be defined, but their structures
194

CHAPTER 15. FILE 15: CONTINUOUS PHOTON ENERGY SPECTRA
will probably closely parallel those in File 5 for LF=5, 7, 9, and 11. When histogram
representations are used (interpolation scheme, INT=1), 0.25 to 0.5MeV photon energy
bands should be used. The incident energy ranges must agree with data given in File 12
and/or 13. Other procedures are the same as those recommended for File 5 data (tabulated
distribution).

15.3

Procedures

1. Photon energies, Eγ , within a subsection are given in order of increasing magnitude.
2. The TAB1 records for the g(Eγ ← Ei ) within a subsection are given in increasing
order of neutron energy, Ei .
3. The tabulated functions, g(Eγ ← Ei ), should be normalized to unity within four
significant figures.
4. The interpolation scheme for pj (E) must be either linear-linear or log-linear (INT=1, 2,
or 3) to preserve probabilities upon interpolation. Likewise, the interpolation scheme
for (Eγ ← E) must be linear-linear or log-linear with respect to E.
5. The neutron energy mesh should be a subset of that used for the yN K (E) tabulation
γ
in File 12 or for the σN
K (E) tabulation in File 13, and the energy ranges must be
identical. However, the neutron energy mesh for pj (E) need not be the same as that
for g(Eγ ← E), as long as they span the same range.
6. For an MT number appearing in both File 12 and File 13, a continuous photon energy
distribution (LF=1) can appear in only one of those files. Otherwise the distribution
as given in File 15 could not in general be uniquely associated with a corresponding
multiplicity or production cross section.
7. Use the minimum amount of data that will accurately represent the energy distribution
as a function of both Eγ and E. However, do not use too coarse a mesh for E, even if
the distributions are slowly varying functions of E, since the interpolated distribution
will always have a non-zero component up to the maximum energy at which either of
the original distributions has a non-zero component.
8. The limit on the number of neutron energy points for either pj (E) or g(Eγ ← E) is
given in Appendix G.
9. The limit on the number of photon energy points for g(Eγ ← E) is also given in
Appendix G.

195

Chapter 23
File 23: SMOOTH PHOTON
INTERACTION CROSS SECTIONS
23.1

General Comments on Photon Production

Photon interaction data are divided into two files. File 23 is analogous to File 3 and contains
the ”smooth” cross sections. File 27 contains the coherent scattering form factors and
incoherent scattering functions (see Chapter 27).
Electron interaction data are divided into two different files. The smooth cross sections
for elastic scattering, bremsstrahlung, excitation, and the ionization of different atomic
subshells are given in File 23. File 26 is used to give the angular distribution for elastically
scattered electrons, the outgoing photon spectra and energy loss for bremsstrahlung, the
energy transfer for excitation, and the spectra of the scattered and recoil electrons associated
with subshell ionization.
Both photo-atomic and electro-atomic reactions can leave the atom in an ionized state.
See Section 28 for a description of the atomic relaxation data needed to compute the outgoing
X-ray and electron spectra as an ionized atom relaxes back to neutrality.

23.2

General Description

This File is for the integrated photon and electron interaction crosssections. The reaction
type (MT) numbers for photon and electron interaction are in the 500 series. Several common
photon and electron interactions have been assigned MT numbers:
MT
523
526
527
528
533
534-572

Reaction Description
Photo-excitation cross sections
Electro-atomic elastic scattering
Electro-atomic bremsstrahlung
Electro-atomic excitation cross section
Atomic relaxation data (see Section 28)
Photo-electric or electro-atomic subshell ionization

196

CHAPTER 23. FILE 23: PHOTON INTERACTION
Photon cross sections, such as the total cross section, coherent elastic scattering cross
section, and incoherent (Compton) cross section, are given in File 23, which has essentially
the same structure as File 3. These data are given as a function of energy, Eγ , where Eγ is
the energy of the incident photon (in eV). The data are given as energy-cross section pairs.
Similarly, electron cross sections, such as elastic scattering, bremsstrahlung, ionization,
and excitation, are given in File 23. These data are given as a function of the electron energy
in eV, and they are also given as energy-cross section pairs.
Each section in File 23 contains the data for a particular reaction type (MT number).
The sections are ordered by increasing MT number.

23.3

Formats

The following quantities are defined:
ZA,AWR Standard material charge and mass parameters.
EPE Subshell binding energy (equal to the photoelectric edge energy) in eV. The
value is zero if MT is not in the 534-599 range.
EFL Fluorescence yield (eV/photoionization).
Value is zero if not a photoelectric subshell ionization cross section.
NR,NP,Eint Standard TAB1 interpolation parameters (see Section 1.3.1).
σ(E) Cross section (barns) for a photon or electron of energy E given as a table
of NP energy-cross section pairs.
The structure of a section is:
[MAT, 23, MT/ ZA, AWR, 0, 0, 0, 0] HEAD
[MAT, 23, MT/ EPE, EFL, 0, 0, NR, NP/ Eint / σ(E)] TAB1
[MAT, 23, 0/ 0.0, 0.0, 0, 0, 0, 0] SEND

23.4

Procedures

1. Values are usually for elements; hence, except for mono-isotopic elements,
ZA=Z × 1000.0; also, AWR should be for the naturally occurring element.
2. Photoelectric edges must not be multi-valued. The edges are defined by two energy
points differing in the fourth or fifth significant figure.
3. Interpolation is normally log-log (INT=5).
4. Kerma factor (energy deposition coefficients) libraries will normally be local because
there is no universal definition. The application will determine whether annihilation
or other radiation fractions are subtracted.
197

Chapter 26
File 26: SECONDARY
DISTRIBUTIONS FOR PHOTOAND ELECTRO-ATOMIC DATA
26.1

General Description

This file is used to represent the secondary photons or electrons emitted after electro-atomic
reactions, energy given to the residual atom, and the energy transfer associated with excitation. It is based on File 6 formats with appropriate simplifications.
Elastic scattering is represented by the normalized angular distribution for the scattered
electron given in tabulated form (LAW=2, LANG=12) for scattering cosines ranging from
-1 to .999999. Because of the very large mass of the residual atom with respect to the mass
of the electron, it is assumed that the electron scatters without a change of energy, and there
is no energy transfer to the residual atom.
Bremsstrahlung is represented using two subsections. The electron is assumed to scatter
straight ahead with an energy loss described using the LAW=8 format. The photon is
assumed to be emitted isotropically with spectra given as tabulated distributions (LAW=1,
LANG=1, NA=0). Energy transfer to the residual atom is ignored.
Excitation occurs when the incident electron loses some of its energy by exciting the
outer electrons of the atom to higher energy states. The energy transfer to the residual
atom is represented using LAW=8. The electron is assumed to continue in the straightahead direction.
Ionization is represented by giving a section of File 26 for each sub-shell (MT=534,535,...).
There are two electrons coming out of each ionization reaction: the scattered electron and
the recoil electron. Because these two particles are identical, it is arbitrarily assumed that
the particle with the lower energy is the ”recoil” electron, and the one with the higher
energy is the ”scattered” electron. If Ek is the binding energy for the sub-shell, the energy
of the recoil electron varies from 0 to (E − Ek )/2, and the energy of the scattered electron
varies from (E − Ek )/2 to E − Ek . Only the distribution for the ”recoil” electron is given
in File 26. The user can select a recoil energy Er from the distribution and then generate
the corresponding scattered electron with energy E − Ek − Er . The value of Ek is given
198

CHAPTER 26. FILE 26: SECONDARY DISTRIBUTIONS FOR PHOTO- AND
ELECTRO-ATOMIC DATA
in the corresponding section of File 3. It is assumed that both the scattered and the recoil
electrons continue in the direction of the incident electron, and that no kinetic energy is
transferred to the residual atom.
The relaxation of the residual atom left after ionization results in the emission of additional X-rays and electrons. Those spectra can be computed using the atomic relaxation
data described in Section 28.

26.2

Formats

The following quantities are defined:
ZA,AWR Standard material charge and mass parameters
NK Number of subsections in this section (MT).
Each subsection describes one reaction product (in this case, photons or
electrons), or a subsection can describe the energy transfer associated with
excitation or bremsstrahlung.
ZAP Product identifier: zero for photons, and 11 for electrons.
LAW Distinguishes between different representations of the product distributions:

LAW=1 continuum distribution (used for bremsstrahlung and ionization);
LAW=2 two-body angular distribution (used for elastic); and
LAW=8 energy transfer for excitation (used for excitation and
bremsstrahlung).
NR,NP,Eint Standard TAB1 interpolation parameters.
y(E) Yield for the particle being described (the yield is always 1 in File 26, but
the general format is retained for consistency with File 6).
ET(E) Energy transfer during electro-atomic excitation or bremsstrahlung (eV).
The structure of a section is:
[MAT,26, MT/ ZA, 0.0, 0,
0, NK, 0]HEAD
[MAT,26, MT/ZAP, 0.0, 0, LAW, NR, NP/Eint / y(E)] TAB1

----------------------------repeat TAB1 and LAW-dependent structures
for the rest of the NK subsections
----------------------------[MAT,26,MT/ 0.0, 0.0, 0,
0, 0, 0]SEND
199

CHAPTER 26. FILE 26: SECONDARY DISTRIBUTIONS FOR PHOTO- AND
ELECTRO-ATOMIC DATA
The subsections for bremsstrahlung are given in the order photons, then electrons. The
contents of the subsection for each LAW are given below.

26.2.1

Continuum Distribution (LAW=1)

This law is the same as LAW=1 for File 6, except that only LANG=1, NA=0, representing
a simple tabulated energy distribution without angle dependence, is allowed.

26.2.2

Two-Body Angular Distribution (LAW=2)

This law is the same as LAW=2 for File 6, except that only LANG=11-15 for linear-linear
tabulated angular distributions, is allowed. It is only used for the electro-atomic elastic
scattering reaction, and the cosine range is -1 to 0.99999.

26.2.3

Energy Transfer for Excitation (LAW=8)

This law is used to give only the energy transfer during excitation and the energy loss for
bremsstrahlung:
[MAT,26, MT/ 0.0, 0.0, 0, 0, NR, NP/ Eint / ET(E)] TAB1

200

Chapter 27
File 27: ATOMIC FORM FACTORS
OR SCATTERING FUNCTIONS
27.1

General Description

The ENDF system for neutron and photon production data allows two alternatives for
storing angular distribution data. One is by probability per unit cos(θ) vs. cos(θ), and
the other is by Legendre coefficients. Actually, neither of these is a ”natural” method
for photons. The natural method would be atomic form factors or incoherent scattering
functions. These are discussed briefly below.

27.1.1

Incoherent Scattering

The cross section for incoherent scattering is given by:
dσi (E, E ′ , µ)
dσi (E, E ′ , µ)
= S(q; Z)
,
dµ
dµ

(27.1)

where:
dσi /dµ the Klein-Nishina cross section1 which can be written in a closed form.
S(q; Z) the incoherent scattering function. At high momentum transfer (q), S approaches Z. In the other limit, S(0, Z) = 0.
q the momentum of the recoil electron (in inverse angstroms2 ).
"

q =α 1+
1



α′
α

2

− 2µ



α′
α

#1/2

(27.2)

O. Klein and Y. Nishina, Z. Phys. 52, 853 (1929).
In ENDF, q is given in inverse angstroms, as customarily reported in the literature. The above equation
shows q in m0 c units. See Appendix H for unit conversions.
2

201

CHAPTER 27. FILE 27: ATOMIC FORM FACTORS OR SCATTERING FUNCTIONS
α = Eγ /m0 c2 ,
Eγ′ scattered photon energy,
µ cosθ.
The angular distribution can then easily be calculated. Values of S(q; Z) are tabulated as
a function of q in File 27. The user presumably will have subroutines available for calculating
q for energies and angles of interest and for calculating Klein-Nishina cross sections. The
user will then generate the cross sections for the appropriate cases by calculating q’s, looking
up the appropriate values of S, and substituting them in the above formula.

27.1.2

Cooherent Scattering

The coherent scattering cross section is given by:
o

n
dσcoh (E, E ′ , µ)
2
= πr02 1 + µ2 [F (q; Z) + F ′ (E)] + F ′′ (E)2 ,
dµ

where:

(27.3)

q = α [2(1 − µ)]1/2 , the recoil momentum of the atom (in inverse angstroms),
r0 = e2 /m0 c2 , the classical radius of the electron.
F ′ (E) the real part of the anomalous scattering factor.
F ′′ (E) the imaginary part of the anomalous scattering factor.
The quantity F (q; Z) is a form factor, which can be easily tabulated. At high momentum
transfer (q), F approaches zero. In the other limit F (0; Z) tends to Z. The anomalous
scattering factors are assumed to be isotropic. In addition, they smoothly approach zero at
1.0 MeV and can be assumed to be zero at higher energies.
An alternative way of presenting the photon scattering data would be to tabulate incoherent scattering functions and form factors. Users could then provide processing codes to
generate the cross sections from this information. The calculation is quite straightforward
and allows the user to generate all his scattering data from a relatively small table of numbers. The incoherent and coherent scattering data should always be presented as scattering
functions and form factors, respectively, whether or not data are included in File 6.

27.2

Formats

The structure of a section is very similar to that of File 3 (and 23) and is:
[MAT, 27, MT/ ZA, AWR, 0, 0, 0, 0] HEAD
[MAT, 27, MT/ 0.0,
Z, 0, 0, NR, NP/ qint / H(q; Z)]TAB1
[MAT, 27, 0/ 0.0, 0.0, 0, 0, 0, 0] SEND
202

CHAPTER 27. FILE 27: ATOMIC FORM FACTORS OR SCATTERING FUNCTIONS
The general symbol H(q; Z) is used for either F (q; Z) or S(q; Z) for coherent and incoherent scattering, respectively, or for the anomalous factors:
[MAT, 27, MT/ ZA, AWR, 0, 0, 0, 0] HEAD
[MAT, 27, MT/ 0.0,
Z, 0, 0, NR, NP/ Eint / F (E)]TAB1
[MAT, 27, 0/ 0.0, 0.0, 0, 0, 0, 0] SEND

27.3

Procedures

1. Values of H(q; Z) should be entered in each case for the entire energy range for which
integrated coherent and incoherent cross sections are given in File 23. This is true
even though the respective values may be 0.0 or Z over most of the (higher) energy
range.
2. The value of Z is entered in floating-point format.

203

Chapter 28
File 28: ATOMIC RELAXATION
DATA
28.1

General Description

An atom can be ionized due to a variety of interactions. For example, due to photon or
electron interactions the probability of ionizing a particular subshell of the atomic structure
(K, L1, L2, etc.) is determined by using the subshell cross sections (MT=534-599). For
example, if an incident photon of energy E ionizes the K subshell with binding energy EK ,
the atom will emit an electron with energy E − EK , and the atomic structure will be left
ionized, with a ”hole” in the K subshell. One way the atom can proceed to fill this hole is
to bring down an electron from a higher energy level, for example L1, with the simultaneous
emission of an X-ray of energy EK − EL1 . This is a radiative transition. An alternative
path is to bring down an electron from a higher level with the simultaneous emission of an
electron from that level or a higher one. As an example, you might see an electron of energy
EK −EL1 −EM 1 , which fills the vacancy in the K shell and leaves new holes in the L1 and M1
shells. These are called non-radiative transitions. The process will then continue by filling
the new holes from higher levels, etc., until all the ionization energy has been accounted for
by the emission of X-rays and electrons.
The electrons produced by this atomic relaxation can be used as a source for a subsequent
electron transport calculation, or their energy can just be added to the local heating. The
X-rays can be transported elsewhere to cause additional photo-atomic reactions. In general,
the use of File 28 is indicated when high-Z materials are present and photon energies of less
than 1 MeV are of interest.
This file is provided to give the information necessary to compute the emission of Xrays and electrons associated with atomic relaxation cross section. It is based on EADL, the
Evaluated Atomic Data Library developed by D. E. (Red) Cullen at the Lawrence Livermore
National Laboratory (LLNL).
This file gives the subshell energies, emission energies, transition probabilities, and other
quantities needed to compute the X-ray and electron spectra from ionized atoms. It always
uses MT=533. It works together with the photoelectric subshell cross sections from MF=23,
MT=534-599.
204

CHAPTER 28. FILE 28: ATOMIC RELAXATION DATA

28.2

Formats

The following quantities are defined:
ZA, AWR Standard material charge and mass parameters
NSS Number of subshells
SUBI Subshell designator (see the table below)
SUBJ Secondary subshell designator
SUBK Tertiary subshell designator
(if SUBK is zero for a particular transition, it is a radiative transition; otherwise, it is a non-radiative transition.)
EBI Binding energy for subshell (eV)
ELN Number of electrons in subshell when neutral (given as a floating-point value)
NTR Number of transitions
FTR Fractional probability of transition
ETR Energy of transition (eV)
Table
Designator
1.
2.
3.
4.
5.
6.
etc.

of Subshell Designators
MT
Subshell
K
(1s1/2)
534
L1 (2s1/2)
535
L2 (2p1/2)
536
L3 (2p3/2)
537
M1 (3s1/2)
538
M2 (3p1/2)
539

The structure of a section of File 28 is as follows:
[MAT,28,533/
ZA, AWR, 0, 0, NSS,
0]HEAD
[MAT,28,533/SUBI1 , 0.0, 0, 0, NW, NTR/ NW=6*(1+NTR)
EBI1 ,
ELN1 , 0.0,
0.0, 0.0, 0.0,
SUBJ1 , SUBK1 , ETR1 , FTR1 , 0.0, 0.0,
-----------------------------------SUBJN T R ,SUBKN T R ,ETRN T R ,FTRN T R , 0.0, 0.0]LIST
-----------------------------------[repeat LIST for the rest of the NSS subshells]
[MAT,28,533/ 0.0, 0.0, 0, 0, 0, 0] SEND
205

CHAPTER 28. FILE 28: ATOMIC RELAXATION DATA

28.3

Procedures

Sections with MF=28, MT=533 are used together with either photo-atomic or electro-atomic
data evaluations. The value of NSS must be consistent with the number of subshell ionization
cross sections given in File 23 (MT=534,535, ...). Note that the subshell cross section MT
value equals the subshell designator number SUBI plus 533. Subshell LIST records are
given in order of increasing SUBI. Similarly, transitions are given in order of increasing
SUBJ first, and increasing SUBK second. This means that radiative transitions appear
before non-radiative ones for each subshell.
It is possible to have NTR=0 if there are no allowed transitions from higher subshells to
a particular subshell.

206

CHAPTER 28. FILE 28: ATOMIC RELAXATION DATA
Example 1: Atomic Relaxation Data
6.000000+3 11.9078164
-1
0.000000+0 0.000000+0
0
0.000000+0 0.000000+0
0
0.000000+0 0.000000+0
0
6-C - 0 LLNL
EVAL-DEC90 CULLEN
DIST----ENDF/B-VI
MATERIAL 600
-----ATOMIC RELAXATION DATA
------ENDF-6 FORMAT
converted from EADL
1
28

6.000000+3
1.000000+0
2.910100+2
5.000000+0
6.000000+0
3.000000+0
3.000000+0
3.000000+0
5.000000+0
5.000000+0
6.000000+0
3.000000+0
1.756000+1
5.000000+0
8.990000+0
6.000000+0
8.980000+0

11.9078164
0.000000+0
2.000000+0
0.000000+0
0.000000+0
3.000000+0
5.000000+0
6.000000+0
5.000000+0
6.000000+0
6.000000+0
0.000000+0
2.000000+0
0.000000+0
6.700000-1
0.000000+0
1.330000+0

0
0
0.000000+0
2.820200+2
2.820300+2
2.558900+2
2.644600+2
2.644700+2
2.730300+2
2.730400+2
2.730500+2
0
0.000000+0
0
0.000000+0
0
0.000000+0

0
0
0
0

0
0
6
6

451
533

12
17

0
0
0.000000+0
5.614880-4
1.120600-3
4.136090-1
1.361900-1
2.710990-1
4.207480-3
1.100120-1
6.320080-2
0
0.000000+0
0
0.000000+0
0
0.000000+0

4
54
0.000000+0
0.000000+0
0.000000+0
0.000000+0
0.000000+0
0.000000+0
0.000000+0
0.000000+0
0.000000+0
6
0.000000+0
6
0.000000+0
6
0.000000+0

207

10 0
0 6001451
6 6001451
6 6001451
2 6001451
6001451
6001451
6001451
6001451
6001451
6001451
0 6001451
0 6001451
6001 0
6000 0
060028533
860028533
0.000000+060028533
0.000000+060028533
0.000000+060028533
0.000000+060028533
0.000000+060028533
0.000000+060028533
0.000000+060028533
0.000000+060028533
0.000000+060028533
060028533
0.000000+060028533
060028533
0.000000+060028533
060028533
0.000000+060028533
60028 0
600 0 0
0 0 0
-1 0 0

0
1
2
3
4
5
6
7
8
9
10
11
12
13
14
15
16
17
18
19
20
21
22
23
24
25
26
27
28
29
30
31
32
33
34
35

CHAPTER 28. FILE 28: ATOMIC RELAXATION DATA
Example 2: Electron Interaction Data
electron interaction data converted from EEDL
6.000000+3 11.9078164
-1
0
0.000000+0 0.000000+0
0
0
5.438673-4 1.00000+11
0
0
0.000000+0 0.000000+0
0
0
6-C - 0 LLNL
EVAL-DEC89 CULLEN
DIST----ENDF/B-VI
MATERIAL 600
-----ELECTRO-ATOMIC DATA
------ENDF-6 FORMAT
converted from EEDL
1
451
23
526
23
527
23
528
23
534
23
536
23
538
23
539
26
526
26
527
26
534
26
536
26
538
26
539

6.000000+3
0.000000+0
101
1.000000+1
1.995260+1
3.981070+1
7.943280+1
1.584890+2
3.162280+2
6.309570+2

11.9078164
0.000000+0
2
3.063510+9
1.660950+9
9.008210+8
4.889770+8
2.659610+8
1.447630+8
7.900840+7

0
0
113
6

24
37
31
64
12
14
15
15
353
197
120
151
155
155

0
0

0
0

0
1

1.258930+1
2.511890+1
5.011870+1
1.000000+2
1.995260+2
3.981070+2
7.943280+2

2.498030+9
1.354370+9
7.347880+8
3.991460+8
2.171000+8
1.183030+8
6.458900+7

1.584890+1
3.162280+1
6.309570+1
1.258930+2
2.511890+2
5.011870+2
1.000000+3

11.9078164
0
0
5.438673-4
0
2
2
1.000000+0 1.00000+11 1.000000+0
0.000000+0
0
0
2

1
1

10 0
0 6001451
6 6001451
6 6001451
14 6001451
6001451
6001451
6001451
6001451
6001451
6001451
0 6001451
0 6001451
0 6001451
0 6001451
0 6001451
0 6001451
0 6001451
0 6001451
0 6001451
0 6001451
0 6001451
0 6001451
0 6001451
0 6001451
6001 0
6000 0
060023526
10160023526
60023526
2.036940+960023526
1.104370+960023526
5.993570+860023526
3.258180+860023526
1.772160+860023526
9.667950+760023526
5.282720+760023526

0
1
2
3
4
5
6
7
8
9
10
11
12
13
14
15
16
17
18
19
20
21
22
23
24
25
26
27
28
29
30
31
32
33
34
35
36

060026526
260026526
60026526
60026526
1660026526
60026526

223
224
225
226
227
228

...
6.000000+3
1.100000+1
2
1.000000+1
0.000000+0
16

208

1

CHAPTER 28. FILE 28: ATOMIC RELAXATION DATA
0.000000+0
-1.000000+0
0.000000+0
-1.000000+0
-7.500000-2
4.300000-1
6.925000-1
8.325000-1
9.080000-1
9.510000-1
9.760000-1
9.915001-1
0.000000+0
-1.000000+0

1.000000+1
12
5.000000-1 9.999990-1
1.000000+3
12
7.349540-3-6.350000-1
1.996880-2 1.350000-1
5.580130-2 5.350000-1
1.559570-1 7.500000-1
4.339260-1 8.625000-1
1.212620+0 9.250000-1
3.432260+0 9.610000-1
9.598280+0 9.820000-1
2.744700+1 9.956001-1
2.000000+3
12
3.008670-3-9.000000-1

0
4
5.000000-1
0
54
1.014680-2-3.300000-1
2.832790-2 3.000000-1
7.828660-2 6.225000-1
2.204030-1 7.950000-1
6.085020-1 8.875000-1
1.713720+0 9.390000-1
4.874960+0 9.690000-1
1.360430+1 9.870000-1
4.014720+1 9.999990-1
0
64
3.301630-3-5.350000-1

260026526
60026526
2760026526
1.408430-260026526
3.982680-260026526
1.108190-160026526
3.078220-160026526
8.591520-160026526
2.413930+060026526
6.802370+060026526
1.914880+160026526
6.259920+160026526
3260026526
4.906250-360026526

229
230
231
232
233
234
235
236
237
238
239
240
241
242

060026534
260026534
60026534
60026534
760026534
60026534
260026534
60026534
260026534
60026534
1460026534
6.229690-360026534
3.138130-360026534
1.061790-360026534
3.895820-460026534
60026534
3560026534
5.837310-360026534
2.990940-360026534
1.378130-360026534

775
776
777
778
779
780
781
782
783
784
785
786
787
788
789
790
791
792
793
794

...
6.000000+3
1.100000+1
2
2.910100+2
0.000000+0
7
0.000000+0
1.000000-2
0.000000+0
3.152220-2
0.000000+0
1.000000-1
3.162280+1
1.258930+2
3.162280+2
6.309570+2
0.000000+0
1.000000-1
3.162280+1
1.258930+2

11.9078164
5.438673-4
2
1.000000+0
0.000000+0
2
2.910100+2
1.111110+1
2.973140+2
3.265060-1
1.995260+3
7.771360-3
5.440240-3
2.429030-3
7.491710-4
2.631530-4
3.981070+4
7.237010-3
5.119990-3
2.325440-3

0
0

0
1

1
1

1.00000+11 1.000000+0
1
2

1

0
1.000000-1
0
3.152220+0
0
6.309570+0
6.309570+1
1.778280+2
3.758380+2
8.521260+2
0
6.309570+0
6.309570+1
1.496240+2

0
1.111110+1
0
3.143760-1
0
7.133700-3
4.054250-3
1.605960-3
5.767510-4
2.098710-4
0
6.655140-3
3.846900-3
1.886340-3

4
4
28
1.412540+1
8.912520+1
2.511890+2
4.869680+2
70
1.412540+1
8.912520+1
1.938660+2

...
2.06836+10 4.46563-21 2.51189+10 3.13693-21 3.16228+10 2.30613-2160026539 1357
3.98107+10 1.69536-21 5.00000+10 1.51523-21
60026539 1358
60026 0 1359
600 0 0 1360
0 0 0 1361
-1 0 0 1362

209

Chapter 29
INTRODUCTION TO DATA
COVARIANCE FILES
29.1

General Comments

The inclusion of uncertainty estimates is intrinsic to any evaluation of physical constants because
the practical utility of a ”constant” depends on whether the true magnitude of the quantity is
sufficiently close to the quoted best value. The need is now accepted to include uncertainties in
evaluated nuclear cross section files in order that the propagated uncertainties in nuclear analytic
results can be estimated. The resulting files are called ”covariance files” as shorthand for a more
complete name such as ”files of nuclear variance and covariance data.” The priority for development
of formats and evaluation of covariance data is highest where the sensitivity of important calculated
results to the quantities in the associated cross section file is high.
Until ENDF/B-IV, the only means available to evaluators for communicating the estimated uncertainties in the evaluated data was through publication of the documentation of the evaluations.
During the preparation of ENDF/IV and -V, a Data Covariance Subcommittee of CSEWG was
formed to coordinate the efforts at standardizing statements made about the data uncertainties
and correlations.
One of the important aspects of nuclear data and of cross sections in particular is that the
various data tend to be correlated to an important degree through the measurement processes and
the different corrections made to the observable quantities to obtain the microscopic cross sections.
In many applications when one is interested in estimating the uncertainties in calculated results
due to the cross sections, the correlations among the data play a crucial role.
In principle, the uncertainties in the results of a calculation due to the data uncertainties can be
calculated, provided one is given all of the variances in and covariances among the data elements.
In practice, in addition to the uncertainties due to the basic data, the results of calculations have
uncertainties due to imperfections in the calculational models used. In some situations ”modeling
uncertainties” may dominate the uncertainties in computed results; in others they are negligible
compared to the effects of microscopic data uncertainties. In principle improving the models may
reduce ”modeling uncertainties”, although sometimes at a large cost. The data uncertainties may
also be reduced, often at large costs, by performing better measurements, new kinds of measurements, or sometimes a more refined analysis of existing data.
One of the requirements of the uncertainty information is that it be easily processed to yield
the (variances and) covariances for the multigroup or other ”data” used directly in the calculations.

210

CHAPTER 29. INTRODUCTION TO COVARIANCE FILES
For ENDF/B-IV, the principle of having the uncertainty information on the data file was adopted
and a trial formalism was developed. This formalism has the virtue that the information is in such
a form that it can be easily processed with minor modification to existing processing codes. Only a
few evaluations of ENDF/B-IV were issued with data covariance information in this format. Since
then, considerably more work has been done in trying to quantify data covariances within the
ENDF formalism and using the information for purposes of sensitivity studies. These sensitivity
studies have been made in three different areas where the data covariances play a crucial role:
propagation of uncertainties to final calculated results, adjustment of data sets incorporating information from some integral measurements, and determination of data accuracies needed to meet
targeted uncertainties in results. The formalism and formats for representing data covariances in
ENDF/B-V were extended to cover all neutron cross section data in the files.
Formats and procedures exist in ENDF-6 for representing the data covariances in fission neutron
multiplicity (File 1), resonance parameters (File 2), neutron cross sections (Files 3 and 10), angular
distributions (File 4) and energy distributions (File 5). There is also the capability to represent data
covariances obtained from parameter covariances and sensitivities. The ability to represent cross
section uncertainties is rather complete, while for other parameters in the ENDF files there may be
restrictions. In cases such as inelastic scattering one may employ the subterfuge of pseudo-discrete
levels to treat a continuum using the formats and procedures of Files 3 and 33.
Since covariance files may be incomplete, the absence of covariance data in a file in ENDF-6
formats does not imply that the uncertainty component of interest has been evaluated as zero.
Evaluators should not unintentionally enter explicit zero covariance components into a file, since
these would imply to a user that the uncertainty or correlation has been evaluated as negligibly
small.
The dominant reason for the inclusion of covariance files in the ENDF system is to enable estimation of nuclear data contributions to the uncertainties in calculated results for nuclear systems
having broad (neutron) spectra. Therefore, in developing the ENDF formats the highest priority
was given to attaining this goal. The ENDF covariance files are structured to enable processing
them to any energy group structure. As is explained most fully in Chapter 33, except for LB=8
and 9 sub-subsections, the stored quantities are defined to yield the covariances between point
cross sections. To simplify processing, the magnitudes of these components are constant between
the points on the defined energy grid.
The files have a histogram appearance, but the quantities have a precise definition that can
lead to incorrect inferences if the encoded values are used for other than the primary purpose
of uncertainty propagation with broad particle energy spectra. For example, File 33 (except for
LB=8 and 9) sub-subsections literally implies that the cross sections at any two points within the
same energy grid interval are perfectly correlated, and that the uncertainty is no larger for a cross
section averaged over a tiny energy interval than if it were averaged over the whole interval between
grid points. The LB=8 and 9 formats allow the evaluator to avoid this unrealistic implication. A
broadly spaced energy grid was usually chosen in the past to achieve the primary purpose without
attempting to provide greater covariance data detail than is warranted by the available information.
As indicated above, the main purpose of the covariance information in ENDF-6 formats is to
permit the propagation of nuclear data uncertainties for applications with broad neutron spectra.
Users of the file should interpret the files as they were designed. If modifications to the covariance
data must be made by users to place the data on a finer grid without reconsideration of the
uncertainties in the underlying data, those modifications should be designed so that the original
evaluator’s covariance data is recovered if the modified results are collapsed to the evaluator’s
energy grid.

211

CHAPTER 29. INTRODUCTION TO COVARIANCE FILES
Modifications of covariance files to a finer grid have been required in the past by users who
employ the adjustment equations to update an existing evaluation by ”adding” new data and
their associated covariances. To minimize the extent to which such users will be tempted to make
ad hoc changes to covariance files, covariance evaluators for reactions of particular importance
should employ relatively fine energy meshes to reduce the difficulties to be encountered by future
evaluators/users of the covariance files. Overlapping structures in energy and other techniques
should be used to reduce the occurrence of large changes in correlation as one crosses any arbitrary
energy boundary. The File 30 format provides an alternate way to avoid the effects of artificial
energy boundaries.
It is appropriate to define uncertainty quantities1 . Each cross section or related quantity in
an ENDF file represents a physical quantity that has a definite, though unknown true magnitude.
The knowledge of each such quantity X is summarized by its density function defined so that
f (X) ∆X is the probability that the true numerical value of X lies in the range ∆X about X.
The marginal density function f (X) is the average over all other independent variables Y , Z, etc.
of the overall multivariate density function for the cross section data base. The shape of a density
function depends on the experiments that have been performed, relevant to estimating the true
values of the data elements. The density function has unit normalization for each variable.
The ”expected value”, hg(X)i, of any function g(X) is given by the average value of that
function over the marginal density function. The simplest example is the expected value of the
quantity itself:
Z
hXi =

X f (X) dX .

(29.1)

In practice, one often uses the same symbol for a physical quantity, its expected value, and its
value in a particular data set. In the notation used in this Chapter, the true value of the physical
quantity is equal to X = hXi + δX, where hδXi = 0. In this language the cross section, etc.,
quantities in ENDF files are expected values.
The width of the density function reflects the scatter among experimental cross section results
and/or the uncertainties ascribed to the values by the experimenters. That width is a property
of the experiments, not of the cross section quantity, so one cannot in the usual sense ”measure”
nuclear covariance data. The width arises from the ambiguity with which each underlying experimental result defines the true value. These ambiguities are quantified as ”errors” with modifiers like
”systematic” or ”statistical” to indicate the origin of the ambiguities and modifiers like ”standard”
or ”relative” to indicate the normalization of the uncertainty quantities. Since both systematic errors and statistical counting errors broaden the density functions of evaluated quantities, evaluated
uncertainty data must combine both types. The systematic uncertainties are harder to estimate,
and are larger than statistical counting uncertainties in most modern nuclear experiments.
The ENDF-6 formats deal only with the expected values of quantities and the second-degree
moments of the joint density function describing the evaluator’s knowledge of the true value of the
nuclear data vector. It is not necessary to assume that the density functions are normal in shape,
or otherwise, unless one must estimate the probability that the true value lies within a certain
range of the expected value. The ENDF-6 covariance quantities are not intended to represent, and
cannot well represent, any known difference between ENDF-6 formatted cross section and some
more-recently realized ”better” evaluation, or any cross-section imprecision induced by ENDF-6
1
The treatment below is paraphrased from R. Peelle, Sensitivity and Uncertainty Analysis of Reactor
Performance Parameters, Advances in Nuclear Science and Technology, Vol. 14, pp 11, Lewins and Becker,
Eds., Plenum Press, New York, 1982.

212

CHAPTER 29. INTRODUCTION TO COVARIANCE FILES
procedures, or the widths of any physical distributions such as the fission neutron multiplicity
distribution P (ν).
The following quantities are defined that relate to the second moments of the density function.
Here hXi and hY i are cross section or related quantities in a file using ENDF-6 formats. The
quantity f (X, Y ) is the full density function averaged over all variables other than X and Y .
Recall that δX = X − hXi.
The frequently used quantities are the covariance between X and Y :
Cov(X, Y ) = hδX δY i
Z Z
=
(X − hXi)(Y − hY i) F (X, Y ) dX dY,
the variance of X:
Var(X) = Cov(X, X) = hδX 2 i ,
the standard error or uncertainty in hXi:
s(X) = [V ar(X)]1/2
and the correlation coefficient between X and Y
ρ(X, Y ) = Cov(X, Y )/ [s(X) s(Y )] .
The relative standard error, s(X)/hXi, the relative variance Var(X)/hXi2 and the relative covariance, Rcov(X, Y ) = Cov(X, Y )/(hXihY i) are also often used.
Knowledge of the covariance is crucial to the joint application of the quantities X and Y ;
for example, the standard error in the sum X + Y can lie anywhere between s(X) + s(Y ) and
|s(X) − s(Y )| depending upon the degree of correlation between X and Y . A nonzero covariance
between two quantities can arise from a partial dependence of one upon the other or from a common
dependence upon some third uncertain quantity.

213

Chapter 30
File 30. DATA COVARIANCES
OBTAINED FROM PARAMETER
COVARIANCES AND
SENSITIVITIES
30.1

General Comments

File 30 is provided as a means of describing the covariances of tabulated cross sections, multiplicities, and energy-angle distributions that result from propagating the covariances of a set
of underlying parameters (for example, the input parameters of a nuclear model code) using an
evaluator-supplied set of parameter-covariances and sensitivities. Whenever nuclear data are evaluated primarily through the application of nuclear models, the covariances of the resulting data
can be described very adequately, and compactly, by specifying the covariance matrix for the underlying nuclear parameters, along with a set of sensitivity coefficients giving the rate of change of
each nuclear datum of interest with respect to each of the model parameters. Although motivated
primarily by these applications of nuclear theory, use of File 30 is not restricted to any one particular evaluation methodology. It can be used to describe data covariances of any origin, so long
as they can be formally separated into a set of parameters with specified covariances and a set of
data sensitivities.
The need for a covariance format of this type became clear in connection with the R-matrix
analysis of the ENDF/B-VI light-element standards. The key parameters here are the parameters
of a few high-energy resonances in the relevant compound systems. Another area where this
format is expected to find early application is in representing the covariances of cross sections and
secondary-particle emission spectra and angular distributions due to neutron interactions in the
0.1 - 20 MeV range, when the data are obtained primarily from the optical model and statistical
pre-equilibrium theory. Relevant parameters here include the optical parameters, level-density
prescription, pre-equilibrium matrix elements, and gamma-ray strength functions.
It is shown below that multigroup averages of parameter sensitivities are identical to the parameter sensitivities of the corresponding multi-group data. It is the latter that are actually needed in
most applications. (See Section 30.4) To take maximum advantage of this equivalence, sensitivity
information is represented in File 30 in a format that is as close as possible to the format for the
actual data, so that the sensitivities can be retrieved and integrated by processing codes with the

214

CHAPTER 30. FILE 30. COVARIANCES OF MODEL PARAMETERS
least possible modification.
It should be emphasized that File 30 is not intended as a repository for complete ”evaluations
of parameters.” In fact, to limit the bulk of the files and to minimize processing costs, evaluators
are encouraged to reduce the number of parameters and the number of sensitivities per parameter
to the minimum necessary to describe data uncertainties of practical importance. In defining the
format for File 30, no attempt is made to pre-judge the parameter definitions or types of nuclear
theory that may be most appropriate or useful. Discussion of such points is obviously encouraged in
the printed documentation, but the format itself is deliberately kept totally general. One advantage
of this generality is that the results of a wide variety of evaluation methodologies can be described
using a single format. As discussed in Section 30.3 below, this generality also facilitates various
mathematical operations, such as diagonalizing the parameter covariance matrix.

30.1.1

Definitions

In the context of File 30 the word ”sensitivity” is defined as the derivative of an evaluated quantity,
call it σ, with respect to the logarithm of one of the parameters, αi ,
σi′ ≡

∂σ
∂σ
= αi
.
∂ (ln αi )
∂αi

(30.1)

An advantage of employing such derivatives is that σi′ is expressed in exactly the same units as
σ, whether it be an actual cross section or a distribution (energy distribution, angle distribution,
double-differential quantity, etc.). This means, among other things, that integrations over energy
and angle can be performed with minimal changes in multi-group processing codes. The use of
derivatives with respect to the logarithms of the parameters also meshes nicely with the use of
relative parameter covariance matrices, as shown below in equation (30.7).
As discussed in detail in Section 30.2.1, a subsection of one section of File 30 is employed to
store the sensitivities of the data in one section (called the referenced section) of a file elsewhere
in the material of interest.
It should be emphasized that normally there will not be a direct, one-to-one correspondence
between the energy or angular grid in a subsection of File 30 and that used in the referenced
section. This follows from the fact that the derivatives in File 30 are not actually the derivatives
of individual data values. Rather, the collection of data in one such subsection should form an
adequate representation of the energy and angle-dependence of the relevant derivative function,
making effective use of the standard interpolation laws.
File 30 does not permit the representation of the uncertainty in independent variables (the
floating-point numbers that define the energy and angle grids of an ENDF section). This would
seriously complicate the calculation of the uncertainty in averaged quantities, as discussed below.
Further, if σ is thought of as the output of a model calculation, quantities such as the incident energy
or the outgoing angle are specified by the model-code user and have no meaningful uncertainty.
In addition, File 30 may not be used to represent uncertainty of any integer, nor the uncertainty
of standalone (untabulated) quantities that affect energy or angle grids, such as masses, Q-values,
and the boundaries of energy ranges. Thus, it is understood that the data fields normally used to
store probability information (cross sections, multiplicities, or normalized distributions) are used in
File 30 to record sensitivity information, but that other quantities have standard (MF6=30) ENDF
definitions.

215

CHAPTER 30. FILE 30. COVARIANCES OF MODEL PARAMETERS

30.1.2

Treatment of Various Data Types

Following the general guidelines stated above, subsections of File 30 describing cross-section (as
opposed to multiplicity or distribution) sensitivities would have the same mechanical structure as
sections of File 3. Of course, since sensitivities are derivatives, many more negative numbers would
appear in the floating-point data fields than one normally expects to see in File 3. One can treat
ν data in File 1 in the same way as cross sections.
Some interesting points arise with respect to distributions, for example tabulated angular distribution data in File 4 for elastic scattering. If the derivatives of the normalized angular distribution
p(θ) with respect to a given parameter are large, they should be described in a subsection with
(MFSEN,MTSEN)=(4,2) (see Section 30.2.1).
Note that since p(θ) is normalized to unity by definition, the angle integral of the sensitivities
(equal to the parameter-derivative of the angle integral) should be zero. A second important aspect
of the use of two separate functions to build the actual desired data is that, in order to build the
corresponding sensitivities, the product rule is employed. For example, the differential elastic cross
section γ(θ) (barns/steradian) at angle θ is formed as a product,
γ(θ) ≡
so that

dσ
= σp(θ) ,
dΩ

∂p(θ)
∂σ
∂γ(θ)
=σ
+ p(θ)
.
∂αi
∂αi
∂αi

(30.2)

(30.3)

Multiplying both sides of equation (30.3) by αi , and recalling the notation of equation (30.1), one
gets,
γi′ (θ) = σp′i (θ) + σi′ p(θ) .
(30.4)
Equation (30.4) shows, then, how the sensitivity γi′ (θ) is constructed from the data in two dif-

ferent subsections (σi′ and p′i ) of File 30, plus data (σ and p) from Files 3 and 4, respectively.
The generalizations needed to treat three or more separate factors are obvious.
Both to reduce the bulk and to reduce processing costs, evaluators should simply omit
reference to sections in the main evaluation that exhibit little sensitivity to a given parameter. Such omissions will be treated as if zeroes had been entered explicitly. For example,
if the angular distributions are omitted from File 30, then the first term on the right in
equation (30.4) will be omitted.
Just as one is permitted to employ a Legendre representation of p(θ) in File 4, one is
permitted in File 30 to use a Legendre expansion to represent p′i (θ). In fact, if it reduces
the size of the files, it is preferable to use Legendre moments for p′i (θ), even if p(θ) itself is
given in tabular form. As mentioned above p′i (θ) must integrate to zero, so the magnitude
of the implied zeroth Legendre moment of p′i (θ) is zero, not unity. These considerations of
File 4-type sensitivities can be extended in an obvious way to treat neutron spectra in File 5,
isomer ratios in File 9, photon-production multiplicities in File 12, fission-product yields,
etc.
No fundamental new problems are introduced by considering double-differential data, as
represented in File 6. In that case, p becomes a function p(E ′ , θ) of both the final energy
and angle of the outgoing particle. The only complication that this adds is that p′i in
equation (30.4), for example, is also doubly differential, p′i = p′i (E ′ , θ). It is conceivable
that p′i for some parameter will exhibit more severe angle-energy correlations than p(E ′ , θ)
216

CHAPTER 30. FILE 30. COVARIANCES OF MODEL PARAMETERS
itself, so it is permitted to represent the emission sensitivities for a given reaction in File 6
format in File 30, even though the angle and energy distributions for that reaction are given
separately in Files 4 and 5. In this case, the entry MFSEN=6 in the File 30 dictionary
really points to both File 4 and File 5 in the main evaluation. Since the File 6 type matrix
information will in general occupy more space than the approximate treatment in Files 4
and 5, this option should be exercised only on those parameters (i.e., in those sections)
where it is crucial.

30.1.3

Multigrouped Sensitivities

Multigroup operations on the data in an evaluation can be summarized as the performance
of certain weighted integrations over incident energy, secondary particle type, secondary
energy, and secondary angle. Although these operations are very complicated, there is no
commonality between variables (or limits) of integration and the parameters of concern in
File 30. One can take advantage of this in calculating the derivatives of multigroup-averaged
data with respect to the parameters. If we introduce g as a generic group-averaged quantity
(such as a single Legendre moment of one element of a multigroup scattering matrix), which
corresponds to a differential quantity γ, then:
Z
g = dE dE ′ dΩ γi (E, E ′ , θ) ω(E, E ′ , θ) ,
(30.5)
where ω is some weighting function. As discussed below, one frequently is interested in the
uncertainty in such multigroup quantities, and to obtain this uncertainty, one will first need
to calculate the derivative of g with respect to the parameter αi :
Z
∂g
∂γ(E, E ′ , θ)
′
gi =
= αi dE dE ′ dΩ
ω(E, E ′ , θ) ,
∂(ln αi )
∂αi
or
gi′

=

Z

dE dE ′ dΩ γi′ (E, E ′ , θ) ω(E, E ′ , θ) .

(30.6)

Comparing equations (30.5) and (30.6), we obtain the useful result that the sensitivity of a
multigroup value to a given parameter is equal to the multigroup average of the (energy and
angle dependent) parameter sensitivity. Thus an ENDF processing program that calculates
multigroup cross sections can use equation (30.5), with few modifications, to calculate the
parameter sensitivity of multigroup constants, given by equation (30.6). As mentioned in
the General Description above, this is the motivation for storing the sensitivities γi′ in a
format that is as close as possible to the format of the data γ.

30.2

Formats

File 30 is divided into sections identified by the value of MT. (In File 30, MT does not refer
to a reaction type). Each section of File 30 begins with a HEAD record and ends with a
SEND record.
217

CHAPTER 30. FILE 30. COVARIANCES OF MODEL PARAMETERS

30.2.1

Directory and Correspondence Table (MT=1)

The first section, MT=1, of File 30 consists of a ”directory” that displays the contents
and ordering of information in other sections of the file, plus an optional, cross-material
”correspondence table”, described below. The following quantities are defined.
NP Total number of distinct parameters.
NDIR Number of CONT records in the MF=30 directory, including the internal
datablock ”marker” records described below, but excluding both the correspondence table and the SEND record.
NCTAB Number of CONT records in the correspondence table, excluding the SEND
record.
MPi Parameter index.
MFSENi , MTSENi If nonzero, location of a section of data in the main body of the
evaluation (the referenced data) that are sensitive to parameter MPi .
MFSEN and MTSEN determine the formats to be used to represent the energy and angle-dependence of the sensitivities. For example, if the referenced
section describes a normalized angular distribution, MFSEN=4, then any of
the formats described in Chapter 4 of this manual may be employed to describe the sensitivity of the distribution in (MFSEN,MTSEN) to parameter
MPi .
NCi Number of records used to represent this sensitivity information. These NC
records constitute a single subsection of a later section of File 30.
LIBFi Sublibrary number.
MATFi Material number.
MPFi Parameter number.
A section with MT=1 has the following structure:
[MAT,30,1/ ZA, AWR,
0,
0,
0,
NP] HEAD
[MAT,30,1/ 0.0, 0.0,
0,
0, NDIR,NCTAB] CONT
[MAT,30,1/ 0.0, 0.0, MP1 , MFSEN1 , MTSEN1 , NC1 ] CONT
[MAT,30,1/ 0.0, 0.0, MP2 , MFSEN2 , MTSEN2 , NC2 ] CONT
------------------------------------------------------------------------[MAT,30, 1/ 0.0, 0.0, MPN DIR ,MFSENN DIR ,MTSENN DIR , NCN DIR ] CONT
[MAT,30, 1/ 0.0, 0.0, MP1 , LIBF1 , MATF1 , MPF1 ] CONT
[MAT,30, 1/ 0.0, 0.0, MP2 , LIBF2 , MATF2 , MPF2 ] CONT
------------------------------------------------------------------------[MAT,30, 1/ 0.0, 0.0,MPN CT AB ,LIBFN CT AB , MATFN CT AB ,MPFN CT AB ] CONT
[MAT,30, 1/ 0.0, 0.0, 0, 0, 0, 0] SEND
218

CHAPTER 30. FILE 30. COVARIANCES OF MODEL PARAMETERS
The directory serves as a guide for the processing codes and provides, in addition, a
detailed, eye-readable list of the files and sections elsewhere in the current evaluation that
are significantly sensitive to the parameters under consideration. As shown above, this
information is presented in a format that is similar to the main index for this material in
(MF,MT) = (1,451).
In general, a given parameter will affect the data in several different sections, so the same
value of MP will appear in several consecutive entries in the dictionary. MP is higher in the
ENDF hierarchy than MFSEN, which is in turn higher that MTSEN. Within the File 30
framework, then, MP can be considered an index to a ”sub-material”. The first value of
MP1 must be 1, the next new, non-zero value must be 2, and so on. Except for marker
records, MP, MFSEN, and MTSEN must occur in normal ENDF ascending order.
Unlike the main directory in (MF,MT)=(1,451) the File 30 directory contains internal
file-end and sub-material-end ”markers”. That is, within the range of records describing
a given parameter MP, and following the final reference to a given value of MFSEN, an
explicit directory entry with MFSEN=0 is given in order to indicate the end of information
concerning MFSEN-type sensitivities for parameter MP.
[MAT,30, 1/ 0.0, 0.0, MP, 0, 0, 0] CONT
Similarly, following the final reference to a given value of MP in the directory, a directory entry with MP=0 is given to indicate the end of information concerning the current
parameter.
[MAT,30, 1/ 0.0, 0.0, 0, 0, 0, 0] CONT
It may occur that the evaluated data for two different materials are sensitive to the
same parameter, or to a common set of parameters. Here ”sensitive to the same parameter” means that the same numerical value of some particular quantity was employed in
generating both evaluations. If, in addition, the numerical value thus employed has a substantial uncertainty, then this would imply substantial cross-material and/or cross-library
data covariances. These covariances may be important in some applications, for example,
in uncertainty analyses involving physical mixtures of the materials in question. In order
to represent these cross-library or cross-material covariances, the evaluator may include a
correspondence table in the first section of MF=30 to identify the common set. The covariances of these parameters must be given in both evaluations, and the covariances must be
identical. However, since the parameter-numbering scheme need not be the same the two
evaluations, the correspondence table is also used to specify the relationship of the numbers
assigned to these parameters in the two evaluations.
The index parameter NCTAB indicates the number of CONT records appearing in the
correspondence table of the current evaluation. NCTAB may be zero, in which case the
table is omitted. If present, the table includes, in the format shown above, the sub-library
number LIBF, the material number MATF, and the parameter number MPF of a parameter
in some external, or ”foreign” evaluation that is identical to parameter MP of the current
evaluation. A value of LIBF=0 is entered if the foreign sub-library is the same as that of
the current evaluation. The correspondence table should be ordered (in ascending order)
first on MP, then on LIBF, then MATF, and then MPF. No internal ”marker” records are
included in the correspondence table.
219

CHAPTER 30. FILE 30. COVARIANCES OF MODEL PARAMETERS

30.2.2

Covariance Matrix (MT=2)

The second section of File 30, MT=2, contains the NP(NP+1)/2 unique, relative covariances
Rcov(I, J) of the I th parameter with the J th parameter in the form of NP separate LIST
records. This structure permits the inclusion of a large number of parameters without
requiring excessive computer storage during routine data handling. There is one such LIST
record for each MP value. The structure of MT=2 is as follows:
[MAT,30, 2/
ZA, AWR, 0, 0,
0, NP]HEAD
[MAT,30, 2/ PARM1 , 0.0, 0, 0, NCS1 , 1/ {Rcov(1,K ), K=1,NCS1 }]LIST
[MAT,30, 2/ PARM2 , 0.0, 0, 0, NCS2 , 2/ {Rcov(2,K+1), K=2,NCS2 }]LIST
[MAT,30, 2/ PARM3 , 0.0, 0, 0, NCS3 , 3/ {Rcov(3,K+2), K=3,NCS3 }]LIST
--------------------------------------------------------------------------------------------[MAT,30, 2/ PARMN P , 0.0, 0, 0, 1, NP/ {Rcov(NP,NP)}] LIST
[MAT,30, 2/
0.0, 0.0, 0, 0, 0, 0] SEND
Since the filling of the MPth row of covariance matrix begins with the diagonal element,
Rcov(MP,MP), the number of matrix elements NCSM P explicitly given in the list must be
less than or equal to (NP-MP+1). If the number given is smaller than this, the remaining
covariances in that row are taken to be zero. Evaluators can take maximum advantage of
this zero-suppression feature by assigning consecutive MP-values to members of groups of
strongly correlated parameters. The numerical value PARMM P of the MPth parameter (or
optionally just a zero) is entered in the first floating-point field of the LIST.

30.2.3

Sensitivities (MT=11-999)

Sections MT≥11 contain the sensitivities. A single section in this range of MT-values is the
collection of all sensitivities (or subsections) relevant to a given parameter MP. The section
number is determined by the parameter index, using the relation (MT=MP+10). While
evaluators should employ the minimum number of parameters necessary, no particular limit
is placed on MP, other than the obvious one that MT may not exceed 999. The structure
of a section with MT≥11 is as follows:
[MAT,30,MT/ ZA, AWR,
0,
0,
0, NL]HEAD
MT=MP+10


--------------------------------------------------------
[MAT,30, 0/ 0.0, 0.0,
0,
0,
0,
0]SEND
NL in the HEAD record is the number of subsections in the current section. In other
words, NL is the number of referenced sections for the current parameter. The format of a
subsection of a section with MT≥ 11 is, with very few exceptions, the same as the format
of the referenced section in the main body of the evaluation. Certain minor ”bookkeeping”

220

CHAPTER 30. FILE 30. COVARIANCES OF MODEL PARAMETERS
changes are unavoidable; for example, the MF and MT positions of a data record will contain
30 and (MP+10), respectively, not MFSEN and MTSEN.
Of necessity, the subsections of a section of File 30 are simply abutted to one another
without intervening SEND of FEND records. In a sense, the roles of the usual SEND and
FEND records in defining data-type boundaries are taken over here by the contents of the
File 30 directory. (See Section 30.2.1) For example, by reading a copy of the directory in
parallel with the reading of the subsections of a single File 30 ”source section” with MT≥11,
a processing code could create a new ENDF-formatted evaluation on a third file from the
information encountered, with MFSEN and MTSEN written into the usual MF and MT
positions, and with the required SEND, FEND, and MEND records inserted.
Each subsection of the source section must be constructed so that the sensitivity information in section (MFSEN,MTSEN) of a new evaluation created in this way will comply, in
all mechanical details, with the correct, current ENDF formats, as described in the chapter
of this manual devoted to data of the type (MFSEN,MTSEN). Of course, requirements of
completeness (for example, the requirement that MT=2 must appear in File 4 if MT=2
appears in File 3) do not apply in this context, since the absence of such information simply
indicates small sensitivities.
Because of the application of the product rule, as described in Section 30.1.2 above, each
subsection of a section with MT≥ 11 leads in principle to a complete multigroup, multiLegendre table ”transfer” matrix in which the sensitivities corresponding to the referenced
section are combined with regular data from the other sections of the evaluation. These NL
matrices, when summed, give the net sensitivity of all multigroup data to parameter MP,
as in equation (30.4).

30.3

Additional Procedures

30.3.1

Relation of MP-values to Physical Parameters

Since the actual parameter definitions will vary from one evaluation to the next, it is clear
that choices concerning:
a) the assignment of particular MP-values to different physical parameters, and
b) what physical parameters to omit altogether,
are left to the evaluator.

30.3.2

Parameter Values

Because many models are non-linear, the actual numerical values of the parameters PARMn
may be included in the file, in order to record the point in parameter space where the
sensitivities were calculated. See the discussion of this item in Section 30.2.2 The value of
PARMn has no effect on propagated data uncertainties, so the units of PARMn are given
only in the printed documentation. At the evaluator’s option, a zero may be entered in place
of the actual parameter value.
221

CHAPTER 30. FILE 30. COVARIANCES OF MODEL PARAMETERS

30.3.3

Eigenvalue Representation

By use of eigenvalue methods1 , it is straightforward to find a linear transformation that diagonalizes a given covariance matrix. This is a useful method of locating blunders (indicated
by the existence of negative eigenvalues) and redundancies (indicated by zero eigenvalues)
and is recommended as a general procedure prior to submission of any covariance evaluation.
Moreover, once having performed such a diagonalization of a parameter covariance matrix,
one could report in MT=2 of File 30 only the eigenvalues of the matrix and, in MT=11
and above, sensitivities of the data to variations in the effectively-independent linear combinations of the parameters (as summarized in the eigenvectors). If it proves feasible in
individual evaluation situations, and if it leads to a substantial reduction in the overall size
of the file, evaluators are encouraged to employ this technique.

30.3.4

Thinning of Sensitivity Information

The collection of sensitivities in one subsection should form an adequate representation of
the energy- and angle-dependence of the relevant derivative function, making effective use of
the standard interpolation laws. ”Thinning” the sensitivity information (that is, removing
intermediate grid points) is encouraged, in order to reduce the size of the file, but, as a
general guide, such thinning should not induce changes greater than about 10% in the
reconstructed covariances.

30.3.5

Cross-file Correlations

The information in File 30 is considered to describe sources of uncertainty that are independent of those described in Files 31-40. Thus, for a given set of multigroup cross sections, the
multigroup covariance matrix obtained from File 30 should be added, in a matrix addition
sense, to such a matrix derived from the other files. This is the only level on which File 30
”communicates” with the other files.
A complication that can occur with respect to cross-file correlations is that there may
exist strong correlations (due to normalization procedures, for example) between certain lowenergy cross sections that are evaluated directly from measurements and the parameters
employed to calculate the evaluated data at higher energies. If the evaluator wishes to
describe these correlations, the covariances for the low-energy normalization reaction (and
those for other reactions strongly correlated to it) can be ”moved” from File 33 to File 30. A
possible method for accomplishing this is to consider the moved data to have been evaluated
by multiplying a well-known reference cross section by an uncertain, energy-dependent,
correction factor. The correction factor can be assumed to have been evaluated on some
fixed, coarse energy grid, with linear interpolation applied between grid points. In this case
the ”parameters” would be the values of the correction factors at the coarse-grid points
EGi . The sensitivities [see Eq. (30.1)] of the ”experimentally evaluated” cross sections σ
to these new parameters would be a series of triangular ”hat” functions, with peak values
σ(EGi ). (Alternative approaches exist.)
1

For example, the SSIEV routine described in B.T. Smith, et. al., Matrix Eigenvalue Routines EISPACK
Guide, 1976.

222

CHAPTER 30. FILE 30. COVARIANCES OF MODEL PARAMETERS

30.4

Multigroup Applications of Parameter Covariances

Given the relative covariances, Rcov(αi , αj ) ≡ Cov(αi , αj )/(αi αj ), from (MF30, MT2), and
′
the multigrouped sensitivities gmi
from equation (30.6), it is straightforward to obtain the
covariance between one multigroup datum gm and another gn . It is necessary to add the
additional index to keep track of the multiplicity of data types, as well as the possible
multiplicity of materials. (See discussion of the latter point at the end of Section 30.2.1)
Making the usual approximation that gm is not an extremely nonlinear function of the
parameters, we expand in a Taylor series and retain only the first term,
X ∂gm ∂gn
Cov(αi , αj )
∂α
∂α
i
j
ij
X
∂gm ∂gn
=
αi αj
Cov(αi , αj )/(αi αj )
∂α
∂α
i
j
ij
X
′
′
=
gmi
gnj
Rcov(αi , αj ) .

Cov(gm , gn ) =

(30.7)

ij

Equation (30.7) gives the desired multigroup covariance matrix in terms of the multigrouped
(logarithmic) sensitivities from equation (30.6) and data read directly from the second section of File 30.
In addition to providing a direct route to the calculation of the uncertainty of multigroup cross sections due to parameter uncertainties, data provided in File 30 format have
the potential for additional kinds of application, not involving straightforward application of
equation (30.7). Since these issues relate to computing requirements, it is necessary to deal
with specific examples. In situations presently foreseen, the number of nuclear parameters
might be in the range of 10 to 100, so we take 50 as typical. On the other hand, it is easy
to imagine neutronics applications where the number of individual multigroup constants exceeds 10000. For example, if there are 3 high-threshold neutron-emitting reactions for a given
material, the number of individual cross section items might be 3 reactions × 10 ”source”
groups × 80 ”sink” groups × 4 Legendre tables = 9600. In such cases, the data covariance
′
matrix Cov(gm , gn ) becomes prohibitively large (108 items), while the sensitivity matrix gmi
(containing 500 000 items) and parameter covariance matrix Rcov(αi , αj ) (with 2500 items)
remain fairly manageable. Since, according to equation (30.7), all covariance information
content is already contained in the latter two items, it seems likely that multigroup libraries
for high-energy neutronics applications will store these items separately, rather than in the
expanded product form.
Further efficiencies are possible if the ultimate aim is to calculate the uncertainties in a
set of predicted integral quantities (dose, radiation damage, fuel-breeding ratio, etc.), which
can be denoted by a column vector, y. A typical number of such quantities might also be
around 50. The covariance matrix D(y) for the integral quantities is related (again in the
first-order approximation) to the cross section covariance matrix D(g), with elements given

223

CHAPTER 30. FILE 30. COVARIANCES OF MODEL PARAMETERS
by equation (30.7), by the familiar propagation of errors relation,
D(y) = S D(g) ST

(30.8)

where S is the 50 × 10000 sensitivity matrix relating the integral quantities y to the multigroup cross sections g. Matrix S can be obtained from standard neutronics analyses. If we
′
introduce a 10000×50 matrix R, having elements gmi
that define the sensitivities of multigroup constants to a set of parameters α in the file, equation (30.7) can be rewritten in
matrix form,
D(g) = R D(α)RT
(30.9)
Equation (30.8) then becomes


D(y) = S R D(α) RT ST = T D(α) TT

(30.10)

The product matrix T = SR, which contains the direct sensitivity of the integral data y
to the nuclear-model parameters α, is very compact, having about the same size as the
covariance matrix D(α). Note that in evaluating the matrix products in equation (30.9)
one actually never needs to calculate the full 10 000 × 10 000 cross section covariance matrix.
In cases where the evaluator chooses to use File 33 for certain data and File 30 for others, there is no logical problem with adding together integral covariances D33 (y) based on
conventional sensitivity and uncertainty analysis (i.e., based on Files 3 and 33 only) with
analogous data D30 (y) obtained from File 30, using equation (30.9), because the data covariances due to the parameter covariances are, by definition, independent of those described
in the other covariance files.

224

Chapter 31
File 31. COVARIANCES OF THE
AVERAGE NUMBER OF
NEUTRONS PER FISSION
31.1

General Comments

For materials that fission, File 31 contains the covariances of the average number of neutrons
per fission, given in File 1. MT=452 is used to specify ν, the average total number of neutrons
per fission. MT=455 and MT=456 may be used to specify the average total number of
delayed neutrons per fission, ν d , and the average number of prompt neutrons per fission, ν p ,
respectively.
The average number of neutrons per fission is given as a function of incident energy for
particle-induced fission. This energy dependence may be given by tabulating the values as a
function of incident particle energy or (if MT=452 alone is used) by providing the coefficients
for a polynomial expansion as a function of incident neutron energy. Whichever method is
used, the result is that the quantities are specified as a function of incident neutron energy
and in this sense are similar to the data given in File 3. Therefore, the problems associated
with representing the covariances of the average number of neutrons per fission are identical
to those in File 33.
For spontaneous fission, in the sub-library for a radioactive decay, the average multiplicities are given by zero-order terms in polynomial expansions, and the lack of any energy
dependence is recognized in the formats.

31.2

Formats

Particle-induced fission formats for fission neutron multiplicity in File 31, MT=452, 455,
and 456, are directly analogous to those for File 33 given in Section 33.2.
Spontaneous fission formats for neutron multiplicity in File 31, MT=452, 455 and 456, are
modified from those in Section 33.2 because there is no energy dependence to express for
ν or its covariance components. There, NC and NI-type subsubsections for spontaneous
225

CHAPTER 31. FILE 31. COVARIANCES OF FISSION ν
fission neutron multiplicity have no energy variables present.
”NC-type” subsubsections for spontaneous fission ν, MT=452, 455, or 456, have the
following structure, using definitions given in Section 33.2:
For LTY=0:
[MAT,31, MT/ 0.0, 0.0,
[MAT,31, MT/ 0.0, 0.0,

0, LTY,
0,
0]CONT
(LTY=0)
0,
0, 2*NCI, NCI/ {CI,XMTI}]LIST

For LTY=1, 2, or 3:
[MAT,31, MT/ 0.0, 0.0,
0, LTY,
0,
0] CONT
[MAT,31, MT/ 0.0, 0.0, MATS, MTS,
4,
1/
------(XMFS,XLFSS)(0.0, Weight) ]LIST
”NI-type” subsubsections for spontaneous fission neutron multiplicity are allowed with
LB=0 and LB=1, and have the structure:
[MAT,31, MT/ 0.0, 0.0, 0, LB, 2, 1/(0.0,F )] LIST
where F gives the absolute or relative covariance component depending on whether LB=0
or LB=1. (See Section 33.2 of this manual for notation.)

31.3

Procedures

All procedures given in Section 33.3 concerning the ordering and completeness of sections of
File 33 apply to sections of File 31: ν (MAT,31,452), ν d (MAT,31,455) and ν p (MAT,31,456).
Note that in File 1 ν (MT=452), ν d (MT=455) and ν p (MT=456) satisfy the relation:
ν(E) = ν d (E) + ν p (E) .

(31.1)

Therefore, if one of these quantities is ”derived” in terms of the other two, it is permissible
to use NC-type subsections with LTY=0 to indicate that it is a ”derived redundant cross
section”. See Section 33.2.2.1 for an explanation of this format.
If a section of File 31 is used with MT=456, there must also be a section of File 31 with
MT=452.
When a section of File 31 for either MT=452, 455 or 456 is used for induced fission,
there must be a section in File 33 for the fission cross sections, i.e., section (MAT,33,18).
Note:
1. Since ν d is much smaller than ν p , it should never be evaluated by subtracting ν p from
ν.
2. When a polynomial representation is used to describe the data in File 1 MT=452, the
covariance file applies to the tabular reconstruction of the data as a function of energy
and not to the polynomial coefficients.

226

CHAPTER 31. FILE 31. COVARIANCES OF FISSION ν
3. The ENDF-6 formats do not provide for covariance references between different sublibraries except by use of File 30. Therefore, it is not possible to express the covariances
between, for example, the ν for spontaneous fission of 252 Cf and the ν(E) for the major
fissile materials by use of NC-type subsubsections with LTY=1, 2, and 3.
4. In ENDF-6 formats there is no provision to express uncertainty in MT=455 for the
decay constants for the various precursor families.

227

Chapter 32
File 32. COVARIANCES OF
RESONANCE PARAMETERS
32.1

General Comments

File 32, MT=151, contains the variances and covariances of the resonance parameters given
in File 2, MT=151. The resonance parameters, used with the appropriate resonance formulae, provide an efficient way to represent the complicated variations in the magnitudes of
the different resonant partial cross sections, compared to the use of File 3 alone. Similarly
for File 32, the use of the covariances of the resonance parameters of individual resonances
provides an efficient way of representing the rapid variation over the individual resonances
of the covariances of the partial cross sections. The covariance data of the processed cross
sections include the effects of both File 32 and File 33 within a given energy region, similar
to the way the cross sections themselves are the sum of contributions from File 2 and File 3.
In the resonance region, the covariances of the partial cross sections are often characterized by
a) ”long-range” components that affect the covariances over many resonances, and
b) ”short-range” components affecting the covariances of the different partial cross sections in the neighborhood of individual resonances.
The former often can best be represented in File 33, while the latter can be given in File 32.
When the material composition is dilute in the nuclide of concern and the cross sections
are to be averaged over an energy region that includes many resonances, the effects of ”shortrange” components are unimportant and the covariances of the averaged cross sections can
be well represented by processing the long-range components given in File 33. Therefore, the
covariances of the cross sections in the unresolved resonance energy region should be given
entirely by means of File 33 unless resonance self-shielding in this energy region is thought
to be of practical significance for a particular nuclide. For many nuclides these conditions
may also be valid in the high-energy portion of the resolved resonance energy region.
In the resolved resonance region it may be necessary to calculate covariances for the
resonance self-shielding factors to obtain the uncertainty in the Doppler effect. As another
228

CHAPTER 32. FILE 32. COVARIANCES OF RESONANCE PARAMETERS
example, one may require group cross-section covariances where the groups are narrow compared to the resonance width or where only a few resonances are within a group. In these
cases File 32 should be used. Because this situation may be important only in the lower
energy portion of the resolved resonance region, File 32 need not include the whole set of
resonances given in File 2. File 33 remains available for use in combination with File 32.
The ENDF-6 formats for File 32 are structured to maintain compatibility with those of
ENDF/B-V, and there are new features to permit representation of covariance components
among the parameters of different resonances. Covariances between resonance energies and
widths are now also allowed. While File 32 was limited in ENDF/B-V to the Breit-Wigner
representations (LRF=1 or 2), in ENDF-6 formats covariances may also be given for the
Reich-Moore (LRF=3), Adler-Adler (LRF=4) and R-Matrix Limited (LRF=7) formulations.
A limited representation is offered for unresolved resonance parameter covariance data. The
conventions are retained that the cross section covariances for the resonance region are
combined from covariance data in File 32 and File 33, and that relative covariances given
in File 33 apply to the cross sections reconstructed from File 2 plus File 3 when that option
is specified in File 2. (There is a partial exception for LRF=4.) Since the ENDF-6 formats
do not allow a continuous range of values for the total angular momentum J of a resonance,
covariance data for the resonance spin are no longer recognized.
Note that File 32 formats retain many restrictions. For example, there is no provision for
representing covariances between the parameters of resonances in two different materials or
in two isotopes within the same elemental evaluation. Since File 32 can become cumbersome
if many resonances are treated, evaluators will do so only for nuclides of greatest practical
importance. (The LCOMP=2 format is available for use in the case of very many resonances;
this format provides a compact but approximate representation for the resonance parameter
covariance matrix. Details are given in Section 32.2.3).
The strategy employed for File 32 is similar to that for smooth cross sections in that the
variance of a resonance parameter or the covariance between two such parameters can be
given as a sum of several components. Some contributions can be labeled by resonance energy
and parameter type, while others can arise from long-range covariances among parameters
of the same type for different resonances in the same isotope. The latter are labeled by
energy bands, and the same uncertainty characteristic is applied to the indicated parameter
of all the File 2 resonances in a given band.
The idea of assigning the same relative covariance to all parameters of a given type in
an energy region has limited validity. One limit arises because long-range uncertainties in
reaction yields don’t generally carry over proportionately to uncertainties in the corresponding reaction widths. However, gamma-ray widths are sometimes known only for resonances
at low neutron energy, and then the average of these values is used for the resonances at
higher energies. This situation motivates formats resembling File 33 except that a relative
uncertainty in the file applies to the indicated parameter (e.g., Γγ ) of every resonance in the
indicated energy range. The approach here is to allow the long range correlations to extend
over any energy interval in which consistent resolved resonance formulations are utilized.
The definitions of common quantities are as given in Chapter 2.
Caveat: Resonance parameter covariance matrices reported in File 32 are written in terms
of parameters given in File 2 listing; this is true even when a given parameter is not the
229

CHAPTER 32. FILE 32. COVARIANCES OF RESONANCE PARAMETERS
actual quantity used in the calculation. In particular, if the value GAM given in File 2
for the partial width is negative, the negative sign is to be associated with the reduced
width amplitude γλc rather than with Γλc (since Γp
λc is always a positive quantity). More
specifically, Γλc = |GAM| and γλc = sign(GAM × |GAM|/2P ), with P evaluated at the
energy of the resonance. In this case the covariance matrix is written in terms of GAM
rather than Γλc . [See Section 2.2.1 for details; the rule applies for LRF=1, 2 and 3, as well
as for LRF=7].

32.2

Formats

The format for File 32, MT=151, parallels the format for File 2, MT=151, with the restriction to LRF=1,2,3,4 or 7 for LRU=1 (resolved parameters) and to LRF=1 for LRU=2
(unresolved parameters). The File 32 format for LRU=1, LRF=7 may omit some of the
File 2 information. The general structure of File 32 is as follows:
[MAT,32,151/ ZA, AWR,
0,
0, NIS,
0]HEAD
[MAT,32,151/ ZAI, ABN,
0, LFW, NER,
0]CONT (isotope)
[MAT,32,151/ EL, EH, LRU, LRF, NRO, NAPS]CONT (range)

[MAT,32,151/ EL, EH, LRU, LRF, NRO, NAPS]CONT (range)

----------------------------------------------------------[MAT,32,151/ EL, EH, LRU, LRF, NRO, NAPS]CONT (range)

----------------------------------------------------------[MAT,32,151/ ZAI, ABN,
0, LFW, NER,
0]CONT (isotope)
[MAT,32,151/ EL, EH, LRU, LRF, NRO, NAPS]CONT (range)

----------------------------------------------------------[MAT,32,151/ EL, EH, LRU, LRF, NRO, NAPS]CONT (range)

--------------------------------------------------------[MAT,32, 0/ 0.0, 0.0, 0, 0, 0, 0] SEND
[MAT, 0, 0/ 0.0, 0.0, 0, 0, 0, 0] FEND
Data are given for all ranges for a given isotope, then for successive isotopes. The data for
each isotope start with a CONT (isotope) record; those for each range with a CONT (range)
record. File segments need not be included for all isotopes represented in the corresponding
File 2.
If the ”range” record preceding a subsection has NRO6=0, indicating that the energy
dependence of the scattering radius is given in File 2, the initial file segment within the
230

CHAPTER 32. FILE 32. COVARIANCES OF RESONANCE PARAMETERS
subsection has the form:
[MAT,32,151/ 0.0, 0.0, 0, 0, 0, NI]CONT

The next record of a subsection (the first record if NRO=0) has the form:
[MAT,32,151/ SPI, AP, 0, LCOMP, NLS, 0]CONT.
If the compatibility flag, LCOMP, is zero, NLS is the number of L-values for which lists
of resonances are given in a form compatible with that used in ENDF/B-V, as described
in Section 32.2.1. If LCOMP=1, then NLS=0 and subsection formats are as shown in
Sections 32.2.2. If LCOMP=2, the covariance matrix is expressed in compact form as
described in Section 32.2.3.

32.2.1

Compatible Resolved Resonance Format (LCOMP=0)

This format differs from that used for ENDF/B-V only in that covariances of the resonance
spin are all zero. It is applicable only for resolved parameters (LRU=1) and for the BreitWigner formalisms (LRF=1 or 2). No long-range covariances can be defined. The following
covariance quantities are defined:
DE2
Variance of the resonance energy in units eV2 .
2
DN
Variance of the neutron width GN in units eV2 .
DNDG
Covariance of GN and GG in units eV2 .
DG2
Variance of the gamma-ray width in units eV2
DNDF
Covariance of GN and GF in units eV2 .
DGDF
Covariance of GG and GF in units eV2 .
2
DF
Variance of the fission width GF in units eV2 .
DJDN
(null) covariance of resonance J-value and GN.
DJDG
(null) covariance of resonance J-value and GG.
DJDF
(null) covariance of resonance J-value and GF.
2
DJ
(null) variance of resonance J-value.
A complete subsection for LCOMP=0 has the following form:

[MAT,32,151/ SPI, AP, 0, LCOMP,
NLS,
0]CONT (LCOMP=0)
[MAT,32,151/AWRI, 0.0, L,
0, 18*NRS, NRS/
ER1 , AJ1 , GT1 , GN1 , GG1 , GF1 ,
DE21 , DN21 , DNDG1 , DG21 , DNDF1 , DGDF1 ,
DF21 , DJDN1 , DJDG1 , DJDF1 , DJ21 ,
(DJDN1 ,DJDG1 ,DJDF1 =0.0)
 ]LIST

Note that in this compatible format no covariance can be given between parameters of
different resonances even if they overlap. This format therefore permits only a relatively
crude approximation to the true resonance parameter covariance matrix.
231

CHAPTER 32. FILE 32. COVARIANCES OF RESONANCE PARAMETERS

32.2.2

General Resolved Resonance Formats (LCOMP=1)

Following the record that starts with the target spin, SPI, if it states LCOMP=1, the next
card of a subsection defines for that isotope and energy range, how many (NSRS) subsections
will occur for covariances among parameters of specified resonances and how many (NLRS)
subsections are to contain data on long-range parameter covariances. The complete structure
of an LCOMP=1 subsection is as follows:

[MAT,32,151/ SPI, AP, IFG, LCOMP, NLS,
0]CONT (LCOMP=1,NLS=0)
[MAT,32,151/AWRI, 0.0,
0,
0, NSRS, NLRS]CONT


The formats here differ from the LCOMP=0 formats of Section 32.2.1 in that covariance
between parameters of different resonances appear, resonance representations LRF=1,2,3,4
and 7 are allowed, there is no segregation by L-value (or by J and parity for LRF=7),
and the number of parameters considered per resonance is declared in each sub-subsection
primarily to avoid tabulating zero covariances for fission widths in files concerning structural
materials. The listed resonances must be present in File 2, but there is no requirement that
all resonances be included in File 32 that are given in File 2.
The formats for the short-range covariances (NSRS-type) depend on the resonance formalism, while the format for the long-range covariances (NLRS-type) is general.
The following paragraphs cover the formats for the NSRS-type covariances for the various
resonance formulations (LRF=1,2,3,4 and 7) for resolved parameters (LRU=1). For LRF=7,
parameter NLS is is to be interpreted as NJS. Parameter IFG is relevant only for LRF=7
and its meaning is the same as in File 2, described in Section 2.2.1.6.
32.2.2.1

SLBW and MLBW (LRF=1 or 2)

These cases have the same formats. All the resonances for which covariances are to be
included are divided into blocks. Covariances between parameters can only be included for
resonances in the same block. (A given resonance can appear in more than one block.) A subsubsection covers the data for a block. Within each such block, the parameters are given of
those resonances for which covariances are included, and the upper triangular representation
of the parameter covariance matrix is included for the whole block of resonances. For each
block one specifies the number MPAR of parameters to be included for each listed resonance
in the block, since for most cases there is, for example, no fission width and the size of the
covariance matrix can therefore be minimized. The NSRS-type subsections have the form
below for LRF=1 and 2:

232

CHAPTER 32. FILE 32. COVARIANCES OF RESONANCE PARAMETERS
[MAT,32,151/ 0.0, 0.0, MPAR, 0, NVS+6*NRB, NRB/

ER1 , AJ1 , GT1 , GN1 , GG1 , GF1
--------------------ERN RB , AJN RB , GTN RB , GNN RB , GNN RB , GGN RB , GFN RB ,

V1,1 , V1,2 , .... V1,M P AR∗N RB, V2,2 ,
---, V2,M P AR∗N RB , V3,3 ,
---, VM P AR∗N RB,M P AR∗N RB , 0.0, 0.0] LIST
Note that the record first lists all the parameters of resonances in the group, for positive
identification, and then lists the covariance terms. The parameters have the following
meaning:
MPAR Number of parameters per resonance in this block which have covariance data
(In order: ER, GN, GG, GF, GX, indices 1-5)
NVS Number of covariance elements listed for this block of resonances,
NVS = [NRB×MPAR×(NRB×MPAR+1)]/2.
NRB Number of resonances in this block and for which resonance parameter and
covariance data are given in this subsection.
ER Lab system energy of the k th resonance (in this block).
AJk Floating-point value of the spin for the k th resonance.
GTk Total width (eV) for the k th resonance. GT=GN+GG+GF+GX.
GNk Neutron width of the k th resonance.
GGk Gamma width for the k th resonance.
GFk Fission width of the k th resonance.
Vmn Variance (eV2 ) or covariance matrix element, row m and column, n ≥ m.
If j ≤MPAR is the parameter index (j = 4 for fission width) for the k th
resonance in the block, m = j + (k − 1) × MPAR. The indexing is the order
defined above under MPAR.

233

CHAPTER 32. FILE 32. COVARIANCES OF RESONANCE PARAMETERS
32.2.2.2

Reich-Moore (LRF=3)

The representation for the Reich-Moore resonance formulation is very similar. The listing
of the parameters among which covariance terms are allowed takes the form used in File 2,
namely:
ER, AJ, GN, GG, GFA, GFB.
Similarly, in interpreting the indices for the covariance matrix of the parameters in a given
block of resonances, the order when LRF=3 is ER, GN, GG, GFA, GFB and the largest
possible value of MPAR is 5.
32.2.2.3

Adler-Adler (LRF=4)

The Adler-Adler resonance representation includes background constants as well as resonance parameters. The uncertainty in the background constants is to be treated indirectly
just as the smooth background given in File 3. (That is, in the energy region EL, EU for
LRF=4 any relative uncertainty data in File 33 applies to the sum of the cross section in
File 3 and the contribution of the cross sections computed from the Adler-Adler background
constants. Great care will be required if this uncertainty representation for Adler-Adler fits
is used in other than single-isotopes evaluations.) The inherent assumption is that covariance data will be detailed for the largest resonances, and those representing the Adler-Adler
backgrounds. It is assumed that LI=7.
An LRF=4 NSRS subsubsection takes the following form, using the previous definitions
where possible:
[MAT,32,151/ 0.0, 0.0, MPAR, 0, NVS+6*NRB, NRB/
DET1 , DWT1 , GRT1 , GIT1 , DEF1 , DWF1

GRF1 , GIF1 , DEC1 , DWC1 , GRC1 , GIC1
--------------------DETN RB , DWTN RB , GRTN RB , GITN RB , DEFN RB , DWFN RB ,
GRFN RB , GIFN RB , DECN RB , DWCN RB , GRCN RB , GICN RB ,

V1,1 , V1,2 , .... V1,M P AR∗N RB, V2,2 ,
---, V2,M P AR∗N RB , V3,3 ,
---, VM P AR∗N RB,M P AR∗N RB , 0.0, 0.0] LIST
The Adler-Adler parameters of the selected list of resonances are given in the same redundant
style indicated in Section 2.2.1.3. For LRF=4 the maximum value of MPAR is 8 and for a
given resonance the covariance matrix indexing is in the order:
µ = DET = DEF = DEC,

32.2.2.4

ν = DWT = DWF = DWD,

GRT, GIT, GRF, GIF, GRC, GIC

R-Matrix Limited Format (LRF=7)

No long-range subsections are allowed for LRF=7. For each short-range section, it is not
sufficient to give merely the number of resonances in the block (as is done for LRF=3, for
234

CHAPTER 32. FILE 32. COVARIANCES OF RESONANCE PARAMETERS
example). Instead, because the LRF=7 format permits different numbers of channels for
different spin-and-parity (J π ) groups, both the number of spin groups in the block and the
number of channels for each spin group must be specified. As with the other LRF values,
not all resonances need to be included in this listing; further, not all J π groups need to
be included here. NJSX is the number of J π groups that are included in this short-range
section; the format for specifying NJSX is:
[MAT,32,151] 0.0, 0.0, NJSX, 0, 0, 0]CONT
For each spin group the number of channels (NCH), the number of included resonances
(NRB), and the number of lines per resonance (NX) must be specified. In addition, the
number of values to be read (6 times the number of lines) must also be given in the LIST
record, along with the resonance parameters:
[MAT,32,151] 0.0, 0.0, NCH, NRB, 6*NX, NX/
ER1 , GAM1,1 , GAM2,1 , GAM3,1 , GAM4,1 , GAM5,1 ,
GAM6,1 , ------ GAMN CH,1 ,
ER2 , GAM1,2 , GAM2,2 , GAM3,2 , GAM4,2 , GAM5,2 ,
GAM6,2 , ------ GAMN CH,2 ,
ERN RB , GAM1,N RB , GAM2,N RB , GAM3,N RB , GAM4,N RB , GAM5,N RB ,
GAM6,N RB , GAMN CH,N RB ]LIST
The above two (CONT and LIST) records are repeated, once for each included J π group.
Parameters are numbered in the order in which they appear in the listing; the total
number of parameters (NPARB) is the sum (over all included J π values) of (NCH+1)*NRB,
where NRB is the number of resonances from this J π group in the covariance block. Following
the final resonance for the final J π group in this block, the triangular half of the covariance
matrix is presented as a LIST record with N entries, where N=(NPARB*(NPARB+1))/2.
The covariance matrix is given in the following format:
[MAT,2,151] 0.0, 0.0,
0,
0,
N, NPARB/
V1,1 , V1,2 , .... V1,N P ARB, V2,2 ,
---, V2,N P ARB , V3,3 ,
------------------, VN P ARB,N P ARB ] LIST
32.2.2.5

Format for Long-Range Covariance Subsubsections (LRU=1)

Here are described the forms that the formats may take to represent long-range (in energy)
covariances among parameters of a given type. The strategy is to use formats that resemble
those for File 33 but refer to a particular parameter and equally to all resonances of a given
isotope within the indicated energy regions.
Each subsubsection must identify the resonance parameter considered (Γn , Γγ , etc.) via
the parameter IDP, indicate the covariance pattern via a value of LB, and give the energy
regions and covariance components. The list below defines the permitted values of LB. Note
the definition of one LB value not defined in File 33, and that LB=3 and 4 are not employed
in File 32 because of LB=5 is typically much more convenient. The term Cov[Γα(i) , Γα(j) ] is
defined as the covariance between the Γα parameters for two different resonances (indexed
i and j) in the same nuclide.
235

CHAPTER 32. FILE 32. COVARIANCES OF RESONANCE PARAMETERS
LB=-1 Relative variance components entirely uncorrelated from resonance-toresonance but having constant magnitude within the stated energy intervals:




Cov Γα(i) Γα(j) = δij

NE−1
X
k=1

(−1)

Sik Fα(k) Γ2α(i)

LB=0 Absolute covariance components correlated only within each energy interval.




Cov Γα(i) Γα(j) =

NE−1
X
k=1

(0)

i,k
Pj,k
Fα(k)

LB=1 Relative covariance components correlated only within each energy interval.
X i,k (1)
 NE−1

Pj,k Fα(k) Γα(i) Γα(j)
Cov Γα(i) Γα(j) =
k=1

LB=2 Fractional covariance components fully correlated over all energy intervals
with variable magnitude.
X i,k (2) (2)
 NE−1

Cov Γα(i) Γα(j) =
Pj,k′ Fα(k) Fα(k′ ) Γα(i) Γα(j)
k,k′ =1

LB=5 Relative covariance components in an upper triangular representation of a
symmetric matrix.
 X i,k (5)

Pj,k′ Fα(k,k′ ) Γα(i) Γα(j)
Cov Γα(i) Γα(j) =
k,k′

Note that the F (−1) , F (1) , and F(5) parameters have the dimensions of relative covariances,
the F (0) are absolute covariances, and the F (2) are relative standard deviations.
The S and P are dimensionless operators defined in Section 33.2.

The format for LB=-1, 0, 1, or 2 is:

[MAT,32,151/ 0.0, 0.0, IDP, LB, 2*NE, NE/ { Ek , F(LB) }] LIST
For LB=5 the requisite format is:
(5)

[MAT,32,151/ 0.0, 0.0, IDP, LB, NT, NE/ { Ek }, {Fkk′ }] LIST

(LB=5)

In the above, the parameters have the following meaning:
LB Indicator of covariance pattern as defined above.
NE Number of energies in the parameter table for a given subsection.
236

CHAPTER 32. FILE 32. COVARIANCES OF RESONANCE PARAMETERS
NT = NE*(NE+1)/2.
IDP Identification number of a resonance parameter type. This index depends
on the resonance formulation used, as summarized in the table below. This
table is consistent with the order of parameters used in the NSRS-type subsubsections to define the parameter covariances included, the first NPAR
parameters in the list for the given LRF value.

IDP
1
2
3
4
5
6
7
8

1
Er
Γn
Γγ
Γf
Γx

LRF
3
Er
Γn
Γγ
Γf
Γx

2
Er
Γn
Γγ
Γf
Γx

4
Er = µ
ν
GRT
GIT
GRF
GIF
GRC
GIC

For LB=0, ±1, 2, and 5 the energy values in the tables are monotonically increasing and cover
the range EL to EH. For LB=5 the F -values are given in the upper triangular representation
of a symmetric matrix by rows, i.e. F1,1 , F1,2 , ..., F1,N E−1 ; F2,2 , F2,N E−1 ; FN E−1,N E−1

32.2.3

Resolved Resonance
(LCOMP=2)

Compact

Covariance

Format

This format was developed in order to provide a means of communicating the actual
resonance-parameter covariance matrix (as determined during the data-evaluation process)
in a compact, legible, and accurate form, for those situations in which it is not practical to
provide the entire covariance matrix using the LCOMP=1 format. The covariance matrix is
specified as uncertainties plus correlation matrix. The correlation coefficients (whose values
range from -1 to 1) are presented in a compact representation as an integer with NDIGIT
digits plus a sign; each approximated correlation coefficient differs from the original value by
at most 0.5×10−NDIGIT (0.005 if NDIGIT = 2). Allowed values for NDIGIT are 2 through 6.
If the covariance matrix element connecting parameter number i with parameter number
j is denoted by Vij , the uncertainty on parameter i by Di , and the correlation coefficient by
Cij , then these quantities are related by
Di2 = Vii

;

Vij = Di Cij Dj

Values for Cij range from -1 to +1; values for Di are always positive. Note that the diagonal elements of Cij (i.e. those for which i = j) are always exactly 1.0 and therefore are never
specified explicitly. Compacting the off-diagonal correlation coefficients is accomplished as
follows:
237

CHAPTER 32. FILE 32. COVARIANCES OF RESONANCE PARAMETERS
1. Drop (set to zero) all values of Cij between between −10−NDIGIT and +10−NDIGIT .
2. Multiply the remaining coefficients by 10NDIGIT .
3. Map all positive values greater than K and less than or equal to K+1 to the integer
K.
4. Map all negative values less than –K and greater than or equal to –K–1 to the integer
–K.
The reverse mapping takes the integer to the center of the range. For example, with
NDIGIT=2, the positive integer 87 (which corresponds to original correlation coefficients
in the range from 0.87 up to and including 0.88) maps to 0.875. The negative integer –12
(which corresponds to original correlation coefficients in the range from –0.12 inclusive down
to –0.13 exclusive) maps to correlation coefficient −0.125.
In the LCOMP=2 format, parameters for one resonance are given first (on one line in
a LIST record), followed directly by the uncertainties on those parameters, followed by
values and uncertainties for the next resonance. After parameters and uncertainties for all
resonances have been specified, the correlation coefficients are given.
Two types of records are used for the mapped correlation coefficients. A CONT record
reads NDIGIT (the number of digits), NNN (the number of parameters), and NM (the
number of lines of INTG records that follow). NDIGIT can have any of the values 2, 3, 4,
5, or 6; the corresponding number NROW of correlation coefficients that can be displayed
on one INTG record is 18, 13, 11, 9, and 8, respectively.
The information stored in the INTG records may be summarized as follows: Let i and
j represent two of the parameters in the numbering scheme defined above, with i > j. All
those correlation coefficients mapped to non-zero integers Kij are printed in File 32; zerovalued Kij are printed (as blanks or zeros) only when they occur on the same record with
non-zero values. Each line (record) in the file begins by specifying the location (i.e., by
specifying i and j, with i > j); other K’s on the same line correspond to (i, j + 1), (i, j + 2),
... (i, j + N ROW − 1) [so long as j + N ROW − 1 < i ]. If there are more non-zero K’s for
the same i, they are given on another line, again beginning with the next non-zero K.
The following FORTRAN statement sequence can be used to read the mapped correlation
coefficients and reconstruct the correlation matrix:
C CORR is the full correlation matrix of dimensions NNN x NNN
C MXCOR is the maximum dimension of CORR (NNN.LE.MXCOR)
PARAMETER (MXCOR=1000)
DIMENSION KIJ(18), CORR(MXCOR*MXCOR)
C Read the CONT record:
C NNN is the dimension of CORR(NNN,NNN),
C NM is the number of lines to follow in the file
C NDIGIT is the number of digits for the covariance matrix
READ (LIB,10) C1,C2,NDIGIT,NNN, NM, NX, MAT, MF, MT, NS
10 FORMAT (2F11.0, 4I11, I4, I2, I3,I5)
IF(NNN.GT.MXCOR) STOP ’MXCOR Limit exceeded’
238

CHAPTER 32. FILE 32. COVARIANCES OF RESONANCE PARAMETERS
C

Preset the correlation matrix to zero
NN2=NNN*NNN
DO I=1,NN2
CORR(I)=0
END DI
C
Preset the diagonal to one
DO I=1,NNN
CORR(I+(I-1)*NNN)=1
END DO
DO M=1,NM
C Read the INTG record
IF(NDIGIT.EQ.2) THEN
NROW=18
READ (LIB,20) II, JJ, (KIJ(N),N=1,NROW), MAT, MF,
20
FORMAT (I5, I5, 1X, 18I3, 1X, I4, I2, I3, I5)
ELSE IF(NDIGIT.EQ.3) THEN
NROW=13
READ (LIB,30) II, JJ, (KIJ(N),N=1,NROW), MAT, MF,
30
FORMAT (I5, I5, 1X, 13I4, 3X, I4, I2, I3, I5)
ELSE IF(NDIGIT.EQ.4) THEN
NROW=11
READ (LIB,40) II, JJ, (KIJ(N),N=1,NROW), MAT, MF,
40
FORMAT (I5, I5, 1X, 11I5,
I4, I2, I3, I5)
ELSE IF(NDIGIT.EQ.5) THEN
NROW= 9
READ (LIB,50) II, JJ, (KIJ(N),N=1,NROW), MAT, MF,
50
FORMAT (I5, I5, 1X, 9I6, 1X, I4, I2, I3, I5)
ELSE IF(NDIGIT.EQ.6) THEN
NROW= 8
READ (LIB,20) II, JJ, (KIJ(N),N=1,NROW), MAT, MF,
60
FORMAT (I5, I5,
8I7,
I4, I2, I3, I5)
ELSE
STOP ’ERROR - Invalid NDIGIT’
END IF
C Interpret the INTG record and fill the covariance matrix
JP = JJ - 1
Factor =10**(NDIGIT)
DO N=1,NROW
JP = JP + 1
IF(JP.GE.II) GO TO 30
IF(KIJ(N).NE.0) THEN
IF(KIJ(N).GT.0) THEN
CIJ = ( KIJ(N)+0.5)/Factor
ELSE

239

MT, NS

MT, NS

MT, NS

MT, NS

MT, NS

CHAPTER 32. FILE 32. COVARIANCES OF RESONANCE PARAMETERS
CIJ =-(-KIJ(N)+0.5)/Factor
END IF
JJII = JJ + (II-1)*NNN
IIJJ = II + (JJ-1)*NNN
CORR(JJII)= CIJ
CORR(IIJJ)= CIJ
END IF
END DO
30
CONTINUE
END DO
The general structure of the formats has already been described at the beginning of this
section. Formats for the individual subsections differ depending on the value of LRF, and
are discussed below.
In the format descriptions to follow, the notation is as defined in Chapter 2. Uncertainties are denoted by parameters whose names begin with ”D”, but are otherwise identical
to the parameter of Chapter 2. For example, ER represents the resonance energy in the
laboratory system, in units of eV. Therefore, DER represents the uncertainty (square root
of the variance) associated with the resonance energy; DER has the same units as ER.
Parameters which are not variables (that is, which are not searchable parameters in the
analysis process) do not have uncertainties associated with them. For example, the total spin
of a resonance, denoted by parameter AJ, has no associated uncertainty DAJ. In File 32,
the value 0.0 is given instead of DAJ (for LRF = 1,2,3).
For LRF = 1 and 2, the redundant parameter GT (which is equal to GN + GG + GF)
has no corresponding uncertainty DGT specified explicitly. Instead, if needed, DGT may be
calculated using values for DGN, DGG, and DGF, and the correlation matrix.
During the evaluation process, not all potential variables are treated as searchable parameters. For example, if no capture cross section data were available, the evaluator might
choose to set all capture widths GG to a constant value. The associated uncertainty DGG
is then specified in File 32 as 0.0, indicating that this parameter’s uncertainty is not known.
(The proper procedure to be used in evaluating the effect of these unvaried parameters on the
final covariance matrix remains an open question, and is not addressed in this document.)
In order to express the correlation matrix as compactly as possible, the resonance parameters (those which may be varied during the evaluation process) are implicitly numbered,
in the order in which they occur in the listing of File 2. For LRF = 1, 2, or 3, the nonsearchable parameter AJ is included in the list but is NOT included in this numbering, nor
(for LRF=1 or 2) is the redundant parameter GT. Parameters whose value is given but
whose uncertainty is unknown (as described in the previous paragraph) are nevertheless
included in the numerical ordering. For completely non-fissile nuclides, fission widths are
not included in the numbering scheme; likewise, for fissile or fissionable nuclides for which
the evaluator chose to use only one fission width, the second fission width (for LRF = 3)
would not be counted.
Notation used here but not previously defined:

240

CHAPTER 32. FILE 32. COVARIANCES OF RESONANCE PARAMETERS
NRSA For LRF=1,2, or 3, the total number of resonances (for all L values) to be
included in the covariance matrix.
For LRF=7, the number of resonances from a particular J π group to be
included in the covariance matrix.
NJSX For LRF=7 only, the number of J π groups with resonances to be included in
the covariance matrix.
NDIGIT Number of digits to be used in writing the LCOMP=2 type INTG record.

32.2.3.1

SLBW and MLBW (LRF=1 or 2)

The structure of the subsection (assuming NRO=0) is as follows:
[MAT,32,151/ SPI,
AP,
0, LCOMP,
0,
0]CONT
[MAT,32,151/AWRI,
QX,
0,
LRX, 12*NRSA,
NRSA/
ER1 ,
AJ1 ,
GT1 ,
GN1 ,
GG1 ,
GF1 ,
DER1 ,
0.0,
0.0,
DGN1 ,
DGG1 ,
DGF1 ,
ER2 ,
AJ2 ,
GT2 ,
GN2 ,
GG2 ,
GF2 ,
DER2 ,
0.0,
0.0,
DGN2 ,
DGG2 ,
DGF2 ,
-----------------------------------------------ERN RSA , AJN RSA , GTN RSA , GNN RSA , GGN RSA , GFN RSA ,
DERN RSA ,
0.0,
0.0, DGNN RSA , DGGN RSA , DGFN RSA ]LIST
[MAT,32,151/ 0.0,
0.0, NDIGIT,
NNN,
NM,
0 ]CONT
[MAT,32,151/ II,
JJ,
KIJ]INTG
[MAT,32,151/ II,
JJ,
KIJ]INTG
< Continue until a total of NM of INTG-type records are read>
Note that NNN is the total number of unique resonance parameters included in this listing, not including AJ or GT. For fissile nuclides, NNN=NRSA×4; for non-fissile nuclides,
NNN=NRSA×3.

241

CHAPTER 32. FILE 32. COVARIANCES OF RESONANCE PARAMETERS
32.2.3.2

Reich-Moore (LRF=3)

The structure of the subsection is:
[MAT,32,151/ SPI,
AP,
LAD,
LCOMP,
0,
0]CONT
[MAT,32,151/AWRI,
APL,
0,
0, 12*NRSA,
NRSA/
ER1 ,
AJ1 ,
GN1 ,
GG1 ,
GFA1 ,
GFB1 ,
DER1 ,
0.0,
DGN1 ,
DGG1 ,
DGFA1 ,
DGFB1 ,
ER2 ,
AJ2 ,
GN2 ,
GG2 ,
GFA2 ,
GFB2 ,
DER2 ,
0.0,
DGN2 ,
DGG2 ,
DGFA2 ,
DGFB2 ,
----------------------------------------------------ERN RSA ,AJN RSA , GNN RSA , GGN RSA , GFAN RSA , GFBN RSA ,
DERN RSA ,
0.0, DGNN RSA ,DGGN RSA, DGFAN RSA ,DGFBN RSA ]LIST
[MAT,32,151/ 0.0,
0.0, NDIGIT,
NNN,
NM,
0 ]CONT
[MAT,32,151/ II,
JJ,
KIJ]INTG
[MAT,32,151/ II,
JJ,
KIJ]INTG
< Continue until a total of NM of INTG records are read>
Note that NNN is the total number of resonance parameters in this listing (again not including AJ). For non-fissile nuclides, NNN=NRSA×3. For fissile nuclides, NNN=NRSA×4
if only one fission channel is used, and NNN=NRSA×5 if both are used.

242

CHAPTER 32. FILE 32. COVARIANCES OF RESONANCE PARAMETERS
32.2.3.3

R-Matrix Limited Format (LRF=7)

The structure of the subsection is:
[MAT,32,151/ 0.0,
0.0,
IFG,
LCOMP,
NJS,
KRL ]CONT
[MAT,32,151/ 0.0,
0.0,
NPP,
NJSX,
12*NPP, 2*NPP/
MA1 ,
MB1 ,
ZA1 ,
ZB1 ,
IA1 ,
IB1 ,
Q1 ,
PNT1 ,
SHF1 ,
MT1 ,
PA1 ,
PB1 ,
MA2 ,
MB2 ,
ZA2 ,
ZB2 ,
IA2 ,
IB1 ,
Q2 ,
PNT2 ,
SHF2 ,
MT2 ,
PA2 ,
PB1 ,
-----------------------------------------------MAN P P , MBN P P ,
ZAN P P ,
ZBN P P ,
IAN P P , IBN P P ,
QN P P , PNTN P P ,
SHFN P P ,
MTN P P ,
PAN P P , PBN P P ]LIST
[MAT,32,151/ AJ,
PJ,
0,
0,
6*NCH,
NCH/
IPP1 ,
L1 ,
SCH1 ,
BND1 ,
APE1 ,
APT1 ,
IPP2 ,
L2 ,
SCH2 ,
BND2 ,
APE2 ,
APT2 ,
-----------------------------------------------IPPN CH ,
LN CH ,
SCHN CH , BNDN CH ,
APEN CH , APTN CH ]LIST
------------------------------------------------[MAT,32,151/0.0,
0.0,
0,
NRSA,
12*NX,
NX/
ER1 ,
GAM1,1 ,
GAM2,1 ,
GAM3,1 ,
GAM4,1 ,
GAM5,1 ,
GAM6,1 , ... ... ... ... ... ... ... . , GAMN CH,1 ,
DER1 ,
DGAM1,1 ,
DGAM2,1 ,
DGAM3,1 ,
DGAM4,1 ,
DGAM5,1 ,
DGAM6,1 , ... ... ... ... ... ... ... . , DGAMN CH,1 ,
ER2 ,
GAM1,2 ,
GAM2,2 ,
GAM3,2 ,
GAM4,2 ,
GAM5,2 ,
GAM6,2 , ... ... ... ... ... ... ... . , GAMN CH,2 ,
DER2 ,
DGAM1,2 ,
DGAM2,2 ,
DGAM3,2 ,
DGAM4,2 ,
DGAM5,2 ,
DGAM6,2 , ... ... ... ... ... ... ... . , DGAMN CH,2 ,
------------------------------------------------ERN RSA ,GAM1,N RSA , GAM2,N RSA , GAM3,N RSA , GAM4,N RSA , GAM5,N RSA ,
GAM6,N RS , ... ... ... ... ... ... ... . , GAMN CH,N RS ,
DERN RSA ,DGAM1,N RSA ,DGAM2,N RSA ,DGAM3,N RSA ,DGAM4,N RSA ,DGAM5,N RSA ,
DGAM6,N RSA , ... ... ... ... ... ... . , DGAMN CH,N RSA ]LIST

[MAT,32,151/ 0.0,
0.0, NDIGIT,
NNN,
NM,
0 ]CONT
[MAT,32,151/ II,
JJ,
KIJ ]INTG
[MAT,32,151/ II,
JJ,
KIJ ]INTG
< Continue until a total of NM of INTG records are read >
For LRF=7 (unlike other formats), the number of channels may vary from one spin group
to another. The number of resonance parameters NNN is therefore given by the sum (from
1 to NJS) of (NCH×NRSA).

243

CHAPTER 32. FILE 32. COVARIANCES OF RESONANCE PARAMETERS

32.2.4

Unresolved Resonance Format (LRU=2)

For the unresolved resonance region a simplified covariance formulation is permitted. For
the purposes of covariance representation, no energy dependence is specified for the average
parameter relative covariances, even though in File 2 the unresolved region may be represented with energy-dependent average Breit-Wigner parameters using LRF=2. Relative
covariance elements are tabulated, unlike the cases above.
If the evaluator wishes to represent the relative covariance of the unresolved resonance
parameters, the subsection for a given isotope has the following form.
[MAT,32,151/ SPI, AP,
0,
0,
NLS,
0]CONT
[MAT,32,151/ AWRI, 0.0,
L,
0, 6*NJS, NJS/
D1 , AJ1 , GNO1 , GG1 ,
GF1 , GX1 ,
-----------------------------------DN JS , AJN JS , GNON JS , GGN JS , GFN JS , GXN JS ]LIST

[MAT,32,151/ 0.0, 0.0, MPAR, 0,(NPAR*(NPAR+1))/2,NPAR/
RV11 , RV12 , -------------------------,
-----------, RV1,N P AR ,--------, RNN AP R,N P AR ]LIST
MPAR is the number of average parameters for which relative covariance data are given for
each L and J, in the order D, GNO, GG, GF, and GX, for a maximum of 5. That is, relative
covariance values for the first MPAR of these are tabulated for each (L,J) combination. If
MPAR is given as 4 when LFW=0 on the CONT (isotope) record, then the four covariance
matrix indices per (L,J) combination represent D,GNO, GG, and GX.
NPAR=MPAR*(sum of the values of NJS for each L).
The LSSF flag is defined in Section 2.3.1.
RVij , the relative covariance quantities among these average unresolved parameters for
the given isotope. The final LIST record contains the upper triangular portion of the symmetric relative covariance matrix by rows.

244

CHAPTER 32. FILE 32. COVARIANCES OF RESONANCE PARAMETERS

32.3

Procedures

As indicated earlier, it is desirable to utilize File 32 when self-shielding is important or
when only a few resonances fall within an energy group of the processed cross sections. It
is believed that in most cases the covariances the evaluator needs to represent will not use
many of the available File 32 options. One does not expect to find covariance data in File 32
for all the resonance parameters in File 2.
Correspondence Between Files 2 and 32. Completeness of File 32.
1. The overall energy range for a given isotope in File 32 is nested within the corresponding range of File 2. In either case, there may be several energy range control records.
The following rules apply separately for each isotope:
(a) The smallest lower range limit EL for Files 2 and 32 must agree.
(b) The highest upper range limit EH for File 32 may be smaller than or equal to
that for File 2.
(c) In File 32 as in File 2, the energy ranges of the subsections may not leave gaps.
Subsections with LRU=1 may overlap if consistent resonance formulations are
referenced (LRF=1 or 2).
(d) An unresolved energy region (LRU=2) may be used in File 32 if one is employed
in File 2. If one is used, its lower energy range limit must equal the corresponding
limit for File 2.
2. In a File 32 LCOMP=0 subsection, any selection of the resonances shown in File 2
with LRF=1 or 2 must be listed in order of increasing energy. The resonance energy
given in File 32 shall agree with that in File 2 for the same resonance to a relative
tolerance of ≤ 10−5 to assume positive identification.
3. In LCOMP=1, LRU=1 subsections, any selection of the resonances shown in File 2 for
consistent resonance formulations may be listed in NSRS-type subsections. The listed
resonances should be in order of increasing energy within each block, and the blocks
should be arranged by order of increasing energy for the lowest-energy resonance in the
block. A given resonance may appear in more than one block. The resonance energy
given in File 32 shall agree with that in File 2 for the same resonance to a relative
tolerance of ≤ 10−5 .
4. The energy identifiers of every long-range (NLRS-type) subsection must cover the
entire energy range (EL,EH) of the subsection. However, any desired covariance components may be null.
5. Long-range subsections may be included in any order in LCOMP=1, LRU=1 subsections.

245

CHAPTER 32. FILE 32. COVARIANCES OF RESONANCE PARAMETERS
Obtaining Cross Section Covariances From File 32 and File 33.
1. Outside the combined energy range covered by File 32, whether in part or all of the
energy range covered by File 2, the covariance data in File 33 refers to the cross sections
reconstituted from File 2 plus File 3 according to the value of the LSSF flag.
2. In LCOMP=0 subsections a nonzero variance or covariance involving the resonance
spin J (AJ) will be treated as null.
3. When File 32 is present for a resolved resonance range (LRU=1), the covariances
among the effective cross sections in the region are obtained by combining data from
Files 32 and 33.
(a) A resonance parameter covariance matrix is developed by summing all the contributions for each resonance from the File 32 subsections with LRU=1.
(b) The resonance parameter covariance file is processed to obtain the covariance
data of effective cross sections implied by File 32. (This data will in general
be a function of isotopic dilution and temperature.) Covariance for the various
isotopes are summed with appropriate weights if the evaluation is an elemental
one.
(c) Covariances derived from File 32 are summed with those given in File 33. With
one exception, relative covariances given in File 33 apply to the effective cross
sections reconstituted from File 2 and File 3. In that exception, when the AdlerAdler (LRF=4) resonance formulation is employed, any relative covariances in
File 33 for that energy region apply to the sum of the cross section from File 3
with the smooth background given for the LRF=4 resonance data.
4. When File 32 is present for the unresolved resonance range (LRU=2), covariance data
for the region are obtained as follows:
(a) File 33 covariance data for this energy region are taken to represent the covariances of evaluated average cross sections for ”infinite isotopic dilution.”
(b) To obtain covariances among effective cross sections for material dilutions such
that uncertainties in self shielding can become important, the effects of the unresolved resonance parameter covariances in File 32 and the evaluated average cross
section covariances in File 33 are to be combined. A means for this combination
was described by deSaussure and Marable [Ref. 1]. (Note that covariances of the
effective cross sections in one test case were not much affected by uncertainties
in average parameters. See B.L. Broadhead and H.L. Dodds [Ref. 2].)
Example For Fictitious MAT 3333, ZA 99280
The nucleus of concern in this example has resonances represented by LRF=2 for the energy
range 1 to 50 eV. Five (5) resonances are given in File 2, one of which lies at negative
energy, but the resonance at 15 eV is considered only relative to long-range uncertainties.
Example 32.1 shows the File 2 and Example 32.2 shows the corresponding File 32.
246

CHAPTER 32. FILE 32. COVARIANCES OF RESONANCE PARAMETERS
In File 32, three NSRS-type subsections and four NLRS-type subsections are included. Of
the former, the first refers only to the negative energy resonance and indicates the following
parameter covariance contribution for that resonance, in eV2 .
r
Er
Γn
Γγ
Γf

1
2
3
1.0 -1.5 0
-1.5 4.0 0
0
0
0
-0.2 -1.0 0

4
-0.2
-1.0
0
0.8

This block is mostly to express the negative correlation between neutron width and apparent
resonance energy for this guessed resonance.
The second subsection refers only to the 5 eV resonance, and conveys the following
covariance contribution:
r
Er
Γn
Γγ
Γf

1
0.5 × 10−6

2
0
1.0 × 106
1.0 × 106
0.2 × 106

3
0
1 × 106
0.0
2 × 106

4
0
0.2 × 106
2.0 × 106
4.0 × 106

The third block covers Γn covariances for the 30 and 40 eV resonances.
The first NLRS subsection gives the Γγ uncertainty correlated among all the resonances.
The value is assumed to have been determined solely by analysis of the 5 eV resonance. Note
that the variance for the 5 eV resonance generated by the subsection fills the ”hole” in the
table just above (with the value 16 × 10−6 ). The second subsection expresses the expected
resonance-to-resonance uncorrelated fluctuation in Γγ . The third and fourth subsections
cover long-range uncertainties in Γn and Γf the example for the former might be hard for
the evaluator to defend for a real nuclide.
Example 32.1. File 2 for Sample File 32
99.280+3
99.280+3
1.00+0
1.00+0
2.70+2
1.00+0
5.00+0
1.50+1
3.00+1
4.00+1
0.00+0
0.00+0

2.70+2
1.00+0
5.00+1
0.60+0
0.00+0
1.50+0
1.50+0
0.50+0
1.50+0
1.50+0
0.00+0
0.00+0

0
0
LRU=1
0
L=0
4.04 +0
0.07 +0
0.08 +0
7.04 +0
6.04 +0
0
0

0
NIS=1
LFW=0
NER=1
LRF=2
NRO=0
0
NLS=1
LRX=0 6*NRS=30
3.0 +0 0.04 +0
0.01 +0 0.04 +0
0.01 +0 0.04 +0
1.0 +0 0.04 +0
5.0 +0 0.04 +0
0
0
0
0

Example 32.2. Sample File 32
247

0333332151
0333332151
NAPS=0333332151
0333332151
NRS=5333332151
1.0 +0333332151
0.02+0333332151
0.03+0333332151
6.0 +0333332151
1.0 +0333332151
0333332 0
03333 0 0

HEAD
CONT
CONT
CONT

LIST
SEND
FEND

CHAPTER 32. FILE 32. COVARIANCES OF RESONANCE PARAMETERS
99.280+3 2.700+2
0
0
99.280+3 1.000+0
0
LFW=0
1.0000+0 5.000+1
LRU=1
LRF=2
1.0000+0 0.600+0
0 LCOMP=1
2.7000+2 0.000+0
0
0
0.0000+0 0.000+0
MPAR=4
0
-1.0000+0 1.500+0 4.040+0 3.000+0
1.0000+0 -1.500+0
0.00+0 -0.200+0
-1.0000+0 0.000+0
0.00+0 0.800+0
0.0000+0 0.000+0
MPAR=4
0
5.0000+0 1.500+0 0.070+0
0.01+0
0.5000-6 0.000+0 0.000+0
0.00+0
-0.2000-6 0.000+0 -2.000-6
4.00-6
0.0000+0 0.000+0
MPAR=2
0
3.0000+1 1.500+0 7.040+0
1.00+0
4.0000+1 1.500+0 6.040+0
5.00+0
2.0000-3 0.000+0
0.00+0
0.00+0
-0.5000-4 2.000-3
0.00+0
2.00-3
0.0000+0 0.000+0
IDP=3
LB=1
-2.0000+0 1.000-2
5.00+1
0.00+0
0.0000+0 0.000+0
IDP=3
LB=1
-2.0000+0 1.000-2
1.00+0
0.00+0
5.0000+1 0.000+0
0.00+0
0.00+0
0.0000+0 0.000+0
IDP=2
LB=5
1.0000-5 2.000+1
5.00+1
1.00-4
0.0000+0 0.000+0
IDP=4
LB=1
1.0000-5 4.000-4
5.00+1
0.00+0
0.0000+0 0.000+0
0
0
0.0000+0 0.000+0
0
0

NIS=1
0333332151 HEAD
NER=1
0333332151 CONT
NRO=0 NAPS=0333332151 CONT
NLS=0
0333332151 CONT
NSRS=3 NLRS=4333332151 CONT
M*16
NRB=1333332151
0.040+0 1.00+0333332151
4.000+0 0.00+0333332151
0.000+0 0.00+0333332151 LIST
M*16
NRB=1333332151
0.04+0 0.02+0333332151
1.00-6 -1.00-6333332151
0.00+0 0.00+0333332151 LIST
M*22
NRB=2333332151
0.04+0 6.00+0333332151
0.04+0 1.00+0333332151
1.00-3 0.00+0333332151
0.00+0 0.00+0333332151 LIST
2*NE=4
NE=2333332151
0.00+0 0.00+0333332151 LIST
2*NE=8
NE=4333332151
1.00+1 1.00-2333332151 LIST
0.00+0 0.00+0333332151 LIST
NT=6
NE=3333332151
1.00-4 2.00-4333332151 LIST
2*NE=4
NE=2333332151
0.00+0 0.00+0333332151 LIST
0
0333332 0 SEND
0
03333 0 0 FEND

M =NVS+6*NRB
NVS=NRB*MPAR*(NRB*MPAR+1)/2.

References for Chapter 32
1. G. DeSaussure and Marable, Nucl. Sci. Eng. 101, 285 (1989)
2. B. L. Broadhead and H. L. Dodds, Trans. Am. Nucl. Soc. 39, 929 (1981)

248

Chapter 33
File 33, COVARIANCES OF
NEUTRON CROSS SECTIONS
33.1

General Comments

File 33 contains the covariances of neutron cross section information appearing in File 3.
It is intended to provide a measure of the ”uncertainties and their correlations” and does
not indicate the precision with which the data in File 3 are entered. However, it should be
stressed that for most practical applications to which the files are intended, the data will be
processed into multigroup variance-covariance matrices. When generating File 33, it should
be remembered that major aims are to represent adequately:
i. the variances of the group cross sections,
ii. the correlations between the cross sections of the several adjacent groups, and
iii. the long-range correlations among the cross sections for many groups.
Table 33.1 illustrates a typical relation of these three covariances with experimental
uncertainties1 .
Table 33.1: Analogies Between File 33 Covariances Within One Section and Uncertainties
in a Hypothetical Experiment.
File 33
Short range
Medium range

Long range

Experimental
Statistical
Detector Efficiency, Multiple
Scattering, In/Out Scattering
Geometry Background Normalization

1

Energy Dependence
Rapid variation
Slowly varying

More or less constant

As with all analogies, this should be used with care. It is designed to show in a familiar way of thinking
how the covariances within a section are related.

249

CHAPTER 33. FILE 33, COVARIANCES OF NEUTRON CROSS SECTIONS
These primary considerations and the inherent difficulties associated with quantifying
uncertainties should dictate the level of detail given in File 33.
In the resolved resonance region, some of the covariances of the cross sections may be
given through the covariances of the resonance parameters in File 32. In this case, the
long-range components of the covariance matrix of the cross sections, which span many
resonances, may be given in File 33, since often the most important components of the
matrix are long-range.
Example:
Consider a case with the following sources of uncertainty:
2% uncertainty due to statistics (short-range),
2% due to multiple scattering (medium-range), and
1% due to geometry (long-range),
we would cite the uncertainties as:
3% uncertainty for a discrete measurement (one group covering a small energy
range,
calculated as the square root of the squares of individual contributions
√
2
2 + 22 + 12 );
≈2.5% over an energy range encompassing several measurements (several groups
which together cover a 1 to 2 MeV range, calculated as above, but the statistical component becomes smaller than 2% due to the averaging procedure
over the energy interval); and
≈ 1% average over the entire energy range (long- and medium-range terms disappear).

33.2

Formats

File 33 is divided into sections identified by the values of MT. Within a section defined by
(MAT,33,MT), several subsections may appear. Each section of File 33 starts with a HEAD
record, ends with a SEND record. The following quantities are defined:
ZA,AWR Standard material charge and mass parameters.
MTL Non-zero value of MTL is used as a flag to indicate that reaction MT is
one component of the evaluator-defined lumped reaction MTL, as discussed
in paragraphs at the end of Section 33.2 and 33.3 below; in this case, no
covariance information subsections are given for reaction MT and NL=0.
NL Number of subsections within a section.
250

CHAPTER 33. FILE 33, COVARIANCES OF NEUTRON CROSS SECTIONS
The structure of a section is:
[MAT, 33, MT/ ZA, AWR,

2>
NL>
0, 0] SEND

Subsections

Each subsection of the section (MAT,33,MT) is used to describe a single covariance matrix.
It is the covariance matrix of:
- 1st set of energy-dependent cross sections given in section (MAT,3,MT) and
- 2nd set of energy-dependent cross sections given in section (MAT1,MF1,MT1) when
MF1=3 or (MAT1,MF1,MT1.LFS1) when MF1=10 (see definitions below).
The values of MAT1, XMF1, MT1 (and XLFS1, if MF=10) are given in the CONT record
which begins each subsection. Each File 33 subsection is therefore identified with a unique
combination of values (MAT,MT) and (MAT1,MF1,MT1[.LFS1]), and we may use the notation (MAT,MT; MAT1,MF1,MT1[.LFS1]) to specify a subsection.
Each subsection may contain several sub-subsections. Two different types of subsubsections may be used; they are referred to as ”NC-type” and ”NI-type” subsections.
Each sub-subsection describes an independent contribution (called a component) to the
covariance matrix given in the subsection. The total covariance matrix given by the subsection is made up of the sum of the contributions from the individual sub-subsections. The
following quantities are defined:
XMF1 Floating point equivalent of the MF for the 2nd energy-dependent cross section of the pair, for which the correlation matrix is given. If MF1=MF,
XMF1=0.0 or blank.
XLFS1 Floating point equivalent for the final excited state of the 2nd energydependent cross section. For MF1=10, XLFS1 = 10; if MF16=10, XLFS1=0.0
or blank.
MAT1 MAT for the 2nd energy-dependent cross section
MT1 MT for the 2nd energy-dependent cross section
NC Number of NC-type sub-subsections which follow the CONT record.
NI Number of NI-type sub-subsections which follow the NC-type subsubsections.

251

CHAPTER 33. FILE 33, COVARIANCES OF NEUTRON CROSS SECTIONS
The structure of a subsection describing the covariance matrix of the cross sections given in
(MAT,3,MT) and (MAT1,MF1,MT1.[LFS1]) is:
[MAT,33,MT/ XMF1, XLFS1, MAT1, MT1, NC, NI]CONT


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


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

33.2.2

Sub-Subsections

NC-type and NI-type sub-subsections have different structures.
NC-type sub-subsections may be used to indicate that some or all of the contributions to
the covariance matrix described in the subsection are to be found in a different subsection
of the ENDF tape. The major purpose of the NC-type sub-subsections is to eliminate from
the ENDF tape a large fraction of the mostly redundant information, which would otherwise
be needed if only NI-type sub-subsections were used.
NI-type sub-subsections are used to describe explicitly various components of the covariance matrix defined by the subsection.
33.2.2.1

NC-type Sub-Subsections

NC-type subsections may be used to describe the covariance matrices in energy ranges where
the cross sections in (MAT,3,MT) can be described in terms of other evaluated cross sections
in the same energy range. In the context of File 33, and for purposes of discussing NC-type
sub-subsections, we define an ”evaluated” cross section in a given energy range as one, for
which the covariance matrix in that energy range is given entirely in terms of NI-type subsubsections. The covariance matrices involving the ”derived” cross sections may be obtained
in part in terms of the covariance matrices of the ”evaluated” cross sections given elsewhere
in File 33. The following quantity is defined:
LTY Flag used to indicate the procedure used to obtain the covariance matrix.
The values of LTY can be from 0 to 4, as explained below.
LTY=0, ”Derived Redundant Cross Sections”
In File 33, the evaluator may indicate by means of an LTY=0 sub-subsection that in a given
energy range the cross sections in (MAT,3,MT) were obtained as a linear combination of
252

CHAPTER 33. FILE 33, COVARIANCES OF NEUTRON CROSS SECTIONS
other ”evaluated” cross sections having the same MAT number but different MT values. By
the definition of ”evaluated covariances” the covariances of these cross sections are given in
File 33 wholly in terms of NI-type sub-subsections.2 The following additional quantities are
defined:
E1, E2 Energy range (eV) where the cross sections given in the section (MAT,3,MT)
were ”derived” in terms of other ”evaluated” cross sections given in the sections (MAT,3,MTI)s.
NCI Number of pairs of values in the array {CI, XMTI}3 .
{CI, XMTI} Array of pairs of numbers, each pair consisting of the coefficient CI and a
value of XMTI, (i.e. the MTI number given in floating point representation).
The pair of numbers indicates that, in the energy range E1 to E2, the cross
sections in file (MAT,3,MT), were obtained in terms of the cross sections in
files (MAT,3,MTI), as follows:
MAT
σMT
(E) =

NCI
X
i=1

MAT
Ci × σMT
(E) .
i

[At the minimum, the use of an LTY=0 sub-subsection implies that the
evaluator wishes the corresponding covariance components to be derived as
if this expression were valid.]
In this expression we have written the CI’s as Ci , and XMTI’s as MTi . The
numbers CI are constant numbers over the whole range of energy E1 and E2,
usually ±1.
The structure of an NC-type sub-subsection with LTY=0 is:
[MAT,33,MT/ 0.0, 0.0,
[MAT,33,MT/ E1, E2,

0, LTY=0,
0,
0]CONT
0,
0, 2*NCI, NCI/{CI, XMTI}]LIST

Note: In general, each subsection describes a single covariance matrix. However, when an
NC-type sub-subsection with LTY=0 is used in a subsection, portions of NCI+1 covariance
matrices are implied and these are not explicitly given as subsections in the File 33 (see
procedure in Section 33.3.2 item a.3). In such cases the subsection may be thought of as
describing in part several covariance matrices.
LTY=1, 2 and 3, ”Covariances of Cross Sections Derived via Ratios to Standards”
Many important cross sections in ENDF files are based on measurements of cross sections
ratios to standard cross sections. When an evaluated cross section is obtained from such
measurements, covariances so generated between the cross sections for the two reactions may
2

In general the linear relationship given in an LTY=0 sub-subsection applies not only to the range of
energies specified, but also over the whole range of the file.
3
The Notation {AI,BI} stands for A1, B1; A2, B2; ...; Ai, Bi in a list record.

253

CHAPTER 33. FILE 33, COVARIANCES OF NEUTRON CROSS SECTIONS
become important. This is the primary origin of ENDF covariances linking cross sections
for different materials. As seen below, these covariances depend on the covariances of the
standard cross section and on the covariances of the evaluated cross section ratios. When
the resulting multigroup covariance files are utilized, the covariances of the ratios themselves
play an important role if the performance of some system depends on the relative magnitude
of two cross sections.
In order to represent efficiently in File 33 the covariances that depend on ”absolute” ratio
measurements to standards, evaluators may use NC-type sub-subsections with LTY=1, 2 or
3 in appropriate File 33 subsections. [In other cases of covariance components induced by
ratio measurements, it is necessary for the evaluator to explicitly represent the covariance
components that arise and may be found in the literature.4 ]
First we identify the covariances induced if an evaluator obtains cross section σa (E) for
(MAT,MT) within the interval [E1,E2] entirely from absolute ratio measurements to a
cross section standard σS (E) for (MATS,MTS). That is,
σa (E) = α(E) σS (E) , for (E1 ≤ E < E2) .
The evaluated ratio α(E) itself is assumed to be independent of the standard cross section
evaluation because the relevant measurements are similarly independent. If so, then:
Rvar[σa (E)] = Rvar[α(E)] + Rvar[σS (E)] , for (E1 ≤ E < E2)
Rcov[σa (E), σa (E ′ )] = Rcov[α(E), α(E ′ )] + Rcov[σS (E), σS (E ′ )]
for (E1 ≤ E < E2) and (E1 ≤ E ′ < E2)
Rcov[σa (E), σS (E ′ )] = Rcov[σS (E), σS (E ′ )]
for (E1 ≤ E < E2) and (E1 ≤ E ′ < E2) .
Rvar and Rcov are the relative variance and relative covariance defined in Chapter 30, and
the values generated by the specified ratio evaluation are zero outside the specified ranges.
The variance and covariance terms in (MAT,MT) depend on those for both the standard
and the ratio, and only for energies in ranges where the cross section is specified to depend
on this standard. However, the covariance matrix between the cross sections (MAT,MT)
and (MATS,MTS) does not depend on the covariances of the ratio determination but spans
all values of the energy E ′ .
A cross section σa (E) may be obtained from measurements relative to the standard
(MATS,MTS) in the energy region (E1 ≤ E ≤ E2) and cross section σb (E ′ ) relative to another standard (MATS’,MTS’) in the energy region (E1’ ≤ E ′ ≤ E2’). If the cross sections
for the two standards are correlated and the two sets of ratio measurements are uncorrelated,
then one obtains the additional results:
4

W. P. Poenitz, ”Data Interpretation, Objective Evaluation Procedures, and Mathematical Techniques
for the Evaluation of Energy-Dependent Ratio, Shape, and Cross Section Data”, BNL-NCS-51363, Conf.
on Nuclear Data Evaluation and Techniques, p. 264, B. A. Magurno and S. Pearlstein, eds. (March, 1981)

254

CHAPTER 33. FILE 33, COVARIANCES OF NEUTRON CROSS SECTIONS

Rcov[σa (E), σb (E ′ )] = Rcov[σS (E), σS ′ (E ′ )] for (E1 ≤ E ≤ E2) , (E1’ ≤ E ′ ≤ E2’)
zero otherwise;
Rcov[σa (E), σS ′ (E ′ )] = Rcov[σS (E), σS ′ (E ′ )] for (E1 ≤ E ≤ E2) and all E ′ ,
zero otherwise;
′
Rcov[σS (E), σb (E )] = Rcov[σS (E), σS ′ (E ′ )] for (E1’ ≤ E ′ ≤ E2’) and all E,
zero otherwise.
Note that the above expressions apply when σa (E) and σb (E ′ ) refer to the same (MAT,MT)
for which cross sections are obtain by ratios to different standards in the two energy regions.
The most far-reaching relationship correlates σa (E) with all cross sections correlated to
the standard relative to which it was measured. That is:
Rcov[σa (E), σx (E ′ )] = Rcov[σS (E), σx (E ′ )] for (E1 ≤ E < E2) and all σx (E ′ ),
zero otherwise.
The right side may be non-zero for many (MATX,MTX), and discretion may be required
to avoid in cross section processing the generation of negligible but non-zero multigroup
covariance matrices.
Let the cross sections in (MAT,3,MT) be strictly ”derived” in the energy range E1 to
E2 through the evaluation of ratio measurements to the ”evaluated” cross sections given in
(MATS,MFS,MTS[.LFSS]), referred to also as the ”standard” cross sections for this ”ratio
evaluation”. Then, in the subsection (MAT,3,MT;MAT,3,MT) of the File 33 for the material
MAT, an LTY=1 sub-subsection must be used to describe, in part, the covariance matrix
in the energy range E1 to E2. (LFSS=0 when MF=3). The part, or component, of the
covariance matrix represented by the LTY=1 sub-subsection is obtained by the user from
the covariance matrix of the ”standard” cross sections in the File (MFS+30) subsection
(MATS,MTS[.LFSS]; MATS,MFS,MTS[.LFSS]) of the material MATS. The other part, or
component, of the covariance matrix comes from the evaluation of the ”ratios” and is given
explicitly, over the range E1 to E2, by means of NI-type sub-subsections in the File 33
subsection (MAT,3,MT;MAT,3,MT).
This method of evaluation introduces a covariance of the ”derived” cross sections in
(MAT,3,MT) over the energy range E1 to E2 and the ”standard” cross section over its complete energy range. Therefore, in File 33 of the material MAT, in subsection containing
the covariance of the ”standard” cross section, there must be an LTY=2 sub-subsection
to represent this covariance matrix. This LTY=2 sub-subsection [which contains the same
information as the previously given LTY=1 sub-subsection in the subsection (MAT,3,MT;
MAT,3,MT)] refers to a different covariance matrix than the LTY=1 sub-subsection previously mentioned, but it can also be derived from the covariance matrix of the ”standard”
cross sections in File MFS+30 subsection of the standard material MATS.
Finally, as a consequence of the evaluation of the cross sections in (MAT,3,MT) in
the energy range E1 to E2, as a ”ratio” to the ”standard” cross sections, there must be
in the subsection (MATS,MFS,MTS[.LFSS]; MAT,MF,MT) of the File (MFS+30) of the
255

CHAPTER 33. FILE 33, COVARIANCES OF NEUTRON CROSS SECTIONS
”standard” material MATS and LTY=3 sub-subsection in the File 33 subsection (MAT,MT;
MAT,3,MT)] which serves in the material MATS the same role as the LTY=2 sub-subsection
in the material MAT since they describe the same covariance matrix. In addition, the LTY
value of 3 serves as a ”flag” to the user, and the processing codes, to indicate existence of
any additional covariances among cross sections using the same ”standard” cross sections,
covariances not explicitly given in the covariance files. These additional covariance matrices
can be derived from the appropriate LTY=3 sub-subsections and the covariance matrix of
the ”standard” cross sections in the File (MFS+30) subsection. The following quantities are
defined:
E1, E2 Energy range where the cross sections given in the section (MAT,3,MT) were
obtained to a significant extent in terms of ratio measurements to ”standard”
cross sections.
NEI Number of energies that demarcate (NEI1) regions where this standard was
employed in measurements with WEI.
WEI Fractional evaluated weight.
The structure of NC-type sub-subsections with LTY=1,2 and 3 is:
[MAT,33,MT/ 0.0, 0.0,
0, LTY,
0,
0]CONT
[MAT,33,MT/ E1, E2, MATS, MTS, 2*NEI+2, NEI/
(XMFS,XLFSS), {EI,WEI}
]LIST
The number of items in the list record is 2*NEI+2. The file number that contains the
standard cross sections is the integer equivalent of XMFS, except that XMFS=0.0 is entered
when MFS=3. The value of XLFSS is always zero unless MFS=10.
The use of the format LTY=1, 2 and 3 is allowed when the cross sections given in
(MAT,3,MT) are only partially determined from ratio measurements to the ”standard”
cross section. In such cases the list {EI,WEI} indicates the fractional weight of the ratio
measurements to this standard in the evaluation of the cross sections in (MAT,3,MT). That
fractional weight is Wi of WEI in the interval Ei ≤ E < Ei+1 of EI. The first value of Ei in
the sub-subsection shall equal the E1 given, and EN EI =E2.
Note A: LTY=1, 2 and 3 sub-subsections are all used as flags in subsections to represent
relative covariance matrix components obtained from the relative covariance matrix of the
”standard” cross sections that is given in a File 33, 31, or 40. There is, however, as seen in the
formulae above, a major difference between covariance matrices obtained with LTY=1 subsubsections and those obtained from LTY=2 and 3 sub-subsections. This difference results
from the definition of their use given above. LTY=2 and 3 sub-subsections are always used
in subsections where one of the cross sections involved is the ”standard” cross section used.
The LTY=2 subsection appears in the File 33 [in the present case] of the material whose cross
sections are ”derived,” whereas the LTY=3 sub-subsection appears in the File (MFS+30) of
the material whose cross sections are the ”standard”; LTY=1 sub-subsections always appear
in subsections describing covariance matrices of cross sections ”derived” from a ”standard”
256

CHAPTER 33. FILE 33, COVARIANCES OF NEUTRON CROSS SECTIONS
and no LTY=2 or 3 sub-subsections may appear in such subsections. An LTY=1 subsubsection represents a covariance matrix which in principle is a ”square matrix” covering
the ranges E1 to E2. An LTY=2 or 3 subsection describes in principle a ”rectangular
matrix”: the covariance matrix of the ”derived” cross sections over the energy range E1 or
E2 and of the ”standard” cross sections over their complete energy range.
In general, if cross sections in (MAT,3,MT) are ”derived,” over an energy range E1 to
E2, by ”ratios” to ”standard” cross sections, there will be three NC-type sub-subsections
with LTY=1, 2 and 3 generated in the covariance files. The LTY=1 sub-subsection is
given in the subsection (MAT,MT; MAT,3,MT); the LTY=2 sub-subsection is given in the
subsection (MAT,MT; MATS,MFS,MTS[.LFSS]). Both of these subsections are given in the
File 33 of the material MAT of the ”derived” cross sections (MAT,3,MT). The LTY=3 subsubsection is given in the subsection (MATS,MFS,MTS[.LFSS] ; MAT,3,MT) which is in
the File (MFS+30) of the material MATS of the ”standard” cross sections.
There are, however, some instances, such as the one taken in the Example 33.1, where
other cross sections, such as those in (MAT,3,MTI), are ”indirectly derived” from the cross
sections in (MATS,3,MT) through evaluation of ratios of the cross sections in (MAT,3,MTI)
to those in (MAT,3,MT). In such cases, an LTY=1 sub-subsection will also be used in the
subsections (MAT,MT1,MAT,MT1) and (MAT,MT; MAT,MT1) and LTY=2 sub-subsection
will also be used in the subsection (MAT,MT1; MATS,MFS,MTS[.LFSS]). All three of these
subsections are in File 33 of the material MAT. Corresponding to the LTY=2 sub-subsection
in the subsection (MAT,MT1; MATS,MFS,MTS[.LFSS]) of the File 33 of the material
MAT, there will also be an LTY=3 sub-subsection in the subsection (MATS,MTS[.LFSS];
MAT,MT1) of the File (MFS+30) of the material MATS.
Note B: For purposes of discussing the covariance matrices of cross sections derived through
evaluation of ratio measurements, the label ”standard” cross sections is used for the cross
sections relative to which the ratio measurements were made. The cross sections for which
the label ”standard” was used may be any ”evaluated” cross sections of an ENDF library
and are not restricted to the special set of ”standard cross sections” library. The ”standard
cross sections” are the preferred ones to use for ratio measurements in order to minimize the
magnitude of the covariance matrix elements obtained from LTY=1, 2 and 3 sub-subsections.
However, they may not always be the ones that were used in the data available to evaluators
to perform evaluations.
33.2.2.2

NI-type Sub-Subsections

NI-type sub-subsections are used to describe explicitly the various components of the covariance matrix given in the subsection. In each NI-type sub-subsection there is an LB
flag. The numerical value of LB indicates whether the components are ”relative” or ”absolute,” the kinds of correlations as a function of energy represented by the components in the
sub-subsection, and the structure of the sub-subsection. The following quantity is defined:
LB Flag whose numerical value determines the meanings of the numbers given
in the arrays {Ek , Fk }{El , Fl }.

257

CHAPTER 33. FILE 33, COVARIANCES OF NEUTRON CROSS SECTIONS
Flag LB= 0 to 4
The following additional quantities are defined:
NP Total number of pairs of numbers in the arrays {Ek , Fk }{El , Fl }.
NT Total number of numbers in the LIST record; NT=2*NP.
LT Number of pairs of numbers in the second array, {El , Fl }.
If LT=0, the table contains a single array {Ek , Fk }.
If LT6= 0, the table contains two arrays; the first array, {Ek , Fk }, has
(N P − LT ) pairs of numbers in it.
{Ek , Fk }{El , Fl } two arrays of pairs of numbers; each array is referred to as an ”E-table,”
so the ”Ek -table” and the ”El -table” are defined.
In each E-table the first member of a pair is an energy, En ; the second
member of the pair, Fn , is a number associated with the energy interval
between the two entries En and En+1
For values of the LB flag from 0 through 4, NI-type sub-subsections have the following
structure:
[MAT,33,MT/ 0.0, 0.0, LT, LB, NT, NP/ {Ek , Fk }{El , Fl }]LIST
The Ek -table, and the El -table, when present, must cover the complete energy range of
the File 3 for the same (MAT,MT). The first energy entry in an E-table must therefore be
10−5 eV, or the reaction threshold, and the last one 20 × 106 eV unless a large upper-energy
limit has been defined for the evaluation. Some of the Fk ’s, or Fl ’s, may be zero, and the
last value of F in an E-table must be zero.
We now define the meaning of the F values entered in the E-tables for different values
of LB. [Note that the units of F vary]. Let Xi refer to the cross sections in (MAT,3,MT)
at energies Ei and Yj refer to the cross sections in (MAT1,3,MT1) at energies Ej . The
contribution of the sub-subsection to the covariance matrix Cov(Xi , Yj ), having the units of
”barns squared,” is defined as follows for the different values of LB:
LB=0 Absolute components correlated only within each Ek interval
X i;k
Cov (Xi , Yj ) =
Pj;k Fxy;k
k

LB=1 Fractional components correlated only within each Ek interval
X i;k
Cov (Xi , Yj ) =
Pj;k Fxy;k Xi Yj
k

LB=2 Fractional components correlated over all Ek intervals
X i;k
Pj;k′ Fxy;k Fxy;k′ Xi Yj
Cov (Xi , Yj ) =
k,k′

258

CHAPTER 33. FILE 33, COVARIANCES OF NEUTRON CROSS SECTIONS
LB=3 Fractional components correlated over Ek and El intervals
X i;k
Cov (Xi , Yj ) =
Pj;l Fx;k Fy;l Xi Yj
k,l

LB=4 Fractional components correlated over all El intervals within each Ek interval
X i;k,l
Pj;k,l′ Fk Fxy,l Fxy,l′ Xi Yj
Cov (Xi , Yj ) =
k,l,l′

For LB=0, 1 and 2 we have LT=0, i.e., only one Ek table. For LB=3 and LB=4 we have
LT6= 0, i.e., two E-tables, the Ek and the El -tables.
The dimensionless operators P in the above definitions are defined in terms of the operator S as follows:
i;k,l,...
Pj;m,n,...
≡ Sik Sil ...Sjm Sjn ...,
where
Sik ≡ 1 when Ek ≤ Ei ≤ Ek+1 and
Sik ≡ 0 when the energy Ei is outside the range of Ek to Ek+1 of an Ek -table.
Flag LB=5
It is often possible during the evaluation process to generate the relative covariance matrix
of some cross sections averaged over some energy intervals. Such relative covariance matrices may be suitable for use in File 33. Although the use of LB=3 sub-subsections allows
the representation of such matrices one row (or one column) at a time, this method of representation is very inefficient. One sub-subsection must be used for each row (or column)
and the same energy mesh is repeated in the Ek -table (or El -table) of every sub-subsection.
Often, in addition, such relative covariance matrices are symmetric about their diagonal and
there is no way to avoid repeating almost half of the entries with LB=3 sub-subsections. In
order to allow such relative covariance matrices to be entered into the files efficiently, LB=5
sub-subsections may be used. The following definition applies for LB=5 sub-subsections:
X i;k
Pj;k′ Fxy;k,k′ Xi Yj
Cov (Xi , Yj ) =
k,k′

A single list of energies {Ek } is required to specify the energy intervals labeled by the
indices k and k ′ . The numbers Fxy;k,k represent fractional components correlated over the
energy intervals with lower edges Ek and Ek′ .
Since there is no need for Ek -tables with pairs of numbers (Ek , Fk ) like those found in
sub-subsections with LB<5, a new structure is required for LB=5 sub-subsections. The
following quantities are defined:
NT Total number of entries in the two arrays {Ek } and {Fk,k′ }.
NE Number of entries in the array {Ek } defining (NE-1) energy intervals.
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LS Flag indicating whether the Fk,k′ , matrix is symmetric or not.
The structure of an LB=5 sub-subsection is:
[MAT,33,MT/ 0.0, 0.0, LS, LB=5, NT, NE/ {Ek }{Fk,k′ }] LIST.
LS=0 Asymmetric matrix:
The matrix elements Fk,k′ , are ordered by rows in the array {Fk,k′ }:
{Fk,k′ } ≡ F1,1 , F1,2 , ..., F1,N E−1 ; F2,1 , ..., F2,N E−1 ; FN E−1,1 , ..., FN E−1,N E−1
There are (NE –1 )2 numbers in the array {Fk,k′ } and
NT = NE + (NE – 1)2 = NE(NE+1) + 1
LS=1 Symmetric matrix:
The matrix elements Fk,k′ are ordered by rows starting from the diagonal term in the array
{Fk,k′ }:
{Fk,k′ } ≡ F1,1 , F1,2 , ..., F1,N E−1 ; F2,2 , ..., F2,N E−1 ; FN E−1,N E−1
There are [NE*(NE – 1)]/2 numbers in the array {Fk,k′ } and
NT = NE + [NE (NE – 1)]/2 = [NE (NE+1)]/2
Flag LB=6
A covariance matrix interrelating the cross sections for two different reaction types or materials generally has different energy grids for its rows and columns. The LB=6 format
described below allows efficient representation of a rectangular (not square) matrix in one
LIST record with no repetition of energy grids. The following definition applies for LB=6
sub-subsections:
X i;k
Cov (Xi , Yj ) =
Pj;l Fxy;k,l Xi Yj
k,l

where

Xi as before refers to the cross sections at Ei in (MAT,3,MT) and
Yj refers to the cross section at Ej in (MAT1,MF1,MT1[.LFS1]).
The dimensionless operator P is as defined for other LB’s. A single ”stacked” list of
energies {Ek,l } is required to specify the energy intervals with lower boundaries labeled by
the indices k and l. That is, a single array contains the energies for the rows (Ek ) and
then the columns (El ) of the matrix with the energies corresponding to the rows given first:
(ER1 , ER2 , ..., ERN ER , EC1 , EC2 , ..., ECN EC ). The numbers Fxy;k,l represent fractional
components correlated over the energy intervals with lower boundaries Ek and El . The
following quantities are defined:
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NT Total number of entries in the two arrays {Ek,l }{Fk,l }. See below.
NER Number of energies corresponding to the rows of the matrix and defining
(NER – 1) energy intervals.
NEC Number of energies corresponding to the columns of the matrix and defining
(NEC – 1) energy intervals. NEC may be inferred from NT and NER.
The structure of an LB=6 sub-subsection is:
[MAT,33,MT/ 0.0, 0.0, 0, LB=6, NT, NER/ {Ekl }{Fk,l }]LIST

The matrix elements Fk,l are ordered by rows in the array {Fk,l }:

{Fk,l } = F1,1 , F1,2 , ..., F1,N EC−1 ; F2,1 , ..., F2,N EC−1 ; ...; FN ER−1,1 , FN ER−1,2 , ..., FN ER−1,N ER−1
There are (NER – 1)(NEC – 1) numbers in the array {Fk,l }. Therefore, the total number
of entries in the two arrays {Ekl } and {Fk,l } is:
NT = NER + NEC + (NER – 1)(NEC – 1) = 1 + NER*NEC.
Hence
NEC = (NT – 1)/NER.
Flag LB=8 or 9, Short Range Variance Representation
A short range self-scaling variance component should be specified in each File 33 subsection
of the type (MAT,MT; 0,MT) by use of an LB=8 or 9 sub-subsection, unless the cross
section is known to be free from unresolved underlying structure. (See section 33.3.3). The
following quantities are defined:
NP Total number of pairs of numbers in the array {Ek , Fk }.
NT Total number of numbers in the LIST record. NT=2*NP
{Ek , Fk } Array of pairs of numbers; the first member of a pair is an energy, En ;
the second member of the pair, Fn , is a number associated with the energy
interval between the two entries En and En+1 .
The format of an LB=8 or 9 sub-subsection is (just as for LB=0):
[MAT,33,MT/ 0.0, 0.0, LT, LB, NT, NP/ {Ek , Fk }]LIST

(LT=0,LB=8 or 9)

Only one Ek table is required. The Fk values for LB=8 or 9 have the dimension of squared
cross sections. The magnitude of the resulting variance component for a processed average
cross section depends strongly on the size of the energy group as well as on the values of
F in the sub-subsection. For the simplest case of a multigroup covariance matrix processed
on the energy grid of this sub-subsection with a constant weighting function, the variance
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elements Varkk are just Fk for the LB=8 component and zero for the LB=9 component; the
off diagonal elements are zero in both cases.
In general, each Fk characterizes an uncorrelated contribution to the absolute variance of
the indicated cross section averaged over any energy interval (subgroup) ∆Ej that includes
a portion of the energy interval ∆Ek . Sub-subsections corresponding to LB=8 and LB=9
differ in the definition of the contribution to the processed group variance for the energy
group (Ej , Ej+1 ).
The variance contribution Var(Xjj ) from an LB=8 sub-subsection to the processed group
variance for the energy group (Ej , Ej+1 ) is inversely proportional to its width ∆Ej when
(Ej , Ej+1 ) lies within (Ek , Ek+1 ) and is obtained from the relation:
Var(Xjj ) = Fk

∆Ek
,
∆Ej

where Ek ≤ Ej ≤ Ej+1 ≤ Ek+1 . This form is applicable in the resonance range where the
covariances in the other sub-subsections define ”average” coarse energy-grid uncertainties,
while the actual pointwise cross-section values may fluctuate by orders of magnitude. The
evaluator must be aware that the actual uncertainty in the cross sections depends on the
user’s energy-grid. The user should be aware of possible processing problems, for example
when the union grid of the user’s energy group structure and the covariance grid nearly
coincide at some energy. The LB=8 variances should not be used to specify the uncertainty
on the actual value of pointwise cross sections.
The variance contribution Var(Xjj ) from an LB=9 sub-subsection to the processed group
variance for the energy group (Ej , Ej+1 ) is directly proportional to the width ∆Ej when
(Ej , Ej+1 ) lies within (Ek , Ek+1 ). It is obtained from the relation:


∆Ej
Var(Xjj ) = Fk 1 −
,
∆Ek

where Ek ≤ Ej < Ej+1 ≤ Ek+1 . This form is applicable in cases when experimental
evidence suggests the possibility of structure in the cross sections, but the experimental
resolution is not sufficient to determine the detailed shape, which is then approximated by
a smooth curve. The LB=9 option defines the maximum uncertainty in the cross section
due to possible fine-structure, which vanishes when the user’s energy grid is equal or coarser
than the covariance grid. The physical consequence of this property is that it increases the
absolute variance when the user defines an energy grid more refined than the covariance
grid (thus avoiding zero-eigenvalue problems), but will not affect any coarse energy-group
uncertainties. The increase in the variance remains finite (limited to Fk ) and is applicable
to defining the uncertainties of pointwise cross sections.
Note that the Var(Xjj ) are variances in average cross sections. This rule suffices for
arbitrary group boundaries if subgroup boundaries are chosen to include all the Ek . No
contributions to off-diagonal multigroup covariance matrix elements are generated by LB=8
or 9 sub-subsections.

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33.2.3

Lumped Reaction Covariances

A lumped reaction is an evaluator-defined ”redundant” cross section, defined in File 33 for
the purpose of specifying the uncertainty in the sum of a set of cross sections, such as those
for a set of neighboring discrete inelastic levels. The uncertainty in a lumped-reaction cross
section, as well as its correlations with other reactions, are given in the usual way using
the formats described above. On the other hand, the uncertainties and correlations of the
individual parts or components of a lumped reaction are not given.
The File 33 section for one component of a lumped reaction consists of a single HEAD
record that contains, in the second integer field, the section number MTL of the lumped
reaction to which the component contributes. (See definition of the HEAD record at the
beginning of this Section.)
[MAT, 33, MT/ ZA, AWR, 0, MTL, 0, NL=0]HEAD
The value of MTL must lie in the range 851-870, which has been reserved specifically
for covariance data for lumped reactions. These MT-numbers may not be used in Files 3,
4, 5 or 6, so the net cross section and net scattering matrix for a lumped reaction must be
constructed at the processing stage by summing over the reaction components.
A list of the components of a given lumped reaction is given only indirectly, namely,
on the above-mentioned HEAD records. These special HEAD records, with MTL6= 0 and
NL=0, form a kind of index that can be scanned easily by the processing program in order
to control the summing operation.
Except for the need to sum the cross-section components during uncertainty processing,
lumped reactions are ”normal” reactions, in that all covariance formats can be used to
describe their uncertainties in MF=33, MT=MTL. For example, one expects in general that
the covariances of a lumped reaction with other reactions, including other lumped reactions,
will be given by the evaluator. Also, a lumped reaction may be represented, using an
NC-type subsection with LTY=0, as being ”derived” from other reactions, including other
lumped reactions. However, since uncertainties are not provided for the separate component
reactions, a lumped reaction may not be represented as being ”derived” from its components.

33.3

Procedures

Although it is not necessary to have a section in File 33 for every section in File 3, the most
important values of MT for the applications to which the evaluation was intended should
have a section in File 33.

33.3.1

Ordering of Sections, Subsections and Sub-Subsections

a.) Sections
The sections in File 33 are ordered by increasing value of MT.
b.) Subsections
Within a section, (MAT,33,MT), the subsections are ordered in a rigid manner. A subsection of File 33 is uniquely defines the covariances by the set of numbers: (MAT,MT;
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MAT1,MF1,MT1[.LFS1]); the first pair of numbers indicates the section to which the subsection refers and the second set of numbers is the one that appears in the appropriate
fields, XMF1, XLFS1, MAT1 and MT1, of the CONT record which begins every subsection.
[When MT6= 10, XLFS1=0.0.]
1. The subsections within a section are ordered by increasing values of MAT1. In order
to have the covariance matrices of the cross sections for which MAT1=MAT appear
first in a section, the value MAT1=0 shall be used to mean MAT1=MAT in the CONT
record which begins the subsection.
2. When there are several subsections with the same value of MAT1 in a section, these
subsections shall be ordered by increasing values of XMF1. When MF1=MF – 30, the
XMF1 field shall be entered as blank or zero. Therefore, within a given section and
for a given MAT1, the subsections for MF1=MF – 30 will always appear before those
for other MF1 values.
3. When there are several subsections with the same value of MAT1 and MF1 in a section,
these subsections shall be ordered by increasing values of MT1 given in CONT record
which begins the subsections.
4. According to the procedure in Section 33.3.1, item b.2, MAT1=0 means that
MAT1=MAT; similarly, XMF1=0.0 means MF1=MF – 30. Only subsections for which
MT1>MT need to be be given.
5. When there are several subsections with the same values of MAT1, MF1=10, and MT1
in a section, these subsections shall be ordered by increasing values of LFS1.
When both NC-type and NI-type sub-subsections are present in a subsection, the format
requires that the NC-type sub-subsections be given first.
NC-type subsections: Several NC-type sub-subsections may be given in a subsection.
When more than one is given, these must be ordered according to the value of the
energy range lower endpoint E1 given in the LIST record. We note that by definition,
if several LTY=0 NC-type sub-subsections are given in a subsection, the energy ranges
E1 to E2 of the these different sub-subsections cannot overlap with each other or with
any LTY=1 sub-subsections. However, in ENDF-6 formats it is permitted to have
overlapping (E1,E2) ranges for LTY≥ 1, provided that the sum of the WEI values in
one subsection is no greater than unity at any energy. The value of the LTY flag of
NC-type sub-subsections does not affect the ordering of the sub-subsections within a
subsection.
NI-type sub-subsections: There is no special ordering requirement of NI-type subsubsections within a subsection. However, it often happens that the full energy range
of the file is covered by different sub-subsections, the F -values being set to zero in the
E-tables outside the different ranges. The readability of the files is enhanced if these
different sub-subsections are grouped together by the energy range effectively covered
in the sub-subsections.
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33.3.2

Completeness

As previously stated, there is presently no minimum requirement on the number of sections
and subsections in File 33. Lack of a File 33 for a reaction does not imply zero uncertainty.
However, the presence of some subsections in a File 33, as well as the presence of some
sub-subsections in a subsection, implies the presence of other subsections either in the same
File 33 or the File 33 (or 31 or 40) of another material. In what follows we shall identify the
File 33 subsections by their value of the sextet: (MAT,MT; MAT1,MF1,MT1[.LFS1])
a.) Subsections for which MAT1=0
By subsections for which MAT1=0, we mean the subsections of the class:
(MAT,MT; 0,MF1,MT1[.LFS1]),
which according to the procedure in Section 33.3.1, item b.1 implies that MAT1=MAT.
1. If there is a File 33 subsection (MAT,MT; 0,MF1,MT1[.LFS1]) with MT16=MT,
there must be within the same material the two subsections: (MAT,MT; 0,3,MT)
in File 33 and (MAT,MT1; 0,MF1,MT1[.LMF1]) in File MF1+30. Note that
the converse is not necessarily true since the two cross sections (MAT,3,MT) and
(MAT,MF1,MT1[.LMF1]) may have zero covariances between them, which are not required to be explicitly stated in the files. (However, see the discussion in Section 33.3.2
item b. below concerning the desirability of explicitly representing some zero covariances.) This procedure and procedure in Section 33.3.1 item b.4 guarantee that every
section of File 33, (MAT,33,MT), starts with the subsection (MAT,MT; 0,3,MT).
2. In a subsection (MAT,MT; 0,3,MT,0), if there is an NC-type sub-subsection with
LTY=0, it contains a list of MTI given in the NC-type sub-subsections.
3. NC-type sub-subsections with LTY=0 must be given only in subsections of the type
(MAT,MT; 0,3,MT), i.e. with MT1=MT. NC-type sub-subsections with LTY=0, for
derived redundant cross sections, imply many covariance matrices of the ”derived”
cross sections and of the ”evaluated” cross sections. It is a task of the processing code
to generate these covariance matrices from the information given in the File 33.
4. In a subsection (MAT,MT; 0,3,MT) if there is an NC-type sub-subsection with LTY=1,
this sub-subsection contains values of (MATS,MFS,MTS[.LFSS]). In the same File 33,
there must be a sub-subsection (MAT,MT; MATS,MFS,MTS[.LFSS]). There must be
another material MATS with a File (MFS+30) containing the subsection (MATS,
MTS[.LFSS]; 0,MFS,MTS[.LFSS]). Note that according to procedure in Section 33.3.3
item a., given below, MATS must be different from MAT in an ”NC-type subsubsection with LTY=1.
5. In a subsection (MAT,MT; 0,3,MT), if there is an NC-type sub-subsection with LTY=1
which covers the energy range E1 to E2, in the same subsection there must be at least
one NI-type subsections represent the relative covariance matrix of the evaluated ratio
measurements. In the energy range where WEI is the relative weight given to the
evaluated ratio to the indicated standard cross section, the processing code takes into
account the value of WEI when it applies the standards covariances. The evaluator
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CHAPTER 33. FILE 33, COVARIANCES OF NEUTRON CROSS SECTIONS
is responsible for multiplying the covariances of the evaluated ratios by WEI2 before
entry into the NI-type sub-subsections. Note that, where the weight is not unity in
a given energy region, the NI-type sub-subsections that represent the covariance data
for the ratios are mixed together without identification in File 33.
b.) Subsections for MAT16=0
If there is a File 33 subsection (MAT,MT; MAT1,MF1,MT1[.LFS1]) with MAT16= 0, by
analogy with the procedure in Section 33.3.2, item a.1., there must also be a subsection (MAT,MT; 0,3,MT) in the same File 33. There must also be two sub-subsections,
(MAT1,MT1[.LFS1]; 0,MF1,MT1[.LFS1]) and (MAT1,MT1[.LFS1]; MAT,3,MT) in the
File (MF1+30) for material MAT1.

33.3.3

Other Procedures

1. NC-type sub-subsections with LTY=1 shall only be used with MATS=MAT. The use
of LTY=1 sub-subsections is reserved for covariance matrix components arising out of
ratio measurements of cross sections of different nuclides, i.e., different values of MAT.
2. If a single NC-type sub-subsection with LTY=0 is used in a subsection and there are no
NI-type sub-subsections, the value of E1 must be 10−5 eV, or the reaction threshold,
and the value of E2 must be the highest energy for which the corresponding cross
section is given, at least 2×107 eV.
3. As a consequence of the definition of NC-type sub-subsections with LTY=0, if there
are any NI-type sub-subsections in the same subsection, the F -values in their E-tables
must be zero within the range E1 to E2 of these NC-type LTY=0 sub-subsections.
4. NI-type sub-subsections with LB=0 shall in general be avoided and forbidden in
the case of cross sections relative to which ratio measurements have been evaluated.
[Therefore, the acknowledged ”standard cross sections” shall not have LB=0, NI-type
sub-subsections.] The use of LB=0 NI-type sub-subsections should be reserved for
the description of covariance matrices of cross sections which fluctuate rapidly and for
which details of the uncertainties in the deep valleys of the cross sections are important.
5. The formats of File 33 allow for the possibility of great details to be entered in the files
if needed. The number of NI-type sub-subsections and the number of energy entries
in their Ek and El -tables will be a function of the details of the covariance matrices
available and the need to represent them in such detail. However, good judgement
should be used to minimize as much as possible the number of different entries in the
Ek and El -tables. An important quantity to note is the union of all of the E values
in the Ek and El -tables of a File 33. A reasonable upper limit of the order of a few
hundred different E values for the union of all energy entries in all of the Ek and
El -tables in a File 33 should be considered.
Note that the evaluator’s covariance values will be most readily recognized in a processed multigroup covariance matrix when the energies in the Ek and El -tables can be
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chosen from the set of standard multigroup energy boundaries. If in the File 33 the
uncertainties in a cross section are represented using LB=0 or 1 in regions of width
∆Ei , and if the File is processed to give a multigroup covariance library with group
width in that energy region ∆Ea such that ∆Ei > ∆Ea , the correlation patterns
in ENDF-6 are defined so that the processed group uncertainties are lowest and the
inter-group correlations greatest when an energy group of the processed covariance
matrix is evenly split by a covariance file E value. This behavior has alarmed some
users. However, because of the correlation pattern set up, no big anomalies arise in the
uncertainty projected for an integral quantity that is sensitive to a broad spectrum
of incident particle energies. The magnitude of the effect can be reduced by using
narrower intervals in LB=0, 1 files, or more favorably by using overlapping files with
staggered energy edges.
6. The ground rules above (see 33.2, under LT=1 sub-subsections) state that if cross
sections are obtained by evaluating ratio measurements to a ”standard cross section,”
the latter cross section should be ”evaluated” in the sense that there are no NC-type
sub-subsections with LTY=0 or 1 describing the covariance data for that cross section.
This leads to procedural requirements.
(a) Evaluators of established standard cross sections should endeavor to avoid the
use of any LTY=0 or LTY=1 sub-subsections. If the physics of an evaluation
problem should require that this rule be broken, the CSEWG should be informed
and the text documentation should call attention to the situation.
(b) If an evaluated cross section is best obtained as the ratio to a cross section in
another material that is not an established standard, than an evaluator needs
to contact the evaluator of that material to be reassured that NC-type subsubsections will not appear in the files for the reference cross section for the
energy region of concern.
Such communication is required in any case to encourage the evaluation of the covariance data for that reference cross section, without use of LB=0 sub-subsections, and
to assure that the required LTY=3 sub-subsection will be entered. To aid the review
process, a written form of this communication should be sent to the CSEWG.
7. The ENDF-6 formats allow the evaluator to recognize the partial dependence of a cross
section on a standard cross section. This means that it is possible to recognize in the
covariance files an evaluation that utilizes some absolute data as well as ratio data to
one or more standard cross sections. Use of this capability is expected to be necessary
to properly represent some covariance information. Since processing complexity is
thereby induced, evaluators are urged to use this capability with caution, and in no
case to represent the dependence of a cross section in a given energy region on ratio
measurements to more than two standard cross sections.
8. The discussion of the covariance terms that arise from evaluated cross section ratio
measurements was based entirely on so-called absolute ratio measurements. Evaluators
should note that various types of cross section ”shape” measurements induce additional
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covariance terms that can be derived for specific situations. Within ENDF-6 formats
the evaluator must include them in NI-type sub-subsections in the covariance files of
the derived cross section.
9. When cross section A is correlated to B, and cross section C is also correlated to B
even though A is uncorrelated to C, evaluators should include the file segments that
express this zero correlation to signify to reviewers and users that an unusual case is
recognized. This is the exception to the general rule that zero covariances need not be
openly expressed in the covariance files.
10. The lumping of reactions for uncertainty purposes will be useful mainly in connection
with discrete-level inelastic scattering cross sections. However, other reactions, such
as (n,n’p), (n,n’), and (n,n’continuum ), may also be treated in this way.
11. In order not to lose useful uncertainty information, reactions lumped together should
have similar characteristics. Ordinarily, the level energies of discrete inelastic levels lumped together should not span a range greater than 30-40%, and the angular
distributions should be similar.
12. The components of a lumped reaction need not have adjacent MT-numbers.
(a) Lumped-reaction MT-numbers must be assigned sequentially, beginning at 851.
The sequence is determined by ordering the lumped reactions according to the
lowest MT-number included among their respective components. Thus, the first
value of MTL encountered on any component-reaction HEAD record will be 851.
The next new value of MTL encountered will be 852, and so on.
(b) Lumped reactions with only a single component are permitted. This is recommended practice when, for example, an important discrete inelastic level is treated
individually, while all of its neighbors are lumped. Covariances for both the individual level and the nearby lumped levels can then be placed together in sections
851-870.
13. An LB=8 or 9 sub-subsection should be included in each (MAT,MT; 0, MT) subsection
unless the cross section is known to be free from unresolved underlying structure5 . The
Doppler effect in reactor applications smooths cross sections on an energy scale too
narrow to be of concern for LB=8 or 9 covariance evaluation. This sub-subsection
must cover the entire energy range of the section (threshold to at least 20 MeV). Use
of zero values for Fk in any part of this energy range should be avoided because such
usage could induce the problem of non-physical full correlation between neighboring
fine-group cross sections, the problem that the LB=8 or 9 formats was designed to
solve. The LB=8 or 9 formats may not be used for cross-reaction covariances.
Note that the law for processing LB=8 sub-subsections directly references the variance
of an average cross section rather than the variance of a pointwise cross section. If
5

For example, if covariance data for neutron scattering from hydrogen were to be represented in File 33,
one would expect no component that could be properly represented by an LB=8 or 9 subsection. Covariance
data in such cases may best be represented in File 30.

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a fine-grid covariance matrix is developed and then collapsed to the evaluator’s Ek
grid with constant weighting, the resulting variance components are just the Fk . (A
complete multigroup covariance matrix cannot in general be correctly ”reprocessed”
to a finer energy mesh; one must process the ENDF covariance files directly).
The values of Fk may be chosen by the evaluator to account for statistical fluctuations
in fine-group average cross sections that are induced by the width and spacing distributions of the underlying resonances. Values may also be chosen to represent the
uncertainty inherent in estimating the average cross sections for small energy intervals
where little or no experimental data exist and smoothness is not certain.
The LB=8 or 9 sub-subsections help prevent mathematical difficulties when covariance
matrices are generated on an energy grid finer than that used by the evaluator, but Fk
values must be chosen carefully to avoid accidental significant dilution of the evaluated
covariance patterns represented in the other Subsections. If no physical basis is apparent for choosing the Fk values, they may be given values about 1% as large on the
evaluator’s grid as the combined variance from the other sub-subsections. Such values
would be small enough not to degrade the remainder of the covariance evaluation and
large enough to assure that the multigroup covariance matrix will be positive definite
for any energy grid if the matrix on the evaluator’s energy grid is positive definite.
Since LB=8 and 9 specify absolute covariances, they should not be employed near
reaction thresholds. In particular, for threshold reactions having an effective threshold
above 0.1 MeV, LB=8 or 9 should not be employed for incident energies less than 1
MeV above the effective threshold.

33.3.4

Examples

We illustrate here the use of File 33 by means of two concrete examples.
33.3.4.1

Use of LTY=1 and LTY=2 NC-type Subsections

Let us consider a hypothetical evaluation of 239 Pu, MAT=1264. Assume that the decision is
made that in File 33 only the fission cross sections and the capture cross sections shall have
covariances represented. The following methods were used in performing the hypothetical
evaluation:
1. Fission cross sections, MT=18
Let Xi stand for the fission cross sections of

239

Pu at energies Ei .

(a) From 10−5 eV to an energy Es , Xi were evaluated in terms of ”direct” or ”absolute” measurements, Ai . By this we mean that in this energy range, Xi and its
uncertainties are independent of any other cross sections. In this energy range
Xi ≡ Ai .

(b) From Es to 20 MeV, Xi was evaluated by means of ratio measurements to Yi , the
fission cross section of 235 U, to which we assign the MAT number 1261. In this
energy range Xi = Ri Yi , where Ri is the evaluated ratio at energy Ei .
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2. Capture cross sections, MT=102
Let Zi stand for the capture cross sections of 239 Pu at energies Ei . In this evaluation,
Zi were obtained by the evaluation of ai , the ratio of capture to fission cross sections,
over the complete range of the file. Therefore we have Zi = ai Xi .
In this evaluation then, only 3 quantities were evaluated: Ai from 10−5 eV to Es , Ri from
Es to 20 MeV, and ai from 10−5 eV to 20 MeV. The evaluation of these quantities resulted in
the evaluation of three covariance matrices: Cov(Ai , Aj ), Cov(Ri , Rj ) and Cov(ai , aj ). Let
us now assume that in addition it has been determined that these three different quantities
are uncorrelated, i.e., covariances such as Cov(Ai , aj ) can be neglected.
Let us denote relative covariance matrices such as Cov(Ai Aj )/(Ai Aj ) as hdAi dAj i, and
similarly for the other quantities.
From 10−5 to Es (since Xi = Ai and Zi = ai Xi we have:
hdXi dXj i = hdAi dAj i
hdXi dZj i = hdAj dAj i
hdZi dZj i = hdai daj i + hdAi dAj i
From Es to 20 MeV (since Xi = Ri Yi and Zi = ai Xi we have
hdXi dXj i
hdXi dZj i
hdXi dYj i
hdZi dZj i
hdZi dYj i

=
=
=
=
=

hdRi dRj i + hdYi dYj i
hdRi dRj i + hdYi dYj i
hdYi dYj i
hdai daj i + hdRi dRj i + hdYi dYj i
hdYi dYj i

We note that in the above we have expressed all of the covariance matrices of the cross
sections in terms of the covariance matrices of the evaluated quantities and the covariance
matrix of the 235 U fission.
For purposes of illustrating the use of the formats we need not know the details of how
the covariance matrices hdAi dAj i, hdRi dRj i and hdai daj i are represented. They must
be represented by one or more NI-type sub-subsections having an Ek table, or could be so
represented. For our purposes, we symbolically represent each one of them in terms of a
single NI-type sub-subsection with a single Ek table:
hdAj dAj i → {EkA , FkA }
hdRj dRj i → {EkR , FkR }
hdaj daj i → {Eka , Fka }
Whether one or more NI-type sub-subsections are used, each one of the E-tables used in the
sub-subsection can be written as:
{EkA , FkA } = {10−5 , F1A ; ...; EkA , FkA ; ...; Es , 0.0; 2 × 107 , 0.0},
{EkR , FkR } = {10−5 , 0.0; Es , FkR ; ...; EkR , FkR ; ...; 2 × 107 , 0.0},
{Eka , Fka } = {10−5 , F1a ; Fka ; ...; Eka , Fka ; ...; 2 × 107 , 0.0},
270

CHAPTER 33. FILE 33, COVARIANCES OF NEUTRON CROSS SECTIONS
the E and F values explicitly shown must have the values indicated above for this example.
In the listing given in Example 33.1 for the File 33 of MAT=1264, corresponding to
our example, we have shown with only one sub-subsection each of the matrices hdAi dAj i,
hdRi dRj i and hdai daj i. ES is taken as 2 × 105 eV.
Note: In the File 33 of MAT 1261 in the subsections (1261,18; 1264,18) and (1261, 18;
1264,102) an LTY=3 NC-type sub-subsection corresponding to the LTY=2 sub-subsections
of Example 33.2 must be inserted.
Example 33.1. File 33 with NC-type LTY=1 sub-subsections
9.423900+4
0.000000+0
0.000000+0
2.000000+5
0.000000+0
0.000000+0
1.000000-5
3.000000+4
2.000000+7
0.000000+0
1.000000-5

2.369990+2
0.000000+0
0.000000+0
2.000000+7
0.000000+0
0.000000+0
0.000000+0
4.900000-3
0.000000+0
0.000000+0
0.000000+0

0
MAT1=0
0
MATS=1261
2.000000+5
LT=0LB=1
1.000000+0
1.000000+5

0
T1=18
LTY=1
MTS=18
1.00000+0
NY=14
2.50000-3
6.40000-3

0
NC=1
0
NT=6
2.00000+7
NE=7
3.00000+2
2.00000+5

LT=0
2.000000+5

LB=1
4.00000-4

NT=6
2.00000+7

NL=3126433
NI=2126433
0126433
NE=2126433
0.00000+0126433
126433
3.60000-3126433
0.00000+0126433
126433
NE=3126433
0.00000+0126433

18 HEAD
18 CONT
18 CONT
18 LIST
18
18 LIST
18
18
18
18 LIST
18

(1264,18; 0,102)
0.00000+0
0.00000+0
2.00000+5
0.00000+0
0.00000+0

0.00000+0 MAT1=1261
MT1=18
NC=1
NI=0126433
0.00000+0
0
LTY=2
0
0126433
2.00000+7 MATS=1261
MTS=18
NT=6
NE=2126433
0.00000+0 2.00000+5 1.00000+0 2.00000+7 0.0000+0126433
0.00000+0
0
0
0
0126433

18 CONT
18 CONT
18 LIST
18
0 SEND

(1264,102; 0,102)
9.42390+4
0.00000+0
0.00000+0
2.00000+5
0.00000+0
0.00000+0
1.00000-5
3.00000+4
2.00000+7
0.00000+0
1.00000-5

2.36999+2
0.00000+0
0.00000+0
2.00000+7
0.00000+0
0.00000+0
0.00000+0
4.90000-3
0.00000+0
0.00000+0
0.00000+0

0
0
0
NL=2126433102 HEAD
MAT=0
MT1=102
NC=1
NI=2126433 18 CONT
0
LTY=1
0
0126433 18 CONT
MATS=1261
MTS=18
NT=6
NE=2126433 18 LIST
2.00000+5 1.00000+0 2.00000+7 0.00000+0126433 18
LT=0
LB=1
NT=14
NE=7126433 18 LIST
1.00000+0 2.50000-3 3.00000+2 3.60000-3126433 18
1.00000+5 6.40000-3 2.00000+5 0.00000+0126433 18
126433 18
LT=0
LB=1
NT=6
NE=3126433 18 LIST
2.00000+5 4.00000-4 2.00000+7 0.00000+0126433 18

(1264,18; 1261,18)

271

CHAPTER 33. FILE 33, COVARIANCES OF NEUTRON CROSS SECTIONS
0.00000+0
0.00000+0
2.00000+5
0.00000+0
0.00000+0
1.00000-5
3.00000+4
2.00000+7
0.00000+0
1.00000-5
0.00000+0
1.00000-5
2.21000-3
3.10000-4
2.21000-3

0.00000+0
0.00000+0
2.00000+7
0.00000+0
0.00000+0
0.00000+0
4.90000-3
0.00000+0
0.00000+0
0.00000+5
0.00000+0
2.53000-2
4.84000-4
3.04000-4
0.00000+0

MAT1=0
MT1=102
NC=1
NI=3126433102 CONT
0
LTY=1
0
0126433102 CONT
MATS=1261
MTS=18
NT=6
NE=2126433102 LIST
2.00000+5 1.00000+0 2.00000+7 0.00000+0126433102
LT=0
LB=1
NT=14
NE=7126433102 LIST
1.00000+0 2.50000-3 3.00000+2 3.60000-3126433102
1.00000+5 6.40000-3 2.00000+5 0.00000+0126433102
126433102
LT=0
LB=1
NT=6
NE=3126433102 LIST
2.00000+5 4.00000-4 2.00000+7 0.00000+0126433102
LS=1
LB=5
NT=21
NE=6125433102 LIST
9.00000-2 2.50000-1 1.00000+0 2.00000+7126433102
3.62000-4 3.56000-4 0.00000+0 4.84000-4126433102
0.00000+0 6.25000-4 2.30000-4 0.00000+0126433102
0.00000+0 0.00000+0 0.00000+0 0.00000+0126433102

(1264,102; 1261,18)
0.00000+0
0.00000+0
2.00000+5
0.00000+0
0.00000+0

33.3.4.2

0.00000+0 MAT1=1261
MT1=18
NL=1
NI=0126433102 CONT
0.00000+0
0
LTY=2
0
0126433102 CONT
2.00000+7 MATS=1261
MTS=18
NT=6
NE=2126433102 LIST
0.00000+0 2.00000+5 1.00000+0 2.00000+7 0.00000+0126433102
0.00000+0
0
0
0
0126433 0 SEND

Use of LTY=0, NC-type Sub-Subsections

Let us consider a hypothetical evaluation of 12 C, MAT=1274. The decision is made that
in File 33 the MT values 1, 2, 4, 102 and 107 shall have covariances represented. We shall
use the notation developed in the previous example. The following method was used in this
evaluation:
1. Total cross sections (MT=1),
σiT , were evaluated over the complete energy range, with the covariance matrix obtained, and:
hdσiT dσjT i → {EkT , FkT },
with {EkT , FkT } = {10−5 , F1T ; ...; EkT , FkT ; ... ; 2 × 107 , 0.0},

2. Elastic cross sections (MT=2),
σiE , were ”derived” up to 8.5 MeV from the ”evaluated” cross sections:
σiE = σiT − σiI − σiC − σiα .
Above 8.5 MeV the elastic cross sections were evaluated and:
hdσiE dσjE i → {EkE , FkE },
with {EkE , FkE } = {10−5 , 0.0; 8.5 × 106 , FkE ; ... ; EkE , FkE ; ... ; 2 × 107 , 0.0}.
272

CHAPTER 33. FILE 33, COVARIANCES OF NEUTRON CROSS SECTIONS
3. Inelastic cross sections (MT=4),
σiI , were evaluated from threshold at 4.8 MeV, to 8.5 MeV and:
hdσiI dσjI i → {EkI , FkI },
with {EkI , FkI } = {10−5 , 0.0; 4.8 × 106 , FkI ; ... ; EkI , FkI ; ... ; 8.5 × 106 , 0.0; 2 × 107 ,
0.0},
Above 8.5 MeV the inelastic cross sections were ”derived” and:
σiI = σiT − σiE − σiC − σiα .
4. Capture cross sections (MT=102),
σiC , were evaluated over the complete energy range and:
hdσiC dσjC i → {EkC , FkC },
with {EkC , FkC } = {10−5 , F1C ; ... ; EkC , FkC ; ... ; 2 × 107 , 0.0},
5. The (n,α) cross sections (MT=107),
σiα , were evaluated from threshold at 6.18 MeV to 20 MeV and:
hdσiα dσjα i → {Ekα , Fkα },
with { Ekα , Fkα } = {10−5 , 0.0; 6.1 × 106 , Fkα ; ... ; Ekα , Fkα ; ... ; 2 × 107 , 0.0 },
In the listing given in Example 33.2 for File 33 of MAT=1274, corresponding to our
example, we have shown only one NI-type sub-subsection for each evaluated covariance
matrix. Again it is assumed that there are no correlations among the directly evaluated
quantities.
The above example has great similarity to the way the evaluation of 12 C was made, the
major difference being that instead of MT=4 being evaluated, the evaluation was made for
MT=51 and MT=91. Since it will illustrate some of the procedures of File 33, let us now
consider adding to the above File 33 for MAT=1274 the covariance matrices for MT=51
and MT=91.
1. Discrete inelastic scattering cross section to the first excited state (MT=51)
σi51 , up to 8.5 MeV is identical to σiI . Therefore, we may consider up to 8.5 MeV that
σi51 is a ”derived” cross section with: σi51 = σiI . This is permissible because MT=4 has
only NI-type sub-subsections in this energy range.
From 8.5 MeV to 20 MeV, MT=51 was evaluated and:
hdσi51 dσj51 i → {Ek51 , Fk51 },
with {Ek51 , Fk51 } = {10−5 , 0.0; 8.5 × 106 , Fk51 ; ... ; Ek51 , Fk51 ; ... ; 2 × 107 , 0.0},

273

CHAPTER 33. FILE 33, COVARIANCES OF NEUTRON CROSS SECTIONS
2. Continuum inelastic cross section (MT=91)
From 8.5 to 20 MeV, the continuum inelastic, σi91 , was ”derived” as: σi91 = σiI − σi51 .
However, we cannot use this relationship for the purposes of File 33 because in this
energy range σiI is indicated in the File as being already ”derived.”
Therefore, for the purposes of File 33, we must write:
σi91 = σiT − σiE − σi51 − σiC − σiα ,
which now only refers to cross sections having exclusively NI-type sub-subsections.
Therefore, we may now add the sections to the File 33, MAT=1274, shown in Example
33.3, to have a more complete File 33.
Example 33.2. File 33 with NC-type LTY=0 sub-subsections.
(1274.1; 0.1)
6.01200+3
0.00000+0
0.00000+0
1.00000-5
0.00000+0

1.18969+1
0
0
0
1112733
0.00000+0
0
1
0
1112733
0.00000+0
0
1
6
3112733
5.00000-5 2.00000+6 2.50000-5 2.00000+7 0.00000+0112733
0.00000+0
0
0
0
0112733

1
1
1
1
0

HEAD
CONT
LIST

2
2
2
2
2
2
2
2
2
0

HEAD
CONT
CONT
LIST

4
4
4
4
4
4
4
4
4
0

HEAD
CONT
CONT
LIST

SEND

(1274,2; 0,2)
6.01200+3
0.00000+0
0.00000+0
1.00000-5
1.00000+0
-1.00000+0
0.00000+0
1.00000-5
2.00000+7
0.00000+0

1.18969+1
0
0
0
1112733
0.00000+0
0
2
1
1112733
0.00000+0
0
0
0
0112733
8.50000+6
0
0
8
4112733
1.00000+0 1.00000+0 4.00000+0-1.000000+0 1.0200+2112733
1.07000+0
112733
0.00000+0
0
1
8
4112733
0.0000000 8.50000+6 8.00000-6 1.50000+6 2.500005112733
0.00000+0
112733
0.00000+0
0
0
0
0112733

LIST

SEND

(1274,4; 0,4)
6.01200+3
0.00000+0
0.00000+0
1.00000+0
-1.00000+0
0.00000+0
4.80000+6
8.50000+6
2.00000+7
0.00000+0

1.18970+1
0
0
0
1112733
0.00000+0
0
4
1
1112733
0.00000+0
0
0
0
0112733
1.00000+0-1.00000+0 2.00000+0-1.00000+0 1.02000+2112733
1.07000+2
112733
0.00000+0
0
1
8
4112733
1.00000-3 6.00000+6 1.00000-4 8.50000+6 0.0000000112733
2.00000+7
0
0
8
4112733
0.00000+0
112733
0.00000+0
0
0
0
0112733

(1274,102; 0,102)
274

LIST

SEND

CHAPTER 33. FILE 33, COVARIANCES OF NEUTRON CROSS SECTIONS
6.01200+3
0.00000+0
0.00000+0
1.00000-5
0.00000+0

1.18969+1
0
0
0
1112733102 HEAD
0.00000+0
0
102
0
1112733102 CONT
0.00000+0
0
1
6
3112733102 LIST
3.60000-3 1.00000+3 4.00000-2 2.00000+7 0.00000+0112733102
0.00000+0
0
0
0
0112733 0 SEND

(1274,107; 0,107)
6.01200+3
0.00000+0
0.00000+0
6.18000+6
2.00000+7
0.00000+0

1.18969+1
0
0
0
1112733107 HEAD
0.00000+0
0
107
0
1112733107 CONT
0.00000+0
0
0
8
4112733107 LIST
1.00000-5 6.32000+6 1.00000-4 7.36000+6 2.00000-4112733107
0.00000+0
112733107
0.00000+0
0
0
0
0112733 0 SEND

Example 33.3.
Additional sections of File 33 which could be added to File 33 are given in Example 33.3
(1274,51; 0.51)
6.01200+3
0.00000+0
0.00000+0
1.00000-5
1.00000+0
0.00000+0
1.00000-5
0.00000+0

1.18969+1
0
0
0
1127433
0.00000+0
0
51
1
1127433
0.00000+0
0
0
0
0127433
8.50000+6
0
0
2
1127433
4.00000+0
127433
0.00000+0
0
1
6
3127433
0.00000+0 8.50000+6 2.50000-3 2.00000+7 0.00000+0127433
0.00000+0
0
0
0
0127433

51
51
51
51
51
51
51
0

HEAD
CONT
CONT
LIST
LIST
SEND

(1274,91; 0,91)
6.01200+3
0.00000+0
0.00000+0
8.50000+6
1.00000+0
-1.00000+0
0.00000+0

1.18969+1
0
0
0
1127433 91 HEAD
0.00000+0
0
91
1
0127433 91 CONT
0.00000+0
0
0
0
0127433 51 CONT
2.00000+7
0
0
10
5127433 51 LIST
1.00000+0-1.00000+0 2.00000+0-1.00000+0 5.10000+1127433 51
1.02000+2-1.00000+0 1.07000+2
127433 51
0.00000+0
0
0
0
0127433 0 SEND

275

Chapter 34
File 34. COVARIANCES FOR
ANGULAR DISTRIBUTIONS OF
SECONDARY PARTICLES
34.1

General Comments

File 34 contains covariances for angular distributions of secondary particles. It is assumed
that uncertainties will not be required on all quantities in File 4.
A central question is whether quantities in File 3 may have important correlations with
those in File 4, or whether one needs to be concerned only with correlations of angular
distribution parameters as a function of incident energy. It is judged that covariances between the magnitude and shape are likely to be important only when theory plays a strong
role in an evaluation. When such covariances occur, the idea, developed below, is that one
expresses covariances with the a0 Legendre coefficients even though a0 ≡ 1 in the ENDF
system.
Because of the simplicity of representing the covariances of Legendre coefficients rather
than normalized probability components, only the former is considered here even for cases
where File 4 has tabulated p(µ). Furthermore, the covariance matrix in File 34 may refer to
Legendre coefficients in the LAB coordinate system even when the data in File 34 are given in
the CM coordinate system. This is done for convenience, since transport calculations involve
Legendre moments of the cross sections, which are related to the Legendre coefficients of
the angular distribution expansion in the LAB system, therefore the covariance matrix in
the LAB system may be more easily generated (and used).
In ENDF-6 formats there is no provision for covariance components linking the angular
distribution parameters for different materials, though a MAT1 field is provided, but is
normally zero.

34.2

Formats

The general structure of File 34 follows the normal pattern, with sections by increasing
MT values. The LTT flag definition is modified from its meaning for File 4. The following
276

CHAPTER 34. FILE 34. COVARIANCES FOR ANGULAR DISTRIBUTIONS
quantities are defined:
ZA,AWR

Standard material charge and mass parameters

LTT Flag to specify the representation used, and it may have the following values
in File 34.
LTT=1 the data are given as Legendre coefficient covariances as a
function of incident energy, starting with a1 or higher order
coefficients.
LTT=2 the data are given as Legendre coefficients covariances as a
function of incident energy, starting with a0 . (This information is redundant in the formats, as specified below, but is
considered desirable as an alarm flag.)
LTT=3 if either L or L1=0 anywhere in the Section.
NMT1 Number of subsections present in File 34 for various MT1≥ MT.
A section of File 34 for a given MT has the form:
[MAT,34,MT/ ZA, AWR,
0, LTT, 0, NMT1]HEAD

--------------------------------------------------------
[MAT,34,0/ 0.0, 0.0,
0,
0, 0,
0]SEND
Each subsection begins with a control record that identifies the related MT1 and indicates
how many Legendre coefficients are covered for the angular distributions for reaction types
MT(NL) and MT1(NL), MT1≥MT. The following quantities are defined:
MT1 ”Other” reaction type; this subsection contains data for the covariances
Cov[aL (E1 ), aL1 (E2 )] between Legendre coefficients for two reaction types
at incident energies E1 and E2 for various L and L1.
NL Number of Legendre coefficients for which covariance data are given for the
reaction MT. (This value must be the same for each subsection.)
(The first coefficient is a0 if LTT=3, a1 ≥ 1 if LTT=1).
NL1 Number of Legendre coefficients for which covariance data are given for reaction MT1.
L Index of the Legendre coefficient for reaction MT for this sub-subsection.
Note that sub-subsections need not be given for all values of L and L1.
277

CHAPTER 34. FILE 34. COVARIANCES FOR ANGULAR DISTRIBUTIONS
L1 Index of the Legendre coefficient for reaction MT1 for this sub-subsection.
LCT Flag to specify the frame of reference used
LCT=0 the data are given in the same coordinate system as used in
File 4
LCT=l the data are given in the LAB system
LCT=2 the data are given in the CM system.
NI Number of LIST records contained in this sub-subsection.
LS Flag, recognized when LB=5, to indicate whether the matrix is symmetric
(1=yes, 0=no).
LB Flag to indicate the covariance pattern as a function of incident energy. LB
values 0,1,2,5 & 6 are allowed, and are defined as for File 33 in Section 33.2.
NT Total number of items in the list,
For LB=0,1,2, NT=2*NE;
for LB=5, NT is dependent on LS as given in Section 33.2;
for LB=6, NT=1+NER*NEC.
{Data} For LB=5, sequence {Ek } {Fk,k };
for LB=6, sequence {Ek } {Fk,l } (as in File 33).
A subsection has the following form:
[MAT,34,MT/ 0.0, 0.0, MAT1, MT1, NL, NL1]CONT (MAT1=0)
[MAT,34,MT/ 0.0, 0.0, L1 ,
L11 , LCT, NI1 ]CONTa
[MAT,34,MT/ 0.0, 0.0, LS1 , LB1 , NT1 , NE1 / {Data1 }] LIST
------------------------------------------------------------------[MAT,34,MT/ 0.0, 0.0, LSN I1 , LBSN I1 , NTSN I1 , NEN I1 /{DataN I1 }] LIST
------------------------------------------------------------------[MAT,34,MT/ 0.0, 0.0, LN SS , L1N SS , 0, NIN SS ] CONT
[MAT,34,MT/ 0.0, 0.0, LS1 , LB1 , NT1 , NE1 / {Data1 } LIST
------------------------------------------------------------------[MAT,34,MT/ 0.0, 0.0, LSN IN SS , LBN IN SS , NTN IN SS , NEN IN SS / {DATAN IN SS }]
LIST
a

In this first subsection, L and L1 are the smallest values present of NL and NL1.

The number of sub-subsections NSS for a given MT1 is NL*NL1, and they are ordered
as (L,L1) = (1,1),(1,2),...,(NL,NL1). (Not all L-values need be included). When MT1=MT,
redundancy is avoided by giving each sub-subsection only once, when L1≥ L. In this case
NSS=NL*(NL+1)/2.
278

CHAPTER 34. FILE 34. COVARIANCES FOR ANGULAR DISTRIBUTIONS

34.3

Procedures

It is strongly recommended that the maximum order of the Legendre expansion for uncertainty representation be minimized.
If there are important cases (e.g., n-p scattering) where the shape of the angular distribution is correlated with the magnitude of the scattering cross section, the convention is that
the covariances among scattering (integrated) cross sections must be in File 33 and must not
be repeated, so all sub-subsections in File 34 with L=L1=0 would contain null covariance
components. This procedure would maintain the convention that covariance components
are summed from various portions of the ENDF file corresponding to a particular material.
(The information contained in File 34 for L or L1 non-zero is the motivation for the present
procedure). Note that, in the case of correlation between shape and magnitude of a scattering cross section, it is possible for an absorption cross section MT-value to show up in
File 34 (with L=0 only).

279

Chapter 35
File 35. COVARIANCES FOR
ENERGY DISTRIBUTIONS OF
SECONDARY PARTICLES
35.1

General Comments

File 35 contains covariance matrices for the energy distribution of secondary particles given in
File 5. The data in File 5 are normally given in the Laboratory system, and are expressed as
normalized probability distributions. If the spectral distributions are correlated with angular
distributions and given in File 6, the covariance information in File 35 refers to the angleintegrated distributions. At present, no formats are defined for correlated distributions; for
the purpose of uncertainty estimations the separability assumption in the outgoing energy
and angle is implied.
Since there is usually very fragmentary experimental information with which to construct
the data given in File 5, the uncertainties in the secondary distributions are highly correlated
as a function of incident particle energy. It is therefore proposed that only a few covariance
matrices be used in each MT value in File 35 to cover the complete incident energy range.
Each covariance matrix applies to the complete secondary energy distributions for the broad
incident energy range specified, regardless of how these secondary energy distributions are
specified, or broken down into various components, in File 5. No covariances between the
different incident energy ranges are allowed. Also, no covariances linking different materials
or reaction types are allowed. Furthermore, no covariances with information in other files,
for instance File 3 and ν(E) in File 1 are allowed in File 35.

35.2

Formats

Each subsection covers a covariance matrix for one incident particle energy range, and the
complete incident energy range is covered by the NK subsections.
A new type of LB subsection is defined (LB=7). Covariances in File 35 refer to normalized
probabilities, therefore it is natural to specify the covariance matrices as absolute covariances
of the normalized probabilities rather than the corresponding relative covariances. The
280

CHAPTER 35. FILE 35. COVARIANCES FOR ENERGY DISTRIBUTIONS
LB=7 subsection is similar to an LB=5 subsection, but with entries that are absolute rather
than relative. The following quantities are defined:
NK Number of subsections
E1 Lowest incident neutron energy to which the covariance matrix in the subsection applies.
E2 Highest incident neutron energy to which the covariance matrix in the subsection applies. The value of E2 in a subsection becomes the value of E1 in
the next subsection.
LS=1 Flag indicating that the covariance data matrix Fk,k′ , is symmetric.
LB=7 Flag indicating that the elements of the covariance matrix Fk,k′ are absolute.
NT Total number of entries in the list. NT=[NE*(NE+1)]/2.
NE Number of entries in the array {Ek′ }
{Ek′ } Array containing outgoing particle energies, and defining NE-1 energy intervals for outgoing particles. The value of E1′ in the array must be the lowest
′
outgoing particle energy possible at E1 ; EN
E in the array must be the highest
outgoing particle energy possible at E2 and represented in File 5.
{Fk,k′ } covariance matrix. The Fk,k′ ’s are ordered by rows, starting from the diagonal
term
{Fk,k′ } ≡ F1,1 , F1,2 ; F2,2 , F2,3 , ..., ...; FN E−1,N E−1 .
The structure of a section of File 35 is as follows:
[MAT,35,MT/ ZA, AWR, 0, 0, NK, 0] HEAD


[MAT,35, 0/ 0.0, 0.0, 0, 0, 0, 0] SEND
The structure of a subsection is:
[MAT,35,MT/ E1 , E2 , LS, LB, NT, NE/{Ek′ },{ Fk,k′ }] LIST

(LS=1,LB=7)

In terms of the dimensionless operators defined in Section 33.2, the covariance between
two bin-probabilities (i.e. yields) in the energy-intervals Ei and Ej , respectively, of the
normalised probability distribution is:


 X i,k
Pj,k′ Fk,k′
Cov p (E → Ei′ ) , p E → Ej′ =
k,k′

where
E1 ≤ E ≤ E2 , and the operator P is defined so that only the term in the sum for
′
Ek′ ≤ Ei′ ≤ Ek+1
and Ek′ ′ ≤ Ej′ ≤ Ek′ ′ +1 is non-zero.
281

CHAPTER 35. FILE 35. COVARIANCES FOR ENERGY DISTRIBUTIONS

35.3

Procedures

Because probability distributions must remain normalized to unity, covariance matrices of
the bin probabilities in File 35 must satisfy the constraint (in addition to being symmetric)
that the sums of the elements in any row of the matrix (hence also in any column) must be
zero. Therefore, one of the covariance matrix elements in each row (or column) is redundant,
but this redundancy is kept in the file to cross-check the consistency of the covariance matrix.
The Fk,k′ are the covariance matrix elements defined above for the normalized spectral
yields Yk for the outgoing particle energy intervals on the evaluator’s energy grid {Ek′ }
(i.e. bin probabilities and not bin-averaged probability distributions). Normalization of
the covariance matrix is considered adequate if for each row (or column) k the following
condition is satisfied:
X
Sk
Fk,k′ .
(35.1)
< 10−5 , where Sk =
Yk
k′
If the above constraint has not been applied in the evaluation process, corrected values
Fbk,k′ may be obtained from the following relation:
X
Fbk,k′ = Fk,k′ − Sk Yk′ − Sk′ Yk + Yk Yk′
Sj .
(35.2)
j

The correction is applicable not only to the covariance matrix in the ENDF file, but also
to derived covariance matrices (for example, after processing and condensation on the user’s
energy grid).
This correction procedure is approximate and therefore appropriate only in case of relatively small deviations of the matrix from the normalization constraint given by equation (35.1).
The secondary energy distribution uncertainty analysis using the ”hot-cold” technique
of Gerstl, et al., (see References 1 and 2) can be based on spectral uncertainty data stored
in this form. In this case the covariance matrix is a 2 × 2 matrix for each incident energy
range.

References for Chapter 35
1. S. A. W. Gerstl, ”Uncertainty Analyses for Secondary Energy Distributions”, A Review
of the Theory and Application of Sensitivity and Uncertainty Analysis: Proceedings of a
SeminarWorkshop, August 22-24, 1978, C. R. Weisbin, et al., Eds, Radiation Shielding
Information Center report, Oak Ridge National Laboratory ORNL/RSIC42 (1978)
p. 219.
2. S. A. W. Gerstl, R. J. LaBauve, and P. G. Young, A Comprehensive Neutron CrossSection and Secondary Energy Distribution Uncertainty Analysis for a Fusion Reactor,
Los Alamos Scientific Laboratory report LA-8333-MS (1980).

282

Chapter 40
File 40. COVARIANCES FOR
PRODUCTION OF RADIOACTIVE
NUCLEI
40.1

General Comments

File 40 contains the covariances of neutron activation cross-section information appearing
in File 10. This file is based on File 33, which should be consulted for further information,
and on File 10.

40.2

Formats

The following quantities are defined:
ZA,AWR Standard material charge and mass parameters.
LIS Level number of the target.
NS Number of subsections; one for each LFS.
QM Mass-difference Q-value based on the ground state of the residual nucleus
QI Reaction Q-value (eV). (See Chapter 10.)
LFS Level number of the nuclide (ZAP) produced in the neutron reaction of type
MT.
NL Number of subsections.
Sections
File 40 is divided into sections identified by the value of MT. Each section of File 40 starts
with a HEAD record, ends with a SEND record, and has the following structure:
283

CHAPTER 40. FILE 40. COVARIANCES FOR RADIONUCLIDE PRODUCTION
[MAT,40,MT/ ZA, AWR, LIS,
0, NS, 0] HEAD

[MAT,40,MT/ 0.0, 0.0,
0,
0, 0, 0] SEND
Subsections
Each subsection has the following structure:
[MAT,40, MT/ QM, QI,
0, LFS, 0, NL]CONT

--------------------------------------------------
Sub-subsections
Each sub-subsection is used to describe a single covariance matrix, the covariance matrix of
the energy-dependent cross section given in section (MAT,10,MT.LFS) with given final state
(LFS) and the energy-dependent cross sections given in section (MAT1,MF1,MT1[.LFS1])
The values of MAT1, MF1, MT1, and LSF1 (if MF1=10) are given in the CONT record
that begins every sub-subsection.
Each sub-subsection may contain several sub-sub-subsections. Each sub-sub-subsection
describes an independent contribution to the covariance matrix given in the sub-subsection.
The total covariance matrix in the sub-subsection is made up of the sum of the contributions
of the individual sub-sub-subsections. The following additional quantities are defined:
XMF1 Floating point form of MF1 (the file number for the 2nd cross section to
which the covariance data relates).
XLFS1 Floating point form of LFS1 (the index of the final state for the 2nd cross
section to which the covariance data relates).
MAT1 MAT for the 2nd cross section to which the covariance data relates.
MT1 MT for the 2nd cross section to which the covariance data relates.
NC Number of ”NC-type” subsections which follow the CONT record.
NI Number of ”NI-type” subsections which follow the ”NC-type” subsections.
The structure of a sub-subsection describing the covariance matrix of the cross sections
defined by (MAT,10,MT.LFS] and (MAT1,MF1,MT1[.LFS1] is:

284

CHAPTER 40. FILE 40. COVARIANCES FOR RADIONUCLIDE PRODUCTION
[MAT,40,MT/ XMF1, XLFS1, MAT1, MT1, NC, NI] CONT

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

------------------------------------------------------------
The formats of the sub-sub-subsection for File 40 are exactly the same as the formats
for the sub-subsections for File 33.

40.3

Procedures

The procedures for File 40 are the same as for File 33 except that File 40 has one more level
of indexing corresponding to the LFS and LFS1 flags and as noted below.

40.4

Ordering of Sections,
Subsections,
subsections, and Sub-sub-subsections

Sub-

1. Sections: The sections in File 40 are ordered by increasing value of MT.
2. Subsections: Within a section, (MAT,40,MT), the subsections are ordered by increasing value of LFS.
3. Sub-subsections: A sub-subsection of File 40 is uniquely identified by the set of numbers (MAT,MT,LFS; MAT1,MT1,MF1[.LFS1]); the first two numbers indicate the section, the third indicates the subsection, while the last four indicate the sub-subsection.
(a) The sub-subsections within a subsection are ordered by increasing value of MAT1.
The value of MAT1=0 shall be used to mean MAT1=MAT.
(b) When there are several sub-subsections with the same value of MAT1 in a subsection, these sub-subsections shall be ordered by increasing values of XMF1. When
MF1=MF, the XMF1 field shall be entered as blank or zero. Therefore, within
a given subsection, the sub-subsections for MF1=MF will always appear before
those for other MF1 values.
(c) When there are several sub-subsections with the same value of MAT1 and MF1 in
a subsection, these sub-subsections shall be ordered by increasing values of MT1.
If MAT1=MAT and MF1=MF, then only those sub-subsections for MT≥MT
shall be given.
(d) When there are several sub-subsections with the same value of (MAT1,MT1,MF1)
in a subsection, these sub-subsections shall be ordered by increasing values
285

CHAPTER 40. FILE 40. COVARIANCES FOR RADIONUCLIDE PRODUCTION
of LFS1. If MAT1=MAT, MF1=MF, and MT1=MT, then only those subsubsections for LFS1≥LFS shall be given. (LFS1=0 implies the ground state
of the product and does not imply LFS1=LFS).

40.5

Completeness

There are no minimum requirements on the number of sections, subsections, and subsubsections in File 40. However, the presence of certain data blocks in File 40 implies
the presence of others, either in File 33 or 40 of a referenced material. In what follows, we
shall identify the sub-subsections of File 40 by
(MAT,MT,LFS; MAT1,MT1,MF1[.LFS1]).
The presence of this data block with MAT16=MAT or MF16=10 implies the presence of
appropriate data in:
a. (MAT,MT,LFS;MAT,MF=10,MT.LFS), in File 40 of MAT;
b. (MAT1,MT1[.LFS1]; MAT1,MF1,MT1.LFS1), in File (MF1+30) of MAT1;
c. (MAT1,MT1[.LFS1]; MAT ,MF=10,MT.LFS), also in File (MF1+30) of MAT1.

286

Acknowledgments
The following people have made significant contributions to the writing and editing of the
formats and procedures contained in this document: Ch. Dunford(NNDC), M. Greene
(ORNL), R. LaBauve (LANL), D. Larson and N. Larson (ORNL), C. Lubitz (KAPL), R.
MacFarlane (LANL), V. McLane (NNDC), D. Muir (LANL), S. Pearlstein (BNL), R. Peele
(ORNL), F. Perey (ORNL), R. Roussin (ORNL), R. E. Seamon (LANL), L. Stewart (LANL),
A. Trkov (JSI, Ljubljana).
The NNDC would like to thank the people who contributed to reviewing of this version,
particularly Toshihiko Kawano, Nancy Larson, Cecil Lubitz, Douglas Muir, Donald Smith,
Patrick Talou.

287

Appendix A
Glossary
Terms are given in alphabetical order with numbers preceding letters, lower-case preceding
upper-case letters, and Greek Letters following.
Parameter
a
al
Ak,l

Definition
Parameter used in the Watt spectrum.
lth Legendre coefficient.
Probability of emission of a γ ray of energy Eγ = εk − εl
as a result of the residual nucleus having a transition
from the k th to the lth level.
Al
Legendre coefficients (LANG=0) or µ, pi pairs for tabulated angular distribution (LANG¿0).
An
Mass of the nth type atom; A0 is the mass of the principal scattering atom in a molecule.
ABN
Abundance (atom fraction) of an isotope in this material.
AC
Channel radius
AC1, AC2, AC3, Background constants for the Adler-Adler radiative capAC4, BC1, BC2 ture cross section.
AF1, AF2, AF3, Background constants for the Adler-Adler fission cross
AF4, BF1, BF2 section.
AG
Reduced-width amplitude.
AJ
Floating-point value of compound nucleus spin, J (resonance spin).
AL
Floating point value of the angular momentum, l.
ALAB
Mnemonic of laboratory originating evaluation.
ALRE1
Exit-l-value for reaction.
. . . ALRE4
AMUF
Number of degrees of freedom used in fission-width distribution.
AMUG
Number of degrees of freedom used in radiation-width
distribution.

288

Chapter
5
4,6,14
11

6
7
2,32
2
2
2
2
2,32
2
1
2
2
2

APPENDIX A. GLOSSARY
Parameter
AMUN

Definition
Number of degrees of freedom used in neutron-width
distribution.
AMUX
Number of degrees of freedom used in competitive-width
distribution.
AP
Scattering radius.
APL
l-dependent scattering radius.
APSX
Total mass in neutron units of the n particles being
treated by LAW=6.
AS
Floating point value of channel-spin s.
AT1, AT2, AT3, Background constants for the Adler-Adler total cross
AT4, BT1, BT2 section.
AUTH
Author of evaluation.
AVGG
Statistical R-matrix parameter.
AWD
Atomic mass (not a ratio) of the daughter nucleus
(amu).
AWI
Projectile mass in neutron units.
AWP
Product mass in neutron units.
AWR
Ratio of mass of atom (or molecule) to that of the neutron.
AWRI
Ratio of mass of particular isotope to that of the neutron.
AWRIC
Mass-ratio for charged-particle exit channel (see page
D.40).
AWT
Nuclear mass (not a ratio) of outgoing particle.
AZD
Atomic number of daughter nucleus.
AZP
Atomic number of outgoing particle.
b
Parameter used in Watt spectrum function definition.
Bi (N )
List of constants.
BC
Boundary-condition parameter.
BR
Branching ratio for production of a particular nuclide
and level.
Cn
Coefficients of a polynomial; NC coefficients are given.
Cn (Ei )
Array of yield data for the ith energy point; contains 4
parameters per fission product.
CI
Channel index.
CI
Coefficient of the cross section for a reaction contributing to the value of a ”derived” cross section (usually
1.0).
CONT
Smallest possible ENDF record, a ”control” record.
CT
Chain indicator.
D
Mean level spacing for a particular J-state.
DDATE
Original distribution date of the evaluation.
2
DE
Variance of the resonance energy ER.

289

Chapter
2
2
2,32
2
6
2
2
1
2
2
1
6
all
2,32
2
2
2
2
5
7,12
2
8
1
8
2
31,33

all
8
2,32
1
32

APPENDIX A. GLOSSARY
Parameter
DECr
DEFr
DETr
DF2
DG2
DGDF
DJ2
DJDF
DJDG
DJDN
DN2
DNDF
DNDG
DWCr
DWFr
DWTr
DYC
DYI
{ Ek }
E
E′
E1, E2
Eavail
Eint
Eth
E”x”
EB
EBAR
EBI
ED
EDATE
EFH
EFL

EFR
EGk
EGD
EGP

Definition
Resonance energy for the radiative capture cross section.
Resonance energy for the fission cross section.
Adler-Adler resonance energy for the total cross section.
Variance of GF.
Variance of GG.
Covariance of GG and GF.
Variance of AJ.
Covariance of AJ and GF.
Covariance of AJ and GG.
Covariance of AJ and GN.
Variance of GN.
Covariance of GN and GF.
Covariance of GN and GG.
Value of Γ/2, (ν), used for the radiative capture cross
section.
Value of Γ/2, (ν), used for the fission cross section.
Value of Γ/2, (ν), used for the total cross section.
1-σ uncertainty in cumulative fission-product yield.
1-σ uncertainty in fractional independent fissionproduct yield.
List of energies for a covariance file energy grid
Energy of the incident neutron (eV).
Secondary neutron energy (eV).
Range of neutron energies.
Available energy.
Interpolation scheme for each energy range.
Threshold energy (eV).
Average decay energy (eV) of ”x” radiation for decay
heat applications.
Total energy released by delayed β’s.
Statistical R-matrix parameter
Binding energy for subshell (eV)
Logarithmic parameter for a R-matrix element
Date of evaluation.
Constant in energy-dependent fission spectrum
Constant in energy-dependent fission spectrum
Fluorescence yield (eV/photo-ionization). Value is zero
if not photoelectric subshell ionization cross section.
Kinetic energy of the fragments.
Photon energy or Binding Energy.
Total energy released by the emission of delayed γ rays.
Total energy released by the emission of ”prompt” γrays.

290

Chapter
2
2
2
32
32
32
32
32
32
32
32
32
32
2
2
2
8
8
32-40
all
5,6,7
31,33
5
all
3
8
1
2
28
2
1
5
5
23
1, 5
12, 13, 14
1
1

APPENDIX A. GLOSSARY
Parameter
EH
EI
EL
ELFS
ELIS
ELN
EMAX
END
ENDATE
ENP
ENU
EPE
ER

ERAV
ESi
ESk
ES(N)
ET
ET(E)
ETR
EU
fi (µ, E, E ′ )
fk (E → E ′ )
F (q; Z)
Fx,k,k′ (LB)

FC
FD
FPS
FTR
g(Eγ ← E)

Definition
Upper limit for a resonance region energy range.
Energy points where the weighting of the standard cross
is given.
Lower limit for a resonance region energy range.
Excitation energy of the reaction product.
Excitation energy of the target nucleus.
Number of electrons in subshell when neutral.
Upper limit of energy range for evaluation
Kinetic energy of the delayed fission neutrons.
End-point energy of the particle or quantum emitted.
Master file entry date (yyyymmdd).
Kinetic energy of the ”prompt” fission neutrons.
Energy carried away by neutrinos.
Sub-shell binding energy (equal to photoelectric edge
energy) in eV. The value is zero if MT6=534-599.
Total energy release due to fission minus neutrino energy.
Resonance energy (in the laboratory system).
Energy (eV) of radiation produced.
Average decay energy of radiation produced.
Energy of the ith level.
Energy of the level from which the photon originates.
Energy of Nth point used to tabulate energy-dependent
widths.
Total energy release due to fission.
Energy transfer during electro-atomic excitation or
bremsstrahlung
Energy of transition (eV)
Logarithmic parameter for a R-matrix element
Normalized product energy-angle distribution
k th partial energy distribution; definition depends on LF
value.
Form factor for coherent photon scattering.
Covariance components correlated over the energy interval with lower edges Ek and Ek′ (exact definition depends on LB value.
Continuum spectrum normalization factor.
Discrete spectrum normalization factor.
Floating-point value of state designator for a fission
product nuclide.
Fractional probability for transition.
Particular class of the functions gj (Eγ ← E) tabulated
in File 15; in units eV-1.

291

Chapter
2, 32
31, 33
2, 32
8
1
28
1
1
8
1
1
1
23
1
2,32
8
8
12
12, 13, 14
2
1
26
28
2
6
5
27
31-40

8
8
8
28
15

APPENDIX A. GLOSSARY
Parameter
gj (Eγ ← E)
GE
GF

Definition
j th normalized partial distribution; in units eV-1.
Eliminated width.
Fission width Γf evaluated at resonance energy ER.
Average fission width - may be energy dependent.
GFA
First partial fission width
GFB
Second partial fission width
GG
Radiation width Γγ evaluated at resonance energy ER.
Average radiation width - energy dependent if LRU=2.
GICr
Asymmetrical capture parameter.
GIFr
Asymmetrical fission parameter.
GITr
Related to the asymmetrical total cross section parameter.
GN
Neutron width Γn evaluated at resonance energy ER.
GN0
Average reduced neutron width; energy dependent.
GPj,i ≡ Gpi
Conditional probability of photon emission in a direct
transition from level j to level i, i < j.
GRCr
Symmetrical capture parameter.
GRE1 . . . GRE4 Partial widths.
GRFr
Symmetrical fission parameter.
GRTr
Related to symmetrical total cross section parameter.
GT
Resonance total width Γ evaluated at the resonance energy ER.
GX
Competitive width Γx evaluated at resonance energy
ER, or, average competitive reaction width.
H(q; Z)
Form factor or incoherent scattering function; either
F (q; Z) or S(q; Z), respectively.
H(N)
Array containing text information that describes evaluated data set.
HEAD
First record in a section
HL
Half-life of the reaction product.
HSUB
Library identifier (eye-readable)
I
Normalizing denominator (see 5.3).
IDP
Resonance parameter identification number.
INT
Statistical parameter for R-matrix element (LRU=1,
LRF=5), or, interpolation scheme used for interpolating
between cross sections obtained from average resonance
parameters (LRU=2).
INT(m)
Interpolation scheme identification number used in mth
range.
IPS
Imaginary part of a non-hard-sphere phase shift.
IPS(e)
Real part of PS(E).
IR0(e)
Imaginary part of R0(E).
IRP
Imaginary part of background-R-matrix element.

292

Chapter
15
2
2, 32
2
2
2
2, 32
2
2
2
2
2, 32
2
12
2
2
2
2
2, 32
2
27
1
all
8
1
5
2, 32
2

0
2
2
2
2

APPENDIX A. GLOSSARY
Parameter
ISG
ISH
k
L
LA
LAD
LANG
LASYM
LAT
LAW
LB
LBK
LCOMP
LCON
LCOV
LCT
LDRV
LE
LEP
LF
LFI
LFS
LFW
LG
LI

LIBF

LIDP
LIP

Definition
Spin group index.
Shift function flag.
Boltzmann’s constant.
Value of the l-state (neutron angular momentum).
Value of l (for the lth coefficient).
Angular distribution flag
Angular distribution indicator
Flag indicating whether asymmetric S(α, β) is given.
Flag indicating which temperature has been used to
compute α and β.
Distinguishes between different representations of fi , the
normalized energy-distribution of a reaction product
Flag which determines meanings of the F -numbers in
the arrays {Ek , Fk }{El , Fl } .
Background R-matrix parameter (LRU=1, LRF=5), or,
background-R-function flag (LRU=1, LRF=6).
Indicates ENDF-5 compatible format.
Continuum spectrum flag.
Indicates whether covariance data are given.
Indicates which reference frame is used for both secondary angles and energies.
Distinguishes between different evaluations with the
same material keys
Indicates whether energy-dependent fission-product
yields are given.
Selects interpolation scheme for secondary energy.
Specifies the energy distribution law that is used for a
particular subsection (partial energy distribution).
Indicates whether this material is fissionable.
Indicator that specifies the final excited state of the
residual nucleus produced by a particular reaction.
Indicates whether average fission widths are given in the
unresolved resonance region for this isotope.
Transition probability array flag for distinguishing between doublet and triplet arrays in File 12.
Indicates kind of Adler-Adler parameters given.
Isotropy flag.
Temperature interpolation flag.
Sub-library where some data are sensitive to the
same model parameters as data in present sublibrary/material.
Identifies identical particles for LAW=5.
Product modifier flag.

293

Chapter
2
2
7

7
7
6
8, 31, 33,
34, 40
2
32
8
8
4,6
1
8
6
5, 6, 12,
13, 15
1
3, 8, 9, 10,
40
2, 32
12
2
4, 14
7
30

6
8

APPENDIX A. GLOSSARY
Parameter
LIS
LISO
LIST
LLN
LMF
LNU
LO
LP
LPS
LR
LREL
LRF

LRP
LRU
LRX
LS
LSSF
LT

LTHR
LTP
LTT

LTY
L1
L2
Mn
MAT

Definition
State number of the target nucleus (for materials that
represent nuclides).
Isomeric state number of the target nucleus.
Record used to list a series of numbers.
Indicates form for storing S(α, β).
File number for this MT containing multiplicity or cross
section.
Indicates representation of ν(E) used.
Indicates whether multiplicities or transition probability
arrays given.
Indicates whether particular photon is a primary.
Optical model phase shift flag
Defines x in (n,n’x); used in the reactions MT=51, 52,
53, . . . , 90, and 91. (See Section 3.4.4)
Release number
Indicates which resonance parameter representation
used for energy range; definition depends on value of
LRU for range.
Indicates whether resolved and/or unresolved resonance
parameters given in File 2.
Indicates whether energy range contains data for resolved or unresolved resonance parameters.
Indicates whether a competitive width is given.
Indicates whether Fk,k′ matrix is asymmetric or symmetric (LB=5 or 7).
Indicates how File 2 and File 3 are to be combined.
Temperature dependence (see also Appendix F.1).
Specifies whether temperature-dependent data are
given.
Number of pairs of numbers in the array {El , Fl }.
Thermal data flag.
Specifies representation used for LAW=5.
Specifies whether Legendre or probability representation
used.
Specifies whether Legendre coefficient covariance data
start with a0 coefficient.
In ”NC-type” sub-subsections, indicates the procedure
used to obtain the covariance matrix.
Integer to be used as a flag or a test.
Number of Legendre coefficients.
Integer to be used as a flag or a test.
Number of atoms of the nth type in the molecule.
Material number.

294

Chapter
1, 3, 8, 9,
10, 40
1, 8
all
7
8
1
12
12, 13
2
3
1
2, 32

1
2, 32
2
31, 33, 35
2, 32
0
3, 4, 5, 6,
7
31, 33
7
6
4, 6, 14
34
31, 33
1
34
1
7
0

APPENDIX A. GLOSSARY
Parameter
MAT1
MATF

Definition
Referenced material for covariance data.
MAT in which some data are sensitive to the same parameter.
MATP
Material number for the reaction product.
MATS
MAT in which a pertinent standard cross section (MTS)
exists.
MF
File number.
MFn
MF of the nth section.
MFSEN,MTSEN MF,MT of a section in which data are sensitive to the
indicated parameter (MP).
MODn
Modification indicator for section MFn and MTn .
MP
Model parameter index.
MPAR
Number of parameters for which covariance data is
given.
MPF
Model parameter index given the same parameter (MP)
in another sublibrary/material
MT
Reaction type number, or, covariance file section identifier.
MTn
MT of the nth section.
MT1
Referenced reaction type for covariance data.
MTL
Indicates MT that is a component of the lumped reaction.
MTS
Reaction type number for relevant standard cross section.
MUF
Number of degrees of freedom for fission widths.
MTRE1
MT values for inelastic or charged particle reactions
. . . TRE4
N0
Identifies reaction product that has radioactive ground
state.
N1
Number of items in a list to follow (except for MT 451).
N2
Number of items in a second list to follow.
NA
Number of angles (cosines) at which secondary distributions given.
NAC
Number of channel radii
NAPS
Controls use of channel radius a and scattering radius
AP.
NB
Total number of β value given.
NBC
Number of boundary-condition parameters.
NBK
Background-R-matrix parameter.
NBT(n)
Value of N separating the mth and (m+1)th interpolation
rangers.
NC
Number of terms used in the polynomial expansion.
Total number of decay energies (eV) given (NC = 3 or
17).
295

Chapter
33
30
8
31, 33
0
1
30
1
30
32
30
all
1
33
33
31, 33
2
2
8
1
1
6
2
2, 32
7
2
2
0
1
8

APPENDIX A. GLOSSARY
Parameter

NCn
NCH

NCI

NCP
NCRE
NCS
NCT
NCTAB
ND
NDIGIT
NDIR
NDK
NE

NEI
NEP
NER

NF
NFOR
NFP
NFRE
NGRE
NHS

Definition
Number of partial distributions used to represent
f (Eγ ← E).
Number of ”NC-type” sub-subsections.
Number of physical records in the nth section.
Number of channels using particular background Rmatrix element, phase shift, penetrability, channel radius, or boundary condition (LRU=1, LRF=5).
Number of reactions summed to obtain the reaction of
interest.
For a ”derived” cross section, number of reaction types
for which cross sections are combined in the derivation.
Number of channels.
Number of charged-particle reactions.
Number of channels in a particular spin group.
Total number of channels.
Number of CONT records in the corresponding table.
Number of discrete energies (File 6);
Number of branches into which nuclide ZAP decays.
Number of digits for compact resonance covariance representation
Number of CONT records in the MF=30 directory.
Total number of decay modes given.
Number of incident-energy points at which widths given.
Number of points at which tabulated distributions
given.
Number of points at which θ(E) (File 5) are given.
Number of energy points given in a TAB2 record.
Number of energy points in EI,WEI list.
Number of secondary energy points.
Number of energy ranges given for this isotope.
Total number of discrete energies for given spectra type
(STYP).
Number of energies corresponding to the rows of LB=6
covariance matrix.
Number of channels not requiring a phase shift.
Number of secondary energy points in tabulation.
Library format.
Number of fission-product nuclide states to be specified
at each incident-energy point.
Number of fission reactions.
Number of capture reactions.
Number of channels that require hard-sphere phase
shifts.

296

Chapter
15
31, 33
1
2

1
31, 33
2
2
2
2
30
6
8
32
30
8
2
4
5
14, 15
33
6
2, 32
8
33
2
5
1
8
2
2
2

APPENDIX A. GLOSSARY
Parameter
NI

NIRE
NIS
NJS
NK

NL
NLG
NL1
NLIB
NLJ
NLRS
NLS
NLSC
NLSJ
NM
NML
NMOD
NMT1
NMU
NN
NNF
NO
NP

Definition
Total number of items in the B(N) list; NL=6*(NS+1).
Number of isotropic angular distributions given in section (MT) for which LI=0, i.e., with at least one
anisotropic distribution).
Number of ”NI-type” sub-subsections.
Number of inelastic reactions.
Number of isotopes in this material.
Number of sets of resonance parameters (each having
the same J state) for a specified l-state.
Number of elements in transformation matrix;
NK=(NM+1)2.
Number of partial energy distributions (one subsection
for each partial distribution).
Number of partial energy distributions (LCON=5).
Number of discrete photons plus the photon continuum.
Number of subsections in this section (MT).
Number of incident-neutron energy ranges for covariance representation, each with a subsection.
Highest order Legendre polynomial given at each energy.
Number of subsections within a section.
Number of logarithmically parameterized elements.
Number of Legendre coefficients.
Library identifier.
Count of the number of levels for which parameters will
be given.
Number of subsections containing data on long-range
resonance parameter covariance.
Number of l-values considered; a set of resonance parameters is given for each l-value.
Number of l-value for convergence.
Number of resonances specified by l, s, and J.
Maximum order Legendre polynomial required to describe the angular distributions.
Number of entries in MT list.
Modification number.
Number of subsections, for MT1≥MT.
Number of emission cosine values for LAW=7.
Number of elements in the LIST record
Number of precursor families considered.
Decay information flag.
Number of points in a tabulation of y(x) that are contained on the same record.
Number of Bragg edges.

297

Chapter
7
14

31, 33
2
2, 32
2, 32
4
5, 6
8
12, 13, 14
26
35
4, 6, 14
33
2
34
1
2
32
2, 32
2
2
4
2
1
34
6
8
1
8
all but ↓
7

APPENDIX A. GLOSSARY
Parameter

NPE
NPP
NPS
NPSX
NR
NRB
NRM
NRO
NRP
NRS
NRT
NS

NSG
NSP
NSRS
NSS
NST
NSUB
NT

NTP
NTR
NVER
NVS
NW
NWD
NX
NXC
Pj (E)

Definition
Total number of distinct model parameters.
Total number of pairs of numbers in the arrays
{Ek , Fk }{El , Fl }.
Number of charged-particle penentrabilities.
Number of pairs of numbers in the {Ek , Fk } array.
Number of non-hard-sphere phase shifts.
Number of particles distributed by LAW=6.
Number of different interpolation intervals in a tabulation of y(x) that are contained in the same record.
Number of resonances in block.
Number of interpolation intervals for emission cosine for
LAW=7.
Energy dependence of the scattering radius.
Number of interpolation intervals for emission energy
(LAW=7).
Number of resolved resonances for a given l-state.
Total number of resonances.
Number of non-principle scattering atom types.
Number of states of the radioactive reaction product.
Number of levels below the present one, including
ground state.
Number of spin groups.
Total number of spectra radiation types (STYP) given.
Number of subsections for covariances among parameters of specific resonances.
Number of different s-values.
Number of subshells.
Number of statistically parameterized background Rmatrix elements.
Sub-library number.
Number of transitions for which data given.
Number of entries for each discrete energy ER.
Total number of items in LIST.
Control flag for background-R-matrix or penetrability
list.
Number of transitions.
Library version number.
Number of covariance elements for a block of resonances.
Number of words in LIST record.
Number of elements in the text section.
Number of sets of background constants to be given.
Number of the sections to be found in the dictionary
Probability or weight given to j th partial distribution,
gj (Eγ ← E).
298

Chapter
30
31, 33
2
8
2
6
all
32
6
2, 32
6
2, 32
2
7
8, 9, 10
12
2
8
32
2
28
2
1
2
8
33, 34, 35
2
28
1
32
6
1
2
1
15

APPENDIX A. GLOSSARY
Parameter
Pk (EN )

Definition
Chapter
Fractional part of cross section that can be described 5
by the k th partial distribution of the nth incident-energy
point.
p(µ, E)
4
PAR
Parity π of target nuclide.
8
PCP(E)
Charged-particle penetrability.
2
PMT
Floating point value for MT.
2
PS(E)
Complex phase shift.
2
Q
Reaction Q-value (eV); Q=(rest mass of initial state - 3, 9, 10
rest mass of final state).
Total decay energy (eV) available in corresponding de- 8
cay process, not necessarily the same as maximum energy of emitted radiation).
QI
Reaction Q-value.
3, 9, 10
QM
Mass-difference Q-value.
3, 9, 10
QRE1 . . . QRE4 Q-values.
2
QX
Effective Q-value for the competitive width.
2, 32
Rmjα
Probability of de-excitation.
11
R0, R1, R2
Logarithmic parameters for an R-matrix element.
2
R0(E)
Complex background R-function.
2
RCOV
Relative covariance of model parameters.
30
RDATE
Date and number of last revision.
1
REF
Reference to evaluation.
1
RFS
Isomeric state flag for daughter nuclide.
8
RI
Resonance index.
2
Intensity of radiation produced (relative units).
8
RICC
Total internal conversion coefficient.
8
RICK
K-shell internal conversion coefficient.
8
RICL
L-shell internal conversion coefficient.
8
RIN
Statistical R-matrix parameters.
2
RIS
Internal pair formation coefficient (STYP=0.0); 8
positron intensity (STYP=2.0).
RNPM
Number of particular sections (MT’s).
2
RNSM
Number of summed sections (MT’s).
2
RP
Spectrum
of the continuum component of the radiation 8
R
RP(E) dE = 1.
RPB
Real part of background R-matrix element.
2
RPS
Real part of a non-hard-sphere phase shift.
2
RPS(E)
Real part of PS(E).
2
RR0(E)
Real part of R0(E).
2
RTYP
Mode of decay of the nuclide in its LISO state.
8
RVij
Relative covariance quantities among average unre- 32
solved parameters.

299

APPENDIX A. GLOSSARY
Parameter
S(α, β, T )
S0, S1
SF
SMT
SPD
SPI
SPP
STA
STYP
SUBI
SUBJ
SUBK
T
T1/2
TAB1
TAB2
TEMP
TM
TPi
TYPE
U
Vmn
WEI
x
x(n)
XLFS1
XLFSS
XMF1
XMFS
XMTI

y(E)

Definition
Defined (for a moderating molecule) by equation (7.6).
Logarithmic parameters for an R-matrix element.
Statistical R-matrix parameter.
Floating-point value for MT.
Spin and parity for the daughter nucleus.
Nuclear spin of the target nucleus, I (positive number).
Spin and parity.
Target stability flag.
Decay radiation type (defined in Section 8.4).
Subshell designator.
Secondary subshell designator
Tertiary subshell designator.
Temperature (K) at which temperature dependent data
given.
Half-life of the original nuclide (seconds).
Control record for one-dimensional tabulated functions.
Control record for two-dimensional tabulated functions.
Target temperature.
Maximum temperature parameters.
Probability of a direct transition from level NS to level
I, I=0,1,2, . . . (NS-1).
Indicates the type of transition for beta and electron
capture.
Defines the upper energy limit for the secondary neutron, so that 0 ≤ E ≤ E − U (given in the Lab system).
Variance-covariance matrix element among resonance
parameters.
Weight of the standard cross section at a given EI relative to the next given energy.
E ′ /θ(E).
nth value of x.
Floating-point form of final excited state number references for covariance data.
Floating-point form of LFSS, final excited state number
of a reaction with a standard cross section
Floating-point form of file number reference for covariance data.
Floating-point form of MFS, file number in which pertinent standard cross section (MTS) may be found.
Floating-point equivalent of MT number of the reaction
for which the cross section contributes to a ”derived”
cross section.
Yield for particle described.

300

Chapter
7
2
2
2
2
2, 8, 32
2
1
8
28
28
28
4, 5, 6, 7
8
all
all
1
5
12
8
5
32
31, 33
5
0
33, 40
31, 33
33, 40
31, 33
31, 33

26

APPENDIX A. GLOSSARY
Parameter
y(n)
yi (E)
Y (E)

YC
YI
ZA
ZAI
ZAFP
ZAN
ZAP
ZSYMA
ZSYMAM
α
β
δ(Eγ − εj + εi )
∆
λi
θ

ν(E)
σ(E)
σbn
σf n
σkγ (E)
σm0 (E)
σs (E)

σT (background)

Definition
nth value of y.
Product yield or multiplicity.
Total multiplicity at energy E(eV); given as energymultiplicity pairs.
Partial multiplicity at energy E(eV).
Cumulative yield for a particular fission product.
Fractional independent yield for a particular fission
product.
(Z,A) Designation of the original nuclide. ZA=1000.Z +
A
(Z,A) designation for an isotope. ZA=1000.Z + A
(Z,A) identifier for a particular fission product.
ZA=1000.Z + A
(Z,A) designation of the next nuclide in the chain.
ZA=1000.Z + A
(Z,A) designation of the product nuclide. ZA=1000.Z +
A
Text representation of material: Z-chemical symbol-A
Text representation of material Z-chemical symbol-Astate.
√
Momentum transfer, α = (E ′ + E − 2µ EE ′ )/A0 kT .
Energy transfer, β = (E ′ − E)/kT .
Delta function, with εj , εi being energy levels of the
residual nucleus.
Uncertainty in quantity.
Decay constant (sec−1 ) for the ith precursor.
Parameter describing secondary energy distribution;
definition of θ depends on the energy distribution law
(LF).
Total average number of neutrons formed per fission
event.
Cross section (barns) for a particular reaction type at
incident energy point, E, in (eV).
Bound atom scattering cross section of nth type atom,
Free atom scattering cross section of nth type atom.
Photon production cross section for a discrete photon or
photon continuum specified by k.
Neutron cross section for exciting moth level with neutron energy E.
Scattering cross sections, e.g., elastic scattering at energy E as given in File 3 for the particular reaction type
(MT).
AT1 + AT2 /E + A3 /E 2 + AT4 /E 3 + BT1 E + BT2 E 2

301




√C
E

Chapter
0
6
9, 12
12
8
8
all
2, 32
8
8
8
8
1
7
7
11
1, 8
1
5

1
3, 10, 23
7
7
13
11
4

2

APPENDIX A. GLOSSARY
Parameter
σW
σ(ΩE)
dΩ
σkγ
dΩ

µ

Definition
Wick’s limit cross section in units of barns/steradian.
Differential scattering cross section in units of
barns/steradian.
Differential photon production cross section in
barns/steradian.
Cosine of scattered angle in either laboratory or centerof-mass system.

302

Chapter
4
4
14
4,6,14

Appendix B
Definition of Reaction Types
Reaction types (MT) are identified by an integer number from 1 through 999. Version
ENDF-6 of the ENDF format supports incident charged particles and photons in a manner
consistent with the definitions of MT’s used in previous versions of the ENDF format to the
extent possible. Users should be aware of the few differences. In the following table, those
MT numbers restricted to incident neutrons are labeled (n,xxx); those that are limited to
incident charged particles and photons are labeled (y,xxx) and those that allow all particles
in the entrance channel are labeled (z,xxx), where x can represent any exit particle. See
Section 0 for complete descriptions of MT numbers. Refer to Sections 3.4 (incident neutrons)
and 3.5 (incident charged particles and photons) for the list of MT numbers that should be
included in each evaluation.
For the ENDF-6 format, all particles in the exit channel are named (within the parenthesis) except for the residual. The identity of this residual can be specified explicitly in File 6
or determined implicitly from the MT number. In cases where more than one MT might
describe a reaction, the choice of MT number is then determined by the residual which is the
heaviest of the particles (AZ,A) in the exit channel. For example, 6 Li(n,t)α is represented by
MT=700, rather than my MT=800; and MT=32 represents the 6 Li(n,nd)α reaction rather
than MT=22. Sequential reaction mechanism descriptions can be used, where necessary, for
reactions such as X(n,np)Y. These are described in Sections 0.4.3.3 and 0.4.3.4.

B.1

Reaction Type Numbers MT

MT
1

(n,total)

2

(z,z0)

Description
Neutron total cross sections. (See sum
rules for cross sections in Section 0.5.1 Table 14).
Elastic scattering cross section for incident
particles.

303

Comments
Redundant.
Undefined for incident
charged particles.

APPENDIX B. DEFINITION OF REACTION TYPES
MT
3

4

5

Description
(z,nonelas.) Nonelastic neutron cross section. (See sum
rules for cross sections in Section 0.5.1 Table 14).
(z,n)
Production of one neutron in the exit channel. Sum of the MT=50-91.

(z,anything) Sum of all reactions not given explicitly in
another MT number. This is a partial reaction to be added to obtain MT=1.

6-9

Not allowed in Version 6.

10

(z,contin.)

11

(z,2nd)

12-15
16

(z,2n)

17

(z,3n)

18

(z,fission)

19
20
21
22

(n,f)
(n,nf)
(n,2nf)
(z,nα)

23

(n,n3α)

24

(z,2nα)

25

(z,3nα)

Total continuum reaction; includes all continuum reactions and excludes all discrete
reactions.
Production of two neutrons and a
deuteron, plus a residual 1 .
Unassigned.
Production of two neutrons and a residual1. Sum of MT=875-891, if they are
present.
Production of three neutrons and a residual.
Particle-induced fission (sum of MT 19, 20,
21 and 38, if present).
First-chance neutron-induced fission 2 .
Second-chance neutron-induced fission2 .
Third-chance neutron-induced fission2 .
Production of a neutron and an alpha particle, plus a residual.
Production of a neutron and three alpha
particles, plus a residual1 .
Production of two neutrons and an alpha
particle, plus a residual1 .
Production of three neutrons and an alpha
particle, plus a residual1 .

1

Comments
Redundant.
For
photon production
only.
Redundant. For incident neutrons, this
is inelastic scattering (MT=50 is undefined).
Each particle can
be identified and its
multiplicity given in
File 6. Not allowed
in Files 4, 5.
9
Be(n,2n) in format
Version 5.
Redundant; to be
used for derived files
only.

The ”residual” is the remainder after the reaction specified by has taken place (e.g. isotope of the target
nucleus with mass A − 1 after an (n,2n) reaction). This residula may break up further if LR¿0.
2
Note that the partial fission cross sections are not defined for charged particles.

304

APPENDIX B. DEFINITION OF REACTION TYPES
MT
26

Description
Not allowed in Version 6.

27

(n,abs)

28

(z,np)

29

(z,n2α)

30

(z,2n2α)

31
32

(z,nd)

33

(z,nt)

34

(z,n3 He)

35

(z,nd2α)

36

(z,nt2α)

37
38
39

(z,4n)
(n,3nf)

40
41

(z,2np)

42

(z,3np)

43
44

(z,n2p)

45

(z,npα)

46-49

Comments
Version 5: (n,2n)
isomeric state; used
in file 8 and 6, 9, or
10.
Absorption; sum of MT=18 and MT=102 Rarely used.
through MT=117
Production of a neutron and a proton, plus
a residual.
Production of a neutron and two alpha particles, plus a residual.
Production of two neutrons and two alpha
particles, plus a residual.
Not allowed for Version 6.
Used only as an LR
flag.
Production of a neutron and a deuteron,
plus a residual.
Production of a neutron and a triton, plus
a residual.
Production of a neutron and a 3 He particle,
plus a residual.
Production of a neutron, a deuteron, and
2 alpha particles, plus a residual.
Production of a neutron, a triton, and 2 alpha particles, plus a residual.
Production of 4 neutrons, plus a residual.
Fourth-chance fission cross section.
Not allowed for Version 6.
Used only as an LR
flag.
Not allowed for Version 6.
Used only as an LR
flag.
Production of 2 neutrons and a proton,
plus a residual.
Production of 3 neutrons and a proton,
plus a residual.
(Unassigned)
Production of a neutron and 2 protons,
plus a residual.
Production of a neutron, a proton, and an
alpha particle, plus a residual.
Not allowed in Version 6.
Version 5: description of 2nd neutron
from 9 Be(n,2n) reactions to excited
states.
305

APPENDIX B. DEFINITION OF REACTION TYPES
MT
50

(y,n0 )

51

(z,n1 )

52

(z,n2 )

90

...
(z,n40 )

91

(z,nc )

92-100
101

(n,disap)

102
103

(z,γ)
(z,p)

104

(z,d)

105

(z,t)

106

(z,3 He)

107

(z,α)

108

(z,2α)

109

(z,3α)

110
111

(z,2p)

Description
Comments
Production of a neutron, leaving the resid- Not allowed for inual nucleus in the ground state.
cident neutrons; use
MT=2.
Production of a neutron, with residual in
the 1st excited state.
Production of a neutron, with residual in
the 2nd excited state.
Production of a neutron, with residual in
the 40th excited state.
Production of a neutron in the continuum
not included in the above discrete representation.
(Unassigned)
Neutron disappearance; equal to sum of Rarely used.
MT=102-117.
Radiative capture.
Production of a proton, plus a residual. For incident proSum of MT=600-649, if they are present. tons, this is inelastic
scattering
(MT=600 is undefined).
Production of a deuteron, plus a residual. For
incident
Sum of MT=650-699, if they are present. deuterons,
this
is inelastic scattering (MT=650 is
undefined).
Production of a triton, plus a residual. For incident tritons,
Sum of MT=700-749, if they are present. this is inelastic scattering (MT=700 is
undefined).
3
Production of a He particle plus a residual. Sum of MT=750-799, if they are
present.
Production of an alpha particle, plus a
residual. Sum of MT=800-849, if they are
present.
Production of 2 alpha particles, plus a
residual.
Production of 3 alpha particles, plus a
residual.
(Unassigned)
Production of 2 protons, plus a residual.

306

APPENDIX B. DEFINITION OF REACTION TYPES
MT
112

(z,pα)

113

(z,t2α)

114

(z,d2α)

115

(z,pd)

116

(z,pt)

117

(z,dα)

118-119
120

121-150
151

(n,RES)

152-200
201

(z,Xn)

202

(z,Xγ)

203

(z,Xp)

204

(z,Xd)

205

(z,Xt)

206

(z,X3 He)

207

(z,Xα)

208

(z,Xπ + )

209

(z,Xπ 0 )

210

(z,Xπ − )

Description
Comments
Production a proton and an alpha particle,
plus a residual.
Production of a triton and 2 alpha particles, plus a residual.
Production of a deuteron and 2 alpha particles, plus a residual.
Production of proton and a deuteron, plus
a residual.
Production of proton and a triton, plus a
residual.
Production of deuteron and an alpha particle, plus a residual.
(Unassigned)
Not allowed for Version 6.
Version 5:
target destruction nonelastic
minus
total (n,n’γ)
(Unassigned)
Resonance parameters that can be used to Incident neutrons
calculate cross sections at different temper- only.
atures in the resolved and unresolved energy regions.
(Unassigned)
Total neutron production.
Redundant; use in
derived files only.
Total gamma production.
Redundant; use in
derived files only.
Total proton production.
Redundant; use in
derived files only.
Total deuteron production.
Redundant; use in
derived files only.
Total triton production.
Redundant; use in
derived files only.
3
Total He production.
Redundant; use in
derived files only.
Total alpha particle production.
Redundant; use in
derived files only.
Total π + production.
For use in highenergy evaluations.
0
Total π production.
For use in highenergy evaluations.
−
Total π production.
For use in highenergy evaluations.

307

APPENDIX B. DEFINITION OF REACTION TYPES
MT
211

(z,Xµ )

Description
Total µ+ production.

212

(z,Xµ− )

Total µ− production.

213

(z,Xκ+ )

Total κ+ production.

214

(z,Xκ0long )

Total κ0long production.

215

(z,Xκ0short )

Total κ0short production.

216

(z,Xκ− )

Total κ− production.

217

(z,Xp− )

Total anti-proton production.

218

(z,Xn− )

Total anti-neutron production.

219-250
251

(n,...)

252

(n,...)

253

(n,...)

254-300
301-450

(z,...)

451

(z,...)

452

(z,...)

453
454
455

(z,...)
(z,...)

456

(z,...)

457

(z,...)

+

(Unassigned)
µ, average cosine of the scattering angle
(laboratory system) for elastic scattering
of neutrons.
ξ, average logarithmic energy decrement
for elastic scattering of neutrons.
γ, average of the square of the logarithmic energy decrement divided by twice the
average logarithmic energy decrement, for
elastic scattering of neutrons.
(Unassigned)
Energy release parameters, , for total and
partial cross sections; MT= 300 plus the
reaction MT number, e.g., MT=302 is the
elastic scattering kerma.
Heading or title information; given in File 1
only.
ν T , average total (prompt plus delayed)
number of neutrons released per fission
event.
(Unassigned)
Independent fission product yield data.
ν d , average number of delayed neutrons released per fission event.
ν p , average number of prompt neutrons released per fission event.
Radioactive decay data.

308

Comments
For use in highenergy evaluations.
For use in highenergy evaluations.
For use in highenergy evaluations.
For use in highenergy evaluations.
For use in highenergy evaluations.
For use in highenergy evaluations.
For use in highenergy evaluations.
For use in highenergy evaluations.
Derived files only.

Derived files only.
Derived files only.

Derived files only.

APPENDIX B. DEFINITION OF REACTION TYPES
MT
458
459
460-464
465-466

(n,...)
(z,...)

467-499
500
501
502
503
504
505
506
507-514
515
516
517
518
519-521
522
523
524-525
526
527
528
529-531
532
533
534

K

535

L1

536

L2

537

L3

538

M1

Description
Comments
Energy release in fission for incident neutrons.
Cumulative fission product yield data.
(Unassigned)
Not allowed in Version 6.
Version 5: delayed
and prompt neutrons from spontaneous fission.
(Unassigned)
Total charged-particle stopping power.
Total photon interaction.
Photon coherent scattering.
(Unassigned)
Photon incoherent scattering.
Imaginary scattering factor.
Real scattering factor.
(Unassigned)
Pair production, electron field.
Pair production; sum of MT=515, 517.
Redundant.
Pair production, nuclear field.
Not allowed in Version 6.
(Unassigned)
Photoelectric absorption.
Version 5:
MT=602.
Photo-excitation cross section.
(Unassigned)
Electro-atomic scattering.
Electro-atomic bremsstrahlung.
Electro-atomic excitation cross section.
(Unassigned)
Not allowed in Version 6.
Version 5: (γ,n).
Atomic relaxation data.
Version 5:
total
photonuclear
(1s1/2) subshell photoelectric or electroatomic cross section.
(2s1/2) subshell photoelectric or elctroatomic cross section.
(2p1/2) subshell photoelectric or elctroatomic cross section.
(2p3/2) subshell photoelectric or elctroatomic cross section.
(3s1/2) subshell photoelectric or elctroatomic cross section.

309

APPENDIX B. DEFINITION OF REACTION TYPES
MT
539

M2

540

M3

541

M4

542

M5

543

N1

544

N2

545

N3

546

N4

547

N5

548

N6

549

N7

550

O1

551

O2

552

O3

553

O4

554

O5

555

O6

556

O7

557

O8

558

O9

559

P1

560

P2

Description
(3p1/2) subshell photoelectric or
atomic cross section.
(3p3/2) subshell photoelectric or
atomic cross section.
(3d3/2) subshell photoelectric or
atomic cross section.
(3d5/2) subshell photoelectric or
atomic cross section.
(4s1/2) subshell photoelectric or
atomic cross section.
(4p1/2) subshell photoelectric or
atomic cross section.
(4p3/2) subshell photoelectric or
atomic cross section.
(4dp3/2) subshell photoelectric or
atomic cross section.
(4d5/2) subshell photoelectric or
atomic cross section.
(4f5/2) subshell photoelectric or
atomic cross section.
(4f7/2) subshell photoelectric or
atomic cross section.
(5s1/2) subshell photoelectric or
atomic cross section.
(5p1/2) subshell photoelectric or
atomic cross section.
(5p3/2) subshell photoelectric or
atomic cross section.
(5d3/2) subshell photoelectric or
atomic cross section.
(5d5/2) subshell photoelectric or
atomic cross section.
(5f5/2) subshell photoelectric or
atomic cross section.
(5f7/2) subshell photoelectric or
atomic cross section.
(5g7/2) subshell photoelectric or
atomic cross section.
(5g9/2) subshell photoelectric or
atomic cross section.
(6s1/2) subshell photoelectric or
atomic cross section.
(6p1/2) subshell photoelectric or
atomic cross section.
310

Comments
elctroelctroelctroelctroelctroelctroelctroelctroelctroelctroelctroelctroelctroelctroelctroelctroelctroelctroelctroelctroelctroelctro-

APPENDIX B. DEFINITION OF REACTION TYPES
MT
561

P3

562

P4

563

P5

564

P6

565

P7

566

P8

567

P9

568

P10

569

P11

570

Q1

571

Q2

572

Q3

573-599
600

(z,p0 )

601

(z,p1 )

602

(z,p2 )

603

(z,p3 )

604

(z,p4 )

649

...
(z,pc )

650

(z,d0 )

Description
Comments
(6p3/2) subshell photoelectric or elctroatomic cross section.
(6d3/2) subshell photoelectric or elctroatomic cross section.
(6d5/2) subshell photoelectric or elctroatomic cross section.
(6f5/2) subshell photoelectric or elctroatomic cross section.
(6f7/2) subshell photoelectric or elctroatomic cross section.
(6g7/2) subshell photoelectric or elctroatomic cross section.
(6g9/2) subshell photoelectric or elctroatomic cross section.
(6h9/2) subshell photoelectric or elctroatomic cross section.
(6h11/2) subshell photoelectric or elctroatomic cross section.
(7s1/2) subshell photoelectric or elctroatomic cross section.
(7p1/2) subshell photoelectric or elctroatomic cross section.
(7p3/2) subshell photoelectric or elctroatomic cross section.
(Unassigned)
Production of a proton leaving the residual Not allowed for innucleus in the ground state.
cident protons; use
MT=2.
Production of a proton, with residual in
the 1st excited state.
Production of a proton, with residual in Version 5: photothe 2nd excited state.
electric absorption;
see MT=522.
Production of a proton, with residual in
the 3rd excited state.
Production of a proton, with residual in
the 4th excited state.
Production of a proton in the continuum
not included in the above discrete representation.
Production of a deuteron leaving the residual nucleus in the ground state.

311

APPENDIX B. DEFINITION OF REACTION TYPES
MT
651

(z,d1 )

652

(z,d2 )

699

...
(z,dc )

700

(z,t0 )

701

(z,t1 )

702

(z,t2 )

749

...
(z,tc )

750

(n,3 He0 )

751

(n,3 He1 )

799

...
(n,3 Hec )

800

(z,α0 )

801

(z,α1 )

849

...
(z,αc )

850
851-870
871-874
875

(z,2n0 )

876

(z,2n1 )

Description
Comments
Production of a deuteron, with the residual
in the 1st excited state.
Production of a deuteron, with the residual
in the 2nd excited state.
Production of a deuteron in the continuum
not included in the above discrete representation.
Production of a triton leaving the residual
nucleus in the ground state.
Production of a triton, with residual in the
1st excited state.
Production of a triton, with residual in the
2nd excited state.
Production of a triton in the continuum
not included in the above discrete representation.
Production of a 3 He particle leaving the
residual nucleus in the ground state.
Production of a 3 He, with residual in the
1st excited state.
Production of a 3 He in the continuum not
included in the above discrete representation.
Production of an alpha particle leaving the
residual nucleus in the ground state.
Production of an alpha particle, with residual in the 1st excited state.
Production of an alpha particle in the continuum not included in the above discrete
representation.
(Unassigned)
Lumped reaction covariances.
(Unassigned)
Production of 2 neutrons with residual in
the ground state.
Production of 2 neutrons with residual in
the 1st excited state.

...

312

APPENDIX B. DEFINITION OF REACTION TYPES
MT
891

(z,2nc )

892-999

B.2

Description
Comments
Production of 2 neutrons in the continuum
not included in the above discrete representation.
(Unassigned)

Residual Breakup Flags LR

Many reactions are sequential in nature. That is, a particle or gamma ray may be emitted
first, then the residual nucleus decays by one or more paths. Most often, the first stage of
the reaction proceeds through a well-defined discrete state of the residual nucleus and the
angular dependence of the first emitted particle must be uniquely described. A simple, twobody reaction is one in which the incident particle is inelastically scattered from the target
nucleus leaving the target in an excited state, which immediately decays by gamma emission.
Other excited states of the same target may, however, decay by particle emission, electronpositron pair formation, or internal conversion. It is often necessary to completely specify
the reaction mechanism, in particular for isotopic depletion and/or build-up calculations.
The following numbers can be used as flags to indicate the mode of decay of the residual
nucleus.
LR
0 or blank
1
22
23
24
25
28
29
30
31
32
33
34
35
36
39
40

Description
Simple reaction. Identity of product is implicit in MT. Only
gamma rays may be emitted additionally.
Complex or breakup reaction. The identity and multiplicity of
all products are given explicitly in File 6.
α emitted (plus residual, if any).
3α emitted (plus residual, if any).
nα emitted (plus residual, if any).
2nα emitted (plus residual, if any).
p emitted (plus residual, if any).
2α emitted (plus residual, if any).
n2α emitted (plus residual, if any).
Residual nucleus decays only by gamma emission.
d emitted (plus residual, if any).
t emitted (plus residual, if any).
3
He emitted (plus residual, if any).
d2α emitted (plus residual, if any).
t2α emitted (plus residual, if any).
Internal conversion.
Electron-positron pair formation.

313

APPENDIX B. DEFINITION OF REACTION TYPES
Examples
1.
2.
3.

T(d,γ)5 He∗ (16.39 MeV) MT=102 LR=24 (5 He decays via n+α)
Li(n,n’)7 Li∗ (0.48 MeV) MT=51 LR=31 (Residual decays by γ emission)
7
Li(n,n’)7 Li∗ (4.63 MeV) MT=52 LR=33 (7 Li∗ decays via t+α).

7

B.3

Summary

Version ENDF-6 formats and procedures are recommended for all new evaluations; this
is the only format allowed for incident charged particles. It must be taken into account,
however, that many old ENDF/B-V materials for incident neutrons will be carried over
without technical changes to the data.
A few files and several MT numbers are defined for the first time for ENDF-6. A few MT
numbers allowed for format Version 5 have now been removed and must be replaced. Other
MT numbers are allowed only in ENDF/B-V and these are not defined here - the reader
is referred to the ENDF/B-V format manual. Many MT numbers above the 600 series are
redefined for 6 and all Version 5 materials must be changed accordingly prior to reissue.
A few of the MT numbers are not defined for certain particles incident: for example,
MT=1 is not defined for incident charged particles: MT=50 is not defined for incident
neutrons: MT=600 is not defined for incident protons; etc. These exceptions are labeled
but should be obvious if one follows the explicit definitions closely.
Several MT numbers cannot be used with File 4 or 5; other MT numbers must have a
File 6 (File 4 and 5 are not allowed). The changes between previous format manuals are
significant, therefore, much effort has been expended to explicitly define the MT numbers
for ENDF-6 and, hopefully to associate them with the proper files. For explicit information
on usage, see Sections 0, 3.4, and 3.5.

314

Appendix C
ZA Designations of Materials and
MAT Numbers
A floating-point number, ZA, is used to identify materials. If Z is the charge number and A
the mass number then ZA is computed from
ZA = (1000.0*Z) + A
For example, ZA for 238 U is 92238.0, and ZA for beryllium is 4009.0. For materials other
than isotopes, the following rules apply. The MAT number is 100*Z+I where I is unique for
the isotope and its isomer state.
1. If the material is an element that has more than one naturally occurring isotope, then
A is set to 0.0. For example, ZA for the element tungsten is 74000.0. The MAT
number is 100*Z.
2. For compounds, the ZA is arbitrary and is calculated from ZA=MAT+100.
The MAT number assignments for compounds have the following structure.
Hydrogen (except organics)
Deuterium
Lithium
Beryllium
Carbon (including organics)
Oxygen
Metals
Fuels

1-10
11-20
21-25
26-30
31-44
45-50
51-70
71-99

315

APPENDIX C. ZA DESIGNATIONS OF MATERIALS AND MAT NUMBERS
The presently recognized assignments are:
Compound
Water
Para Hydrogen
Ortho Hydrogen
H in ZrH
Heavy Water
Para Deuterium
Ortho Deuterium
Be
BeO
Be2 C
Be in BeO
Graphite
Liquid Methane
Solid methane
Polyethylene
Benzene
O in BeO
Zr in ZrH
UO2
UC

MAT Number
1
2
3
7
11
12
13
26
27
28
29
31
33
34
37
40
46
58
75
76

316

Appendix D
Resonance Region Formulae
D.1

The Resolved Resonance Region (LRU=1)

The following resonance formalisms are given for a particular isotope in the laboratory
system, without Doppler broadening.

D.1.1

Single-Level Breit-Wigner (SLBW, LRF=1)

D.1.1.1

Elastic Scattering Cross Sections

Processing codes should sum the cross section, as shown below from l = 0 to l =NLS-1,
including any ”empty” or ”non-resonant” channels, in order to get the potential-scattering
contribution. If higher l-values contribute to the scattering in the resonance region, it is the
responsibility of the evaluator to provide a suitable File 3 contribution. (See Sections 2.4.19
and 2.4.20.)
σn,n (E) =

NLS−1
X

l
σn,n
(E),

(D.1)

l=0

where

4π
sin2 φl
k2
NR
π X XJ Γ2nr − 2Γnr Γr sin2 φl + 2(E − Er′ ) Γnr sin(2φl )
+ 2
gJ
k J
(E − Er′ )2 + 41 Γ2r
r=1

l
σn,n
(E) = (2l + 1)

(D.2)

The hard-sphere phase shifts φl , the wave number k, the primed resonance energy Er′ , the
neutron width Γnr , and through it the total width Γr , are all functions of energy, φl (E), k(E),
Er′ (E), Γnr (E), and Γr (E), but this dependence is not shown explicitly. Also, each resonance
parameter carries the implicit quantum numbers l and J, determined by the appropriate
entries in the ENDF file. In case a given pair (l, J) is compatible with two different values
of the channel spin, s, the width is a sum over the two partial channel spin widths. This
allows one to omit an explicit sum over channel spin when defining the cross sections.
317

APPENDIX D. RESONANCE REGION FORMULAE
D.1.1.2

Radiative Capture Cross Section
σn,γ (E) =

NLS−1
X

l
σn,γ
(E)

(D.3)

l=0

where:
l
σn,γ
(E)

NR
Γnr Γγr
π X XJ
gJ
= 2
k J
(E − Er′ )2 + 14 Γ2r
r=1

(D.4)

and Γγr , the radiative capture width, is constant in energy.
D.1.1.3

Fission Cross Section
σn,f (E) =

NLS−1
X

l
σn,f
(E) ,

(D.5)

l=0

where
l
(E)
σn,f

NR
XJ
π X
Γnr Γf r
= 2
gJ
k J
(E − Er′ )2 + 14 Γ2r
r=1

(D.6)

and Γf r , the fission width, is constant in energy.
D.1.1.4

The Competitive Reaction Cross Section

The competitive reaction cross section, σn,x (E), is given in terms of analogous formulas
involving Γxr , the competitive width. By convention, the cross section for the competitive
reaction is given entirely in File 3, and is not to be computed from the resonance parameters.
The reason for this is that the latter calculation can be done correctly only for a single
competitive channel, since the file can define only one competitive width.
The statistical factor gJ = (2J + 1)/[2(2I + 1)] is obtained from the target spin I and
the resonance spin J given in File 2 as SPI and |AJ|, respectively.
The sum on l extends over all l-values for which resonance parameters are supplied. There
will be NLS terms in the sum. NLS is given in File 2 for each isotope. In general, ENDF
resonance files are limited to l=0, 1, and 2, so that the potential-scattering contribution
will be represented by hard-sphere scattering up to the energy where f -wave (l=3) potential
scattering starts. At that point, the evaluator may have to supply File 3 scattering to
simulate the higher l-values. He or she may also require a File 3 contribution at lower
energies to represent any differences between hard-sphere scattering and experiment.
The sum on J extends over all possible J-values for a particular l-value. NRJ is the
number of resonances for a given pair of l and J values and may be zero. NRS is the total

318

APPENDIX D. RESONANCE REGION FORMULAE
number of resonances for a given l-value and is given in File 2 for each l-value.
NRS =

J=JMAX
X

NRJ , where

J=JMIN

1
2
1
l−I −
2
1
I −l−
2

JMAX = l + I +
JMIN =
=
=

|I − l| −

and
if l ≥ I
if I ≥ l
1
2

in both cases

Γnr (|Er |) ≡ GNr is the neutron width, for the rth resonance for a particular value of l
and J, evaluated at the resonance energy Er . For bound levels, the absolute value |Er | is
used.
Pl (E) Γnr (|Er |)
Γnr =
(D.7)
Pl (|Er |)

Γr = Γnr (E) + Γγr + Γf r + Γxr is the total width, a function of energy through Γnr and Γxr ,
since Γγr and Γf r are constant with respect to energy. The ”competitive” width, Γxr , is not
entered explicitly in File 2. It is calculated from the equation:
Γxr = Γr − Γnr − Γγr − Γf r

at

E = Er

(D.8)

The following quantities are given in File 2 for each resonance:
Er = ER, the resonance energy
J = |AJ|, the angular momentum (”spin”) of the resonance state
I = SPI, the angular momentum (”spin”) of the target nucleus
gJ = statistical factor (2J + 1)/[2(2I + 1)]
Γnr (|Er |) = GN, the neutron width
Γγr = GG, the radiation width
Γf r = GF, the fission width and
Γr (|Er |) = GT, the total width evaluated at the resonance energy.
Since the competitive width Γxr , is not given, Γr should be obtained from File 2 directly,
and not by summing partial widths.

319

APPENDIX D. RESONANCE REGION FORMULAE
For p-, d- and higher l-values, the primed resonance energy Er′ is energy-dependent:
Er′ = Er +

Sl (|Er |) − Sl (|E|)
Γnr (|Er |) .
2Pl (|Er |)

(D.9)

The fact that the resonance energy shift is zero at each Er is an artifact of the SLBW
formalism, and implies a different R-matrix boundary condition for each resonance.
The neutron wave number in the center-of-mass system is given as:
√
2mn AWRI p
k=
|E| ,
(D.10)
h̄ AWRI + 1.0
where

AWRI= ratio of the mass of a particular isotope to that of the neutron.
E= laboratory energy in eV.
The energy is written with absolute value signs so that the same formula can be used for
positive incident neutron energies and for negative (bound state) resonance energies. (When
inelastic scattering can occur, resonances below the level threshold are at ”negative energy”
in the inelastic channel.)
The shift factor Sl is defined as:
S0 = 0,
S1
S2
S3

(D.11)

1
= −
,
1 + ρ2
18 + 3ρ2
= −
,
9 + 3ρ2 + ρ4
675 + 90ρ2 + 6ρ4
;
= −
225 + 45ρ2 + 6ρ4 + ρ6

(the quantity ρ is defined below).
For higher l-values, Sl is defined by Equation (2.9) in Reference 1. In conventional
R-matrix theory, the shift factors are defined differently for negative energies (Reference 1,
Equations 2.11 a-c). In ENDF, the positive-energy formulas are used, but the absolute value
of E is used in SLBW and MLBW. For the R-Matrix Limited format, Section D.1.7, a flag
indicates whether shifts are to be calculated or assumed to be zero for each particle-pair.
The penetration factor Pl is defined:
P0 = ρ,

(D.12)
3

ρ
,
1 + ρ2
ρ5
=
,
9 + 3ρ2 + ρ4
ρ7
;
=
225 + 45ρ2 + 6ρ4 + ρ6

P1 =
P2
P3

320

APPENDIX D. RESONANCE REGION FORMULAE
For higher l-values, the expressions for Pl are defined by Equation (2.9) in Reference 1. In
conventional R-matrix theory, the penetrabilities are zero for negative energies. The theory
uses the ”theoretical” definition of a reduced width, Γ(E) = 2Pl (E)γ 2 , where E is a channel
energy (center-of-mass), and it suffices to say that Pl (E) = 0 if E < 0.
In ENDF, the ”experimental” definition is used, Γ(E) = Γ (|Er |) Pl (E) / Pl (|Er |) , and
it is necessary to make the convention that a penetrability for a negative resonance energy
is evaluated at its absolute value. A negative kinetic energy can occur in an exit channel if
the reaction is exothermic, and in this case Pl (E < 0) is zero.
The φl is the (negative of the) hardsphere phase shift and is given by:
φ0 = ρb ,
φ1 = ρb − tan−1 ρb ,


3b
ρ
−1
φ2 = ρb − tan
,
3 − ρb2
(
)
2
ρ
b
(15
−
ρ
b
)
φ3 = ρb − tan−1
.
15 − 6b
ρ2

(D.13)

For higher l-values, the φl are defined by Equation 2.12 in Reference 1. It is not necessary
to evaluate a phase shift at negative energies.
Parameters ρ and ρb are defined as k× radius, where radius is defined as follows:
a = channel radius

1

in units of 10−12 cm

a = 0.123 × AWRI1/3 + 0.08

(D.14)

AP = energy-independent scattering radius, which determines the low-energy scattering cross section. It is given in File 2 following SPI.
AP(E) = energy-dependent scattering radius, given as a TAB1 card preceding the ”SPI
AP....NLS...” card.

If

NRO =0
NAPS =0
NAPS =1

If

NRO
NAPS
NAPS
NAPS
1

=1
=0
=1
=2

(AP energy-independent)
ρ = k a; ρb = k AP
ρ = ρb = k AP

(AP energy-dependent)
ρ = k a; ρb = k AP(E)
ρ = ρb = k AP(E)
ρ = k AP; ρb = k AP(E).

The channel radius, strictly speaking, involves A1/3 (the target mass in amu), and not (AWRI)1/3 , but
as long as the mass of the incident particle is approximately unity, as it is for neutrons, the difference is not
important. AWRI= A/mn , where mn is the neutron mass (see Appendix H).

321

APPENDIX D. RESONANCE REGION FORMULAE

D.1.2

Multilevel Breit-Wigner (MLBW, LRF=2)

The equations are the same as SLBW, except that a resonance-resonance interference term
l
is included in the equation for elastic scattering of l-wave neutrons, σn,n
(E):
l
σn,n
(E)



NR r−1
2Γnr Γns (E − Er′ )(E − Es′ ) + 14 Γr Γs
π X XJ X
gJ
.
= 2
′ )2 + (Γ /2)2 ] [(E − E ′ )2 + (Γ /2)2 ]
k J
[(E
−
E
s
r
s
r
r=2 s=1

(D.15)

This form, which has ≈ NR2J energy-dependent terms and can involve a great deal of computer time, may be written in the following form with only NRJ terms: (See Section 2.4.12):
l
σn,n
(E)

NR
π X XJ Gr Γr + 2Hr (E − Er′ )
= 2
gJ
k J
(E − Er′ )2 + (Γr /2)2
r=1

(D.16)

where
Gr

Hr

NR
Γnr Γns (Γr + Γs )
1 XJ
,
=
′
2 s=1,s6=r (Er − Es′ )2 + 41 (Γr + Γs )2

(D.17)

NR
XJ

Γnr Γns (Er′ − Es′ )
=
(Er′ − Es′ )2 + 41 (Γr + Γs )2
s=1,s6=r

(D.18)

For the user who does not require ψ- and χ-broadening, the amplitude-squared form of
the equations, which are mathematically identical to the conventional MLBW equations,
require less computing time
σn,n (E) =

NLS−1
X

l
σn,n
(E)

(D.19)

l=0

l
(E)
σn,n

π
= 2
k

1

I+ 2
X

s=|

I− 12

"

|

lsJ
Unn
(E) = e−2iφl 1 +

l+s
X

J=|l−s|
NR
sJ
X

lsJ
(E)
gJ 1 − Unn

Er′
r=1

ΓlsJ
nr

i
− E − i Γr /2

#

2

(D.20)

(D.21)

Caution: The use of this formalism is NOT recommended. Computation time is no longer a
valid issue for modern computers. All new evaluations should be done with the Reich-Moore
formalism (LRF=3 format for very simple cases and LRF=7 for the rest).
It is important to note that the ENDF version of the multilevel Breit-Wigner formalism
does not correspond exactly to the full multilevel formalism because it allows multilevel
computations only for elastic; for other types of partial cross sections the formalism is
single-level.

322

APPENDIX D. RESONANCE REGION FORMULAE

D.1.3

Reich-Moore (R-M, LRF=3)

Evaluators should be aware that the full Reich-Moore formalism is far more general than its
ENDF implementation under LRF=3, which is a severely limited subset of the Reich-Moore
capabilities. For a more general treatment of the Reich-Moore formalism see the section on
R-Matrix Limited format (LRF=7).
This description of the ENDF Reich-Moore formalism differs from previous versions
by using notation in closer agreement with References 1 and 2. The dependence of all
quantities on channel spin has been made explicit, to support a format extension which
permits specifying the individual channel-spin components of the neutron width.
Partial cross sections may be obtained from a collision matrix Uab , which connects entrance channels a with exit channels b. In ENDF, the formalism is applied to neutron
reactions, a = n:
π
σnb = 2 gn |δnb − Unb |2
(D.22)
k
These partial cross sections are not observable, but must be summed over the appropriate
entrance and exit channels to yield observable cross sections. The statistical factor gn is a
result of prior averaging over channels with different magnetic sub-states, since the ENDF
formulae apply to unpolarized particles.
In the Reich-Moore formalism as implemented in LRF=3, the only reactions requiring
explicit channel definitions are elastic scattering and fission; capture is obtained by subtraction (although it is possible to obtain it directly from the collision matrix elements).
Neutron channels are labeled by three quantum numbers, l, s, and J. In the ENDF format,
l runs from zero to NLS-1, the highest l-value that contributes to the cross section in the
energy range of interest. The channel spin s is the vector sum of the target spin I and the
neutron spin i (= 1/2), and takes on the range of values |I − 1/2| to I + 1/2. The total
angular momentum J is the vector sum of l and s, and runs from |l − s| to l + s. The
fission channels do not correspond to individual two-body fission product breakup, but to
Bohr-channels in deformation space, which is why two are adequate for describing many
neutron-induced fission cross sections. It is not necessary to specify the quantum numbers
associated with the two ”ENDF-allowed” fission channels, and they can simply be labeled
f1 and f2.
If one sums over all incident channels n and exit channels b, and invokes unitarity, the
resulting total cross section can be expressed in terms of the diagonal matrix elements as:
1

NLS I+
2π X X2
σT (E) = 2
k l=0
s=|I− 21 |

l+s
X

J=|l−s|

gJ Re [1 − UlsJ,lsJ ]

(D.23)

The elastic cross section is obtained by summing the incident neutron channels over all
possible lsJ values and the exit neutron channels over those quantities l′ s′ J ′ that have the
same ranges as lsJ. Conservation of total angular momentum requires that J ′ = J; the
ENDF format LRF=3 imposes additional ”conservation rules”, namely l′ = l and s′ = s
which are actually just simplifying assumptions, with some basis in theory and experiment
(these assumptions are not required in LRF=7, for example). The six-fold summation then
323

APPENDIX D. RESONANCE REGION FORMULAE
reduces to the familiar form:
1

NLS I+
2π X X2
σnn (E) = 2
k l=0
s=|I− 21 |

l+s
X

J=|l−s|

gJ |1 − UlsJ,lsJ |2 .

(D.24)

The absorption (non-elastic) cross section is obtained by subtraction:
σabs (E) = σT (E) − σnn (E).

(D.25)

Fission is obtained from the collision matrix by summing equation (D.22) over all incident
lsJ values and over the two exit fission channels, b=f1 and b=f2,
1

NLS I+
2π X X2
σf (E) = 2
k l=0
s=|I− 21 |

l+s
X

J=|l−s|

gJ

h

2

lsJ
lsJ
Unf
1 + Unf 2

2

i

.

(D.26)

The Reich-Moore formalism is described in Reference 3. Here we repeat the level-matrix
form of the collision matrix as given in the earlier versions of this manual:
 

J
Unb
= e−i(φn +φb ) 2 (I − K)−1 nb − δnb ,
(D.27)
where:

iX
Γnr Γbr
= δnb −
.
2 r Er − E − i Γγr /2
1/2 1/2

(I − K)nb

(D.28)

Here φb is zero for fission, φn = φl (defined previously), and the summation is over those
resonances r which have partial widths in both of the channels n and b; Er is the resonance
energy; Γγr is the ”eliminated” radiation width; Γnr and Γbr are the partial widths for the
rth resonance in channels n and b.
Caution: While the following equations are correct, they can lead to serious numerical
problems if programmed in this form. For a computationally more stable form the reader is
referred to Section II of the SAMMY manual [Reference 4].
If we define a matrix ρ by the equation


ρnb = δnb − (1 − K)−1 nb
(D.29)
then the various cross sections take the following forms:
Total:
σT (E) =
Elastic:
σnn =




2π X
−2i φl
.
g
(1
−
cos
2φ
)
+
2Re
ρ
e
J
l
nn
k 2 lsJ



π X
gJ 2 − 2 cos 2φl + 4Re(ρnn e−2iφl ) − 4Re(ρnn ) + 4|ρnn |2 .
2
k lsJ
324

(D.30)

(D.31)

APPENDIX D. RESONANCE REGION FORMULAE
Absorption (fission plus capture):
σn,abs (E) =
Fission:
σnf (E) =


4π X
2
g
Re(ρ
)
−
|ρ
|
.
J
nn
nn
k 2 lsJ

(D.32)


4π X
2
2
g
|ρ
|
+
|ρ
|
.
J
nf
1
nf
2
k 2 lsJ

(D.33)

The phase shifts and penetrabilities are evaluated in terms of a and AP as described
earlier. The shift factor has been set equal to zero in the above equations (Er′ → Er );
hence they are strictly correct only for s-wave resonances. Originally, the ENDF ReichMoore format was used for low-energy resonances in fissile materials, which are s-waves.
However, it is believed that the ”no-shift” formulae can be safely applied to higher l-values
also, since the difference in shape between a shifted resonance and one that is not shifted at
the same energy has no practical significance. Evaluations using the correct shift factor can
be reported under the newer LRF=7 format option.
The comments in Section D.1.1 about the summation over channels applies to the ReichMoore formalism also. Until the formats revision approved by CSEWG in 1999, the format
did not permit the specification of channel spin; therefore, if an evaluation includes l > 0
resonances for I > 0 nucleus, it was necessary for the processing codes to include the
potential-scattering contributions from the ”missing” channels. (It is adequate to arbitrarily
assume that the supplied values are for the s = I − 1/2 channels, and to use the same
potential-scattering radius in the missing I + 1/2 channels. See Sections 2.4.19 and 2.4.20.)
Having the ability to specify which channel spin does not solve this problem, unless the
evaluator actually supplies resonances for both channels. In cases where the data can be fit
with all the resonances in the same s-channel, the ”other one” will still be absent from the
ENDF file, since the format stipulates nothing about avoiding missing channels. This is why
it is reasonable for the processing codes to run over the triple lsJ loop, inserting potential
scattering in every channel, and resonances whenever they are supplied.
Note: When both positive and negative AJ values are given in the file, negative AJ implies
s = I − 1/2 and positive AJ implies s = I + 1/2. When AJ=0, one and only one of I − 1/2
or I + 1/2 is possible, so the possible ambiguity of ±0 does not arise. In this case s = l;
parity conservation prevents the occurrence, for a given J, of two s-values differing by one
unit.

D.1.4

Adler-Adler (AA, LRF=4)

The formulae, taken from References 5 and 6, are given for the total, radiative capture, and
fission cross sections. They have been slightly recast to make them conform to the definitions
used earlier in this Appendix. Furthermore, only the l = 0 terms are given, consistent with
current usage of this formalism. Procedures are discussed in Section 2.4.13. Since only
s-waves are considered, higher l-wave contributions to the potential scattering must be put
into File 3 by the evaluator.

325

APPENDIX D. RESONANCE REGION FORMULAE
Total Cross Section:
4π
sin2 φ0
(D.34)
2
k
"




√
NRS
π E X νr GTr cos 2φ0 + HrT sin 2φ0 + (µr − E) HrT cos 2φ0 − GTr sin 2φ0
+
k2
(µr − E)2 + νr2
r=1

AT2 AT3 AT4
2
+ 2 + 3 + BT1 E + BT2 E .
+ AT1 +
E
E
E

σT (E) =

Radiative Capture Cross Section:
√ "NRS
π E X νr Gγr + (µr − E)Hrγ
σn,γ (E) =
k2
(µr − E)2 + νr2
r=1

(D.35)


AC2 AC3 AC4
2
+ 2 + 3 + BC1 E + BC2 E .
+ AC1 +
E
E
E

Fission Cross Section:
√ "NRS
π E X νr Gfr + (µr − E)Hrf
σn,f (E) =
k2
(µr − E)2 + νr2
r=1

(D.36)


AF2 AF3 AF4
2
+ AF1 +
+ 2 + 3 + BF1 E + BF2 E .
E
E
E

Although the format uses different names for µ and ν for each reaction, they are actually
equal:
DETr
DWTr

D.1.5

= DEFr = DECr = µr
= DWFr = DWCr = νr

General R-Matrix (GRM, LRF=5)

The format is no longer available in ENDF-6.

D.1.6

Hybrid R-Function (HRF, LRF=6)

The format is no longer available in ENDF-6.

D.1.7

R-Matrix Limited Format (RML, LRF=7)

In R-Matrix theory, a channel may be defined by c = (α, l, s, J), where
α represents the two particles making up channel; α includes mass (ma and mb with
subscript a indicating the incident particle for an entrance channel), charge (Za and
Zb ), spin (ia and ib ) and parity (πa and πb ) and all other quantum numbers for each
of the two particles, plus the Q-value.
326

APPENDIX D. RESONANCE REGION FORMULAE
l is the orbital angular momentum; the associated parity is (−1)l .
s represents the channel spin (including the associated parity); that is, s is the vector
sum of the spins of the two particles of the pair.
J is the total angular momentum (and associated parity); J is the vector sum of l and
s.
Only J and its associated parity are conserved for any given interaction. The other quantum
numbers may differ from channel to channel, so long as the sum rules for spin and parity
are obeyed.
In the Reich-Moore approximation to R-matrix theory, the radiation width is treated
separately and differently from widths for other channels (which are hereafter referred to
as ”particle channels”). In this LRF=7 format, there is assumed to be an ”eliminated
channel”, which, for the strict interpretation of the Reich-Moore approximation, contains
all the radiation width; in this format, it is possible for some portion of the radiation width to
be treated in the same fashion as the particle widths. In the equations below, the eliminated
width appears only in the denominator of the R-matrix.
In all formulae given below, spin quantum numbers (e.g., J) are implicitly assumed
to include the associated parity. Vector sum rules are implicitly assumed to be obeyed;
readers unfamiliar with these sum rules are referred to the paragraph on Spin and Angular
Momentum Conventions for details.
Let the angle-integrated cross sections from entrance channel c to exit channel c′ with
total angular momentum J be represented by σcc′ . This cross section is given in terms of
the scattering matrix Ucc′ as:
σcc′ =

π
gJα eiwc δcc′ − Ucc′
2
ka

2

δJJ ′

(D.37)

where
kα is the center-of mass momentum associated with incident particle-pair α,
gJα is the spin statistical factor,
wc is zero for non-Coulomb channels. (Details for the charged-particle case are
presented later.)
The spin statistical factor is given by:
gJα =

2J + 1
,
(2ia + 1)(2ib + 1)

(D.38)

2ma m2b
E.
(ma + mb )2

(D.39)

and center-of mass momentum kα by:
kα2 =

327

APPENDIX D. RESONANCE REGION FORMULAE
The scattering matrix U can be written in terms of the matrix W as
Ucc′ = Ωc Wcc′ Ωc′ ,

(D.40)

Ωc = ei(wc −φc ) .

(D.41)

where Ω is given by:
Here again, wc is zero for non-Coulomb channels, and the potential scattering phase shifts
for non-Coulomb interactions φc are defined in many references (e.g., Reference 1). The
matrix W in equation (D.40) is related to the R-matrix (in matrix notation with indices
suppressed) via:
W = P 1/2 (I − RL)−1 (I − RL∗ ) P −1/2 .
(D.42)
The quantity I in this equation represents the identity matrix. The quantity L in equation (D.42) is given by
L = (S − B) + iP
(D.43)

with P the penetration factor, S the shift factor, and B the arbitrary boundary constant at
the channel radius ac . Formulae for P and S are likewise found in many references (see, e.g.,
equation (2.9) in Reference 1); for non-Coulomb interactions see Table D.1 for the appropriate formulae. For fission, the penetrability is unity. For non-eliminated capture channels,
the penetrability is unity. For two charged particles, formulae for the penetrabilities are
provided in Section II.C.4 of the SAMMY Users’ Manual [Reference 4].
Table D.1: Hard sphere penetrability (penetration factor) P , level shift factor S, and potential scattering phase shift φ for orbital angular momentum l, center of mass momentum
k, and channel radius ac , with ρ = kac .
l
0
1
2
3
4

l

Pl
ρ
ρ3 /(1 + ρ2 )
ρ5 /(9 + 3ρ2 + ρ4 )
ρ7 /(225 + 45ρ2
+ 6ρ4 + ρ6 )
ρ9 /(11025 + 1575ρ2
+ 135ρ4 + 10ρ6 + ρ8

Sl
0
-1 / (1 + ρ2 )
−(18 + 3ρ2 )/(9 + 3ρ2 + ρ4 )
−(675+90ρ2 +6ρ4 )/(225+45ρ2 +6ρ4 +ρ6 )

ρ2 Pl−1
2
(l−Sl−1 )2 +Pl−1

ρ2 (l−Sl−1 )
2
(l−Sl−1 )2 +Pl−1

−(44100 + 4725ρ2 + 270ρ4 + 10ρ6 )/
(11025 + 1575ρ2 + 135ρ4 + 10ρ6 + ρ8 )

− l

φl
ρ
ρ − tan−1 ρ
ρ − tan−1 [3ρ/(3 − ρ2 )]
ρ − tan−1 [ρ(15 − ρ2 )/
(15 − 6ρ2 )]
ρ − tan−1 [ρ(105 − 10ρ2 )/
(105 − 45ρ2 + ρ4 )]

φl−1 − tan−1

or

Bl =

h

(Pl−1
(l−Sl−1 )

(Bl−1 +Xl )
(1−Bl−1 Xl )

i

with Bl = tan(ρ − φl )
P
and Xl = (l−Sl−1 )
l−1

In the eliminated-channel approximation, the R-matrix of equation (D.42) [for the spin
group defined by total spin J and implicit parity π] has the form:
#
"
X
′
γλc γλc
+ Rcbkg δcc′ δJJ ′
(D.44)
Rcc′ =
Eλ − E − iΓλγ /2
λ
328

APPENDIX D. RESONANCE REGION FORMULAE
where all levels (resonances) of that spin group are included in the sum. Subscripts λ designate the particular level; subscripts c and c′ designate channels (including particle-pairs
and all the relevant quantum numbers). Again, the width Γλγ occurring in the denominator corresponds to the ”eliminated” non-interfering capture channels of the Reich-Moore
approximation.
The ”background R-matrix” Rcbkg of equation (D.44) will be discussed in the paragraph
on Extensions to R-Matrix Theory, that follows.
The channel width Γλc is given in terms of the reduced width amplitude γλc by:
2
Γλc = 2 γλc
Pc (E)

(D.45)

where Pc is the penetrability, whose value is a function of the type of particles in the
channel, of the orbital angular momentum l, and of the energy E. Note that the reduced
width amplitude γλc is always independent of energy, but the width Γλc may depend on
energy via the penetration factor.
Cross sections may be calculated by using the above expressions for R and L to calculate
W , and from there calculating U and, ultimately, σ. However, while equation (D.42) for W
is correct, an equivalent form which is computationally more stable is:
W = I + 2 i X,

(D.46)

where X is given in matrix notation by:
X = P 1/2 L−1 L−1 − R


−1

R P 1/2 .

(D.47)

When the suppressed indices and implied summations are inserted, the expression of X
becomes
X

1/2
−1
−1
(L
−
R)
(D.48)
R ′′ ′ Pc′ δJJ ′ .
Xcc′ = Pc1/2 L−1
c
cc′′ c c
c′′

The various cross sections are then written in terms of X.

D.1.7.1

Energy-Differential (Angle-Integrated) Cross Sections (Non-Coulomb
Channels)

The observable cross sections are found in terms of X by first substituting equations (D.40,
D.41, and D.46) into equation (D.37), summing over spin groups (i.e., over J π ), and then
summing over all channels corresponding to those particle pairs and spin groups. If X r represents the real part and X i the imaginary part of X, then the angle-integrated (but energydifferential) cross section for the interaction which leads from particle-pair α to particle-pair
α′ has the form:
X
4π X
gJα
σα,α′ (E) = 2
kα J
c


+

 2

i
i
sin φc (1 − 2Xcc
) − Xcc
sin(2φc ) δα,α′
X
c′

329

i
Xcc
′


2

)
2 

r
+ Xcc
.
′

(D.49)

APPENDIX D. RESONANCE REGION FORMULAE
[This formula is accurate only for cases in which one of particles in α is a neutron; however,
both particles in α′ may be charged.]
In equation (D.49) the summations are over those channels c and c′ {of the spin group
defined by J π } for which the particle-pairs are respectively α and α′ . More than one ”incident
channel” c = (α, l, s, J) can contribute to this cross section, e.g., when both l = 0 and l = 2
are possible, or when, in the case of incident neutrons and non-zero spin target nuclei, both
channel spins are allowed. Similarly, there may be several ”exit channels” c′ = α′ , l′ , s′ , J ′ ),
depending on the particular reaction being calculated (elastic, inelastic, fission, etc.).
The total cross section (for non-Coulomb initial states) is the sum of equation (D.49)
over all possible final-state particle-pairs α′ , assuming the scattering matrix is unitary (that
is, assuming that the sum over c′ of |Ucc′ |2 = 1). Written in terms of the X matrix, the total
cross section has the form:

X 1
4π X
2
i
r
gJα
sin φc + Xcc cos(2 φc ) − Xcc sin(2φc ) ,
(D.50)
σα,total (E) = 2
kα J
2
c
where again the sum over c includes only those channels of the J π spin group for which the
particle-pair is α.
The angle integrated elastic cross section is given by
X 
4π X
i
r
σα,α (E) = 2
gJα
sin2 φc (1 − 2Xcc
) − Xcc
sin(2φc )
kα J
c
)
2 

X


2
r
i
+ Xcc
.
(D.51)
Xcc
+
′
′
c′

In this case, both c and c′ are limited to those channels of the J π spin group for which the
particle-pair is α; again, there may be more than one such channel for a given spin group.
Similarly, the reaction cross section from particle-pair α to particle-pair α′ (where α′ is
not equal to α) is

XX

2  r  2
4π X
i
(D.52)
Xcc′ + Xcc′
gJα
σα,α′ (E) = 2
kα J
c
c′

Here c is restricted to those channels of the J π spin group from which the particle-pair is α,
and c′ to those channels for which the particle-pair is α′ .
The absorption cross section has the form
(
)



X
X
X
2


4π
2
r
i
i
σα,absorption (E) = 2
gJα
+ Xcc
.
(D.53)
Xcc
Xcc
−
′
′
kα J
c
c′

Here both the sum over c and the sum over c′ include all incident particle channels (i.e.,
particle-pair α only) for the J π spin group.
The capture cross section for the eliminated radiation channels can be calculated directly
as:
(
)



X
X
X
2


4π
2
r
i
i
gJα
+ Xcc
Xcc
,
(D.54)
Xcc
−
σαγ (E) = 2
′
′
kα J
inc c
all c′
330

APPENDIX D. RESONANCE REGION FORMULAE
or may be found by subtracting the sum of all reaction cross sections from the absorption
cross section. In equation (D.54), the sum over c includes all incident particle channels for
the J π spin group, and the sum over c′ includes all particle channels, both incident and exit,
for that spin group.
D.1.7.2

Angular Distributions

Angular distributions (elastic, inelastic, or other reaction) cross sections for incident neutrons
can be calculated from Reich-Moore resonance parameters. Following Blatt and Biedenharn
[Reference 7] with some notational changes, the angular distribution cross section in the
center-of-mass system may be written
X
dσalphaα′
=
CLαα′ (E) PL (cos β)
dΩCM
L

(D.55)

in which the subscript αα′ indicates which type of cross section is being considered, PL is
the Legendre polynomial of degree L, and β is the angle of the outgoing neutron (or other
particle) relative to the incoming neutron in the center-of-mass system. The coefficients
CLαα′ (E) are given by
CLαα′ (E) =

1 X X
4kα2 J
J
1

2

X

c1 =(αl1 s1 J1 )

× B{l1 s1 l1′ s′1 ,J1 }{l2 s2 l2′ s′2 ,J2 }L

X

c′1 =(α′ l1′ s′1 J1 )

X

c2 =(αl2 s2 J2 )

X

(D.56)

c′2 =(α′ l2′ s′2 J2 )

i
h
1
Re (δc1 c′1 − Uc1 c′1 )(δc2 c′2 − Uc∗2 c′2 )
(2ia + 1)(2ib + 1)

where the various summations are to be interpreted as follows:

J1 sum over all spin groups defined by spin J1 and the implicit associated parity.
J2 sum over all spin groups defined by spin J2 and the implicit associated parity.
c1 sum over all those channels c1 belonging to the J1 spin group and having particle-pair
α [c1 = α, l1 , s1 , J1 ].
c′1 sum over those channels c′1 in J1′ spin group with particle-pair α′ [c′1 = (α′ , l1′ , s′1 , J1 )].
c2 sum over those channels c2 in J2 spin group with particle-pair α [c2 = (α, l2 , s2 , J2 )].
c′2 sum over those channels c′2 in J2 spin group with particle-pair α′ [c′2 = (α′ , l2′ , s′2 , J2 )].
Also note that ia and ib are spins of the two particles in particle-pair α.
The geometric factor B can be exactly evaluated as a product of terms:
B{l1 s1 l1′ s′1 ,J1 }{l2 s2 l2′ s′2 ,J2 }L = Al1 s1 l1′ s′1 ;J1 Al2 s2 l2′ s′2 ;J2 Dl1 s1 l1′ s′1 l2 s2 l2′ s′2 ;LJ1 J2 ,
where the factor Al1 s1 l1′ s′1 ;J1 is of the form:
p
Al1 s1 l1′ s′1 ;J1 = (2l1 + 1)(2l1′ + 1) (2J1 + 1) ∆(l1 J1 s1 ) ∆(l1′ J1 s′1 )
331

(D.57)

(D.58)

APPENDIX D. RESONANCE REGION FORMULAE
and similarly for Al2 s2 l2′ s′2 ;J2 . The expression for D is:
Dl1 s1 l1′ s′1 l2 s2 l2′ s′2 ;LJ1 J2 = (2L + 1) ∆2 (J1 J2 L) ∆2 (l1 l2 L) ∆2 (l1′ l2′ L)

(D.59)
s1 −s′1

× w(l1 J1 l2 J2 , s1 L) w(l1′ J1 l2′ J2 , s′1 L) δs1 s2 δs′1 s′2 (−1)

n! (−1)n
×
(n − l1 )! (n − l2 )! (n − L)!

′

n′ ! (−1)n
,
(n′ − l1′ )! (n′ − l2′ )! (n′ − L)!

in which n is defined by
2 n = l1 + l2 + L;

(D.60)

D is zero if l1 + l2 + L is an odd number. A similar expression defines n′ . The ∆2 term is
given by:
(a + b − c)! (a − b + c)! (−a + b + c)!
,
(D.61)
∆2 (abc) =
(a + b + c + 1)!
for which the arguments a, b, and c are to be replaced by the appropriate values given in
equations (D.58) and (D.59). The expression for ∆2 (abc) implicitly includes a selection rule
for the arguments; that is, the vector sum must hold:
~a + ~b = ~c

(D.62)

The quantity w in equation (D.59) is defined as:
kX
max

w(l1 J1 l2 J2 , s L) =

k=kmin

(−1)k+l1 +J1 +l2 +J2 (k + 1)!
[k − (l1 + J1 + s)]! [k − (l2 + J2 + s)]!

(D.63)

1
[k − (l1 + l2 + L)]! [k − (J1 + J2 + L)]!
1
×
(l1 + J1 + l2 + J2 − k)! (l1 + J2 + s + L − k)! (l2 + J1 + s + L − k)!
×

(and similarly for the primed expression), where kmin and kmax are chosen such that none of
the arguments of the factorials are negative. That is,
kmin = max {(l1 + J1 + s), (l2 + J2 + s), (l1 + l2 + L), (J1 + J2 + L)}
kmax = min {(l1 + J1 + l2 + J2 ), (l1 + J2 + s + L), (l2 + J1 + s + L)}

(D.64)

Single-channel case
For the single-channel case, the coefficients CLαα′ (E) reduce to:

CLαα (E) =

1 X X
4kα2 J
J
1

×

2

X

X

c1 =(αl1 s1 J1 ) c2 =(αl2 s2 J2 )

B{l1 s1 l1 s1 J1 }{l2 s2 l2 s2 J2 } L



1
Re (1 − Uc1 c1 )(1 − Uc∗2 c2 )
(2ia + 1)(2ib + 1)
332

(D.65)

APPENDIX D. RESONANCE REGION FORMULAE
where the existence of only one channel requires that the primed quantities of equation (D.61)
be equal to the unprimed (e.g., α = α′ ). The geometric factor B becomes:
B{l1 s1 l1 s1 J1 } {l2 s2 l2 s2 J2 } L = Al1 s1 l1 s1 ;J1 Al2 s2 l2 s2 ;J2 Dl1 s1 l1 s1 l2 s2 l2 s2 ; L J1 J2

(D.66)

where the factor A reduces to the simple form
Al1 s1 l1 s1 ;J1 = (2l1 + 1)(2J1 + 1) ∆2 (l1 J1 s1 )

(D.67)

and the expression for D reduces to:
Dl1 s1 l1 s1 l2 s2 l2 s2 ;L J1 J2 = (2L + 1) ∆2 (J1 J2 L) ∆4 (l1 l2 L) w2 (l1 J1 l2 J2 , s1 L) δs1 s2

2
n!
×
(D.68)
(n − l1 )! (n − l2 )! (n − L)!
in which n is again defined as in equation (D.60).
D.1.7.3

Kinematics for Angular Distributions of Elastic Scattering

If E represents the laboratory kinetic energy of the incident neutron, E ′ the lab kinetic
energy of the outgoing particle, θ the laboratory angle of the outgoing neutron, and Q the
Q-value for the reaction, the E ′ may be expressed in terms of E, θ, and Q as


s
2
2 2

mb
ma
ma

(D.69)
− sin2 θ
cos θ +
E′ = E 
ma + mb
ma + mb
ma + mb

where ma represents the mass of the incident particle (neutron) and mb , the mass of the sample (target) nucleus. Similarly, the center-of-mass angle β between outgoing and incoming
neutron is found from
s
)
(
ma
m2b
cos β = ±
− sin2 θ − sin2 θ
(D.70)
cos θ
mb
m2a
and the Jacobian of transformation from center-of-mass to laboratory system is
d(cos β)
ma 1 + (2 cos2 θ − 1) m2a /m2b
= 2 cos θ
+ p
d(cos θ)
mb
1 − sin2 θ m2a /m2b

(D.71)

The elastic angular distribution cross section in the laboratory system is then found by
combining equations (D.51 or D.65) with (D.71), using the relationship in equation (D.70),
to give:
dσ
dσ d(cos β)
.
(D.72)
(θ) =
d Ωlab
dΩCM d(cos θ)
Note that the lowest energy into which a neutron may scatter (i.e., the energy of a neutron
after 180-degree scattering) is
2

mb − ma
′
(D.73)
E (cos θ = −1) = E
mb + ma
333

APPENDIX D. RESONANCE REGION FORMULAE
and the energy of 90-degree scattering is:


mb − ma
E (cos θ = 0) = E
mb + ma
′

D.1.7.4



(D.74)

Spin and Angular Momentum Conventions

The spin and angular momentum conventions used in the Reich-Moore Format are described
in Table D.2. Note that the word ”channel” refers to the physical configuration as well as
to the quantum numbers given here. For example, for an incident neutron (intrinsic spin
~
i = 1/2 impinging on a target (sample) whose spin is I, the channel spin is s, where ~s = ~i + I.
The relative orbital angular momentum of this channel (neutron + target) is l, and total
spin is J, where J~ = ~s + ~l. The exit channel might be the same as the entrance channel, or
it might include, for example, two particles whose individual spins are i′ and I ′ and whose
channel spin is s′ , where s~′ = ~i′ + I~′ . The relative angular momentum of the two particles
is l′ , and the total J must satisfy J~ = s~′ + ~l′ .
Table D.2: Spin and angular momentum conventions
Symbol
i or i′
I or I ′
l or l′
s or s′

J

Meaning
Intrinsic spin of incident neutron or outgoing particle
Spin of target or residual nuclei
Orbital angular momentum of incident or outgoing particle
Incident or outgoing channel spin, equal to target
spin plus incident particle spin
(1) Spin of resonance
(2) Spin of excited level in the compound nucleus
(3) Total angular momentum quantum number

Value or range of values
1/2 for incident neutron
integer or half-integer
non-negative integer

~s = I~ + ~i or s~′ = I~′ + ~i′
J~ = ~l + ~s = ~l′ + s~′

For readers unfamiliar with vector summation, the rules are as follows: All quantum
numbers are either integer (0, 1, 2, . . . ) or half-integer (1/2, 3/2, 5/2, . . . ). If vectors of magnitude a and b are to be added, then the sum c has magnitude in the range
|a − b| ≤ c ≤ a + b; c takes on only integer values if a + b is integer, and half-integer values
if a + b is half-integer. The parity associated with c is the product of the parities associated
with a and b. Note also that parity associated with orbital angular momentum l is rarely
expressed explicitly, as it is always (−1)l .
D.1.7.5

Extensions to R-Matrix Theory

As stated in equation(D.44), the R-matrix has the form:
#
"
X
γλc γλc′
+ Rcbkg δcc′ δJJ ′ .
Rcc′ =
Eλ − E − i Γλγ /2
λ
334

(D.75)

APPENDIX D. RESONANCE REGION FORMULAE
The external or background R-matrix Rcbkg can be written in many different ways; four
options are available in the RML format:
Option 0. Rcbkg = 0 (in which case the background is described by ”dummy” resonances
whose energies lie outside the range of validity of this parameterization).
Option 1. Rcbkg is a tabulated complex function of the energy.
Option 2. Rcbkg is a real statistical parameterization of the form available in SAMMY
[Reference 4],
Rcext (E) = Rcon,c + Rlin,c E + Rq,c E 2 − slin,c (Ecup − Ecdown )
 up

Ec − E
− (scon,c + slin,c E) ln
.
(D.76)
E − Ecdown
Option 3. Rcbkg is a complex statistical parameterization of the forms described by
Fröhner [References 8 and 9],




Γγ /Q̃
E−E


Rcext (E) = Rc + 2sc tanh−1
+i
(D.77)

2 
Q̃/2
E−E
1−
Q̃/2

with

Q̃ = Ecup − Ecdown



and E = Ecup + Ecdown /2

(D.78)

The quantity Q̃ is not to be confused with the Q-value for channel c.

D.1.7.6

Modifications for Charged Particles

The penetrabilities Pl , shift factors Sl , and potential-scattering phase shifts φl defined in
Table D.1 apply only to non-Coulomb interactions such as those involving incident neutrons,
where it is possible for the two particles in a channel to both have a positive charge; examples
are the (n,α) or (n,p) interactions. In this case the expressions for penetrabilities, shift factors
and phase shifts must be modified to include the long-range interactions; see for example
the discussion of Lane and Thomas [Reference 8].
Expressions for Pl , Sl and φl for particle pair α involve the parameter ηα , which is defined
as
zα Zα e2 µα
(D.79)
ηα =
h̄kα
where z and Z are the charge numbers for the two particles in the particle pair. The reduced
mass µα is defined in the usual manner as
µα =

m α Mα
m α + Mα
335

(D.80)

APPENDIX D. RESONANCE REGION FORMULAE
where m and M are the masses of the two particles in the channel α. The center-of-mass
momentum h̄kα is defined as


2mα Mα
M
2 2
h̄ kα =
E
+Q
(D.81)
m α + Mα
m+M
where Q is the Q-value for the particle-pair.
If ac is the channel radius for this channel, we again define ρ as
ρ = kα ac

(D.82)

The penetrabilities Pl (η, ρ), shift factors Sl (η, ρ) and phase shifts φl (η, ρ) are then calculated
as functions of Fl (η, ρ) and Gl (η, ρ), the regular and irregular Coulomb wave functions,
respectively. The equations are as follows:
ρ
A2l
ρ ∂Al
Sl =
Al ∂ρ
Gl
cos φl =
Al
where
A2l = Fl2 + G2l
Pl =

(D.83)

In Equation D.41, the Coulomb phase-shift difference wc is required for charged-particle
interactions. From Lane and Thomas [Reference 8], this quantity has the value

0
l=0


(D.84)
W c = Pl
−1 ηα
l 6= 0
n=1 tan
n

336

APPENDIX D. RESONANCE REGION FORMULAE

D.2

The Unresolved Resonance Region (LRU=2)

Average resonance parameters are provided in File 2 for the unresolved region. Parameters
are given for possible l and J-values (up to d-wave, l = 2) and the following parameters may
0
be energy dependent: Dl,J , ΓnlJ , ΓγlJ , Γf lJ , and ΓxlJ . The parameters are for the single-level
Breit-Wigner formalism. Each width is distributed according to a chi-squared distribution
with a designated number of degrees of freedom. The number of degrees of freedom may
be different for neutron and fission widths and for different (l, J) values. These formulae do
not consider Doppler broadening.

D.2.1

Cross Sections in the Unresolved Region

Definitions and amplifying comments on the following are given in Section D.2.2.
a. Elastic Scattering Cross Section
σn,n (E) =

NLS−1
X

l
σn,n
(E),

(D.85)

l=0

4π
l
(2l + 1) sin2 φl
σn,n
(E) =
k2
#
"


NJS
Γn Γ n
2π 2 X gJ
+
− 2Γnl,J sin2 φl .
2
k J Dl,J
Γ
l,J
The asymmetric term in E − Er′ is assumed to average to zero under the energy-averaging
denoted by hi.

b. Radiative Capture Cross Section

σn,γ (E) =

NLS−1
X

l
σn,γ
(E),

(D.86)

l=0

l
σn,γ
(E)



NJS
2π 2 X gJ
Γ n Γγ
=
k 2 J Dl,J
Γ
l,J

c. Fission Cross Section
σn,f (E) =

NLS−1
X

l
σn,f
(E),

(D.87)

l=0

l
σn,f
(E)



NJS
Γn Γ f
2π 2 X gJ
.
=
k 2 J Dl,J
Γ
l,J

The sum over l in the above equations extends up to l = 2 or NLS-1 (the highest l-value
for which data are given). For each value of l, the sum over J has NJS terms. The number
337

APPENDIX D. RESONANCE REGION FORMULAE
of J-states for a particular l-state will depend on the value of l. NLS and NJS are given in
File 2.
The averages are rewritten as:




Γn Γ n
Γnl,J Γnl,J
=
Rn,l,J
Γ
Γl,J
l,J




Γnl,J Γγl,J
Γ n Γγ
Rγl,J ,
=
Γ
Γl,J
l,J




Γn Γ f
Γnl,J Γf l,J
=
Rf l,J
Γ
Γl,J
l,J

(D.88)

where Rγl,J , Rf l,J and Rnl,J are width-fluctuation factors for capture, fission, and elastic
scattering, respectively. Associated with each factor is the number of degrees of freedom for
each of the average widths, and the integrals are to be evaluated using the MC2 -II method.
Data given in File 2 for each (l, J) state
µnl,J = AMUN, the number of degrees of freedom for neutron widths
µf l,J = AMUF, the number of degrees of freedom for fission widths
µxl,J = AMUX, the number of degrees of freedom for competitive widths
µγl,J = AMUG, the number of degrees of freedom for radiation widths
Γx,l,J = GX, the average competitive reaction width
0

Γn,l,J = GN0, the average reduced neutron width
Γγ,l,J = GG, the average radiation width
Γf,l,J = GF, the average fission width
Dl,J = D, the average level spacing
The average neutron widths are defined in Section D.2.2.2, equation (D.98), where
ΓnlJ = hΓn (l, J)i. Degrees of freedom are discussed in Section 2.4.16.
The average total width, at energy E, is:

Γl,J = Γnl,J + Γγl,J + Γf l,J + Γxl,J ,

(D.89)

and all widths are evaluated at energy E. J =AJ, I =SPI, and l=L are given in File 2. The
penetration factors and phase shifts are functions of a or AP, as described earlier.

338

APPENDIX D. RESONANCE REGION FORMULAE

D.2.2

Definitions for the Unresolved Resonance Region

Editions of ENDF-102 prior to ENDF/B-V have had some errors in the ”Definitions” section of Appendix D (previously Section D.2.1). To clarify the points and facilitate parallel
reading with Gyulassy and Perkins, Reference 10, their parenthesized indices will be used.
Section D.2.3 contains a table of equivalences to the notation used in section D.2.1.
D.2.2.1

Sums and Averages

In an energy interval ∆ε, let the resonances be identified by a subscript λ = 1, 2, . . . which
goes over all the resonances. The present discussion is concerned with the combinatorial
aspects of level sequences, hence λ enumerates all the resonances, whether their widths
are observably large or not. One purpose of this section is to permit estimation of missed
resonances by comparing observed level densities or strength functions with the theoreticallyexpected relations. The latter are concerned with the set of all resonances, and not just those
that are observable in a particular experiment.
Let x denote a set of quantum numbers that label a subset of resonances in the interval.
If there are N (x) such resonances, their level density is:
ρ(x) = N (x)/∆ε,

(D.90)

D(x) = 1/ρ(x) .

(D.91)

and their level spacing is:
If yλ is some quantity associated with each resonance, λ, the sum of the y-values over
the subset x is:
x
X
yλ .
(D.92)
λ

In this section, the summation index λ is written as a subscript, and the range of the
summation is indicated by the superscript x. The expression in (D.92) says ”sum the
quantity y over every resonance in the interval ∆ε, which has the quantum numbers x.”
Usually, these resonances will possess other quantum numbers too, but it is the set x which
determines whether they are included or not.
An average of the quantity y over the set x is
x

hyix =
D.2.2.2

1 X
yλ
N (x) λ

(D.93)

Reduced Widths

In this section, reduced widths follow the experimental definition rather than the theoretical
usage Γ = 2P γ 2 . A partial width for the decay of a resonance into a particular channel carries
many quantum numbers, but we need only three, the total and orbital angular momenta J
and l, and the channel spin s. The reduced neutron width, Γlnλ (J, s), is defined by
√
(D.94)
Γnλ (l, J, s) = Γlnλ (J, s) E νl (E)
339

APPENDIX D. RESONANCE REGION FORMULAE
where:

and

νl = Pl /ρ,
ν0 = 1,
ν1 = ρ2 /(1 + ρ2 )
ν2 = ρ4 /(9 + 3ρ2 + ρ4 )
ρ = ka, where a is the channel radius.

Assuming additivity of partial widths,
Γlnλ (J) =

X

Γlnλ (J, s)

(D.95)

s

where Σs is a summation over the 1 or 2 possible channel-spin values.
If we average over resonances, and assume that the average partial width is independent
of channel spin,2
X
Γln (J) =
Γln (J, 6 s) = µl,J Γln (J, 6 s)
(D.96)
s

Equation (D.96) introduces the multiplicity µl,J , which for neutrons can have the value
1 or 2, depending on whether the channel spin has one or two values. For l = 0, or I = 0,
or J = 0, µl,J = 1. In other cases, s can take on the values I ± 1/2 subject to the additional
vector sum
~s = ~l + J~
(D.97)

which may again restrict µl,J to the value one.3
The other new notation is the line through the quantum number s, meaning that the
quantity Γln (J, 6 s) does not depend on the value of s. This is not the same as omitting
s from the parentheses, since that defines the left-hand side quantity. This is the primary
source of confusion in previous discussions. Since νl depends only on l,
√
hΓn (l, J)i = hΓln (J) E νl i
√
(D.98)
= µl,J hΓln (J, 6 s) E νl i
√
= µl,J hΓln (J, 6 s)i E νl
√
where the bar over E νl denotes some average value appropriate to the interval.
D.2.2.3

Strength Function

The pole-strength function was originally introduced as an average over the R-matrix reduced widths for a given channel, γc2 . Using the experimental convention,
l,J,s

Γl (J, s)
S(l, J, s) = n
.
D(l, J, s)
2
3

This is not true for the individual resonances.
For example, if I = 1/2, µ1,2 = 1.

340

(D.99)

APPENDIX D. RESONANCE REGION FORMULAE
Since the channel spin values are uniquely determined by J and l, together with the target
spin I which is common to all the resonances, s is superfluous in defining the subset over
which the average is taken, and:
Γln (J, s)
S(l, J, s) =
D(l, J)

l,J

.

(D.100)

If the parity π were used as an explicit quantum number, l could be dropped:
Γl (J, s)
S(l, J, s) = n
D(J, π)

J,π

,

(D.101)

because l and π are equivalent for labeling resonances. That is, every resonance with a given
J and π will have channels labeled by the same set of l-values, whether their partial widths
are observably large or not. Some authors go one step further and drop π, so that J means
J, π, but that is an invitation to confusion.
Expressing S(l, J, s) as a sum over reduced widths gives
S(l, J, 6 s) =

l,J
X

Γlnλ (J, s)/∆ε,

(D.102)

λ

where we use the assumed independence of hΓln (J, 6 s)i on s to get the same result on the
left-handside.
The strength function S(l, J) is defined as
X
S(l, J) =
S(l, J, s)
(D.103)
s

= µl,J S(l, J, s)

The corresponding sum and average forms are
S(l, J) =

l,J
X

Γlnλ (J)/∆ε

(D.104)

λ

l,J

Γln (J)
D(l, J)

=

The next ”natural” summation would be to collect the different l-contributions to the total
width, to form S(J), but this is not what is observable. Instead one defines S(l) as a
weighted sum of the S(l, J, s):
X
g S(l, J, s)
S(l) =

Js

X

g

(D.105)

Js

This equation occurs in Lynn, Reference 1, as number (6.126), with a confusing typographical
error, namely the index s is missing from S(l, J, s).
341

APPENDIX D. RESONANCE REGION FORMULAE
Actually, the strength function was introduced first in the ”s-wave” form:
S(0) =

X g Γ0
n
D

(D.106)

λ(l=0)

and later generalized by Saplakoglu et al. (Reference 11) to the p-wave form:
S(1) =

X


1
g Γ1n λ .
∆ε(2l + 1)

(D.107)

λ(l=1)

For expository purposes, it is clearer to start from equation (D.105). The sum on J and s
is for fixed l:
I+ 12
l+s
X
X
X
≡
.
(D.108)
Js

s=I− 21 J=|l−s|

It is important to note that the outer sum on channel spin is correct as written. It goes
over the values I ± 1/2 if I ≥ 1/2, and over the single value 1/2, if I = 0. It is not further
constrained by equation (D.97) because now it is the ”independent variable.” The inner
sum on J enumerates some J-values once, and some twice, the latter occurring when both
s-values can produce that J-value. The number of times J occurs is the same µl,J that
appeared previously.
If we are summing a quantity that is independent of s, then equation (D.108) can be
rewritten:
X
X
g
y(6 s) =
µlJ y(6 s)
(D.109)
Js

l,J

The multiplicity µl,J takes care of the sum on s, and the tilde over the sum on J, as
emphasized by Gyulassy and Perkins, Reference 10, reminds us that J goes over its full
range, ”once-only”:4
X
g

l+I+ 21

X

=

J

J=|l−I− 21 |

if l ≥ I

(D.110)

l+I+ 21

X

=

J=|

I−l− 21

|

if I ≥ l

The denominator in Equation (D.105) can be shown to be
X
g = 2l + 1 ,

(D.111)

Js

or, since g is independent of s,

X
g

µl,J g = 2l + 1 .

J

4

Reference 10 has this written incorrectly.

342

(D.112)

APPENDIX D. RESONANCE REGION FORMULAE
Gyulassy and Perkin (Reference 10) assume, and later approximately justify by comparison
to experiment, that S(l, J, 6 s) is also independent of J. With this, Equation (D.105) becomes:
X
1
S(l) =
g S(l, 6 J, 6 s) = S(l, 6 J, 6 s)
(D.113)
(2l + 1) Js
=

S(l, J)
,
µl,J

using Equation (D.103).
Note the peculiar fact that S(l) and S(l, J, s) are independent of J, but S(l, J) is not.
This is a consequence of the fact that more than one channel spin value can contribute to
S(l, J), inducing a ”J-dependence” in the form of a possible factor of two.
As a sum over resonances,
X
1
S(l) =
g S(l, 6 J, 6 s)
(D.114)
(2l + 1) Js
X X
1
g Γlnλ (J, s)/∆ε
=
(2l + 1) Js
λ(l,J)

=

1
(2l + 1)

X
g X
J

g Γlnλ (J)/∆ε

λ(l,J)

The right hand side of equation (D.115) says to sum Γln(J) over all possible values of J, which
is what is meant by Equations (D.106) and (D.107). We can suppress the explicit J’s and
write, as in Equation (D.107),
X


1
S(l) =
g Γlnλ λ
(D.115)
∆ε(2l + 1)
λ(l)

but we have to remember that Γln is still Γln (J), and not a new quantity.
As an average, using the same convention,
l

gΓln
1
.
S(l) =
(2l + 1) D(l)

(D.116)

Otherwise, all the notation is correct: D(l) is the spacing of l-wave resonances without
regard to their J-values, and the average h il goes over all resonances possessing the quantum
number l, again without regard for their J-values. It is worth noting explicitly that although
S(l, J) is ”almost” independent of J, this is not true of Γln (J) . As Equation (D.104) shows,
its J-dependence is canceled by the J-dependence of D(l, J), up to the factor µl,J . This
property is what makes strength functions useful.
D.2.2.4

Level Spacings

Gyulassy and Perkin (Reference 10) emphasize that:
X
g
ρ(l) =
ρ(l, J)
J

343

(D.117)

APPENDIX D. RESONANCE REGION FORMULAE
which, together with the assumption:
ρ(l, J) = K(l)(2J + 1)
leads to
ρ(l) = ρ(l, J)

(D.118)

(2l + 1)
ωI,l ,
g

(D.119)

where
ωI,l = (l + 1)/(2l + 1) for l ≤ I
= (I + 1)/(2I + 1) for l ≥ I

(D.120)

and is unity if l = 0 or I = 0.
The reader is referred to Reference 10 for a fuller discussion but here we can point out
that, for a given parity, ρ(l, J) is independent of l, by definition. As noted, every resonance
with a given J and π has the same set of associated l-channels, whether it has an observable
width or not. Hence:
ρ(0, J) = ρ(2, j) = ρ(4, j) = . . .
ρ(1, J) = ρ(3, j) = ρ(5, j) = . . .

and

The further assumption of parity-independence makes ρ(l, J) totally independent of l. As a
result, Gyulassy and Perkin’s K(l) from Equation (D.118) is independent of l, and
ρ(l) = C(2l + 1) ωI,l ,

(D.121)

where C depends on the nuclear species but not on any quantum numbers.
D.2.2.5

Gamma Widths

In the limited energy range usually covered by the unresolved resonance region, the gamma
width may be assumed to be constant and equal to that obtained from an analysis of the
resolved resonances. If, however, the energy range is rather wide, an energy dependence as
given by some of the well-known theoretical models, Reference 1, may be built in. Since the
observed gamma width is the sum of a large number of primary gamma transitions, each
assumed to have a chi-squared distribution of µ = 1, the sum is found to have a µ ≥ 20. In
effect this implies that the gamma width is a constant, since a chi-squared distribution with
a large number of degrees of freedom approximates a δ-function.
D.2.2.6

Degrees of Freedom

For the reasons enumerated for File 2 in Section 2.4.16, the following values should be used:
1. Neutron width, 1 ≤ AMUN ≤ 2, and specifically, AMUN=µl,J .
2. Radiation width, AMUG=0.

344

APPENDIX D. RESONANCE REGION FORMULAE
3. Fission width, 1 ≤ AMUF ≤ 4, to be determined by comparison with experiment.
Only integer values are permitted, although non-integers occur in some analyses.
4. Competitive width, 1 ≤ AMUX ≤ 2, because only a single inelastic level excitation is
permitted as a competitive reaction. Specifically, AMUX=µl′ ,J , where J is the spin
of the resonance, and l′ is the orbital angular momentum of the inelastically scattered
neutron. Since the daughter nucleus may have a spin Ie different from the target spin I,
l′ may be different from l and the number of channel spin values µl′ ,J may be different
from µl,J .

D.2.3

Equivalent Quantities in Sections D.1 and D.2
Symbol in Section

D.1
r

D.2.1
-

D.2.2
λ

Γnr

Γnλ (l, J)

Γγr , Γf r , . . .

-

Pl
Dl,J
0

D.3
D.3.1

ρνl
D(l, J)

Γnl,J

hΓln (J)il,J

Γnl,J

hΓn (l, J)il,J

Definition
This is a non-equivalence. λ enumerates all
resonances; r enumerates those within a subset and hence implies a set of quantum numbers.
The neutron width, summed over channel
spin.
Not used in D.2.2, but the same implication
of l, J holds.
Penetration factor.
Average level spacing for a subset of resonances with given l and J.
The l-wave reduced width, averaged over all
resonances with given l and J.
The average neutron width. In practice, the
energy-dependence of this quantity is not averaged, but extracted before averaging.

The Competitive Width
Penetrability Factor for the Competitive Width in the Resolved Resonance Region

A. SLBW and MLBW
For these formalisms, the only physical situation which can be handled without approximation is that in which a single inelastic competitive process is possible, because the formalism
presently permits the definition of only one additional quantity. The most common case
will occur when inelastic scattering to the first excited state of the target nucleus is energetically possible. Ignoring, as in the case of elastic scattering, the possibility that the partial
widths depend on channel spin, the penetrability is identical to that for elastic scattering,

345

APPENDIX D. RESONANCE REGION FORMULAE
but the energy is reduced by the excitation energy of the first excited state, corrected for
recoil, so that
Γxr (E) = Γn′ r (E) =
and

Pl (E − E1∗ )Γxr (|Er |)
,
Pl (|Er − E1∗ |)

Γxr = 0 if

if E ≥ E1∗

E < E1∗

(D.122)
(D.123)

where E1∗ is (AWRI+1)/AWRI times the excitation energy of the first excited state,
(E1ex = −QX in File 2).
This definition involves two conventions, both taken over from the elastic case. One is
the way in which an ”experimental” reduced width Γln′ is defined in terms of the theoretical
reduced width γ 2 , and the other is the way in which negative energy levels are treated.
Neither of these problems arises in the theory, where Γ = 2P γ 2 and all quantities are
defined in terms of the channel energy. Note that the l-value to be used in the penetrability
is not that of the incident neutron, but of the ”exit” inelastically scattered neutron.
It is conceivable that an (n,α) or (n,p) reaction to the ground state of the daughter
nucleus could be open, without inelastic competition, in which case the formula for Γxr
would be the same, but the Pl would be a Coulomb penetrability, and the excitation energy
El∗ would be replaced by the approximate Q-value and reduced mass. The R-Matrix Limited
format allows for this possibility (see Section D.1.7), as well as for accurate treatment of the
inelastic channels.
If more than one competitive process is energetically possible, then the SLBW and
MLBW formats are inadequate to give the correct energy dependence of the competitive
width, since they supply only one number, and a partial width is required for each process.
For example, when two inelastic levels can be reached,
Γxr (E) =

Pl1 (E − E1∗ )Γn′1 r (|Er |) Pl2 (E − E2∗ )Γn′2 r (|Er |)
+
,
Pl1 (|Er − E1∗ |)
Pl2 (|Er − E2∗ |)

(D.124)

with appropriate modification below each threshold. Note that the exit l-values are independent of the incident-neutron l-value.
For codes that presently approximate Γ as a constant in the denominator, a possible
procedure is to substitute a step function:
Γxr (E) = 0
if E < E1∗
= Γxr (|Er |) if E ≥ E1∗

and then make some provision to handle the resultant discontinuity in the cross section.
Users who are unable to handle this degree of complexity, and would like to use GTr from
File 2 as the total width without regard for whether the competitive process is energetically
possible or not should at least be aware of the problem.
B. When the Adler-Adler and Reich-Moore formalisms are used for low-energy fissile materials, no recommendation concerning the treatment of Γxr need be given, and users can
presume that it is zero. When Reich-Moore formalism is used above the thermal region, the
same comments apply as for the SLBW/MLBW formalism.
C. The R-matrix Limited format (LRF=7) allows multiple reaction channels and multiple
elastic channels as needed. Detailed formulae are given in Section D.1.7.
346

APPENDIX D. RESONANCE REGION FORMULAE

D.3.2

Penetrability Factor for the Competitive Width in the Unresolved Resonance Region

Since many codes treat the average total width in the denominator of expressions like
hΓn Γγ /Γi as an energy-independent constant, the penetrability factor of the competitive
width needs to be handled by specifying energy-dependent unresolved resonance parameters.
The formalism, which is a simple average over SBLW line shapes, takes account of the
energy-dependence of the neutron widths in the numerator, by extracting their penetrability
factors before the averaging is done. These then contribute to the energy-dependence of the
average cross section. The energy-dependence of the neutron width in the denominator,
i.e., in Γ, is neglected. No such fix is readily available for the energy-dependence of the
competitive width, whose penetrability factor will involve the threshold dependence of an
inelastic cross section. The evaluator can circumvent this difficulty by specifying energydependent parameters and setting hΓx i = 0 below its threshold; then allowing it to build up
according to the formulas given in Section D.3.
The degrees of freedom, AMUX, should be 1.0 or 2.0. (See Section 2.4.16).

D.3.3

Calculation of the Total Cross Section when a Competitive
Reaction is Specified

When a competitive reaction is specified for SBLW or MLBW and Γ exceeds the sum of
the partials Γn + Γγ + Γf , the ENDF convention is that the scattering, capture, and fission
cross sections will be calculated from the sum of File 2 and File 3 contributions, but the
competitive reaction will be contained entirely in File 3, and no File 2 contribution should
be added to it. The reason for this is that users can avoid problems in coding up resonant
competitive widths. In the File 2 calculations, the correct total width Γ must be used in
order to get the correct line shape.
This puts the total cross section in a special category. If it is calculated as the sum of
σn , σγ , σf , σx , then the above prescription works satisfactorily. However, if it is calculated
from the SLBW formula,
σn,t (E) =

Γnr Γr
π X
gJ
2
k J
(E − Er′ )2 + 14



Γr 2
2

(D.125)

as it is in some applications, then it will include the competitive reaction, and the user
should not add the File 3 contribution to it. The ENDF convention presumes that σn,t will
be calculated by summing the partial reactions.
The R-matrix Limited format does not have this problem. The ”competitive” reactions
are treated normally, and File 2 and File 3 are added together for all reactions. That is
because the total width is always the sum of the explicitly given partial widths. If a File 3
contribution were specified for the total cross section, then it would be added to the {1-ReU}
calculation, but not to the sum-of-parts calculation, as the latter would already include the
File 3 contribution for each partial reaction. This assumes that the File 3 total is the sum
of the File 3 partials.
347

APPENDIX D. RESONANCE REGION FORMULAE

References
1. J.E. Lynn, The Theory of Neutron Resonance Reactions, Clarendon Press,
Oxford (1958)
2. A.M. Lane and R.G. Thomas, Rev. Mod. Phys. 30, 257 (1958)
3. C.W. Reich and M.S. Moore, Phys. Rev. 111, 929 (1958)
4. N.M. Larson, Updated Users’ Guide for SAMMY: Multilevel R-Matrix Fits to Neutron
Data Using Bayes’ Equations, ORNL/TM-9179/R7, Oak Ridge National Laboratory,
Oak Ridge, TN, USA (2006). Also available as ENDF-364/R1.
5. F.T. Adler and D.B. Adler, Interpretations of Neutron Cross Sections of the Fissionable
Materials for the Resolved Resonance Region, U.S. Atomic Energy Commission Report
CONF-660303, Conf. on Neutron Cross Section Technology (1966), Book
2, p. 873
6. D.B. Adler and F.T. Adler, Argonne National Laboratory report ANL-6792, 695
(1963)
7. J.M. Blatt and L.C. Biedenharn, Rev. Mod. Phys. 24, 258 (1952)
8. F.H. Fröhner, Applied Neutron Resonance Theory, Kernforschungszentrum, Karlsruhe
report KFK 2669 (1978)
9. F.H. Fröhner, New Techniques for Multi-Level Cross Section Calculation and Fitting,
Brookhaven National Laboratory report BNL-NCS-51363, Proceedings of the
Conference on Nuclear Data Evaluation, Methods and Procedures (1981),
Vol. I, p.375
10. M. Gyulassy and S.T. Perkins, Nucl. Sci. Eng. 53, 482 (1974)
11. A. Saplakoglu, L. M. Bollinger, and R. E. Coté, Phys. Rev. 109, 1258 (1958)

348

Appendix E
Kinematic Formulas
The notation used to describe two-body kinematics is shown in Fig.E-1. The scattering
treatment used for neutrons can be generalized to two-particle reactions by allowing the
emitted particle mass to be different from the incident particle mass (i.e., m3 6=m1 ). Applying conservation of mass, energy, and momentum gives the following non-relativistic
kinematic equations:
A =
A′ =
β =
γ =
ε3
=
ε1
ε1 =
µ3
ε4
ε1
µ4
E3
E1

=
=
=
=

ω3 =
E4
=
E1
ω4 =

m2
m1
m3
m1
1/2


1+A Q
A(A + 1 − A′ )
1+
,
A′
A E1
A′
β,
A + 1 − A′
A′ 2
β ,
A2
2

A
E1 ,
A+1
µ,
ε3
A′
,
′
A + 1 − A ε1
−µ ,


A′
2
β
+
1
+
2βµ
,
(1 + A)2
1 + βµ
p
,
2
β + 1 + 2βµ


A + 1 − A′ 2
γ + 1 − 2γµ ,
2
(1 + A)
1 − γµ
p
.
γ 2 + 1 − 2γµ
349

(E.1)
(E.2)
(E.3)
(E.4)
(E.5)
(E.6)
(E.7)
(E.8)
(E.9)
(E.10)
(E.11)
(E.12)
(E.13)

APPENDIX E. KINEMATIC FORMULAS
If the incident and scattered particles are the same, A′ = 1, and these formulas reduce to
the familiar set used for neutron scattering. The elastic reaction corresponds to A′ = 1 and
Q = O.
According to these formulas, the secondary energy distribution of the emitted particle
and the complete energy-angle distribution of the recoil nucleus are completely determined
by A, A′ , Q, and the angular distribution for the emitted particle.

m3
ǫ3

θ

m1
ǫ1

b

b

m2
ǫ2

µ = cos θ

m4
ǫ4

Figure E.1: Kinematics variables for two-particle reactions in the center-of-mass system.
When the masses are small and the energies large, or when either m1 or m2 is a photon,
it is appropriate to use relativistic kinematics. A good example of the this is calculating the
energy and emission angle for the photon from neutron capture on 1 H as the center of mass
cosine µ varies over its range. Another example is the distributions for the products of the
photo breakup of the deuteron. Define:
AWR target mass ratio to the neutron,
AWRI projectile mass ratio to the neutron,
AWRP product mass ratio to the neutron, and
Q reaction Q-value.

350

APPENDIX E. KINEMATIC FORMULAS

m3
ǫ3

φ3
m1
E1

b

b

ω = cos φ
φ4

m4
ǫ4
Figure E.2: Kinematics variables for two-particle reactions in the laboratory system.
The following equations can be used to compute the quantities shown in Figure E:
m
m1
m2
m3
m4
si
di
sf
df
s
ki2
kf2

mneutron c2 in eV, see Appendix E.
AWRIm
AWRm
AWRPm
m1 + m2 − m3 = Q
m1 + m2
m1 − m2
m3 + m4
m3 − m4
s2i + 2m2 E1
(s − s2i )(s − d2i )
=
(4s)
(s − s2f )(s − d2f )
=
4s
q

=
=
=
=
=
=
=
=
=
=

z2 =

z3 =
z4 =

q

q

(E.14)
(E.15)
(E.16)
(E.17)
(E.18)
(E.19)
(E.20)
(E.21)
(E.22)
(E.23)
(E.24)
(E.25)

ki2 + m22

(E.26)

kf2 + m23

(E.27)

kf2 + m24

(E.28)
351

APPENDIX E. KINEMATIC FORMULAS

E3 =
E4 =
p21 =
p21 =
p21 =
ω3 =
ω3 =

z2 z3 −

q
ki2 kf2 u

m2
q
z2 z4 − ki2 kf2 u

− m3

− m4
m2
E12 + 2m1 E1
E32 + 2m3 E3
E4 + 2m4 E4
√
(E1 + si )(E3 + m3 ) − z3 s
p
p12 p23
√
(E1 + si )(E4 + m4 ) − z4 s
p
p12 p24

(E.29)
(E.30)
(E.31)
(E.32)
(E.33)
(E.34)
(E.35)

where the k quantities are center-of-mass momenta, and the p quantities are lab momenta.
The symbol z is used instead of the more conventional ω because of the conflict with the
symbol used for laboratory cosines in Figure E.

352

Appendix F
Summary of Important ENDF Rules
F.1

General

1. Cross sections for all significant reactions should be included.
2. The data in an ENDF file are specified over the entire energy range 10−5 eV to 20 MeV.
It should be possible to determine values between tabulated points with use of the
interpolation schemes provided.
3. All cross sections are in barns, all energies in eV, all temperatures in degrees Kelvin,
and all times in seconds.
4. Summary documentation and unusual features of the evaluation should appear in the
File 1 comments.
5. Threshold energies and Q-values must be consistent for all data presented in different
files for a particular reaction.

F.2

File 2 - Resonance Parameters

1. Only one energy region containing resolved resonance parameters can be used, if
needed.
2. The cross sections from resonance parameters are calculated only within the energy
range EL to EH, although some of the resonance parameters may lie outside the range.
3. Every ENDF Material has a File 2 even if no resonance parameters are given in order
to specify the effective scattering radius.
4. In the unresolved resonance region interpolation should be done in the cross section
space and not in the unresolved resonance parameter space. Any ENDF interpolation
scheme is allowed.

353

APPENDIX F. SUMMARY OF IMPORTANT ENDF RULES
5. The Breit-Wigner single-level or multilevel formalisms should be used in the resolved
resonance region unless experimental data prove that use of the other allowed formalisms is significantly better.

F.3

File 3 - Tabulated Cross Sections

1. All File 3 data are given in the laboratory system.
2. The total cross section MT=1 is the sum of all partial cross sections and has an energy
mesh that includes all energy meshes for partial cross sections.
3. The following relationships among MT numbers are expected to be satisfied if data
are presented:
1=
2+3
3 (or 1 - 2) = 4 (or 51→ 91) + (6→ 9+16) + 17 + 18
(or 19→ 21+38) + (22→ 25) + (28→ 37)
+ (102→ 114)
4=
sum (51→ 91)
18 =
sum (19→ 21) + 38
101 = sum (102→ 114)
103 =
sum (700→ 718)
104 =
sum (720→ 738)
105 =
sum (740→ 758)
106 =
sum (760→ 778)
107 =
sum (780→ 798)
4. Threshold reactions begin with a zero cross sections at the threshold energy.

F.4

Relation Between Files 2 and 3

1. If there are resonance parameters in File 2, there are contributions to the total
(MT=1) and scattering (MT=2) cross sections and to the fission (MT=18) and capture (MT=102) cross sections if fission and capture widths are also given. These must
be added to the File 3 Sections MT=1, 2, 18, and 102 over the resonance region in
order to obtain summation values for these cross sections.
2. The cross sections in File 3 for MT=1, 2, 18, and 102 in the resonance region are used
to modify the cross section calculated from the resonance formalisms, if necessary.
The File 3 ”background” may be positive or negative or even zero if no modifications
are required. The summation cross section (File 2 + File 3) should be everywhere
positive.
3. Double-valued points (discontinuities) are allowed anywhere but are required at resonance region boundaries. A typical situation for MT=1, 2, 18, and 102 in File 3 is a
tabulated cross section from 10−5 to 1 eV, tabulated ”background” to the cross sections
354

APPENDIX F. SUMMARY OF IMPORTANT ENDF RULES
calculated in the resolved resonance region between EL1 and EH1, tabulated ”background” to the cross sections calculated in the unresolved region between EL2=EH1
and EH2, and tabulated cross sections from EH2 to 20 MeV. Double-value points occur
at EL1, EL2, and EH2.
4. The tabulated ”background” used in File 3 to modify the cross sections calculated
from File 2 should not be highly structured or represent a large fraction of the cross
sections calculated from File 2. It is assumed that the ”background” cross section is
assumed to be at 0 Kelvin. (The ”background” cross section is usually obtained from
room temperature comparisons, but this should be unimportant if the ”background”
cross section is either small or slowly varying).
5. The generalized procedure for Doppler-broadening cross sections from Files 2+3 is to
generate a pointwise cross section from the resolved resonance region on an appropriate
energy mesh at 0K and add it to File 3. This summation cross section can be kernelbroadened to a higher temperature.

F.5

File 4 - Angular Distributions

1. Only relative angular distributions, normalized to an integrated probability of unity,
are given in File 4. The differential scattering cross section in barns per steradian is
determined by multiplying File 4 values by the File 2+3 summation scattering cross
section σs /(2π).
2. Discrete channel angular distributions (e.g., MT=2,51-90,701...) should be given as
Legendre coefficients in the center-of-mass system, with a maximum of 64 higher order
terms, the last being even, in the expansion. If the angular distribution is highly
structured and cannot be represented by a Legendre expansion, a tabular angular
distribution in the CM system must be given.
3. Angular distributions for continuum and other reactions must be given as tabulated
distributions in the Lab system , unless they are given in File 6 .
4. The angular distribution, whether specified as a Legendre expansion or a tabulated
distribution, must be everywhere positive.
5. Angular distribution data should be given at the minimum number of incident energy
points that will accurately describe the energy variation of the distributions.

F.6

File 5 - Secondary Energy Distribution

1. Only relative energy spectra, normalized to an integrated probability of unity, are
given in File 5. All spectra must be zero at the end points. The differential cross
section in barns per eV is obtained by multiplying the File 5 values by the File 2+3
cross section times its multiplicity (2 for the (n,2n) reaction).
355

APPENDIX F. SUMMARY OF IMPORTANT ENDF RULES
2. While distribution laws 1, 3, 5, 7, 9, and 10 are allowed, distribution laws 3 and 5 are
discouraged but can be used if others do not apply.
3. The sum of all probabilities for all laws used for a particular reaction must be unity
at each incident energy.
4. The constant U must be specified, where applicable, to limit the energy range of
emitted spectra to physical limits.

356

Appendix G
Maximum Dimensions of ENDF
Parameters

357

APPENDIX G. MAXIMUM DIMENSIONS OF ENDF PARAMETERS
File
(MF)
1

2

Section
(MT)
451
452
455
456
151

Variable Max.
value
NXC
350
NC
4
NCD
4
NCP
4
NE
250
NER
12
NFRE
1
NGRE
1
NIRE
4
NCRE
4
NIS
10
NRS
5,000
NJS
6
NLS
4
NLCS
20

3
4

All6=4
All

5

All

6

All

7
9,10

All
1
6= 1
All
All

NP
NE
NL
NM
NP
NE
NF
NK
NE
NL
NS
NP
NP
NL
NP

Definition of the number represented by
the variable
Card images in directory
Polynomial terms in expansion of ν
Polynomial terms in expansion of ν d
Polynomial terms in expansion of ν p
Energy mesh in unresolved region
Energy ranges
Fission reactions
Radiative capture reactions
Inelastic scattering reactions
Charged-particle reactions
Isotopes
Resonances for a given l-value
Number of J-values
Number of l-values
Number of l-values which must be given to converge reaction
50,000 Incident energy points
2,000 Incident energy points
64
Highest order Legendre polynomial
64
Maximum order Legendre polynomials required
201
Angular points
200
Incident energy points
1,000 Secondary energy points
2,000 Number of subsections
2,000 Incident energy points
64
Highest order Legendre polynomial
3
Non-principal scattering atom types
50,000 Energy points
5,000
64
Highest order Legendre polynomial
10,000 Mesh size

All

NR

20

14
All
other
All

Interpolation ranges

358

Appendix H
Recommended Values of Physical
Constants to be Used in ENDF
H.1

Sources for Fundamental Constants

The basic source for Fundamental Constants used by CSEWG in evaluating and processing
ENDF data are the values reported in 1998 CODATA internationally recommended
values of the Fundamental Physics Constants (Source 1) as taken from the NIST
Reference on Constants, Units, and Uncertainties Web site1 .
These are supplemented by the mathematical constants from MathSoft2 (Source 2).
Atomic masses not given in the CODATA recommended values should be taken from the
Atomic Mass Tables of G. Audi and A. Wapstra3 (Source 3).

H.2

Fundamental Constants and Derived Data

Values of the Fundamental Physics Constants, as approved by CSEWG, are given in this
Appendix. These values should be used until updates are approved by CSEWG.
Values for quantities which are derived from these fundamental constants, and which
were previously given in the body of this manual, are also presented in this section with the
expressions by which they have been replaced in the body of this Manual. These values may
not appear in subsequent revisions of this Appendix.
1

The information was prepared by P. J. Mohr and B. N. Taylor, ”The 1998 CODATA Recommended
Values of the Fundamental Physics Constants”, Version 3.1, National Institue of Standards and Technology
(December 1999). The Web site is located at http://physics.nist.gov/cuu/Constants/.
2
The information was prepared by Steven Finch of MathSoft, Inc., for their Web site which is located
at http://www.mathsoft.com/asolve/constant/constant.html.
3
G. Audi, A. H. Wapstra and C. Thibault, The AME2003 atomic mass evaluation, Nucl. Phys. A. 729,
337 (2003). The Web site, maintained by the NNDC is located at ”http://www.nndc.bnl.gov/masses/”

359

APPENDIX H. RECOMMENDED VALUES OF PHYSICAL CONSTANTS

Table 1: Fundamental Constants
Expression
Definition
e
natural logarithmic base
π
Archimedes’ constant
e
elementary charge
−1
7
α = 10 h̄/(c e) inverse fine-structure constant
u
atomic mass unit (amu)
h
Planck’s constant
h̄
Planck’s constant/2π
(=107 h̄[erg.s]/e[C])
k
Boltzmann’s constant
c
speed of light (in vacuum)
NA
Avogadro’s number

H.3

Numeric value
2.718 281 828 52
3.141 592 653 59
1.602 176 462 ×10−19 C
137.035 999 76
931.494 013 ×106 eV / c2
4.135 667 27 ×10−15 eV s
6.582 118 89 ×10−16 eV s

Source
2
2
1
1
1
1
1

8.617 342 ×10−5 eV K−1
299 792 458 m s−1
6.022 141 99 × 1023 mol−1

1
1
1

Use of Fundamental Constants by Code Developers

Code developers are encouraged to locate any values for fundamental physics constants
that may be currently buried deep within their codes, and to replace these values by the
expressions given here; the values would then be specified in only one location in the code.
This ensures internal consistency, and expedites any necessary updates.
Code developers should double-check that numerical constants (e.g., π or e) are represented to a degree of accuracy consistent with the precision of the computers on which the
codes are to be run.
In subsequent revisions of this manual, values for derived quantities may not be given.
Instead, code developers should calculate those quantities by directly evaluating the expressions. This will ensure that values are as precise as the computer permits.

H.4

Use of Fundamental Constants by Evaluators

Evaluators should use the fundamental constants, mass numbers, Q-values, etc., as specified
in this section, for evaluations submitted for acceptance by ENDF.
Evaluators are encouraged to specify values for ”hidden” physical constants within the
File 1 comments of the ENDF file in order to prevent future confusion in the event of changes
in the accepted values.
The following table gives values which were previously given in the body of this Manual,
along with the expressions which should be used in the future for these values. These
expressions have been substituted for the values at the appropriate places in the Manual.
These values should not be used in any future applications; instead, please use
the values for the Fundamental Constants as specified in this Appendix. (For
example for mp /mn , do not use 0.99862; use the value derived from the values for mp and
mn ).
360

APPENDIX H. RECOMMENDED VALUES OF PHYSICAL CONSTANTS

Table 2. Masses
Expression Definition
mn
neutron mass
me
electron mass
mp
proton mass
md
deuteron mass
mt
triton mass
3
He mass (hellion)
m3He
mα
α mass

Numeric value
1.008 664 915 78 amu
5.485 799 110 ×10−4 amu
1.007 276 466 88 amu
2.013 553 212 71 amu
3.015 500 713
amu
3.014 932 234 69 amu
4.001 506 1747 amu

Source
1
1
1
1
3
1
1

Table 3. Energies needed to break particles into their constituent nucleons.
d
deuteron
2.22 MeV
t
triton
8.48 MeV
3
3
He
He
7.72 MeV
α
alpha
28.3 MeV

Table 5. Derived quantities whose values were formerly given in the body of the
Manual
Value previously
Expression
Location
Units for value
given in the Manual
0.998 62
mp /mn
1.996 26
md /mn
page 0.18
2.989 60
mt /mn
2.989 03
m3He /mn
3.967 13
ma /mn
−8
Page 4.7
3.0560 ×10
1 / ( eV barn steradian ) mn u/(2h2 c2 ) × 10−28
Eq. (6.9)
4.784 53 ×10−6
(10−12 cm)2 eV / amu
2u/(h̄2 c2 )
Eq. (6.10) 2.480 58 × 104
eV / amu
uα2 /2
√
page D.3
2.196 771 ×10−3
10−12 cm (eV)−1/2
2mn /h̄

361

Appendix I
A History of the ENDF System and
its Formats and Procedures
Abstract
The following information provides a short historical record of the development of the data
formats and procedures of the Evaluated Nuclear Data File (ENDF) from the first proposal
of specifications in May 1966 to the ENDF/B-VI.8 release in 2001.

I.1

Introduction

The process of digesting experimental data, combining it with the predictions of nuclear
model calculations and attempting to extract the true value of a cross section is referred to
as an evaluation.
Historically, individual laboratories around the world had prepared evaluated nuclear
data, e.g., neutron-cross sections, for their use in nuclear reactor calculations. These laboratories stressed their own needs for materials, cross section types and energy ranges depending
upon their specific applications. Each of these laboratories developed their own methods for
the storage and retrieval of these data.
In addition, it was noted that some neutron transport programs had built-in neutron
cross-section libraries that could not be modified. As a result, reactor designers could not
use new cross-section data, which in some cases had been available for more than half a
decade [1].
There were fairly detailed nuclear data libraries available by 1963, the United Kingdom
Nuclear Data Library (UKNDL) from Ken Parker at the Atomic Weapons Research Establishment, Atomic Energy Authority in Aldermaston, UK, the fast reactor data library
from Joe Schmidt at the Institute for Neutron Physics and Reactor Technology, Nuclear
Research Center, Karlsruhe, Germany, the NDA library from Herb Goldstein at Nuclear
Development Associates, in New York, and the Evaluated Nuclear Data Library (ENDL)
from Bob Howerton at the Lawrence Radiation Laboratory in Livermore, California to name
just a few.

362

APPENDIX I. HISTORIC PERSPECTIVE BY NORMAN E. HOLDEN
At the 1961 Vienna Conference on the Physics of Fast and Intermediate Reactors, Ken
Parker [2] indicated some of the requirements for the neutron cross section libraries. They
had to specify reaction processes available or else a zero value cross section would automatically be assumed. There had to be a simple presentation of the data on punched cards,
which would be easy to revise. However, the data could not be revised frequently or the
reactor designers would be unable to perform comparative calculations, as they made design
revisions. There was a need to cross check the data for errors and the best data should
provide reasonable answers for simple systems, such as bare reactor cores.
At the 1964 Geneva Conference on Peaceful Uses of Atomic Energy, John Story [3] from
the Atomic Energy Establishment, Winfrith, UK, defined a data file as a complete set of
evaluated cross section data for a single material and a data library as data files for a number
of materials.
The various data libraries that were available often gave different answers, when the
libraries were used to calculate the same reactor configuration. However, dissimilarities
in the internal formats of the various libraries made it difficult to understand why these
differences occurred.
There was a need for a common file between these existing systems, which would allow
for an inter-comparison of these libraries. The stimulus for action came from a discussion
among Henry Honeck of Brookhaven National Laboratory, Al Henry of Westinghouse and
George Joanou of General Atomics at the Colony Restaurant in Washington, D.C. The
Reactor Mathematics and Computation (RMC) Division of the American Nuclear Society
(ANS) was requested to sponsor two meetings to discuss this common link, as a result of
the above discussion. Honeck as chairman of the Division’s sub-committee on Evaluated
Nuclear Data Files held some meetings. A group of eighteen representatives from fifteen US
laboratories met in New York City on July 19, 1963 to review cross section libraries and
discuss means for interchanging these libraries. A sub-committee was appointed to meet in
Hanford on September 18-20, 1963 to examine library formats in more detail.
The conclusions of these discussions were that there was a need for a standard format
for evaluated nuclear data and the format should be as flexible as possible so that existing
libraries could be translated into the standard format and that future needs could be easily
incorporated into the file. This standard format would serve as a link between a data
library and the processing codes. It was also suggested that a center should be established
and charged with the development and the maintenance of the Evaluated Nuclear Data File
(ENDF) and with the collection and distribution of data.
A preliminary report of the detailed formats for ENDF was sent for review and comment.
Twenty-two people attended a final meeting at Brookhaven on May 4-5, 1964 to discuss
changes and settle on a final version. The description of this system (which was labeled
Version A and referred to as ENDF/A) was documented in the report BNL-8381 [4]. The
ENDF/A file originally contained an updated version of the UKNDL library as well as
evaluated data from a number of different laboratories.
The reactor designer wants evaluated data for all neutron-induced reactions covering the
full range of incident neutron energies for each material in a reactor. However, evaluators
usually supply “bits and pieces”, which are put together to form a fully evaluated set for a
given material. ENDF/A provides a storage system for these “bits and pieces” or partial

363

APPENDIX I. HISTORIC PERSPECTIVE BY NORMAN E. HOLDEN
evaluations.
In addition to the need to allow all nuclear data evaluations to be placed on a common
basis, there was also a need for an evaluated nuclear data file to be used for reactor design
calculations. The description of this system (which was labeled Version B and referred to
as ENDF/B) was documented in the report BNL-50066 [5].
Where the format of ENDF/A was highly flexible in order to accept data in almost any
arrangement or representation, the format of ENDF/B had to be simple to facilitate the
writing of processing programs to use the data. This new ENDF/B library format would
be mathematically rigorous, with specific interpolation schemes between tabulated points,
so that cross section integrals, products and ratios would yield well-defined and repeatable
results. There would be codes developed for plotting, integration and other processing of
cross sections that would be written in FORTRAN for computer interchangeability and
distributed to assist others who wanted to use ENDF data.
A material was defined as either an isotope or a collection of isotopes with a material
number designated by the symbol MAT. The data for a material is divided into files with the
file number designated by the symbol MF. A file is subdivided into sections, each containing
data for a particular reaction, where the reaction type is designated by the symbol MT.
File 1 was for general information. File 2 contained information on resolved and unresolved resonance parameters. File 3 contained information on smooth cross sections. File
4 contained information on secondary angular distributions. File 5 contained information
on secondary energy distributions. File 6 contained information on secondary energy-angle
distributions. File 7 contained information on the thermal neutron scattering law.

I.1.1

ENDF/B-I

Version I of the ENDF/B data file was released in July 1968. As mentioned above, the
description of the formats and procedures are documented in BNL-500665.
File 1 had a Hollerith description in Section 1, neutron multiplicity in Section 2, radioactive decay data in Section 3 and fission product yield data in Section 4.
File 2 used two energy ranges, one for resolved and one for unresolved parameters.
File 3 limited quantities to 500 points, except for the scattering cross section where
2000 points were allowed. Temperature dependence was ignored, except for thermal cross
sections.
File 4 would contain mostly data for elastic scattering. Some data may be given for
inelastic or total reaction in the thermal range and may be temperature dependent.
File 5 should have data for inelastic, (n,2n), (n,3n) and fission reactions if smooth cross
sections for these reactions are given in File 3.
The evaluations for ENDF/B-I were taken from the existing evaluations of neutron interactions for 58 materials and converted into the ENDF format. The emphasis in this version
was to create the necessary infrastructure to support the library system.

I.1.2

ENDF/B-II

Version II of the ENDF/B data file was released in August 1970. The description of the
formats and procedures for the neutron data are documented in BNL-50274 [6]. The formats
364

APPENDIX I. HISTORIC PERSPECTIVE BY NORMAN E. HOLDEN
and procedures for photon production and interaction data are documented in LA-4549 [7].
The File 1 changes included the following. An index was added to MT = 451 (general
information). Each record in the index contained a file number (MF), reaction type number
(MT) and the number of card images required to specify the data for each section for the
material. The format for specifying radioactive decay data (MT = 453) was extensively
modified. The format for specifying fission product yield data (MT = 454) was modified to
allow specification of yield data for meta-stable states. A new section was defined to contain
data for delayed neutrons from fission (MT = 455).
The File 2 changes included the following. The test, LRF, which indicated the type
of resolved resonance formula used, was expended to include, LRF = 1, single level BreitWigner parameters; LRF = 2, multi level Breit-Wigner parameters; LRF = 3, R-Matrix,
Reich-Moore multi level parameters and LRF = 4, Adler-Adler multi level resonance parameters are given. All materials will contain a File 2. For those materials where resolved and/or
unresolved resonance parameters are not given, File 2 will contain the effective scattering radius, AP. The previous test, LIS, has been removed, where LIS = 0 indicated the scattering
cross section should be calculated from resonance parameters plus the smooth cross section
from File 3. This meant that the elastic scattering cross section would always be calculated
using the resolved and unresolved resonance parameters. The constant C (used to calculate
the penetration factor) was replaced by a quantity AWRI. AWRI is defined as the ratio of
the mass of a particular isotope to that of a neutron. An option, LRU=2, was added for
specifying the unresolved resonance parameters, so that all average resonance parameters
could be given as a function of incident neutron energy. Energy dependent parameters could
be given for each l-J state.
The File 3 changes included the reaction Q-value was defined as the kinetic energy (in
eV) released by a reaction (positive Q values) or required by a reaction (negative Q values).
The threshold energy (negative Q only) is given by
Eth = |Q|(AW R + 1)/AW R ,

(I.1)

where AW R is the atomic weight ratio given on the HEAD record. The maximum number
of allowed energy points per section was increased from 2,000 to 5,000. An initial state
indicator was added to the HEAD record. This will allow the inclusion of cross section data
for meta-stable states and more than one section may be given for the same reaction type
(MT number).
No changes in File 4.
File 5 changes included the following. The definition of LF=3 (discrete energy loss law)
was changed to read
F (E → E ′ ) = δ[E ′ − E{(A2 + 1)/(A + 1)2 } + A/(A + 1)]θ ,
where A = AW R and θ is the level excitation energy (positive value). T and LT have been
removed from the TAB1 records that contain p(E) for cases in which LF = 5, 7, or 9. A
value, U, replaces T. U was introduced to define the proper upper limit for the secondary
neutron energy distributions so that
0 ≤ E′ ≤ E − U ,
365

APPENDIX I. HISTORIC PERSPECTIVE BY NORMAN E. HOLDEN
Old MT Number
New MT Number
5
51
6
52
7
53
8
54
9
55
10
56
11
57
12
58
13
59
14
60
15
91
27
No longer used
29
No longer used
51
61
52
62
53
63
80
90
109 (Not assigned)
109 (n,3) cross section
455 (Not assigned)
455 delayed neutron from fission
700-799 (not assigned) 700-799 assigned (see appendix B)
where E ′ , E, U are given in the laboratory system. Further, the normalization constants for
LF = 7 and LF = 9 have been redefined to account for the use of U. LF = 2, 4, 6, and 8
have been deleted.
For all files, certain reaction type (MT) numbers have been changed:
The format for specifying temperature dependent data has been modified so that the
data for the second (and higher) temperatures may be given at a lesser number of points
than was given for the first temperature.
This version included evaluations for neutron reactions with fission product nuclei and
data for various components of the energy release in fission.

I.1.3

ENDF/B-III

Version III of the ENDF/B data file was released in late 1972. There was no formal documentation of the formats and procedures for this version of ENDF prepared and published.
For the first time, the “standard” cross sections for neutron-induced reactions were identified. A special purpose “Standard Library” was created and sent to the IAEA for worldwide
distribution. A special purpose library for Dosimetry cross sections was created. New fission
product cross sections were added to improve calculations of decay heat.

366

APPENDIX I. HISTORIC PERSPECTIVE BY NORMAN E. HOLDEN

I.1.4

ENDF/B-IV

Version IV of the ENDF/B data file was released in February 1975. The description of the
formats and procedures are documented in BNL-NCS-50496 [8].
A general change in this version was in the energy range for general-purpose materials,
where the range covered energies from 10−5 eV up to 20 MeV.
The File 1 changes included a change in the formats for specifying radioactive decay.
Section MT = 453 was changed to include only production of radioactive nuclides and
section MT = 457 was added to include radioactive decay data. Section MT = 456 was
added to supply data for the number of prompt neutrons per fission (νp ).
The File 3 changes included the energy mesh for the total cross section must include the
energy meshes for partial cross sections. Time sequential (n,2n) reactions are described by
using sections MT = 6-9 and MT = 46-49. An LR flag was added to designate x in the (n,n′ )
reactions when x is not a photon. In this case, the temperature field S (formerly T) is used
to designate the Q-value or energy difference of the combined reactions. Sections MT = 718,
738, 758, 778, 798 and MT = 719, 739, 759 779 and 799 are redefined to describe continuum
levels for (n, x′ ) reactions. MT = 718 describes the (n,p′c ) continuum cross sections as part
of the (n,p) cross section and should be included in the total cross section. MT = 719 is
used to describe a continuum cross section for exit protons, whose cross section is already
represented in the total cross section by other types.
Until ENDF/B-IV, the only method available to evaluators to communicate uncertainty
information was through the documentation [9]. Francis Perey designed the first data covariance format, which was approved by CSEWG in May 1973 and revised in December
1973.Three general-purpose evaluations for C, for 14 N and for 16 O, were released with covariance files.

I.1.5

ENDF/B-V

Version V of the ENDF/B data file was released in June 1979, which was subsequently named
Version V.0. Version V.1 was released in 1983 and Version V.2 was released in January 1985.
The description of the formats and procedures are documented in BNL-NCS-50496, edition
2 [10].
The File 1 changes included the following. The HEAD card of MT = 451 was changed.
NXC, the number of dictionary entries, had been moved to the sixth field of the hollerith
LIST record of MT = 451. Field 5 now contained NLIB, the library identifier, and field
6 now contained NMOD, the material modification number. Following the HEAD card of
MT = 451 is a new CONT card, which contains information about the excitation energy,
stability, state number, and isomeric state number of the target nucleus. In the LIST record
of MT = 451, the LDD and LFP flags have been abolished. The number of dictionary
entries, NXC, is now in the sixth field of the first card in this LIST record. The fourth
field on each dictionary card in MT = 451 is now used to indicate the modification status
(MOD) for the section described by the card. Radioactive decay data (MT = 453 and 457)
have been removed from File 1. Entirely new formats have been devised and the radioactive
decay data is given in MF = 8, MT = 457. The fission product yields section (MT = 44)
has been removed from File 1. Fission product yield information is now given in File 8 using
367

APPENDIX I. HISTORIC PERSPECTIVE BY NORMAN E. HOLDEN
new formats. A new section to describe energy release in fission (MF = 1, MT = 458) has
been implemented.
The File 2 changes included the restriction that the Reich-Moore resonance parameter
representation is no longer permitted in ENDF/B, only in ENDF/A.
The File 3 changes included the following. The total “gas production” MT’s have been
defined for H(203), D(204), T(205), He-3(206) and He-4(207). The non-elastic cross section
(MT = 3) is now optional and no longer required since total gamma ray production must
be entered in File 13 and never as multiplicities in File 12.
The file 4 changes include a simplified format using a new flag, LI, has been introduced
to indicate that all angular distributions for an MT are all isotropic.
The File 5 changes include only the distribution laws for LF = 1, 5, 7, 9 and 11 are now
allowed. LF = 11 is a new format for an energy dependent Watt spectrum.
The File 8 changes include the following. Information may be given for any MT specifying
a reaction in which the end product is radioactive. The MT section contains information
about the end product and how it decays. Files 9 and 10 may be used to give the cross
section for the production of the end product. Fission product yield information is given
under MT = 454 and 459. The format has been modified to include the 1 σ uncertainty
of the yields. MT = 454 is for independent yields and MT = 459 is for the cumulative
yields. The spontaneous radioactive decay data is given in MT = 457. This is an entirely
new format.
The file 9 and 10 changes include the following. Isomer production is described in the
new File 9 or File 10. In File 9, the cross sections are obtained by use of multiplicities. In
File 10, the absolute cross sections are given.
The File 17 and 18 changes are the following. Formats for time dependent photon
production data files have been defined. They may be used in ENDF/A only.
The Files 19, 20, 21 and 22 changes are the electron production data files have been
implemented.
The Files 31, 32 and 33 changes are the formats for data co-variance files first introduced
in Version IV have been extensively modified and expanded. They are now included in this
document for the first time. Twenty-four materials and reactions are available in ENDF-V
format containing co-variance files.
The Department of Energy put restrictions on the distribution of the library for the first
time. This version was the first to have the evaluation of standards completed in advance
of the evaluation of other materials.

I.1.6

ENDF/B-VI

Version VI of the ENDF/B data file was released July 1990, which was subsequently named
Version VI.0. Version VI.1 was released in September 1991 and contained corrective revisions. Version VI.2 was released in June 1993 and provided new evaluations for 24 isotopes.
Version VI.3 was released in May 1995 and contained new evaluations for 11 materials,
with some evaluations extending in neutron energy to 40 MeV. This was the first attempt
to include high-energy evaluations. Version VI.4 was released in December 1996 and was
mainly a corrective revision of Version VI.3. Version VI.5 was released over 1997-1998 and

368

APPENDIX I. HISTORIC PERSPECTIVE BY NORMAN E. HOLDEN
included 14 new evaluations, as well as two proton-induced files and one deuteron-induced
file. Version VI.6 was released over 1998-1999 and included 33 evaluations with a neutron
energy range from 20 MeV to 150 MeV, as well as 33 proton-induced evaluations up to 150
MeV. Version VI.7 was released over 1999-2000 and included 17 new evaluations of fission
products, as well as general-purpose evaluations. Finally, Version VI.8 was released in 2001
and included 8 new evaluations and 33 evaluations, which were modified to include thermal neutron photon production data. The description of the formats and procedures are
documented in BNL-NCS-44945 [11].
The File 3 changes include the following. The limit on the number of energy points (NP)
is changed from 10,000 to 50,000 points.
The file 4 changes include the following. The highest order Legendre polynomial (NL),
(given at each energy) is 64.
The File 5 changes include the following. There is a correction of the LF = 1 example,
so that all secondary energy distributions start and end with zero values for the distribution
function.
The File 6 changes include the addition of the reference system for secondary energy
and angle, LCT = 3, for the center of mass system for both angle and energy of light
particles (A ≤ 4), laboratory system for heavy recoils (A > 4). This is for use in continuum
energy-angle distribution when Kalbach-Mann systematics are not used, LANG6=2.

I.1.7

ENDF/B-VII

The general-purpose evaluated nuclear data file, ENDF/B-VII.0, was released in December
2006 [12]. The major advances over the ENDF/B-VI.8 library are new cross sections for
actinide isotopes, more precise standard cross sections, improved thermal neutron scattering,
more extensive neutron fission product cross sections, photonuclear reactions, extension of
evaluations to 150 MeV, new light element reactions, post-fission beta-delayed photon decay
spectra, new radioactive decay data, some sample covariance evaluations and new actinide
fission energy deposition. The library is in the same ENDF-6 format as the earlier ENDF/BVI library.

I.2

Status of ENDF Versions and Formats

In order to summarize the status of the various versions of ENDF that have been released
with the formats and procedures and their documentation, which have been referenced
above, we have the following short list.
• ENDF/B.I was released in ENDF-1 format [5].
• ENDF/B.II and ENDF/B.III were released in ENDF-2 format [6, 7].
• ENDF/B.IV was released in ENDF-4 format [8].
• ENDF/B.V was released in ENDF-5 format [10].
• ENDF/B.VI and ENDF/B.VII were released in ENDF-6 format [11].
369

APPENDIX I. HISTORIC PERSPECTIVE BY NORMAN E. HOLDEN

I.3

CSEWG

The recommendation from the original discussion on ENDF for a center to maintain the File
was met by the assignment of the task to the Brookhaven National Laboratory’s Nuclear
Data Center. The group that would undertake to develop ENDF/B was called the Cross
Section Evaluation Group (CSEWG). It was organized under the sponsorship of the Division
of Reactor Development and Technology of the U. S. Atomic Energy Commission (AEC).
The first meeting was held at BNL on June 9-10, 1966, where representatives of sixteen
laboratories attended the meeting. A list of materials was submitted for which data were
needed. Laboratories volunteered to take responsibility for complete evaluations of materials
important to their own laboratory’s programs.
At the following CSEWG meeting on November 14-16, 1966, material evaluations had
already been submitted and progress had been made to couple ENDF format to data library
preparation programs for reactor physics calculations. Several CSEWG subcommittees were
started that would provide guidance for the development of ENDF/B. One of these first
subcommittees created was that of Codes and Formats. Its mission was to make all necessary
revisions to assure compatibility of edit and retrieval codes with the ENDF/B and provide
guidance for future code development.

I.3.1

Codes and Formats Subcommittee Leadership

• Henry Honeck (SRL), from November 1966 to September 1969
• Robert Dannels (WNES), from November 1969 to December 1973
• Donald Mathews (GA), from December 1973 to May 1976
• Raphael LaBauve (LANL), from May 1976 to May 1980
• Raphael LaBauve (LANL) and Robert Roussin (ORNL), from May 1980 to October
1995.
Starting in 1995, the activity level of CSEWG had dwindled and the subcommittee
structure was abandoned. Individuals were assigned the responsibilities originally given to
the subcommittees.

370

Bibliography
[1] H. Goldstein, “Panel Discussion”, Proc. Conf. Neutron Cross-Sections and Technology,
NBS-299 2 1309 (1968).
[2] K. Parker, “Physics of Fast and Intermediate Reactors”, IAEA, Vienna, 3-11 August
1961 (1962).
[3] J.S. Story, M.F. James, W.M.M. Kerr, K. Parker, I.C. Pull and P. Schofield, Proc. 3rd
Int. Conf. Peaceful Uses of Atomic Energy, Geneva, 31 August - 9 September 1964 (1965).
[4] H.C. Honeck, “ENDF, Evaluated Nuclear Data File, Description and Specifications”,
BNL-8381, Brookhaven National Laboratory (January 1965).
[5] H.C. Honeck,“ENDF/B - Specifications for an Evaluated Nuclear Data File for Reactor
Applications”, BNL-50066, Brookhaven National Laboratory (May 1966), revised by S.
Pearlstein (July 1967).
[6] M.K. Drake, “Data Formats and Procedures for the ENDF Neutron Cross Section Library”, BNL-50274, Brookhaven National Laboratory (October 1970).
[7] D.J. Dudziak, “ENDF Formats and Procedures for Photon Production and Interaction
Data”, LA-4549, ENDF-102, Rev., Vol. II, Los Alamos Scientific Laboratory (July 1971).
[8] D. Garber, C. Dunford and S. Pearlstein, “ENDF-102, Data Formats and Procedures
for the Evaluated Nuclear Data File, ENDF”, BNL-NCS-50496, Brookhaven National
Laboratory (October 1975).
[9] G. Perey, “Expectations for ENDF/B-V Co-variance Files: Coverage, Strength and Limitations”, pp. 311-318 in C.R Weisbin, R.W. Roussin, H.R. Hendrickson and E.W. Bryant,
“A Review of the Theory and Application of Sensitivity and Uncertainty Analysis: Proceedings of a Seminar-Workshop”, Oak Ridge, Tennessee, 22-24 August 1978 (February
1979).
[10] R. Kinsey, “ENDF-102, Data Formats and Procedures for the Evaluated Nuclear Data
File, ENDF”, BNL-50496 2nd Edition, Brookhaven National Laboratory (October 1979),
revised by B. Magurno (November 1983).
[11] P.F. Rose and C.L. Dunford, “ENDF-102, Data Formats and Procedures for the Evaluated Nuclear Data File, ENDF-6”, BNL-NCS-44945, Brookhaven National Laboratory
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BIBLIOGRAPHY
(July 1990), revised (October 1991), revised (November 1995), revised by V. McLane,
C.L. Dunford and P.F. Rose (February 1997).
[12] M.B. Chadwick, P. Obložinský, M. Herman, et al., “ENDF/B-VII.0: Next Generation Evaluated Nuclear Data Library for Nuclear Science and Technology”, Nuclear Data
Sheets 107 2931-3060 (2006).

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