Fit-MB

A Python 3 script for (hassle-free) fitting of Mößbauer (MB) spectra.

It is easy to use but has some limitations. The script is limited to doublets (with no intensity differences) and singlets, cannot handle magnetic properties and is restricted to Lorentzian lines shapes.

External modules

lmfit numpy scipy matplotlib tabulate

Quick start

1. Edit or create a parameter file (e.g. mb-param.text) with a text editor:

   #...
   #
   MB-data = example_data.dat
   # 
   # ⇦ just a comment
   #...
   #----------------------------------------
   # label |   δ    |  ΔEQ  | fwhm  | ratio
   #----------------------------------------
   L1Fe      -0.12    2.92    
   L2Fe      -0.09    2.21   
   # adjust    ⇧       ⇧
   

   example_data.dat is the file that contains data from measurement. First column should contain velocity, second intensity. Recognized delimiters are , or space(s).

   The script can also process WissEl data (.ws5) directly. Three additional parameters from a calibration (folding point: FP, channel in which the velocity is zero: v0, and maximum velocity: vmax) must be included in the parameter file. It is also necessary to change the number of channels directly in the script under N_chan, if the number of channels is different from 512. You can obtain these parameters with mcal or cal-mb, for example.

   #...
   # Note that FP, v0 and vmax must be specified. 
   MB-data = example_data.ws5
   FP = 256.621
   v0 = 125.282
   vmax = -4.2622 
   # 
   # ⇦ just a comment
   #...
   #----------------------------------------
   # label |   δ    |  ΔEQ  | fwhm  | ratio
   #----------------------------------------
   L1Fe      -0.12    2.92    
   L2Fe      -0.09    2.21   
   # adjust    ⇧       ⇧
   

   The number of labels (e.g. L1Fe) is equivalent to the number of MB active species. Furthermore rough estimates of $δ$ (isomer shift) and $ΔE_Q$ (quadrupol splitting) are required.

   If the number of species is insufficient add or remove (or comment #) them in the parameter file (e.g. mb-param.text) and restart the script as described under 2.

2. Start the script with:

   python3 mb-fit.py mb-param.txt

   After the matplotlib window has opened, select a species in the upper legend. Simply click on the label or the radio button. Adjust parameters for each species with the sliders at the bottom of the window. It should roughly agree with what you expect. You can also fix some of the parameters, but in general this is not necessary. Then press the Fit button.

   You can also try to fit the data without adjusting any parameters. Simply click Fit. In this example no alteration of the start parameters was necessary.

3. The result of the fit is displayed in the lower area, parameters and statistics are also displayed in the terminal (console, cmd, std.out). Check if the results are okay. Otherwise adjust the sliders and click the Fit button again.

   

   Terminal output:

   # Fit report for example_data
   ## File statistics:
   MB data     : example_data.dat
   data points : 256
   variables   : 9
   
   χ²          : 8.3411e-05
   red. χ²     : 3.3770e-07
   R²          : 0.9952
   
   ## Fit results:
   data in 1σ  : 30
   data in 3σ  : 101
   y0          : 0.9995±0.0001
   
   |   species |    δ /mm·s⁻¹ |   ΔEQ /mm·s⁻¹ |   fwhm /mm·s⁻¹ |   r (area)/% |   r (int)/% |
   |-----------|--------------|---------------|----------------|--------------|-------------|
   |      L1Fe |  0.669±0.016 |   1.637±0.026 |    0.877±0.049 |   20.26±0.93 |       19.85 |
   |      L2Fe | -0.170±0.002 |   2.240±0.003 |    0.519±0.006 |   79.74±0.93 |       80.15 |
   

4. Finally save the results by clicking on the Save button. The optimized parameters will be saved in mb-param-fit.txt, a fit report in example_data-report.txt (similar to the last console output), raw data and the fitted curves in example_data-fit.dat (a file which you can open in Gnuplot, Excel or Origin for example) and the content of the lower plot area in example_data-fit.png.

Terminal output:

mb-param-fit.txt saved.
example_data-report.txt saved.
example_data-fit.dat saved.
example_data-fit.png saved.

In the case of WissEl data, the folded spectrum is also saved (same output as above, plus):

example_data-fold.dat saved.

7. Exit.

Command-line options

• filename, required: filename, e.g. mb-param.txt. A file, that contains the name (and location) of the file that contains MB data and start parameters for the fit.

Parameter file

Below is a sample parameter file with all necessary information. Important are the name (and location) of the file that contains MB data (MB-data = example_data.dat) and start parameters for the fit. The term MB-data = must not be changed. In case of raw data from a multi-channel analyzer (WissEl .ws5 files for example), the terms FP = , v0 = and vmax = must not be changed. You can obtain these parameters from a calibration with mcal or cal-mb, for example.

Only label, $δ$ and $ΔE_Q$ are essential and the labeling must be unique. Every time something has changed in the parameter file, the script must be restarted.

Simply add or remove # before a label, to include or exclude a MB active species.

It is strongly recommended to have the parameter file and the data file in the same directory.

#==============================================================================
# Example of a MB parameter file for fit-mb.py
#==============================================================================
# This is a comment.
# Lines starting with '#' are ignored by the script.
#
# The MB (raw) data file should contain 'velocity' (1st column) and 
# 'intensity' (2nd column). Further columns and lines starting with '#' 
# are ignored. Recognized delimiters are ',' or ' ' (whitespace(s)).
# WissEl files '.ws5' are also accepted. 
# For WissEl '.ws5' files, 'FP' (folding point), 'v0' (channel in which 
# the velocity is zero) and 'vmax' (maximum velocity) must also be specified.
#
# The terms 'MB-data = ', FP =  ', 'v0 = ', and 'vmax = ' must not 
# be changed since they are recognized by the script. 
#
MB-data = example_data.dat
FP = 256.621
v0 = 125.282
vmax = -4.2622 
#
# The start parameters for the fit must be entered in the following order:
#
# label_1 δ_1 ΔEQ_1 fwhm_1 ratio_1
# label_2 δ_2 ΔEQ_2 fwhm_2 ratio_2
# label_3 δ_3 ΔEQ_3 fwhm_3 ratio_3
# ...
# 
# label =  unique atom / compound name
# δ     =  isomeric shift in mm/s
# ΔEQ   =  quadrupole splitting in mm/s; should be positive
# fwhm  =  full width at half maximum; line width for broadening
# ratio =  ratio of the MB active compound / nucleus
#
# At least one species with label, δ, ΔEQ must be defined.
# The labeling must be unique (e.g. 'Fe1', 'Fe2', ...). Identical labels
# (e.g. 'Fe1', 'Fe1', ...) lead to errors. Labels should be without
# spaces (e.g. 'LFe1' or 'LFe_1' instead of 'LFe 1').
#
# 'fwhm' and 'ratio' are optional. If there is no value for 'fwhm', 
# but there is a value for 'ratio', 'ratio' is considered to be 'fwhm', 
# because the third parameter in the line is assumed to be 'fwhm'.
# 
# If 'fwhm' is not specified it is set to 0.1. If 'fwhm' is not specified 
# then 'ratio' cannot be specified (see above remark).
# 
# If 'ratio' is not specified it is set to 0.1.
# Defining a ratio different from 1 or 100% can be useful if there is 
# a main component and an impurity which is also MB active
# or a mixture of two or more compounds with MB active nuclei. 
# However, setting a 'ratio' as start parameter is just for orientation,
# since the fit starts with a fixed value.
#
#----------------------------------------
# label |   δ    |  ΔEQ  | fwhm  | ratio
#----------------------------------------
#L1Fe      0.24    1.51    0.33    0.40
#L2Fe      0.25    3.12    0.39    0.35
#L3Fe     -0.03    1.24    0.44    0.12
#L4Fe     -0.06    0.61    0.37    0.13
L1Fe       0.264   2.321   0.434   0.492
L2Fe       0.279   1.551   0.427   0.508

MB data file

Below is a sample data file. Lines starting with # are ignored. Only the first two columns are considered. The first column must contain the velocity, the second column must contain the intensity data. Recognized delimiters are , and spaces.

# filename
# sample: sample name
# 80 K : temperature
# 0 T  : field
# 125.69, -4.693
# ---------------------------
 -4.5893,   587126.4,       1
 -4.5525,   588314.7,       2
 -4.5157,   586286.4,       3
 -4.4789,   586907.1,       4
 -4.4421,   587216.4,       5
 -4.4052,   586408.4,       6
....

WissEl ws5 file

Below is a sample WissEl ws5 file. Lines starting with < are ignored. It is necessary to change the number of channels directly in the script under N_chan, if the number of channels is different from 512.

In principle any raw data (not only WissEl) can be processed, as long as FP, v0, vmax are included in the parameter file.

<?xml version="1.0" encoding="UTF-8" standalone="no" ?>
<wissoft version="1.1">
<comment>http://www.wissel-gmbh.de</comment>
<data channels="512" time="0">
132130
132207
131879
132294
132142
....

Pre-fit plot

The pre-fit plot is in the upper part of the matplotlib window. The active species is selected in the legend of plot. The plot is updated if any of the parameters is changed using the sliders at the bottom of the window. The purpose of the pre-fit plot in combination with the sliders is to set the fitting parameters so that they roughly reflect the shape of the measured data.

If errbar_ws5 = True is set in the script, error bars are displayed for folded raw data (e.g. WissEl .ws5).

The most important parameters to be changed are $δ$ and $ΔE_Q$. Fwhm is recognized by the fitting procedure, but is less important. The ratio is only for orientation and is ignored as a start parameter for the fit.

Adjustment sliders and check (fix) buttons

There are four separate sliders for $δ$ (isomer shift), $ΔE_Q$ (quadropule splitting), fwhm (full width at half maximum), and ratio (ratio of the species / component). For the fit, the start value for ratio is always set to 0.1 regardless of the slider value. Changing the ratio is therefore only for orientation.

Three parameters can be fixed at a certain value with the check buttons fix $δ$, fix $ΔE_Q$, and fix fwhm. Fixed values are not changed during fit.

Select the species/component in the legend of the diagram at the top and change the parameters for each component individually.

The most important parameters to be changed are $δ$ and $ΔE_Q$. Fwhm is recognized by the fitting procedure, but is less important. The ratio is only for orientation and is ignored as a start parameter for the fit.

If $ΔE_Q$ is zero or close to zero and the fit fails, $ΔE_Q$ should be fixed around 0. In a subsequent fit, this fix can often be removed.

Curve fitting

After clicking on the Fit button, (raw) data, the fitted curves for each component, the resulting curve and the residuals are displayed in the lower area of the plot window.

The ratio of the components (in %), $δ$ and $ΔE_Q$ (in mm/s) and $R^2$ are given in the legend.

Optionally, the 3σ uncertainty band can be displayed (set plot_3s_band = True in the script).

If the result is not suitable, the parameters should be adjusted with the sliders and the Fit should be restarted.

If a fit fails or is poor, try changing the number of species (components), $δ$, $ΔE_Q$ and fwhm in that order.

The terminal provides a more detailed fit report.

# Fit report for example_data
## File statistics: 
MB data     : example_data.dat     ⇦ name of the file that contains the (raw) data
data points : 256                  ⇦ number of data points
variables   : 9                    ⇦ number of variables

mean σ data : 9.2035e-04           ⇦ in case of .ws5 data (`weights` for χ² and red. χ²)
χ²          : 8.3411e-05           ⇦ Chi square(d); close to the numer of data in case of .ws5 
red. χ²     : 3.3770e-07           ⇦ reduced Chi square(d); close to 1 in case of .ws5 
R²          : 0.9952               ⇦ R square(d)

## Fit results:
data in 1σ  : 30                   ⇦ data points in 1σ (optional)
data in 3σ  : 101                  ⇦ data points in 3σ (optional)
y0          : 0.9995±0.0001        ⇦ y0±error (offset)

|   species |    δ /mm·s⁻¹ |   ΔEQ /mm·s⁻¹ |   fwhm /mm·s⁻¹ |   r (area)/% |   r (int)/% |
|-----------|--------------|---------------|----------------|--------------|-------------|
|      L1Fe |  0.669±0.016 |   1.637±0.026 |    0.877±0.049 |   20.26±0.93 |       19.85 |
|      L2Fe | -0.170±0.002 |   2.240±0.003 |    0.519±0.006 |   79.74±0.93 |       80.15 |
      ⇧             ⇧               ⇧                ⇧               ⇧              ⇧
    label        δ±error       ΔEQ±error         fwhm±error     ratio from      ratio from  
                                                                  area          integral 

A good fit result has a $R²$ close to 1 (> 0.98) and a χ² below 1e⁻³ or less. But this strongly depends on the quality of the measured data.

In case of unfolded data, uncertainties (standard deviations) are calculated from the difference of the intensities from the left-hand side and the right-hand side of the unfolded spectrum. χ² and red. χ² a are then weigthed by the mean standard deviation (or square root of the mean variance of all data pairs). χ² should be close to the number of data and red. χ² should be close to 1 for a good fit result.

χ² from lmfit:
$$\chi^2 = \sum_{i}^N [\rm Residuals_i]^2$$
(weighted in case of .ws5 files)

red. χ² from lmfit $$\chi^2_\nu = \chi^2 / (N-N_{\rm varys})$$
$N$ is the number of data points and $N_{varys}$ is number of variable parameters.
(weighted in case of .ws5 files)

$R²$: coefficient of determination

Data points in 1σ or 3σ are optional (set print_in_sigma = True in the script).

r (area) in % is the ratio of a component calculated by the area of the individual component divided by the sum of all components.
r (int) in % is the ratio of a component calculated by the integral (area under the curve) of an individual component divided by the total area / integral of all components. Integrals or areas under curves are calculated with the trapezoidal rule. r (int) is often closer to the values calculated by the mfit2 program.

The errors are calculated by the lmfit module.

Saving results

After clicking the Save button the following files are saved:

parameterfile-fit.txt          ⇦ new parameter file with fitted parameters
data_filename-report.txt       ⇦ fit report; similar to the last terminal output
data_filename-fit.dat          ⇦ a file that contains all data from fit and raw data
data_filename-fit.png          ⇦ exactly the plot (as PNG) in the lower window
data_filename-fold.dat         ⇦ folded spectrum (only if a WissEl ws5 file has been processed)

In case of the parameter file -fit is added to the filename of the new parameter-file. The filename (without extension) of the file that contains the measured data is the prefix for the report, data and plot files. -fit is added to the prefix in case of the latter two files. The folded spectrum is saved, if if a WissEl ws5 file has been processed (-fold is added to the filename).

The file data_filename-fit.dat contains the following data:

#velocity   data       residuals  fit        L1Fe       L2Fe      
-4.58930    0.99832    1.00224    0.99917    0.99924    0.99942
-4.55250    0.99906    1.00149    0.99916    0.99923    0.99942
-4.51570    0.99900    1.00154    0.99916    0.99923    0.99942
-4.47890    0.99909    1.00145    0.99915    0.99922    0.99942
-4.44210    0.99952    1.00101    0.99914    0.99922    0.99942
...
   ⇧          ⇧           ⇧          ⇧          ⇧          ⇧
velocity   raw data   residuals   best-fit   single fit curves 
                      (+ extra)   curve      for each component

To the residuals some extra in y is added to display them above data and fit curves.

Exit

To exit the script, klick on the Exit button or close the matplotlib window.

Known Issues

• If $ΔE_Q$ is 0 or close to 0, the fitting procedure has sometimes problems finding the correct solution. In this case fix $ΔE_Q$ at a value close to zero, Fit, remove fix $ΔE_Q$ and Fit again.
• If $ΔE_Q$ is 0 or close to 0, the error is very large. This results from the calculation of errors in lmfit. There is no solution for this behaviour.
• The script has not been tested with raw data from 1024 channel multi-channel analyzers.

Remarks

• The script is benchmarked against the mfit2 program from Dr. Eckhard Bill. Within the given restrictions, the results for $δ$ and $ΔE_Q$ match down to the second decimal place.
• Raw spectra (WissEl .ws5 for example) are expected to start at channel 1 and be folded to the right.
• χ² and red. χ² are rather meaningless in case of files that contain only velocity and intensity. However, if the fit is good both values get smaller.
• In case of unfolded data, the error can be estimated from the differences in the intensities of the left-hand side and right-hand side sub-spectra. The weighting for χ² and red. χ² is 1 / (mean standard deviation). The mean standard deviation is the square root of the mean variance of two times the intensities of the left-hand side and right-hand side data pairs which are supposed to be equal. χ² is close to the number of data points and red. χ² is close to 1 in case of a good fit. All values are normalized.
  Please note that parameters like $δ$ or $ΔE_Q$ are mainly derived from channel or velocity data (x-values), while only errors from transmission or intensity data (y-values) are taken into account for the weigths of χ² and red. χ².
• R² is calculated by 1 - variance(residual * mean standard deviation) / variance(intensities), because R² is calculated wrongly by lmfit in case of weights.

Example