Stat STEM3.0 Manual

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StatSTEM
User Guide
For StatSTEM v3.0

StatSTEM is freely available under the GNU public license
(GPLv3).

Contents
1 Getting started
1.1 MATLAB . . . . . . . . . . . . . . . . . . . . . . . .

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2 A first look

3 Load and save files

4 Figure options

5 Preparation
5.1 Peak Finder Routine . . . . . .
5.2 Add/Remove peaks . . . . . . .
5.2.1 Type . . . . . . . . . . .
5.2.2 Add . . . . . . . . . . .
5.2.3 Remove . . . . . . . . .
5.2.4 Remove all . . . . . . . .
5.2.5 Select region . . . . . . .
5.2.6 Remove region . . . . .
5.3 Assign column types . . . . . .
5.3.1 Projected unit cell . . .
5.3.2 Auto assign . . . . . . .
5.3.3 Add missing types . . .
5.3.4 Change type to . . . . .
5.4 Image parameters/actions . . .
5.4.1 Pixel size . . . . . . . .
5.4.2 Cut part from image . .
5.4.3 Flip contrast image . . .
5.4.4 Make image background
5.4.5 Normalise image . . . .
5.5 Import peak locations . . . . .

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6 Fit Model
6.1 Standard procedure . . . . . . . .
6.1.1 Background . . . . . . . .
6.1.2 Width of atomic columns .
6.1.3 Test for convergence . . .
6.1.4 Parallel computing . . . .
6.2 Model selection (MAP) . . . . . .

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7 Analysis
7.1 General options . . . . . . . . . . .
7.1.1 Select new model from MAP
7.1.2 Show models from MAP . .
7.2 Index columns . . . . . . . . . . . .
7.2.1 Projected unit cell . . . . .
7.2.2 Start indexing . . . . . . . .
7.3 Atom counting: statistical method .
7.3.1 Pre-analysis . . . . . . . . .
7.3.2 Post-analysis . . . . . . . .
7.3.3 Create 3D model . . . . . .
7.4 Atom counting - Simulation based .
7.4.1 Match with simulations . . .
7.4.2 Create 3D model . . . . . .
7.5 3D model . . . . . . . . . . . . . .
7.5.1 Save model as XYZ . . . . .
7.5.2 Coordination number . . . .
7.5.3 Save coor number as XYZ .
7.6 Strain and more . . . . . . . . . . .
7.6.1 Lattice of type . . . . . . .
7.6.2 Show shift central atom . .
7.6.3 Calculate octahedral tilt . .
7.6.4 Make displacement map . .
7.6.5 Make strain map . . . . . .

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8 Closing StatSTEM

9 Remarks and suggestions

10 References

Appendix

A Structure of saved files

B Indexing/strain mapping

Getting started

The program can be downloaded from the StatSTEM website. One
can choose between running StatSTEM from MATLAB or installing
a compiled version (executable).

1.1

MATLAB

Once all StatSTEM files are copied to your computer, start MATLAB and go to the folder containing the StatSTEM files. The program can be started by running the main script StatSTEM.m.

A first look

In StatSTEM different options are offered for quantifying your images. In general, the quantification consists of 3 steps:
• Preparation
• Modelling
• Analysis
For each step a tab panel is present on the left hand side. On the
right, loaded images and generated results are shown. Multiple images can be loaded. At the bottom left, one can load and store
files. On the bottom, messages are shown to inform you when computations are finished, changes are made, or things went wrong. A
progress bar indicates the status of the computation. As computations are often iterative, the progress bar cannot be used for estimating how much time computations will take.

Figure 2.1: A tab panel can be selected for each quantification step
4

Load and save files

Once StatSTEM is started, you can load your image into the program
by either clicking on the Load button below or the addition tab
in the right panel. StatSTEM supports files having the MATLAB
(.mat) and text (.txt) format. After loading the image, a dialogue
will appear asking for the pixel size. Images and the corresponding
parameters can be saved in the MATLAB format by clicking on the
Save button next to the Load button. In the saved file, all variables
are stored in as StatSTEM class files. When working with these files,
make sure that the StatSTEM folder is always loaded to the path in
MATLAB. In Appendix A more details are given on the information
that these files hold.

Figure 3.1: Buttons for loading and saving files indicated by respectively the green and red circle(s)

Figure options

StatSTEM offers different options to show the images and the parameters you want. For each loaded file, the Select Image panel allow
you to select an image. Different parameters displayed in Select Options can be shown in the image. By hitting the Export button, the
shown image will be opened in a new MATLAB figure which allows
one to save the image in the desired format.

Figure 4.1: On the right hand side of StatSTEM you can select an
image with the desired parameters. The export button, indicated by
the red circle, can be used to open the image in a new MATLAB
figure.

Preparation

Before fitting a Gaussian model to your loaded experimental images,
starting coordinates for the atomic column positions have to be defined (see Fig. 5.3). For this step, different options are available on
the preparation panel.

5.1

Peak Finder Routine

To quickly find the column positions, a peak finder routine is available which searches for local maxima in the image. When using this
option, a new window will open in which the parameters of the peak
finder routine can be tuned. As noise is often present in experimental images, filters can be inserted for smoothing the image. In total
3 different filters can be selected, a gaussian, averaging and a disk
filer. Furthermore a threshold value can be defined, above which the
pixel values should lie. In this window, the routine can be tested for
the selected parameters. Once appropriate parameters are selected,
the parameters can be used to find and export peak locations to
StatSTEM by hitting the Use values button.

5.2

Add/Remove peaks

In this panel, different routines are available to define, remove or,
change starting coordinates manually. By pressing escape the routines will be aborted.
5.2.1 Type
By using this option, starting coordinates can be labelled. This
is particularly useful when your image consists of different column
types, for example columns containing different elements. When
adding column types, StatSTEM will automatically label different
column types by numbers. In the fitting procedure, labelled columns
may be given a special treatment. This will be explained in Section
6.1.2. For now it is important to stress out that the fitted parameters
are more reliable when identifying different column types in your
image.

5.2.2 Add
This option allows one to add manually starting coordinates in the
image. Each added starting coordinate will have the selected column
type label (indicated next to the button). By pressing escape or
clicking outside the image, the routine will be aborted.
5.2.3 Remove
Starting coordinates, of any column type, can be manually removed
by this option. By clicking in the image, the starting coordinate
closest to the cursor will be removed. By pressing escape or clicking
outside the image, the routine will be aborted.
5.2.4 Remove all
All starting coordinates will be removed.
5.2.5 Select region
Select a region in the image in which the starting coordinates should
be maintained. Outside the selected region, all starting coordinates
will be removed. In this routine, the user defines the corner points
of the selected area one by one. A right click will connect the last
defined point with the starting point. By pressing escape, the routine
will be aborted.
5.2.6 Remove region
Select a region in the image in which the starting coordinates should
be removed. In this routine, the user defines the corner points of the
selected area one by one. A right click will connect the last defined
point with the starting point. By pressing escape, the routine will be
aborted.

5.3

Assign column types

Here, different options are listed to deal with image where different
atomic columns types are present. To obtain the most reliable parameter when fitting, it is important to identify the different column
types in your image.
5.3.1 Projected unit cell
In StatSTEM, automatic routine are available that can identify the
different column types in an image when giving a project unit cell.
Here, the relative location of each column in the projected unit cell

together with lattice parameters are given. You can use the buttons New and Delete to add and remove a column, respectively. All
columns expect the first one can be remove by using the button
Clear. You can also give information on the depth location of each
atom in a column by using the button z-information. The functionality will be explained in 7.2.1. A database is added to StatSTEM
for some common materials and viewing directions. Make sure that
in StatSTEM you fill in the correct pixel size of the image as the
lattice parameters in the projected unit cell should be close to the
experimental values.

Figure 5.1: The column locations in a projected unit cell of Au viewed
along the [100]-direction.
5.3.2 Auto assign
By using this option, StatSTEM will try to identify to different column types present in your image. In this procedure, lattice directions are first determined by using the most central coordinate with
its neighbouring coordinates. Here, the projected unit cell is used.
Then columns are indexed with respect to this central coordinate.
In this manner, the position of each column in the project unit cell
is identified and the different column types can be assigned.
5.3.3 Add missing types
By using this option, StatSTEM will use the projected unit cell to
find and add the locations of the missing column types. In this
procedure, lattice directions are first determined by using the most
central coordinate with its neighbouring coordinates. Here, the projected unit cell is used. Then columns are indexed with respect to
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this central coordinate. In this manner, the position of each column
in the project unit cell is identified and the missing column types are
identified and added.
5.3.4 Change type to
By using this option, a region in the image can be selected to change
the type of the columns to the selected type label (indicated next to
the button). In this routine, the user defines the corner points of the
selected area one by one. A right click will connect the last defined
point with the starting point. By pressing escape, the routine will be
aborted.

5.4

Image parameters/actions

In this panel, general image parameters can be changed and image
operations can be executed
5.4.1 Pixel size
Here, the image pixel size can be changed. By changing the pixel
size, starting coordinates and fitted parameters will be rescaled.
5.4.2 Cut part from image
This option allows one to cut out a rectangular region from the image
manually. Define the new corner points in the image.
5.4.3 Flip contrast image
This function reverses the image contrast
5.4.4 Make image background
By using this function, a part of the image intensities can be made
equal to a background intensity. This function is useful to remove
image contributions from neighbouring nanoparticles for example.
First, you define the background value. This can be done by giving
a value, selecting a region in the current image or in another image.
Then a region is selected in the current image where the image intensities should be equal to the given background value. Note, that
when selecting regions, corner points are defined. To close the region, you can right click on the image. The function can be aborted
by pressing escape.

5.4.5 Normalise image
This function allows one to normalise the image intensities by using
the formula:
Iimage − Ivac
(5.1)
Imax − Ivac
If values are unknown, they can be computed from a recorded detector scan/map. An extra option is to convert the normalised image
to electron counts, by using the incident electron dose. The experimental dose can be calculated by using the dwell time (µs) and beam
current (pA).
Inorm =

Figure 5.2: Parameters to normalise an image. Ivac = 0 and Imax = 1.
The option convert calibrated map to electron counts is enabled, to
multiple the image intensities with the dose value 3120.7553.

5.5

Import peak locations

In this section, a MATLAB (.mat) or text (.txt) file can be loaded
containing starting coordinates. The starting coordinates have to
be defined in Ångström in a (n × 2) vector containing the x- and
y-coordinates. Furthermore, another StatSTEM file can be loaded
to use its starting or fitted coordinates.

Figure 5.3: An experimental HAADF STEM image of a Pt/Ir sample
with defined starting coordinates.

Fit Model

From this panel, a Gaussian model can be fitted to the image. Here,
the standard procedure can be used, when all starting coordinates
are defined. Otherwise, the model selection method can be used.

6.1

Standard procedure

In this standard procedure, each atomic column will be modelled as
a Gaussian peak. Detailed information about the fitting routine can
be found in Ref. [1]. The options in this panel are only available if at
minimum one starting coordinate is defined. The model can be fitted
by hitting the Run fitting routine button. When the model is fitted,
the fitted Gaussian model with the fitted coordinates will be shown
(see Fig. 6.1). The fitting procedure can be aborted by hitting the
Abort fitting routine. This button may not always work as MATLAB
sometimes cancels or postpones functions during a computation. A
second (or third) hit usually works. When the procedure will be
aborted, StatSTEM shows a message.

Figure 6.1: The fitted model of an experimental HAADF STEM
image of a Pt/Ir sample with the fitted coordinates of the atomic
columns.
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Once the model is fitted, the fitted coordinates of the atomic columns
and the total intensity of electron scattered by the atomic columns,
the so-called scattering cross-sections, are calculated. These scattering cross-sections can be seen in the histogram of scattering crosssections (see Fig. 6.2).

Figure 6.2: The histogram of scattering cross-sections of the atomic
columns in an experimental HAADF STEM image of a Pt/Ir sample.
In the panel, different options of the fitting procedure may be defined
(by using the Show options button). The following sections discuss
the available options.
6.1.1 Background
In this options, the user can choose to fit a constant background.
If no background will be fitted, a constant value may be given. By
using the Select button, the mean value of a selected region in the
image can be determined.
6.1.2 Width of atomic columns
In the fitting procedure, one may choose to fit Gaussian peaks to all
columns having the same or a different width. In the Same width
option, the estimated Gaussian peaks will have the same width for
columns of the same atom type (as can be labelled in the preparation
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step discussed in Chapter 5). In the Different width option, a different width is estimated for each Gaussian peak of an atomic column.
In the User defined option, the user may define a width for each
column type which will be used to find the atomic column positions.
Usually, the Same width option is used as this is computationally
less demanding and gives good results. Only when no information
is known about the structure under study (the present elements are
unknown), the Different width option is advisable. More information
can be found in Ref. [1].
6.1.3 Test for convergence
This option may be used for testing the correctness of the starting
coordinates and fitting parameters. The number of iterations will be
limited to 4.
6.1.3.1 Re-use fitted coordinates
After a test fit is done, the fitted intermediate coordinates may be
more close the true values than the starting values. Therefore, this
option may be used to make the fitted coordinates the new starting
coordinates.
6.1.4 Parallel computing
For improving computational speed, the fitting procedure uses parallel computing in which the calculations are divided over the different
CPU cores of your computer. The number of CPU cores used for
parallel computing may be reduced to lower the CPU usage during
the fitting procedure (be aware that total calculation time will most
likely increase).

6.2

Model selection (MAP)

This method can be used to automatically identify the atomic columns
and fit a Gaussian model. More information will be given in an updated manual.

Analysis

In this panel, the fitted parameters of the model may be used for
further analysis. In version 3.0, options are available to do atom
counting, column coordinate based analysis and to create a (simple)
3D model. More options may become available in future versions.

7.1

General options

In this panel, general options that influence a further analysis are
given.
7.1.1 Select new model from MAP
When using the model selection (MAP) method to fit a Gaussian
model with the correct number of columns, models a fitted for a
range of number of columns. By using this function, you can select
a new model (with a specific number of columns) that will be used
for further analysis.
7.1.2 Show models from MAP
By using this option, you can open a new window where you can see
the fitted models by the model selection (MAP) method in function
of the number of columns.
Select columns in image
Select a region in the image. The columns in this region will be taken
into account for further analysis. The columns outside this region
are neglected. By pressing escape, the routine will be aborted.
Select columns in histogram
This option allows one to exclude outliers in the histogram of scattering cross-sections. First the lower limit must be defined, then the
upper limit. The scattering cross-sections of the columns that fall
outside the limit are neglected in a further analysis. By pressing
escape, the routine will be aborted.
Select columns on type
By using this option, columns can be selected based on the column
type labels given in the preparations step (see also Chapter 5).

7.2

Index columns

In this panel, you can index the columns in terms of distance from a
reference point. This operation should be done before performing a
column coordinate based analysis (see section 7.6) or creating a 3D
model (see sections 7.3 and 7.4).
7.2.1 Projected unit cell
A necessary input to index columns is the location of the different
column positions in a projected unit cell, mentioned in section 5.3.1.
If one intends to make a 3D model later on, z-information must be
given as is shown in the Fig. 7.1. Then it becomes possible to
define the lattice parameter in c-direction and per column one can
define the depth location of each atom in this column. Atoms in a
column can be added and removed by the buttons New and Delete.
All atoms except the first one can be deleted by using the Clear
button. Each column can be selected from the pop-up window in
the right-top corner. A database is added to StatSTEM for some
common materials and viewing directions. Make sure that you fill
in the correct pixel size of the image as the lattice parameters in the
projected unit cell should be close to the experimental values.

Figure 7.1: A project unit cell can be made by defining the projected
lattice constants (a and b) and the relative positions of the columns.
Additional information can be given on the depth location of the
atoms in the columns.

7.2.2 Start indexing
By using this option, all coordinates will be indexed as a function of
distance in unit cells from a reference coordinate. For this procedure
first a reference coordinate will be chosen, then the lattice directions are found and the indexing procedure starts. If the automatic
routines in StatSTEM fail, you can guide StatSTEM by using the
advanced options listed below. The details of the automatic routine
are described in Appendix B.
Reference coordinate
In this section a reference coordinate can be chosen. A displacement
map is made from this coordinate. Furthermore, this coordinate gets
the index (0,0) during the creation of the strain map. One can choose
between different column types for selecting a reference coordinate.
Automatically StatSTEM will use the most central coordinate as a
reference, this can be change by manually selecting another coordinate.
Direction a lattice
For finding the direction of the lattice, an automatic routine can be
used or a manual input can be given. The automatic routine used
the projected unit cell parameters to identify the lattice direction.
Here, the distance of the neighbouring coordinate with respect to
the reference coordinate is compared to the given lattice parameters
in the projected unit cell. Once the direction is found, the lattice
parameters are automatically improved by fitting (this can be disabled). Here, a box of N unit cells (standard 3 UC to each side)
around the reference coordinate is used for finding the values of the
a (and b) lattice parameter in the image. This option is advised to
be used, as the pixel size recorded by an electron microscope is never
that accurate. Be however, aware that this option changes the values
of a and b and should only be turned off when you are 100% sure
about the pixel size.

7.3

Atom counting: statistical method

In HAADF STEM imaging, a statistics based method is developed
to count the number of atoms based on the scattered cross-sections
(the total intensities of electrons scattered by the atomic columns),
which increase monotonically with thickness [5]. Be aware that
this method is only reliable when only one column type
is present. In this statistics based method, the scattering crosssections are presented in a histogram. Owing to a combination of
experimental noise an residual instabilities, broadened - rather than
discrete - components are observed in such a histogram. Therefore,
these results cannot directly be interpreted in terms of number of
atoms. By evaluation of the so-called integration classification likelihood (ICL) criterion in combination with Gaussian mixture model
estimation, the number of components and their respective locations
can be found. From the estimated locations of the components, the
number of atoms can be quantified. More information can be found
in Refs. [2, 3, 4].

Figure 7.2: The ICL for an experimental image. A local minimum
appears at 10.
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7.3.1 Pre-analysis
Maximum number of components
In order to evaluate the ICL, an upper limit on the number of components must be given. Up to a given number of components, Gaussian mixture models are fitted to the histogram of scattering crosssections and the ICL criterion is determined. A rough estimate for
this upper limit can be obtained by using the shape of the particle
under study.
7.3.2 Post-analysis
After selecting atomic columns and the maximum number of components, the Gaussian mixture model may be fitted to the histogram
of scattering cross-sections by using the Run ICL button. For each
number of components, the ICL-criterion is determined and displayed. For atom counting, one searches for local minima in the ICLcurve (see for example Fig. 7.2). After the computation, StatSTEM
asks the user to select a local minimum. After a local minimum is selected, the experimental image with the corresponding atom counts
will be shown (see Fig. 7.3). The computation of the ICL values can
be aborted by pressing the Stop function button. The button may
not always work as MATLAB cancels many functions during a com-

Figure 7.3: The atom counts of the different atomic columns in the
experimental HAADF STEM image of a Pt/Ir particle.
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putation. A second (or third) hit usually works. Once the procedure
will be aborted, StatSTEM will show a message asking whether the
user wants to select a local minimum in the current ICL graph.
Counting offset
Once a local minimum in the ICL curve is selected, counting results
may be rescaled by giving the counts an offset. This is particularly
useful for thick particles in which no thin columns are present.
Select new ICL minimum
With this option a new local minimum in the ICL curve can be
selected.
7.3.3 Create 3D model
Once the atom are counted per atomic column and the columns are
indexed, a 3D model can be made. Here, the atom are distributed
symmetrically along the z-direction. Note that this 3D model is
only a simple model, to give an idea about the 3D shape
of the particle. It can not be used a final result! Note that
for this function the project unit cell should contain z-information
(see section 7.2.1). The 3D model that will shown is color coded,
meaning that each atom type will have a different color.

Figure 7.4: A 3D model of a Pt/Ir particle.
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7.4

Atom counting - Simulation based

Another method for doing atom counting is by comparing simulated
library’s of scattering cross-sections with the experimentally measured ones.
7.4.1 Match with simulations
By clicking on this button, first a library should be loaded. Then,
atoms counts are computed by comparing the measured scattering
cross-sections from the fitted model to the loaded library of simulated
scattering cross-sections. A MATLAB (.mat) or text (.txt) file can
be loaded with simulated values of the scattering cross-sections. The
simulated values in function of column thickness must be stored in a
(n × 1) vector. In the figure SCS vs. Thickness, the scattering crosssections obtained by the statistics based method can be compared
to the simulated values (see Fig. 7.6).

Figure 7.5: The atom counts of the different atomic columns in the
experimental HAADF STEM image of a Pt/Ir particle obtained by
a comparison with simulated library values of SCS.

Figure 7.6: A comparison between the scattering cross-sections obtained by the statistics-based method (Experiment) and image simulations (Library).
7.4.2 Create 3D model
Once the atom are counted per atomic column and the columns are
indexed, a 3D model can be made. Here, the atom are distributed
symmetrically along the z-direction. Note that this 3D model is
only a simple model, to give an idea about the 3D shape
of the particle. It can not be used a final result! Note that
for this function the project unit cell should contain z-information
(see section 7.2.1). The 3D model that will shown is color coded,
meaning that each atom type will have a different color.

7.5

3D model

In sections 7.3.3 and 7.4.2, it is mentioned that a 3D model can be
made from atom counting results. Once the model is made, this
panel provide options to export the coordinates as an XYZ file or to
calculate the coordination number.
7.5.1 Save model as XYZ
The 3D model can be saved as an XYZ file that can be loaded into
other software packages such as Vesta or VMD.
7.5.2 Coordination number
The function can be used to determine the coordination number of
each atom in the 3D model. The coordination number is determined
by calculation the number of neighbours each atom√has within a
specific radius. In a fcc material this radius is r = a/ 2.

Figure 7.7: The coordination number for (a) an FCC and (b) an
BCC crystal can be calculated
by√searching for neighbouring atoms
√
within a radius of a/ 2 or 2 ∗ a/ 3, respectively.
Radius
In StatSTEM, the standard
√ radius that is used is a ∗ 0.8, which is
a little bit larger than a/ 2 to compensate for small fluctuations in
the atom positions.
Number of atoms
Standard the coordination number will be determined for all atoms.
As this is a demanding calculation, the can leave out the most central
atoms for this calculation. These atoms are determined based on the
distance from the centre of the particle. In this manner, one can
calculate the coordination number only for a fraction of the atoms
in the particle.
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Figure 7.8: A 3D model of a Pt/Ir particle indication the coordination number per atom.
7.5.3 Save coor number as XYZ
Once the coordination number is determined, it can be saved as a
XYZ file. Here, a specific atom type is given in function of the coordination number. The types as a function of coordination number
are listed below in Table 7.1.
Coordination
number
1
2
3
4
5
6

Atom type

Coordination
number
7
8
9
10
11
12

V
Mg
Au
Na
Se
Zr

Atom type
Lu
Yb
Al
Np
Ho
Co

Table 7.1: Atom type per coordination number that is used when
storing the 3D model with coordination numbers as a XYZ file.
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7.6

Strain and more

In this panel, different methods are available to use the fitted coordinates of the model for a further analysis. Here, the lattice parameter,
displacement of atoms, octahedral tilt and strain map can be made.
7.6.1 Lattice of type
This function determines the lattice parameter per column type. Automatically a coloured plot is made that show the lattice parameter,
as is illustrated in Fig. 7.9. When this function is used, one can also
look at plots where the lattice parameters as a function of distance
from the reference coordinate are shown, both in a- and b-direction.

Figure 7.9: The lattice parameters a and b of a Pt/Ir nanoparticle.
The options to plot the lattice parameter a or b as a function of
distance in the a- or-direction are indicated by the red circle.

7.6.2 Show shift central atom
By clicking on this button a displacement map will be made for
the central atoms in a unit cell based on the column indexing described in section 7.2. By using the values of the projected unit cell,
the expected coordinate are calculated. The displacement map is
generated by comparing the expected coordinate with the measured
coordinates.

Figure 7.10: The displacement of the central atom in PbCsBr3 .

7.6.3 Calculate octahedral tilt
In perovskite particles, there is an oxygen octahedra present surrounding the B-cation (or A-cation). Due to internal strain, the oxygen octahedral can rotate. From the indexed columns, octahedral
tilt can be determined when there are atoms present at the relative
positions in the unit cell: (0,0.5) and (0.5,0). If this condition is satisfied, this function becomes available to determine the octahedral
tilt as a function of distance in the a- and b-direction. The distance
is measured from the reference coordinate selected when indexing the
columns, as described in section 7.2. In the plot, the octahedral tilt
is calculate where it is assumed that the octahedral tilt is alternation
between a clockwise and a anti-clockwise rotation.

Figure 7.11: The octahedral tilt along the b-direction in PtTiO3 .

7.6.4 Make displacement map
By clicking on this button a displacement map will be made based on
the column indexing described in section 7.2. By using the values of
the projected unit cell, the expected coordinate are calculated. The
displacement map is generated by comparing the expected coordinate
with the measured coordinates.

Figure 7.12: A displacement map of a PbCsBr3 particle.

7.6.5 Make strain map
By clicking on this button, the xx , xy , y and ωxy strain maps will
be generated. Here, the derivative of the displacement map is used
[6]. An example is shown in Fig. 7.13.

Figure 7.13: An xx strain map of a Pt/Ir particle.

Closing StatSTEM

StatSTEM can be closed by hitting the red cross in the top-right
corner. When closing StatSTEM, make sure that all files are saved
as no warnings of unsaved files are given.

Remarks and suggestions

When downloading MATLAB, a folder with examples is included. In
this folder MATLAB scripts are present to use StatSTEM without
the graphical user interface. Our StatSTEM website offers a forum
in case you have any questions, remarks or suggestions. New releases
will be announced on this website.

References

[1] Annick De Backer, Karel H. W. van den Bos, Wouter Van den
Broek, Jan Sijbers, and Sandra Van Aert. StatSTEM: An efficient
approach for accurate and precise model-based quantification of
atomic resolution electron microscopy images. Ultramicroscopy,
171:104–116, 2016.
[2] S. Van Aert, K. J. Batenburg, M. D. Rossell, R. Erni, and
G. Van Tendeloo. Three-dimensional atomic imaging of crystalline nanoparticles. Nature, 470:374–377, 2011.
[3] S. Van Aert, A. De Backer, G. T. Martinez, B. Goris, S. Bals,
and G. Van Tendeloo. Procedure to count atoms with trustworthy
single-atom sensitivity. Physical Review B, 87(064107), 2013.
[4] A. De Backer, G. T. Martinez, A. Rosenauer, and S. Van Aert.
Atom counting in HAADF STEM using a statistical model-based
approach: methodology, possibilities, and inherent limitations.
Ultramicroscopy, 134:23–33, 2013.
[5] G. T. Martinez, L. Jones, A. De Backer, A. Bch, J. Verbeeck,
S. Van Aert, and P. D. Nellist. Quantitative STEM normalisation: The importance of the electron flux. Ultramicroscopy,
159:46–58, 2015.
[6] Pedro L Galindo, Slawomir Kret, Ana M Sanchez, Jean-Yves
Laval, Andrés Yáñez, Joaquı́n Pizarro, Elisa Guerrero, Teresa
Ben, and Sergio I Molina. The peak pairs algorithm for strain
mapping from hrtem images. Ultramicroscopy, 107(12):1186–
1193, 2007.

Appendix A

Structure of saved files

StatSTEM files may be saved in a MATLAB (.mat) format. In this
file different structures are created in which all variables are stored. A
stored files belong to a class named StatSTEMfile. For working with
the files, make sure StatSTEM is loaded to the path of MATLAB.
For each step or analysis, a structure is created. In the class files,
a description is given of the parameters (what the hold and in what
the purpose is).

Appendix B

Indexing/strain mapping

Statistical parameter estimation theory is capable of extracting atomic
column positions with high accuracy and precision from (S)TEM images. A direct comparison of the measured column positions with
the expected column positions of an ideal crystal lattice gives the
displacement of the atomic columns (see Fig. B.1(b)). By using a
first derivative, these measured displacement vectors can be used to
compute atomically resolved strain maps (see Figs. B.1(c) - B.1(f)).
For creating a displacement map, a projected unit cell is used which
describes the projected lattice parameters and the relative positions
of the different columns. Next, a reference coordinate is given in
an unstrained area. Here, the distances of the nearest neighbouring
atomic columns are compared to the projected lattice parameters
to find the lattice directions in the image (see Fig. B.1(a)). These
lattice directions are used to predict the column positions of an ideal
structure. Since the pixel size given by the microscope may contain
inaccuracies, the dimensions of the lattice parameters are refined by
fitting the ideal column positions to the measured columns positions
in a selected area of N ×N unit cells around the reference coordinate.
Standard, an area of 3 × 3 unit cells is used, as is indicated by the
blue region in Fig. B.1(a).
After the lattice parameters are found, the column positions of an
ideal crystal are predicted. As the crystal structure under study may
be heavily strained, measured column positions can be shifted by
distances larger than the lattice parameter, hampering a direct identification of the predicted and measured column positions. Therefore, starting from the reference coordinate the neighbouring atomic
columns are identified by using the refined lattice parameters. Next,
the identified columns serve as new starting points to identify their
neighbouring columns. This procedure continues until the boundary of the image or the edge of the particle is reached. During this
process, lattice parameters and their directions are continuously updated as strain is usually non-uniform throughout a particle. Here,
the lattice parameters a = (ax , ay ) and b = (bx , by ) in the a- and
b-lattice directions are updated by using a scaling factor ν:

(new)
(old)
(meas)
(old)
a
=a
+ a
−a
·ν
(B.1)
35

Figure B.1: Strain mapping procedure on an experimental HAADF
STEM image of a CsPbBr3 nanoparticle. (a) The procedure start
by selecting a reference coordinate, indicated by the red cross. Next,
the dimensions of a projected unit cell of the ideal structure are used
to find the lattice directions a and b. As the pixel size in the image
might be inaccurate, a region of 3 × 3 unit cells, indicated in blue, is
selected around the reference coordinate to refine the lattice parameters by using a fitting procedure. (b) A displacement map, created
by comparing the measured column positions to the positions in an
ideal unstrained structure. The first derivatives of the displacement
map to the different lattice directions are used to determine the (c)
xx , (d) yy , (e) xy strain maps and (f) ωxy rotation map

(x2,y2)
v
by

(x1,y1)

a
bx

(x0,y0)

Figure B.2: Diagram of the calculation of displacement vectors u
and v from two positions in the a and b lattice direction.
where a(new) is the new lattice parameter, a(old) is the old lattice
parameter and a(meas) is the measured lattice parameter. When all
the measured atomic columns are identified, the displacement map is
generated by determining the distance between the measured column
positions from the image and the predicted column positions for an
ideal structure (Fig. B.1(b)).
In the next step, the measured displacements of the atomic column
positions with respect to an ideal structure are used to calculate
strain maps. Here, the first derivatives of the displacement vectors
with respect to the lattice parameters give the different strain components [6]:
∂u
∂v
, yy =
,
∂x
∂y

1 ∂u ∂v
1
=
+
= (1 + 2 ) ,
2 ∂y
∂x
2

xx =
xy

ωxy

1
= (1 − 2 )
2

(B.2)

where u = (ux , uy ) and v = (vx , vy ) are the displacement vectors
of an atomic column in the a- and b-lattice directions, respectively
(Fig B.2). In the StatSTEM software, the displacement vectors per
37

atomic column are determined by using the average lattice parameters that are obtained by measuring the distance between the selected
column and its neighbouring columns. The equations of the different
strain components can be rewritten in a set of linear equations [6]:

ux = ax xx + ay 1 


uy = ay yy + ax 2
vx = bx xx + by 1 


vy = bx xx + by 2

−1

xx 2
ax ay
ux uy
=
1 yy
bx by
vx vy

(B.3)

From these linear equations the values of the different strain components at each atomic column position can be computed (Figs. B.1(c)B.1(f)). Furthermore, by using the standard deviation of the measured column positions, the error on the measured strain values can
be computed, which is at maximum 0.008 in the strain maps presented in Figs. B.1(c)-B.1(f).

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