| Larch: Data Analysis Tools for X-ray Spectroscopy

# 9.8. XAFS: Fitting XAFS to Feff Paths

Fitting XAFS data with structural models based on Feff calculations is a primary motivation for Larch. In this section, we describe how to set up a fitting model to fit a set of FEFF calculations to XAFS data. Many parts of the documentation so far have touched on important aspects of this process, and those sections will be referenced here.

## 9.8.1. The Feffit Strategy for Modeling EXAFS Data

The basic approach to modeling EXAFS data in Larch (see [Larch_4]) is to create a model EXAFS \(\chi(k)\) as a sum of scattering paths that will be compared to experimentally derived \(\chi(k)\). The model will consist of a set of FEFF Scattering Paths representing the photo-electron scattering from different sets of atoms. As discussed in 9.7, these FEFF Paths have a fixed set of physically meaningful parameters than can be modified to alter the predicted contribution to \(\chi(k)\). Any of these values can be defined as algebraic expressions of Larch Parameters defined by `_math.param()`. The actual fit uses the same fitting infrastructure as to `_math.minimize()` to refine the values of the variable parameters in the model so as to best match the experimental data. Because \(\chi(k)\) has known properties and \(k\) dependencies, it is common to weight and Fourier transform (as described in 9.5) for the analysis. In general term, the the refinement process will compare experimental and model \(\chi(k)\) after a Transformation.

The model for \(\chi(k)\) used to compare to data is

\[\chi(k) = \sum_{j} \chi_{j}(k, p_j)\]

where \(\chi_j\) is the EXAFS contribution for a FEFF Path, as given in 9.7.5 for path \(j\), where \(p_j\) is the set of adjustable Path Parameters (\(S_0^2\), \(E_0\), \(\delta{R}\), \(\sigma^2\), and so on) listed as `Adjustable` in the Table of Feff Path Parameters.

The number of FEFF Paths used in the sum can be unlimited, and they do not need to originate from a single run of FEFF. This can be important in modeling even moderately complex structures as a single run of FEFF is limited to having exactly one absorbing atom in a cluster of atoms and therefore cannot be used to model multi-site systems without multiple runs.

Because a large number of FEFF Paths could be used to model an XAFS spectrum, and because each Feff Path has up to 8 adjustable parameters, there is the potential of having a very large number of parameters for a fit. In principle, However, there is a limited amount of information in an XAFS spectrum. A simple estimate of how many parameters could be independently measured is given from information theory of Shannon and Nyquist as

\[N_{\rm ind} \approx \frac{2 \Delta k \Delta R}{\pi} + 1\]

where \(\Delta k\) and \(\Delta R\) are the \(k\) and \(R\) range of the usable data under consideration. [Larch_5] argues convincingly that the ‘+1’ here should be ‘+2’, but we’ll remain conservative and keep the ‘+1’, and use this value not only for fit statistics but to limit the maximum number of free variables allowed in a fit. In general, this greatly limits the number of parameters thar can be successfully used in a fit. It should be noted that this limitation is inherent in XAFS (and many other techniques that rely on oscillatory signals), and not a consequence of using Fourier transforms in the analysis.

Because of this fundamental information limit, it is usual to purposely limit the spectra being analyzed. Of course, one usually limits how far out in energy to measure a signal based on the strength of the signal compared to some noise level. This limits the \(k\) range of useful data. In addition, the number of scattering paths that contribute to the XAFS diverges very quickly with \(R\). For any \(R\) interval then, the finite \(k\) range sets an upper limit on the number of parameters available for describing the atomic partial pair distribution function that gives spectral weight in that interval. The distance to which a structural model can determined from real XAFS data is therefore limited to 5 or so Angstroms. Because of these inherent limitations, it is generally preferrable to analyze XAFS data by limiting the \(R\) range of the analysis by using Fourier transforms to convert \(\chi(k)\) to \(\chi(R)\). The `feffit()` function allows several choices of data transformations, including doing 0 (\(k\), unfiltered), 1 (\(R\), or Fourier transformed), or 2 (math:q, or Fourier filtered) Fourier transforms. Fitting in unfiltred \(k\) space is generally not recommended.

## 9.8.2. Fit statistics and goodness-of-fit meassures for `feffit()`

The fit done by `feffit()` is conceptually very similar to the fits described in 8.3. Therefore, many of the statistics discussed in 8.4 are also generated for `feffit()`. In view of the limited amount of information, some of the traditional statistical definitions need to be carefully examid, and possibly altered slightly. For example, the typical \(\chi^2\) statistic (as given in 8.1) is

\[\chi^2 = \sum_{i=1}^{N} \big[\frac{y_i - f(x_i, \vec{\beta})}{\epsilon_i} \big]^2\]

seems simple enough, but actually raises several questions, as we have to decide what each of the terms here is. As above, we’ll typically analyze data in \(R\) space after a Fourier transform, so that the “data” \(y\) actually represents the real and imaginary components of \(\chi(R)\), and the “model” \(f\) will also be the Fourier transform of the parameterized model for \(\chi(k)\).

Perhaps the largest questions are ones that are often dismissed as trivial in standard statistics discussions:

Perhaps surprisingly, the first question is: what is \(N\)? The \(\chi(k)\) data can be measured (or sampled) on an arbitrarily fine grid, and a Fourier transform can use an arbitrary number of points. Thus, the number of points for both \(\chi(k)\) and \(\chi(R)\) can easily be changed without actually changing the quality or quantity of the real data. The best number to use for the sum over the number of data points is then \(N_{\rm ind}\) defined above. Of course, we generally oversample the data, so the value for \(\chi^2\) used and reported is

\[\chi^2 = \frac{N_{\rm ind}}{N}\sum_{i=1}^{N} \big[\frac{y_i - f(x_i, \vec{\beta})}{\epsilon_i} \big]^2\]

where the sum can be over an arbitrary number of samples of \(\chi(k)\) or \(\chi(R)\), but the actual range of the data sets \(N_{\rm ind}\).

A second consideration is what to use for \(\epsilon\), the uncertainty in the “data”. Of course, the uncertainty in the data, \(\epsilon\), depends on the details of the data transformation (for example, whether fitting in \(R\) or \(q\) space). Estimating the noise level in any given spectrum is not at all trivial, and should generally involve a proper statistical treatment of the data. For an individual spectrum, what can be done easily and automatically is to estimate the noise level assuming that the data is dominated by noise that is independent of \(R\): white noise. The function `estimate_noise()` does this [Larch_6], and the estimate derived from this method is used unless you specify a value for `epsilon_k` the noise level in \(\chi(k)\). Though usually \(\epsilon\) is taken to be a scalar value, it can be specfied as an array (of the same length as \(\chi(k)\)) if more accurate measures for the uncertainty of the data is available.

It turns out that \(\chi^2\) is almost always too big, and reduced \(chi^2\) (that is, \(\chi^2/(p - N_{\rm ind})\) where \(p\) is the number of fitted parameters) is far greater than 1. This is partly due to a poor assessment of the uncertainty in the data, and partly due to imperfections in the calculations that go into the model. Together, these are often called “systematic errors” in the EXAFS literature. Because of this issue, an alternative statistic \(\cal{R}\) is often used as a supplement to \(\chi^2\) for EXAFS. The \(\cal{R}\) factor is defined as

\[{\cal{R}} = \frac{\sum_{i=1}^{N} \big[{y_i - f(x_i, \vec{\beta})}\big]^2}{\sum_{i=1}^{N} {y_i^2}}\]

which is to say, the misfit scaled by the magnitude of the data.

## 9.8.3. The Feffit functions in Larch

The function `feffit()` is the principle function to do the fit of a set of Feff paths to XAFS spectra. This essentially runs `_math.minimize()` with a parameter group holding all the variable and constrained parameters, and with a built-in objective function to calculate the fit residual. This objective function defines the residual as the difference of model and experimental \(\chi(k)\) for a list of Datasets. Here, a Feffit Dataset is an important concept that will allow us to easily extend modeling to multiple data sets.

While conceptually fairly simple, the approach is quite general and flexible, and the level of flexibility can sometimes be daunting. A Feffit Dataset has three principle components. First, it has an experimental data, \(\chi(k)\). Second, it has a list of Feff paths – that will be used to calculate the model \(\chi(k)\) (using the same calculation as used by `ff2chi()`). Third, it has a Feffit Transform group which holds the Fourier transform and fitting ranges to select how the data and model are to be compared. Since the fit is done with a single group holding all the parameters, it is important that the Path Parameters for all Feff Paths used in the fits should be written in terms of variable parameters in a single parameter group.

There are then 3 principle functions for setting up and executing `feffit()`:

> 1. `feffit_transform()` is used to create a Transform group, which holds the set of Fourier transform parameters, fitting ranges, and space in which the data and sum of paths are to be compared.
> 2. `feffit_dataset()` is used to create a Dataset group, which consists of the three components described above:
>    1. a group holding experimental data (arrays holding `k` and `chi`).
>    2. a list of Feff paths.
>    3. a Transform group.
> 3. Finally, `feffit()` is run with a parameter group containing the variable and constrained Parameters for the fit, and a dataset or list of datasets groups.

## 9.8.4. `feffit_transform()` and the Feffit Transform Group

`_xafs.``feffit_transform`(fitspace='r', kmin=0, kmax=20, kweight=2, ...)

  create and return Feffit Transform group to be used in a Feffit dataset.

  Parameters: | fitspace – name of FT type for fit: one of (‘k’, ‘r’, ‘q’, ‘w’), default ‘r’. kmin – starting k for FT Window (0). kmax – ending k for FT Window (20). dk – tapering parameter for FT Window (4). dk2 – second tapering parameter for FT Window (None). window – name of window type (‘kaiser’). nfft – value to use for \(N_{\rm fft}\) (2048). kstep – value to use for \(\delta{k}\) (0.05). kweight – exponent(s) for weighting spectra by \(k^{\rm kweight}\) (2). rmin – starting R for Fit Range and/or reverse FT Window (0). rmax – ending R for Fit Range and/or reverse FT Window (10). dr – tapering parameter for reverse FT Window 0. rwindow – name of window type for reverse FT Window (‘kaiser’).
  Returns: | a Feffit Transform group

The parameters stored in the returned Feffit Transform Group will be used to control how the fit is performed, including Fourier transform parameters and fit space for a fit. All the arguments passed in will be stored as variables of the same name in the Feffit Transform group. Additional variables may be stored in this group as well.

The returned Feffit Transform Group will have members listed in Feffit Transform Group Members.

> Table of Feffit Transform Group Memebers Most of these parameters follow the conventions for `xftf()` in section on Fourier Transforms for XAFS.

> member name | description
> --- | ---
> fitspace | space used for fitting – one of `'k'`, `'r'`, `'q'`, or `'w'` (`'r'`)
> kmin | starting \(k\) for FT Window (0).
> kmax | ending \(k\) for FT Window (20).
> kweight | exponent(s) for weighting spectra by \(k^{\rm kweight}\) (2).
> dk | tapering parameter for FT Window (4).
> dk2 | second tapering parameter for FT Window (`None` – equal to `dk`).
> window | name of window type for FT (`'kaiser'`).
> kstep | value to use for \(\delta{k}\) (0.05).
> nfft | value to use for \(N_{\rm fft}\) (2048).
> rmin | starting \(R\) for back FT Window (0).
> rmax | ending \(R\) for back FT Window (10).
> dr | tapering parameter for back FT Window (0).
> dr2 | second tapering parameter for back FT Window (`None` – equal to `dr`).
> rwindow | name of window type for back FT (`'hanning'`).

As an important note, `kweight` can either be a single integer value, or a list or tuple of integers. Supplying more than one value will have the effect of having all the kweights used in the fit. If multiple \(k\) weights are provided, they can be in any order – `kweight=(1, 2, 3)` will produce the same result as `kweight=(3, 1, 2)`. Some functions (such as generating output arrays for plotting) may need to assume 1 \(k\) weight – these will take the first one listed.

## 9.8.5. `feffit_dataset()`

`_xafs.``feffit_dataset`(data=None, pathlist=[], transform=None)

  > create a Feffit Dataset group.

  > param data: | group containing experimental EXAFS (needs arrays `k` and `chi`).
  > param pathlist: | list of FeffPath groups, as created from `feffpath()`.
  > param transform:
  > | Feffit Transform group.
  > returns: | a Feffit Dataset group.

  > A Dataset group is pretty simple, initially consisting of sub-groups `data`, `pathlist`, and `transform`, though each of these can be complex.

  > The value for `data` must be a group containing arrays `k` and `chi` (as determined `_xafs.autobk()` or some other procedure). If it contains a value (scalar or array) `epsilon_k`, that will be used as the uncertainty in \(\chi(k)\) for weighting the fit. Otherwise, the uncertainty in \(\chi(k)\) will be estimated automatically, and stored in this dataset.

  > The `pathlist` is a list of Feff Paths, each of which can have its Path Parameters written in terms of fit parameters (see the final example in the previous section). This list of paths will be sent to `ff2chi()` to calculate the model \(\chi\) to compare to the experimental data. Finally, `transform` is a Feffit transform group, as defined above.

  > A Dataset will also have a few other components, including:

  > > component name | description
  > > --- | ---
  > > epsilon_k | estimated noise in the \(\chi(k)\) data.
  > > epsilon_r | estimated noise in the \(\chi(R)\) data.
  > > n_idp | estimated number of independent points in the data.
  > > model | a group for the model \(\chi\) spectrum.

  The `model` component will be set after a fit, and will contain the standard set of arrays for \(\chi(k)\) and \(\chi(R)\) for the fitting model, and can be directly compared to the arrays for the experimental data.

## 9.8.6. `feffit()`

`_xafs.``feffit`(paramgroup, datasets, rmax_out=10, path_outputs=True)

  execute a Feffit fit.

  Parameters: | paramgroup – group containing parameters for fit datasets – Feffit Dataset group or list of Feffit Dataset group. rmax_out – maximum \(R\) value to calculate output arrays. path_output – Flag to set whether all Path outputs should be written.
  Returns: | a fit results group.

  The `paramgroup` is a group containing all fitting parameters for the model. The `datasets` argument can be either a single Feffit Dataset as created by `feffit_dataset()` or a list of them. If `path_outputs==True`, all Feff Paths in the fit will be separately Fourier transformed.

  When the fit is completed, the returned value will be a group containing three objects:

  > 1. `datasets`: an array of FeffitDataSet groups used in the fit.
  > 2. `params`: This will be identical to the input parameter group.
  > 3. `fit`: an object which points to the low-level fit.

  In addition, the output statistics listed below in Table of Feffit Output Statistics. will be written the `paramgroup` group. Since each varied and constrained parameter will also have best-values and estimated uncertainties, this allows the parameter group to be considered the principle group for a particular fit – it holds the variable parameters and statistical results needed to compare two fits.

  On output, a new sub-group called `model` will be created for each Feffit Dataset. This will parallel the `data` group, in the sense that it will have output arrays listed in the Table of Feffit Output Arrays.

  If `path_outputs==True`, all Feff Paths in the fit will be separately Fourier transformed., with the result being put in the corresponding FeffPath group.

  A final note on the outputs of `feffit()`: the `param` sub-group in the output is truly identical to the input `paramgroup`. It is not a copy but points to the same group of values (see 5.2.4).

> Table of Feffit Output Statistics. These values will be written to the `paramgroup` group. Listed here are the group component name and a description of its content. Many of these are described in more detail in 8.4

> > component name | description
> > --- | ---
> > chi_reduced | reduced chi-square statistic.
> > chi_square | chi-square statistic.
> > covar | covariance matrix.
> > covar_vars | list of variable names for rows and columns of covariance matrix.
> > errorbars | Flag whether error bars could be calculated.
> > fit_details | group with additional fit details.
> > message | output message from fit.
> > nfree | number of degrees of freedom in fit.
> > nvarys | number of variables in fit.

> Table of Feffit Output Arrays. The following arrays will be written into the `data` and `model` sub-group for each dataset. The arrays will be created using the Path Parameters used in the most recent fit and the Feffit Transform group. Many of these arrays have names following the conventions for `xftf()` in section on Fourier Transforms for XAFS.

> > array name | description
> > --- | ---
> > k | wavenumber array of \(k\).
> > chi | \(\chi(k)\).
> > kwin | window \(\Omega(k)\) (length of input chi(k)).
> > r | uniform array of \(R\), out to `rmax_out`.
> > chir | complex array of \(\tilde\chi(R)\).
> > chir_mag | magnitude of \(\tilde\chi(R)\).
> > chir_pha | phase of \(\tilde\chi(R)\).
> > chir_re | real part of \(\tilde\chi(R)\).
> > chir_im | imaginary part of \(\tilde\chi(R)\).

## 9.8.7. `feffit_report()`

`_xafs.``feffit_report`(fit_result)

  return a printable report from a Feffit fit.

  Parameters: | fit_result – output group from `feffit()`.

## 9.8.8. Example 1: Simple fit with 1 Path

We start with a fairly minimal example, fitting spectra read from a data file with a single Feff Path.

    ## examples/feffit/doc_feffit1.lar

    # read data
    cu_data  = read_ascii('../xafsdata/cu_metal_rt.xdi')
    autobk(cu_data.energy, cu_data.mutrans, group=cu_data, rbkg=1.0, kw=2)

    # define fitting parameter group
    pars = group(amp    = param(1.0, vary=True),
                 del_e0 = param(0.0, vary=True),
                 sig2   = param(0.0, vary=True),
                 del_r  = guess(0.0, vary=True) )

    # define a Feff Path, give expressions for Path Parameters
    path1 = feffpath('feffcu01.dat',
                     s02    = 'amp',
                     e0     = 'del_e0',
                     sigma2 = 'sig2',
                     deltar = 'del_r')

    # set tranform / fit ranges
    trans = feffit_transform(kmin=3, kmax=17, kw=2, dk=4, window='kaiser', rmin=1.4, rmax=3.0)

    # define dataset to include data, pathlist, transform
    dset = feffit_dataset(data=cu_data, pathlist=[path1], transform=trans)

    # perform fit!
    out = feffit(pars, dset)
    print(feffit_report(out))

    try:
        fout = open('doc_feffit1.out', 'w')
        fout.write("%s\n" % feffit_report(out))
        fout.close()
    except:
        print('could not write doc_feffit1.out')
    #endtry

    plot_chifit(dset)

    ## end examples/feffit/doc_feffit1.lar

This simply follows the essential steps:

> 1. A group of parameters `pars` is defined. Note that you can include upper and/or lower bounds and mix the use of `_math.guess()` and `_math.param()`.

> 2. A Feff Path is defined with `feffpath()`, as discussed in the previous section. Here we assign each of the Path Parameters to the name of one of the fitting parameters. More complex expressions and relations can be used, but for this example, we’re keeping it simple.

> 3. A Feffit Transform is created with `feffit_transform()`, which essentially sets the Fourier transform parameters and fit ranges.

> 4. A Feffit Dataset is created with `feffit_dataset()`. To begin the fit, this includes a `data` group, a `transform` group, and a `pathlist`, which is a list of FeffPaths.

> 5. The fit is run with `feffit()`, and the output group is saved. This output group is used by `feffit_report()` to generate a fit report (shown below).

> 6. Plots are made from the dataset, using rather long-winded `plot()` commands.

running this example prints out the following report:

    =================== FEFFIT RESULTS ====================
    [[Statistics]]
       nvarys, npts       = 4, 104
       n_independent      = 15.260
       chi_square         = 38.0085
       reduced chi_square = 3.37545
       r-factor           = 0.00279
       Akaike info crit   = 21.9259
       Bayesian info crit = 24.8269

    [[Data]]
       fit space          = 'r'
       r-range            = 1.400, 3.000
       k-range            = 3.000, 17.000
       k window, dk       = 'kaiser', 4.000
       paths used in fit  = ['feffcu01.dat']
       k-weight           = 2
       epsilon_k          = Array(mean=0.001076, std=0.003504)
       epsilon_r          = 0.036161
       n_independent      = 15.260

    [[Variables]]
       amp            =  0.928280 +/- 0.039477   (init=  1.000000)
       del_e0         =  4.356280 +/- 0.517286   (init=  0.000000)
       del_r          = -0.006018 +/- 0.002643   (init=  0.000000)
       sig2           =  0.008668 +/- 0.000311   (init=  0.000000)

    [[Correlations]]    (unreported correlations are <  0.100)
       amp, sig2            =  0.928
       del_e0, del_r        =  0.920
       del_r, sig2          =  0.159
       amp, del_r           =  0.137

    [[Paths]]
       Path pfh2ced6i, Feff.dat file = feffcu01.dat
         atom      x        y        z       ipot
          Cu       0.0000,  0.0000,  0.0000  0 (absorber)
          Cu       0.0000, -1.8016,  1.8016  1
         reff   =  2.54780
         degen  =  12.000000
         n*s02  =  0.928280 +/- 0.039477  'amp'
         e0     =  4.356280 +/- 0.517286  'del_e0'
         r      =  2.541782 +/- 0.002643  'reff + del_r'
         deltar = -0.006018 +/- 0.002643  'del_r'
         sigma2 =  0.008668 +/- 0.000311  'sig2'

    =======================================================

and generates the plots shown below

Figure 9.8.1 Data and Fit in \(k\) space

Figure 9.8.2 Data and Fit in \(R\) space

Results for Feffit for a simple 1-shell fit to a spectrum from Cu metal.

This is a pretty good fit to the first shell of Cu metal, and shows the basic mechanics of fitting XAFS data to Feff Paths. There are several things that might be added to this for modeling more complex XAFS data, including adding more paths to a fit, including multiple-scattering paths, simultaneously modeling more than one data set, and building more complex fitting models. We’ll get to these in the following examples.

But first, a small detour. The plotting commands in the above example for plotting \(\chi(k)\) and \(\chi(R)\) for data and model will be useful for the other examples as well, so we’ll create a slightly generalized function to make such plots and put this and several other plotting functions into a separate file, doc_macros.lar. This will look like this:

    def write_report(filename, out):
        "write report to file"
        try:
            f = open(filename, 'w')
            f.write(out)
            f.close()
        except:
            print( 'could not write %s' % filename)
        #endtry
    #enddef

    ## end of examples/feffit/doc_macros.lar

This defines several new plotting functions `plot_chifit()`, `plot_path_k()`, and `plot_path_r()`, and so on which we’ll find useful in later examples. Using the first of these, we can then replace the plot commands in the script above with:

    run('doc_macros.lar')
    plot_chifit(dset, title='First shell fit to Cu')

and get reproducible plots without having to copy and paste the same code fragment everywhere. We’ll use this in the examples below.

## 9.8.9. Example 2: Fit 1 dataset with 3 Paths

We’ll continue with the Cu data set, and add more paths to model further shells. This is fairly straightforward, but in the interest of space, we’ll limit the example here to 3 paths to model the first two shells of copper. This is a small step, but highlights a main concern with XAFS analysis that we need to address. This is the fact that there simply is not enough freedom in the XAFS signal to measure all the possible adjustable Path Parameters independently. Thus we need to be able to apply constraints to the Path Parameters.

Here, we use two of the most common types of constraints. First, we apply the same amplitude reduction factor and the same \(E_0\) shift to all Paths. These may seem obvious for this example, but for more complicated examples, either including shells of mixed species or Feff Paths generated from different calculation, these become less obvious.

Second, we introduce a scale the change in distance by a single expansion factor \(\alpha\) (`alpha` in the script), and using the built-in value of half-path distance, `reff`, and setting `deltar = 'alpha*reff'` for all Paths. During the calculation of \(\chi(k)\) for each path that happens in the fitting process, the value of `reff` will be updated to the correct value for each path. Thus, as the value of `alpha` varies in the fit, each path will use its proper value for `reff`, so that each `deltar` will be different but not independent. This ensures that all the path lengths change in a manner consistent with one another.

    ## examples/feffit/doc_feffit2.lar

    # read data
    # read data
    cu_data  = read_ascii('../xafsdata/cu_metal_rt.xdi')
    cu_data.mu = cu_data.mutrans
    autobk(cu_data, rbkg=1.1, kw=2)

    # define fitting parameter group
    pars = group(amp    = param(1,    vary=True),
                 del_e0 = param(3,    vary=True),
                 c3_1   = param(.002, vary=True),
                 sig2_1 = param(.002, vary=True),
                 sig2_2 = param(.002, vary=True),
                 sig2_3 = param(.002, vary=True),
                 alpha  = param(0,    vary=True) )

    # define 3 Feff Path, give expressions for Path Parameters
    path1 = feffpath('feff0001.dat',     s02 = 'amp', e0 = 'del_e0',
                     sigma2 = 'sig2_1',  deltar  = 'alpha*reff', third='c3_1')

    path2 = feffpath('feff0002.dat',     s02 = 'amp', e0 = 'del_e0',
                     sigma2 = 'sig2_2',  deltar  = 'alpha*reff')

    path3 = feffpath('feff0003.dat',     s02 = 'amp', e0 = 'del_e0',
                      sigma2 = 'sig2_3', deltar  = 'alpha*reff')

    trans = feffit_transform(kmin=3, kmax=17, kw=2, dk=4, window='kaiser', rmin=1.4, rmax=3.5)

    # define dataset to include data, pathlist, transform
    dset  = feffit_dataset(data=cu_data, pathlist=[path1, path2, path3], transform=trans)

    # perform fit!
    out = feffit(pars, dset)

    report = feffit_report(out)
    print( report)
    run('doc_macros.lar')
    write_report('doc_feffit2.out', report)

    plot_chifit(dset, title='Three-shell fit to Cu', rmax=5)

    ## end examples/feffit/doc_feffit2.lar

Here we simply create `path2` and `path3` using nearly the same parameters as for `path1`. Compared to the previous example, the other changes are that the \(R\) range for the fit has been increased so that the fit will try to fit the second shell, and that `sigma2` is allowed to vary independently for each path.

The output for this fit is a bit longer, being:

    =================== FEFFIT RESULTS ====================
    [[Statistics]]
       nvarys, npts       = 7, 138
       n_independent      = 19.717
       chi_square         = 5.01788
       reduced chi_square = 0.394592
       r-factor           = 0.00248
       Akaike info crit   = -12.9813
       Bayesian info crit = -6.11108

    [[Data]]
       fit space          = 'r'
       r-range            = 1.400, 3.500
       k-range            = 3.000, 17.000
       k window, dk       = 'kaiser', 4.000
       paths used in fit  = ['feff0001.dat', 'feff0002.dat', 'feff0003.dat']
       k-weight           = 2
       epsilon_k          = Array(mean=0.002788, std=0.010181)
       epsilon_r          = 0.093696
       n_independent      = 19.717

    [[Variables]]
       alpha          =  0.002880 +/- 0.002964   (init=  0.000000)
       amp            =  0.932893 +/- 0.035852   (init=  1.000000)
       c3_1           =  0.000147 +/- 0.000082   (init=  0.002000)
       del_e0         =  5.721576 +/- 0.815157   (init=  3.000000)
       sig2_1         =  0.008679 +/- 0.000282   (init=  0.002000)
       sig2_2         =  0.013117 +/- 0.001131   (init=  0.002000)
       sig2_3         =  0.006285 +/- 0.003059   (init=  0.002000)

    [[Correlations]]    (unreported correlations are <  0.100)
       alpha, c3_1          =  0.952
       alpha, del_e0        =  0.951
       amp, sig2_1          =  0.929
       c3_1, del_e0         =  0.837
       amp, sig2_2          =  0.297
       sig2_1, sig2_2       =  0.276
       sig2_1, sig2_3       =  0.213
       amp, sig2_3          =  0.213
       c3_1, sig2_3         = -0.189
       alpha, amp           =  0.186
       alpha, sig2_1        =  0.172
       c3_1, sig2_2         =  0.169
       alpha, sig2_2        =  0.152
       alpha, sig2_3        = -0.148
       amp, del_e0          =  0.148
       amp, c3_1            =  0.145
       del_e0, sig2_1       =  0.143
       c3_1, sig2_1         =  0.125

    [[Paths]]
       Path pfh2ced6i, Feff.dat file = feff0001.dat
         atom      x        y        z       ipot
          Cu       0.0000,  0.0000,  0.0000  0 (absorber)
          Cu       0.0000, -1.8016,  1.8016  1
         reff   =  2.54780
         degen  =  12.000000
         n*s02  =  0.932893 +/- 0.035852  'amp'
         e0     =  5.721576 +/- 0.815157  'del_e0'
         r      =  2.555137 +/- 0.011329  'reff + alpha*reff'
         deltar =  0.007337 +/- 0.011329  'alpha*reff'
         sigma2 =  0.008679 +/- 0.000282  'sig2_1'
         third  =  0.000147 +/- 0.000082  'c3_1'

       Path pe3tkyh7w, Feff.dat file = feff0002.dat
         atom      x        y        z       ipot
          Cu       0.0000,  0.0000,  0.0000  0 (absorber)
          Cu      -3.6032,  0.0000,  0.0000  1
         reff   =  3.60320
         degen  =  6.000000
         n*s02  =  0.932893 +/- 0.035852  'amp'
         e0     =  5.721576 +/- 0.815157  'del_e0'
         r      =  3.613576 +/- 0.011329  'reff + alpha*reff'
         deltar =  0.010376 +/- 0.011329  'alpha*reff'
         sigma2 =  0.013117 +/- 0.001131  'sig2_2'

       Path pkky2qtk6, Feff.dat file = feff0003.dat
         atom      x        y        z       ipot
          Cu       0.0000,  0.0000,  0.0000  0 (absorber)
          Cu       1.8016, -1.8016,  0.0000  1
          Cu       1.8016,  0.0000, -1.8016  1
         reff   =  3.82180
         degen  =  48.000000
         n*s02  =  0.932893 +/- 0.035852  'amp'
         e0     =  5.721576 +/- 0.815157  'del_e0'
         r      =  3.832806 +/- 0.011329  'reff + alpha*reff'
         deltar =  0.011006 +/- 0.011329  'alpha*reff'
         sigma2 =  0.006285 +/- 0.003059  'sig2_3'

    =======================================================

With plots of data and fits as shown below.

Results for Feffit for a 3-shell fit to a spectrum from Cu metal, constraining all path distances to expand with a single variable.

Here, we show both the magnitude and real part of \(\chi(R)\). The fit to the real part shows excellent agreement over the fit \(R\) range of [1.4, 3.4] \(\AA\). It is often useful the contributions from the individual paths. With the macros defined above, this is pretty straightforward, as we can just do:

    plot_modelpaths_k(dset, offset=-1)
    plot_modelpaths_r(dset, comp='re', offset=-1, xmax=6)

to generate the following plots of the contributions of the different paths:

Path contributions to full mode for the 3-shell fit to Cu spectrum.

## 9.8.10. Example 3: Fit 3 datasets with 1 Path each

We’ll extend the above example by adding two more data sets. Since the three data sets have some things in common, we’ll be able to use some a smaller number of total variable parameters for all data sets than if we had fit each of them individually. This further allows us to reduce the number of freely varying parameters in a model of XAFS data, and to better measure the parameters that are varied.

Here, we’ll use data on Cu metal measured at three different temperatures. Since there is no phase change in the material over this temperature range, the structure changes in small and predictable ways that lends itself to simple parameterization. In the interest of brevity, we’ll only use one path, but the example could easily be extended to include more paths.

In this example, we have three distinct datasets, so we’ll have three lists of paths. Each of these will have a single path. Since we’re modeling nearly the same structure, the three paths will use the same Feff.dat file and have many parameters in common, but some parameters will be different for each data set. As with the previous example, we use the same amplitude reduction factor \(E_0\) shift to all data sets. We allow distances to vary, but constrain them so that the change is linear in the sample temperature, as if there were a simple linear expansion in \(R\). To do this, we set `deltar = 'dr_off + T*alpha*reff'`, where `T` is the temperature for the dataset. For \(\sigma^2\) we’ll use one of the built-in models described in Models for Calculating sigma2. Here we’ll use `sigma2_eins()`, but `sigma2_debye()` can be used as well, and does a better job for multiple-scattering paths in simple systems. The model then uses 2 variable parameters for three temperature-dependent distances and 1 variable parameter for three temperature-dependent mean-square displacements. The full script for the fit looks like this:

    ## examples/feffit/doc_feffit3.lar

    # read 3 datasets
    cu_10 = read_ascii('../xafsdata/cu_10k.xmu')
    autobk(cu_10.energy, cu_10.mu, group=cu_10, rbkg=1.0, kw=2)

    cu_50 = read_ascii('../xafsdata/cu_50k.xmu')
    autobk(cu_50.energy, cu_50.mu, group=cu_50, rbkg=1.0, kw=2)

    cu_150 = read_ascii('../xafsdata/cu_150k.xmu')
    autobk(cu_150.energy, cu_150.mu, group=cu_150, rbkg=1.0, kw=2)

    # define fitting parameter group
    pars = group(amp    = param(1, vary=True),
                 del_e0 = guess(2.0),
                 theta  = param(250, min=10, vary=True),
                 dr_off = guess(0),
                 alpha  = guess(0) )

    # define 3 Feff Path, give expressions for Path Parameters
    path1_10  = feffpath('feff0001.dat',  s02 = 'amp', e0 = 'del_e0',
                         deltar = 'dr_off + 10*alpha*reff',
                         sigma2 = 'sigma2_eins(10, theta)')

    path1_50  = feffpath('feff0001.dat',  s02 = 'amp', e0 = 'del_e0',
                         deltar = 'dr_off + 50*alpha*reff',
                         sigma2 = 'sigma2_eins(50, theta)')

    path1_150 = feffpath('feff0001.dat',  s02 = 'amp', e0 = 'del_e0',
                         deltar = 'dr_off + 150*alpha*reff',
                         sigma2 = 'sigma2_eins(150, theta)')

    trans = feffit_transform(kmin=3, kmax=17, kw=2, dk=4, window='kaiser', rmin=1.4, rmax=3.4)

    # define 3 datasets, each with data, pathlist, transform for each
    dset_10   = feffit_dataset(data=cu_10,  pathlist=[path1_10],  transform=trans)
    dset_50   = feffit_dataset(data=cu_50,  pathlist=[path1_50],  transform=trans)
    dset_150  = feffit_dataset(data=cu_150, pathlist=[path1_150], transform=trans)

    # perform fit!
    out = feffit(pars, [dset_10, dset_50, dset_150])
    report = feffit_report(out)

    print( report)
    run('doc_macros.lar')
    write_report('doc_feffit3.out', report)

    plot_chifit(dset_10, title='fit to Cu, 10K', rmax=5, win=1)
    plot_chifit(dset_50, title='fit to Cu, 50K', rmax=5, win=3)
    plot_chifit(dset_150, title='fit to Cu, 150K', rmax=5, win=5)

    # ## end examples/feffit/doc_feffit3.lar

Here we read in 3 datasets for \(\mu(E)\) data and do the background subtraction on each of them. We define 5 fitting parameters, including the characteristic (here, Einstein) temperature which will determine the value of \(\sigma^2\), and two parameters for the linear temperature dependence of \(R\). The output for this fit is:

    =================== FEFFIT RESULTS ====================
    [[Statistics]]
       nvarys, npts       = 5, 390
       n_independent      = 56.476
       chi_square         = 150.382
       reduced chi_square = 2.92139
       r-factor           = 0.01631
       Akaike info crit   = 65.3103
       Bayesian info crit = 75.4794

    [[Datasets (3)]]
     dataset 1:
       fit space          = 'r'
       r-range            = 1.400, 3.400
       k-range            = 3.000, 17.000
       k window, dk       = 'kaiser', 4.000
       paths used in fit  = ['feff0001.dat']
       k-weight           = 2
       epsilon_k          = Array(mean=0.010379, std=0.040095)
       epsilon_r          = 0.348846
       n_independent      = 18.825
     dataset 2:
       fit space          = 'r'
       r-range            = 1.400, 3.400
       k-range            = 3.000, 17.000
       k window, dk       = 'kaiser', 4.000
       paths used in fit  = ['feff0001.dat']
       k-weight           = 2
       epsilon_k          = Array(mean=0.003540, std=0.011780)
       epsilon_r          = 0.118996
       n_independent      = 18.825
     dataset 3:
       fit space          = 'r'
       r-range            = 1.400, 3.400
       k-range            = 3.000, 17.000
       k window, dk       = 'kaiser', 4.000
       paths used in fit  = ['feff0001.dat']
       k-weight           = 2
       epsilon_k          = Array(mean=0.003239, std=0.011981)
       epsilon_r          = 0.108852
       n_independent      = 18.825

    [[Variables]]
       alpha          =  0.000006 +/- 0.000008   (init=  0.000000)
       amp            =  0.869093 +/- 0.030460   (init=  1.000000)
       del_e0         =  5.374313 +/- 0.612595   (init=  2.000000)
       dr_off         = -0.000559 +/- 0.002877   (init=  0.000000)
       theta          =  233.110563 +/- 8.207756   (init=  250.000000)

    [[Correlations]]    (unreported correlations are <  0.100)
       amp, theta           = -0.858
       del_e0, dr_off       =  0.741
       alpha, dr_off        = -0.489
       alpha, del_e0        =  0.105

    [[Paths]]
     dataset 1:
       Path pfh2ced6i, Feff.dat file = feff0001.dat
         atom      x        y        z       ipot
          Cu       0.0000,  0.0000,  0.0000  0 (absorber)
          Cu       0.0000, -1.8016,  1.8016  1
         reff   =  2.54780
         degen  =  12.000000
         n*s02  =  0.869093 +/- 0.030460  'amp'
         e0     =  5.374313 +/- 0.612595  'del_e0'
         r      =  2.547394 +/- 0.002786  'reff + dr_off + 10*alpha*reff'
         deltar = -0.000406 +/- 0.002786  'dr_off + 10*alpha*reff'
         sigma2 =  0.003275 +/- 0.000115  'sigma2_eins(10, theta)'

     dataset 2:
       Path pfh2ced6i, Feff.dat file = feff0001.dat
         atom      x        y        z       ipot
          Cu       0.0000,  0.0000,  0.0000  0 (absorber)
          Cu       0.0000, -1.8016,  1.8016  1
         reff   =  2.54780
         degen  =  12.000000
         n*s02  =  0.869093 +/- 0.030460  'amp'
         e0     =  5.374313 +/- 0.612595  'del_e0'
         r      =  2.548002 +/- 0.002544  'reff + dr_off + 50*alpha*reff'
         deltar =  0.000202 +/- 0.002544  'dr_off + 50*alpha*reff'
         sigma2 =  0.003337 +/- 0.000128  'sigma2_eins(50, theta)'

     dataset 3:
       Path pfh2ced6i, Feff.dat file = feff0001.dat
         atom      x        y        z       ipot
          Cu       0.0000,  0.0000,  0.0000  0 (absorber)
          Cu       0.0000, -1.8016,  1.8016  1
         reff   =  2.54780
         degen  =  12.000000
         n*s02  =  0.869093 +/- 0.030460  'amp'
         e0     =  5.374313 +/- 0.612595  'del_e0'
         r      =  2.549525 +/- 0.002960  'reff + dr_off + 150*alpha*reff'
         deltar =  0.001725 +/- 0.002960  'dr_off + 150*alpha*reff'
         sigma2 =  0.005030 +/- 0.000299  'sigma2_eins(150, theta)'

    =======================================================

Note that an uncertainty is estimated for the Path parameters, including `sigma2`, which is calculated with the `sigma2_eins()` function. Such derived uncertainties do reflect the uncertainties and correlations between variables. For example, a simplistic evaluation for the standard error in one of the `sigma2` parameters using the estimated variance in the `theta` might be done as follows:

    larch> _ave = sigma2_eins(10, pars.theta)
    larch> _dlo = sigma2_eins(10, pars.theta-pars.theta.stderr) - _ave
    larch> _dhi = sigma2_eins(10, pars.theta+pars.theta.stderr) - _ave
    larch> print "sigma2(T=10) = %.5f  (%+.5f, %+.5f)" % (_ave, _dlo, _dhi)
    sigma2(T=10) = 0.00328  (+0.00011, -0.00011)

This gives an estimate about 3 times larger than the estimate automatically derived from the fit. The reason for this is that the simple evaluation ignores the correlation between parameters, which is taken into account in the automatically derived uncertainties. In this case, including this correlation significantly reduces the estimated uncertainty.

The output plots for the fits to the three datasets are given below.

Figure 9.8.7 Fit to Cu \(\chi(k)\) at 10 K

Figure 9.8.8 Fit to Cu \(\chi(k)\) at 50 K

Figure 9.8.9 Fit to Cu \(\chi(k)\) at 150 K

Figure 9.8.10 Fit to Cu \(\chi(R)\) at 10 K

Figure 9.8.11 Fit to Cu \(\chi(R)\) at 50 K

Figure 9.8.12 Fit to Cu \(\chi(R)\) at 150 K

Fit to Cu metal at 10 K, 50 K, and 150 K, from simultaneous fit to all 3 datasets with 5 variables used.

Again, in the interest of brevity and consistency through this chapter, these example are deliberately simple and meant to be illustrative of the capabilities and procedures and should not be viewed as limiting the types of problems that can be modeled.

## 9.8.11. Example 4: Measuring Coordination number

For this and the following example, we switch from Cu metal data to data on a simple metal oxide, FeO. The structure is a basic rock-salt structure, and we’ll model 2 paths for Fe-O and Fe-Fe in this structure. While the data is imperfect, we’ll use it to illustrate a few points in modeling EXAFS data.

For the examples above with Cu metal, we tacitly assumed that the coordination number for the different paths was correct, and we adjusted an \(S_0^2\) parameter. But, as with many analyses on real systems of research interest, we’d like to fit the coordination number for the two different paths here. To do this, we set an \(S_0^2\) parameter to a fixed value, and also force `degen` (the number of equivalent paths in the structure used to generate the Feff.dat files) to be 1 for each path. Instead, we’ll define parameters `n1` and `n2`, and set the Fe-O path’s amplitude to be `s02*n1` and the Fe-Fe path’s amplitude to be `s02*n2`. We’ll allow `n1` and `n2` to vary in the fit, and also define variable parameters for the other path parameters, including separate variables for \(\Delta R\) and \(\sigma^2\). The script for this fit is below:

    ## examples/feffit/doc_feffit4.lar

    feo_dat = read_ascii('../xafsdata/feo_xafs.dat', labels='energy mu')
    autobk(feo_dat, kweight=2, rbkg=0.9)

    # define fitting parameter group
    pars = group(n1     = param(6, vary=True),
                 n2     = param(12, vary=True),
                 s02    = 0.700,
                 de0    = guess(0.1),
                 sig2_1 = param(0.002, vary=True),
                 delr_1 = guess(0.),
                 sig2_2 = param(0.002, vary=True),
                 delr_2 = guess(0.)  )

    # define Feff Paths, give expressions for Path Parameters
    path_feo = feffpath('feff_feo01.dat',
                        degen = 1,
                        s02    = 's02*n1',
                        e0     = 'de0',
                        sigma2 = 'sig2_1',
                        deltar = 'delr_1')

    path_fefe = feffpath('feff_feo02.dat',
                         degen = 1,
                         s02    = 's02*n2',
                         e0     = 'de0',
                         sigma2 = 'sig2_2',
                         deltar = 'delr_2')

    # set tranform / fit ranges
    trans = feffit_transform(kmin=2.0, kmax=13.5, kweight=3,
                             dk=3,  window='kaiser',
                             rmin=1.0, rmax=3.2, fitspace='r')

    # define dataset to include data, pathlist, transform
    dset  = feffit_dataset(data=feo_dat, pathlist=[path_feo, path_fefe],
                           transform=trans)
    out = feffit(pars, dset)
    print( feffit_report(out))
    run('doc_macros.lar')
    write_report('doc_feffit4.out', feffit_report(out))

    plot_chifit(dset, title='Two-path fit to FeO')
    ## end examples/feffit/doc_feffit4.lar

The most important point here is the definitions used in setting up the amplitudes for the paths: first, that we set `degen` to 1, and second that we used the expression `s02*n1` and so forth for the value of the Path’s amplitude. A secondary note is that we gave two different k-weights to `feffit_transform()`, which causes both k-weights to be used in the fit.

The resulting output is

    =================== FEFFIT RESULTS ====================
    [[Statistics]]
       nvarys, npts       = 7, 144
       n_independent      = 17.106
       chi_square         = 19.6759
       reduced chi_square = 1.94686
       r-factor           = 0.00895
       Akaike info crit   = 16.3938
       Bayesian info crit = 22.27

    [[Data]]
       fit space          = 'r'
       r-range            = 1.000, 3.200
       k-range            = 2.000, 13.500
       k window, dk       = 'kaiser', 3.000
       paths used in fit  = ['feff_feo01.dat', 'feff_feo02.dat']
       k-weight           = 3
       epsilon_k          = Array(mean=0.000962, std=0.002704)
       epsilon_r          = 0.207307
       n_independent      = 17.106

    [[Variables]]
       de0            = -1.186615 +/- 0.808632   (init=  0.100000)
       delr_1         = -0.028279 +/- 0.007522   (init=  0.000000)
       delr_2         =  0.049909 +/- 0.005972   (init=  0.000000)
       n1             =  4.550833 +/- 0.738706   (init=  6.000000)
       n2             =  10.986155 +/- 1.043360   (init=  12.000000)
       sig2_1         =  0.010405 +/- 0.001825   (init=  0.002000)
       sig2_2         =  0.012472 +/- 0.000828   (init=  0.002000)

    [[Correlations]]    (unreported correlations are <  0.100)
       n2, sig2_2           =  0.939
       de0, delr_2          =  0.920
       n1, sig2_1           =  0.892
       de0, delr_1          =  0.566
       delr_1, delr_2       =  0.524
       delr_1, sig2_1       =  0.219
       delr_1, n1           =  0.193
       delr_2, sig2_2       =  0.181
       de0, n1              = -0.158
       delr_2, n2           =  0.156
       delr_2, n1           = -0.139
       de0, sig2_1          = -0.110
       n1, n2               =  0.100

    [[Paths]]
       Path prqdqp2pm, Feff.dat file = feff_feo01.dat
         atom      x        y        z       ipot
          Fe       0.0000,  0.0000,  0.0000  0 (absorber)
           O       0.0000,  2.1387,  0.0000  1
         reff   =  2.13870
         degen  =  1.000000
         n*s02  =  3.185583 +/- 0.517094  's02*n1'
         e0     = -1.186615 +/- 0.808632  'de0'
         r      =  2.110421 +/- 0.007522  'reff + delr_1'
         deltar = -0.028279 +/- 0.007522  'delr_1'
         sigma2 =  0.010405 +/- 0.001825  'sig2_1'

       Path pbhrkq5es, Feff.dat file = feff_feo02.dat
         atom      x        y        z       ipot
          Fe       0.0000,  0.0000,  0.0000  0 (absorber)
          Fe       2.1387,  0.0000, -2.1387  2
         reff   =  3.02460
         degen  =  1.000000
         n*s02  =  7.690309 +/- 0.730352  's02*n2'
         e0     = -1.186615 +/- 0.808632  'de0'
         r      =  3.074509 +/- 0.005972  'reff + delr_2'
         deltar =  0.049909 +/- 0.005972  'delr_2'
         sigma2 =  0.012472 +/- 0.000828  'sig2_2'

    =======================================================

with plots:

Figure 9.8.13 Fit to FeO spectra, \(k\) space

Figure 9.8.14 Fit to FeO spectra, \(R\) space

Fits to 2-path fit to FeO EXAFS.

## 9.8.12. Example 5: Comparing Fits in different Fit Spaces

We now turn to comparing fits in unfiltered k-space, R-space, and filter k-space (or “q space”). This is partly to illustrate the preference for using R- or q-space for fitting, and partly to demonstrate how one can run similar fits and compare the results. We’ll use the FeO data from the previous example.

To change fitting models and transform parameters, we’ll make copies of the parameter groups and dataset groups, make a few changes, and re-run the fits. For example, we can change the fitting space with (see examples/feffit/doc_feffit5.lar):

    larch> pars2 = copy(pars)   # copy parameters
    larch> dset2 = copy(dset)   # copy dataset
    larch> dset2.transform.fitspace = 'q'  # fit in backtransformed k-space

Now we can run `feffit()` with the new parameter group and Dataset group, and compare the results either by plotting models from the different copies of the dataset or by viewing the parameter values and fit statistics with:

    larch> out2  = feffit(pars2, dset2)
    larch> print '*** R Space ***'
    larch> print feffit_report(out, with_paths=False, min_correl=0.5)
    larch> print '*** Q Space ***'
    larch> print feffit_report(out2, with_paths=False, min_correl=0.5)

which gives

    =================== FEFFIT RESULTS ====================
    [[Statistics]]
       nvarys, npts       = 7, 144
       n_independent      = 17.106
       chi_square         = 19.6759
       reduced chi_square = 1.94686
       r-factor           = 0.00895
       Akaike info crit   = 16.3938
       Bayesian info crit = 22.27

    [[Data]]
       fit space          = 'r'
       r-range            = 1.000, 3.200
       k-range            = 2.000, 13.500
       k window, dk       = 'kaiser', 3.000
       paths used in fit  = ['feff_feo01.dat', 'feff_feo02.dat']
       k-weight           = 3
       epsilon_k          = Array(mean=0.000962, std=0.002704)
       epsilon_r          = 0.207307
       n_independent      = 17.106

    [[Variables]]
       de0            = -1.186615 +/- 0.808632   (init=  0.100000)
       delr_1         = -0.028279 +/- 0.007522   (init=  0.000000)
       delr_2         =  0.049909 +/- 0.005972   (init=  0.000000)
       n1             =  4.550833 +/- 0.738706   (init=  6.000000)
       n2             =  10.986155 +/- 1.043360   (init=  12.000000)
       sig2_1         =  0.010405 +/- 0.001825   (init=  0.002000)
       sig2_2         =  0.012472 +/- 0.000828   (init=  0.002000)

    [[Correlations]]    (unreported correlations are <  0.500)
       n2, sig2_2           =  0.939
       de0, delr_2          =  0.920
       n1, sig2_1           =  0.892
       de0, delr_1          =  0.566
       delr_1, delr_2       =  0.524
    =======================================================

    =================== FEFFIT RESULTS ====================
    [[Statistics]]
       nvarys, npts       = 7, 230
       n_independent      = 17.106
       chi_square         = 49.3013
       reduced chi_square = 4.87819
       r-factor           = 0.00732
       Akaike info crit   = 32.1071
       Bayesian info crit = 37.9833

    [[Data]]
       fit space          = 'q'
       r-range            = 1.000, 3.200
       k-range            = 2.000, 13.500
       k window, dk       = 'kaiser', 3.000
       paths used in fit  = ['feff_feo01.dat', 'feff_feo02.dat']
       k-weight           = 3
       epsilon_k          = Array(mean=0.000962, std=0.002704)
       epsilon_r          = 0.207307
       n_independent      = 17.106

    [[Variables]]
       de0            = -1.162732 +/- 0.733414   (init= -1.186615)
       delr_1         = -0.028141 +/- 0.006824   (init= -0.028279)
       delr_2         =  0.050134 +/- 0.005426   (init=  0.049909)
       n1             =  4.627901 +/- 0.685285   (init=  4.550833)
       n2             =  11.026974 +/- 0.950967   (init=  10.986155)
       sig2_1         =  0.010592 +/- 0.001674   (init=  0.010405)
       sig2_2         =  0.012508 +/- 0.000753   (init=  0.012472)

    [[Correlations]]    (unreported correlations are <  0.500)
       n2, sig2_2           =  0.939
       de0, delr_2          =  0.920
       n1, sig2_1           =  0.894
       de0, delr_1          =  0.569
       delr_1, delr_2       =  0.527
    =======================================================

We can see that the results are not very different – the best fit values and uncertainties for the varied parameters are quite close for the fit in ‘R’ space and ‘Q’ space.

Now, we can try the fit in unfiltered ‘K’ space:

    larch> pars3 = copy(pars)   # copy parameters
    larch> dset3 = copy(dset)   # copy dataset
    larch> dset3.transform.kweight = 2
    larch> dset3.transform.fitspace = 'k'
    larch> out3 = feffit(pars3, dset3)
    larch  print feffit_report(out3, with_paths=False, min_correl=0.5)

(we need to specify only one k-weight for a k-space fit) which gives:

    =================== FEFFIT RESULTS ====================
    [[Statistics]]
       nvarys, npts       = 7, 230
       n_independent      = 17.106
       chi_square         = 6.47037e+06
       reduced chi_square = 640220
       r-factor           = 0.13907
       Akaike info crit   = 233.703
       Bayesian info crit = 239.58

    [[Data]]
       fit space          = 'k'
       r-range            = 1.000, 3.200
       k-range            = 2.000, 13.500
       k window, dk       = 'kaiser', 3.000
       paths used in fit  = ['feff_feo01.dat', 'feff_feo02.dat']
       k-weight           = 2
       epsilon_k          = Array(mean=0.000995, std=0.002695)
       epsilon_r          = 0.018803
       n_independent      = 17.106

    [[Variables]]
       de0            = -1.452034 +/- 2.585669   (init= -1.186615)
       delr_1         = -0.029206 +/- 0.031253   (init= -0.028279)
       delr_2         =  0.048631 +/- 0.024348   (init=  0.049909)
       n1             =  4.408334 +/- 2.209217   (init=  4.550833)
       n2             =  11.857401 +/- 4.531096   (init=  10.986155)
       sig2_1         =  0.009934 +/- 0.007632   (init=  0.010405)
       sig2_2         =  0.013189 +/- 0.003994   (init=  0.012472)

    [[Correlations]]    (unreported correlations are <  0.500)
       n2, sig2_2           =  0.924
       de0, delr_2          =  0.879
       n1, sig2_1           =  0.873
       de0, delr_1          =  0.633
       delr_1, delr_2       =  0.566
    =======================================================

This has pretty similar best-fit values, but dramatically larger estimates of the errors. The spectrum is really very poorly fit in k-space because we have not accounted for the higher R components. Using R (and Q) space, we’re able to limit the R range used in determining the parameter values, estimated uncertainties, and the goodness-of-fit statistics. But since we can’t place these limits on what portion of the data is being compared to the model spectra in unfiltered k-space fit, the uncertainties reflect the fact that the full experimental spectrum is not well model. This is why it is recommended to not fit in unfiltered k space: the uncertainties in the parameters is too large.

Finally, we can also fit simultaneously in ‘k’ and ‘r’ space by making use of wavelet transforms (see Section 9.6). For this, we specify both k and R ranges, and the fit is done on the wavelet transform. This gives:

    =================== FEFFIT RESULTS ====================
    [[Statistics]]
       nvarys, npts       = 7, 33120
       n_independent      = 17.106
       chi_square         = 514.771
       reduced chi_square = 50.9348
       r-factor           = 0.00659
       Akaike info crit   = 72.235
       Bayesian info crit = 78.1112

    [[Data]]
       fit space          = 'w'
       r-range            = 1.000, 3.200
       k-range            = 2.000, 13.500
       k window, dk       = 'kaiser', 3.000
       paths used in fit  = ['feff_feo01.dat', 'feff_feo02.dat']
       k-weight           = 2
       epsilon_k          = Array(mean=0.000995, std=0.002695)
       epsilon_r          = 0.018803
       n_independent      = 17.106

    [[Variables]]
       de0            = -1.585631 +/- 0.703238   (init= -1.186615)
       delr_1         = -0.030202 +/- 0.007218   (init= -0.028279)
       delr_2         =  0.047365 +/- 0.005778   (init=  0.049909)
       n1             =  4.776567 +/- 0.859693   (init=  4.550833)
       n2             =  11.505340 +/- 0.958908   (init=  10.986155)
       sig2_1         =  0.011025 +/- 0.002265   (init=  0.010405)
       sig2_2         =  0.012943 +/- 0.000820   (init=  0.012472)

    [[Correlations]]    (unreported correlations are <  0.500)
       n2, sig2_2           =  0.945
       de0, delr_2          =  0.933
       n1, sig2_1           =  0.913
    =======================================================

As you can see, the results are all pretty similar.

Of course, for all these examples, we’ve changed only one thing between these fits – the fitting ‘space’. The process of copying the parameter group and dataset, making modifications and re-doing fits can also include changing what parametres are varied, and what constraints are placed between parameters.

## 9.8.13. Example 6: Testing EXAFS sensitivity to \(Z\)

EXAFS is known to be sensitive to the atomic number \(Z\) of the scattering atom. This is due to the \(Z\) dependence of the scattering amplitude and phase shift (\(f(k)\) and \(\delta(k)\) in the EXAFS Equation of Section 9.7.5). A rule-of-thumb that is often stated is that EXAFS is sensitive to \(Z \pm 5\), or perhaps pessimistically to row of the Periodic Table, but not as sensitive as \(Z \pm 1\). Occassionally, you may see work that claims very fine \(Z\) sensitivity, such as being able to distinguish N and O ligands. In those works, XAFS results are typically used with chemical arguments about steric effects or known bond distances.

In this section, we’ll explore the \(Z\) sensitivity of XAFS with real data, by constructing Feff paths with different scatterers and trying to fit data with these different scattering paths. This makes a nice exercise in how to manipulate Feff, how to fit XAFS data, and how to compare XAFS fits. Just to make it a little more complicated, we’ll also explore the idea of Phase corrected Fourier transforms and how they might be used to better determine both \(R\) and \(Z\). The example script here is a little more complex than what we’ve seen above, using many feature of the Larch scripting language to do repeated fits and mathematically manipulating the arrays of data.

Figure 9.8.15 Zn K-edge XAFS data for ZnSe.

For this example, we’ll use Zn K-edge data of ZnSe, with \(\chi(k)\) data shown in Figure 9.8.15. The data is pretty good but not perfect, and is useful for the study here mainly because the structure is simple, with a well-isolated and well-ordered shell of scatterers (the cubic ZnS structure with 1 Zn site surrounded by 4 Se at 2.45 \(\rm\AA\)). While we’re sure that the scatterer should be Se, we’ll try scatterers of Zn, Ge, Se, Br, and Rb, spanning a small but sufficient range of \(Z\).

To construct the scattering paths, we’ll modify a Feff input file. To do this, we start with a feff.inp file generated for crystalline ZnSe by the ATOMS program, and add an IPOT to the list of atomic potentials (here the line `3 35 Br`), and then replace the IPOTs for 1 of the neighboring Se atoms (that is, replacing a 2 with a 3, and changing the tag for convenience). The resulting feff.inp file for Br looks this:

     TITLE  ZnSe (cubic zinc sulfide structure)
     HOLE 1   1.0   *  Zn K edge  (9659.0 eV), second number is S0^2

     CONTROL   1      1     1     1
     PRINT     1      0     0     0
     RMAX      6.0
     NLEG      4

     POTENTIALS
     *    ipot   Z  element
            0   30   Zn
            1   30   Zn
            2   34   Se
            3   36   Kr
          # 3   33   As
          # 3   32   Ge
          # 3   31   Ga

     ATOMS                          * this list contains 35 atoms
     *   x          y          z      ipot  tag              distance
        0.00000    0.00000    0.00000  0 Zn1             0.00000
        1.41690    1.41690    1.41690  2 Se1_1           2.45414
       -1.41690   -1.41690    1.41690  3 Br_1            2.45414
       -1.41690    1.41690   -1.41690  2 Se1_1           2.45414
        1.41690   -1.41690   -1.41690  2 Se1_1           2.45414
        2.83380    2.83380    0.00000  1 Zn1_1           4.00760
       -2.83380    2.83380    0.00000  1 Zn1_1           4.00760
        2.83380   -2.83380    0.00000  1 Zn1_1           4.00760
       -2.83380   -2.83380    0.00000  1 Zn1_1           4.00760
        2.83380    0.00000    2.83380  1 Zn1_1           4.00760
       -2.83380    0.00000    2.83380  1 Zn1_1           4.00760
        0.00000    2.83380    2.83380  1 Zn1_1           4.00760
        0.00000   -2.83380    2.83380  1 Zn1_1           4.00760
        2.83380    0.00000   -2.83380  1 Zn1_1           4.00760
       -2.83380    0.00000   -2.83380  1 Zn1_1           4.00760
        0.00000    2.83380   -2.83380  1 Zn1_1           4.00760
        0.00000   -2.83380   -2.83380  1 Zn1_1           4.00760
       -4.25070    1.41690    1.41690  2 Se1_2           4.69933
       -1.41690    4.25070    1.41690  2 Se1_2           4.69933
        4.25070   -1.41690    1.41690  2 Se1_2           4.69933
        1.41690   -4.25070    1.41690  2 Se1_2           4.69933
       -1.41690    1.41690    4.25070  2 Se1_2           4.69933
        1.41690   -1.41690    4.25070  2 Se1_2           4.69933
        4.25070    1.41690   -1.41690  2 Se1_2           4.69933
        1.41690    4.25070   -1.41690  2 Se1_2           4.69933
       -4.25070   -1.41690   -1.41690  2 Se1_2           4.69933
       -1.41690   -4.25070   -1.41690  2 Se1_2           4.69933
        1.41690    1.41690   -4.25070  2 Se1_2           4.69933
       -1.41690   -1.41690   -4.25070  2 Se1_2           4.69933
        5.66760    0.00000    0.00000  1 Zn1_2           5.66760
       -5.66760    0.00000    0.00000  1 Zn1_2           5.66760
        0.00000    5.66760    0.00000  1 Zn1_2           5.66760
        0.00000   -5.66760    0.00000  1 Zn1_2           5.66760
        0.00000    0.00000    5.66760  1 Zn1_2           5.66760
        0.00000    0.00000   -5.66760  1 Zn1_2           5.66760
     END

Afer running feff.inp, we collect the resulting `feffNNNN.dat` files – either `feff0001.dat` or `feff0002.dat` will have the replaced scatterer, the other will have Se as the scatterer. Once we’ve collected all these files for runs of Feff with feff.inp altered for each scatterer, we’re ready to do the fits.

The script to compare the fits will loop over these different scattering paths, doing a fit for each path, and store the results. This takes advantage of several features of the Larch scripting capabilities, including defining functions, and constructs like loops and dictionaries, as well as mathematical manipulation of the resulting arrays for the phase correction. The full script is:

    ## examples/feffit/doc_feffit6.lar

    # Testing sensitivity to Z of backscatterer, and
    # how well "phase corrected" Fourier transforms predict distance

    def get_zinc_mu(filename, labels='energy dwelltime i0 i1'):
        """read raw data, do background subtraction"""
        dat = read_ascii(filename, labels=labels)
        dat.mu = -ln(dat.i1 / dat.i0)
        autobk(dat, e0 = 9666.0, rbkg=1.25, kweight=2)
        return dat
    #enddef

    def plot_chidata(data, title, win=1):
        kopts = dict(xlabel=r'$k \rm\,(\AA^{-1})$',
                     ylabel=r'$k^2\chi(k) \rm\,(\AA^{-2})$',
                     title=title, xmax=14, new=True, win=win)
        plot(data.k, data.chi*data.k**2, title=title, **kopts)
    #enddef

    def rplot_fit(dset, title, win=1):
        "make R-spae plots of results"
        ropts = dict(xlabel=r'$R \rm\,(\AA)$',
                     ylabel=r'$|\chi(R)| \rm\,(\AA^{-3})$',
                     title=title, xmax=6, win=win, show_legend=True)
        plot(dset.data.r,  dset.data.chir_mag, new=True, label='data', **ropts)
        plot(dset.model.r,  dset.model.chir_mag,  label='Feff Path', win=win)

        chir_pha = dset.model.chir_phcor
        chir_pha_mag = sqrt(chir_pha.real**2 + chir_pha.imag**2)
        plot(dset.model.r,  chir_pha_mag,  win=win, label='phase-corrected data')

        # find region around peak in phase-corrected chi(R)

        irmax = where(chir_pha_mag == max(chir_pha_mag))[0][0]
        x = dset.model.r[irmax-1: irmax+2]
        y = dset.model.chir_im[irmax-1: irmax+2]
        plot(x, y, win=win, color='black', style='short dashed',
             marker='o', label='Im[phase-corrected]')
        return
    #enddef

    def phase_correct(dset):
       "calculate phase-corrected FT, find estimate of R"
       nrpts = len(dset.model.r)
       path1 = dset.pathlist[0]
       # get phase from Feff path
       feff_pha = interp(path1._feffdat.k, path1._feffdat.pha, dset.model.k)
       # phase-corrected Fourier Transform
       chir_pha = xftf_fast(dset.data.chi * exp(0-1j*feff_pha) *
                            dset.data.kwin * dset.data.k**2)[:nrpts]
       chir_pha_mag = sqrt(chir_pha.real**2 + chir_pha.imag**2)
       dset.model.chir_phcor = chir_pha

       # find region around peak in phase-corrected chi(R)
       irmax = where(chir_pha_mag == max(chir_pha_mag))[0][0]
       x = dset.model.r[irmax-1: irmax+2]
       y = dset.model.chir_im[irmax-1: irmax+2]
       # find where Im[chi(R)_phase_corrected] = 0: use linear regression
       o = linregress(y, x)
       dset.model.rphcor = o[1]
       return o[1]
    #enddef

    ###################################################
    znse_data = get_zinc_mu('../xafsdata/znse_zn_xafs.001',
                            labels='energy dwelltime i0 i1')

    # plot_chidata(znse_data, 'Zn K-edge ZnSe')

    # define Feff Paths, storing them in dictionary
    paths = {}
    pathargs = dict(degen=4.0, s02='amp', e0='del_e0', sigma2='sig2', deltar='del_r')
    paths['Zn'] = feffpath('Feff_ZnSe/feff_znzn.dat', **pathargs)
    paths['Ga'] = feffpath('Feff_ZnSe/feff_znga.dat', **pathargs)
    paths['Ge'] = feffpath('Feff_ZnSe/feff_znge.dat', **pathargs)
    paths['As'] = feffpath('Feff_ZnSe/feff_znas.dat', **pathargs)
    paths['Se'] = feffpath('Feff_ZnSe/feff_znse.dat', **pathargs)
    paths['Br'] = feffpath('Feff_ZnSe/feff_znbr.dat', **pathargs)
    paths['Kr'] = feffpath('Feff_ZnSe/feff_znkr.dat', **pathargs)
    paths['Rb'] = feffpath('Feff_ZnSe/feff_znrb.dat', **pathargs)

    # set FT tranform and Fit ranges
    trans = feffit_transform(rmin=1.5, rmax=3.0, kmin=3, kmax=13,
                             kw=2, dk=4, window='kaiser')

    # perform fit for each scatterer
    print( '|Scatterer|RedChi2|    S02     |   sigma2      |      E0     |      R       |R_phcor|')
    fmt = '|  %s     | %5.1f |%s|%s|%s|%s|%7.3f|'

    results = {}
    for ix, scatterer in enumerate(('Zn', 'Ga', 'Ge', 'As', 'Se', 'Br', 'Kr', 'Rb')):
        dset  = feffit_dataset(data=znse_data, pathlist=[paths[scatterer]],
                               transform=trans)
        pars = group(amp    = guess(1.0),
                     del_e0 = guess(0.1, min=-20, max=20),
                     sig2   = param(0.006, vary=True, min=0),
                     del_r  = guess(0.) )

        out = feffit(pars, dset)
        path1 = out.datasets[0].pathlist[0]
        # read scatterer from Feff geometry, to be sure it's correct
        scatt = path1._feffdat.geom[1][0]
        title = 'Path: Zn-%s ' % scatt
        r_phcor = phase_correct(out.datasets[0])
        rplot_fit(out.datasets[0], title, win=1+ix)

        redchi = out.chi_reduced

        _amp   = '%5.2f(%.2f) ' % (out.params['amp'].value, out.params['amp'].stderr)
        _de0   = '%6.2f(%.2f) ' % (out.params['del_e0'].value,  out.params['del_e0'].stderr)
        _ss2   = '%6.4f(%.4f) ' % (out.params['sig2'].value,  out.params['sig2'].stderr)
        _r     = '%6.3f(%.3f) ' % (path1.reff+out.params['del_r'].value, out.params['del_r'].stderr)
        print( fmt % (scatt, redchi, _amp, _ss2, _de0, _r, r_phcor))
        results[scatterer] = out
    #endfor

    ## end examples/feffit/doc_feffit6.lar

After reading in the data, the script builds Feff Paths, putting them into a dictionary. After define the fitting transform range, the script loops over each of the scatterers, doing a simple first-shell fit with each. The the phase-corrected \(\chi(R)\) is then calculated, and the peak value of \(R\) is found for this array. Finally, results are printed out.

> > Table of ZnSe Results. The results from fitting ZnSe XAFS data to scattering paths of Zn-Zn, Zn-Ge, Zn-Se, Zn-Br, and Zn-Rb. All paths started at the nominal distance of 2.454 \(\rm\AA\).

> Scatterer | reduced chi-square | \(S_0^2\) | \(\sigma^2\) | \(E_0\) | \(R\) | \(R_{\rm{ph cor}}\)
> --- | --- | --- | --- | --- | --- | ---
> Zn (30) | 57.5 | 0.73(0.07) | 0.0040(0.0006) | -7.98(1.23) | 2.471(0.006) | 2.451
> Ge (32) | 32.4 | 0.90(0.06) | 0.0050(0.0005) | -2.99(0.93) | 2.461(0.004) | 2.457
> Se (34) | 12.7 | 0.89(0.04) | 0.0059(0.0003) | 0.09(0.58) | 2.448(0.003) | 2.461
> Br (35) | 16.7 | 1.08(0.06) | 0.0070(0.0004) | 1.71(0.79) | 2.444(0.003) | 2.462
> Rb (37) | 88.7 | 1.10(0.14) | 0.0073(0.0009) | 5.27(1.42) | 2.425(0.007) | 2.467

The results of the fits are given in the Table of ZnSe Results, with graphs below showing fits to the 5 different scatterers. The dependence of the results on \(Z\) of the back-scattering atom is seen to be reasonably strong. The values for reduced chi-square definitely indicate that Zn-Se is the best fit, and the values for the fitted parameters have rather clear correlations with \(Z\) – \(S_0^2\), \(\sigma^2\), and \(E_0\) increase with \(Z\), while \(R\) decreases. If anything, the \(Z \pm 5\) rule-of-thumb seems pessimistic, though the fits with Ge and Br look decent enough that it would be easy to conclude that \(Z \pm 2\) is reasonable.

As a curious side note, we note that \(S_0^2\) increases with \(Z\). The scattering factor should also increase with \(Z\), which might lead us to suspect a trend in the opposite direction. The confounding factor here is \(\sigma^2\) – if we fix \(\sigma^2\) to 0.006 (by changing `vary=True` to `vary=False`) and re-run the fits, \(S_0^2\) has a slight negative dependence on \(Z\).

Figure 9.8.16 Fit to ZnSe with Zn back-scatterer

Figure 9.8.17 Fit to ZnSe with Ge back-scatterer

Figure 9.8.18 Fit to ZnSe with Se back-scatterer

Figure 9.8.19 Fit to ZnSe with Br back-scatterer

Figure 9.8.20 Fit to ZnSe with Rb back-scatterer

Fits to ZnSe XAFS data using different back-scattering atoms.

The figures also show phase-corrected Fourier transforms using the total scattering phase from the Feff calculation to correct the data. This correction is done in the `phase_correct()` procedure, where the data from the feffNNNN.dat file (in the `_feffdat` group) is used to construct the Feff phase shift (\(\delta(k)\) in the EXAFS Equation of Section 9.7.5), and apply it to the data. Because this uses complex math, we use the `xftf_fast()` function to the complex Fourier transform and build the magnitude of the transform explicitly.

The phase corrected transforms are shown in black in Figure 9.8.16 through Figure 9.8.20, and show the key benefit of these transforms – the peak in the phase corrected \(\chi(R)\) peaks much closer to an \(R\) that is the bond distance (for Zn-Se, around 2.45 \(\rm\AA\)), whereas the normal XAFS Fourier transform peaks much lower. This suggests a further test on whether the bond distance and \(Z\) of the scattering atom are correct. That is, in order for the phase-correction to give the correct interatomic distance, the total phase-shift has to be correct, which means that the \(Z\) for the scatterer has to be correct (see the EXAFS Equation in Section 9.7.5). So, we can compare the refined distance with the peak in the phase-corrected transform – if they agree, it gives good confidence that the scatterer is correct.

Since the spacing of points in \(R\) is \(\sim 0.03\rm\AA\), using the peak position may not be accurate enough. Instead, we can use the value where \(\rm Im[\chi(R)_{\rm{ph cor}}]\) passes through zero (see dashed lines in Figure 9.8.16 through Figure 9.8.20). These values are reported in the Table of ZnSe Results as \(R_{\rm{phcor}}\).

We see an interesting trend that while the refined distance decreases with \(Z\), the value for \(R_{\rm{phcor}}\) increases slightly. The two values cross between Ge and Se. This is pretty good agreement, especially considering we left out the \(\sigma^2\) contribution to phase-shift in the EXAFS equation from this phase correction, and applied the theoretical phase-shift from Feff to the measured data. The \(Z\) dependence of \(R_{\rm{phcor}}\) is not as strong as the \(Z\) dependence of reduced chi-square, but this analysis also suggests that one can determine \(Z \pm 3\), at least in this rather favorable case.

Again, this is an unusually favorable case – a simple structure with a single well-isolated coordination shell. The phase-correction method will not work at all on a mixed coordination shell, and is likely to give larger errors for a highly-disordered system. But, for simple, well-characterized systems, the ability to do such analysis can be very powerful, and give increased confidence in the refined structure.

References

[Larch_4] | M Newville. Exafs analysis using feff and feffit. Journal of Synchrotron Radiation, 8(Part 2):96–100, 2001. URL: http://dx.doi.org/10.1107/S0909049500016290, doi:10.1107/S0909049500016290.

[Larch_5] | E. A. Stern. Number of relevant independent points in x-ray-absorption fine-structure spectra. Physical Review B, 48(13):9825–9827, 1993. doi:10.1103/PhysRevB.48.9825.

[Larch_6] | M. Newville, B. Boyanov, and D. E. Sayers. Estimation of uncertainties in XAFS data. Journal of Synchrotron Radiation, 6:264–265, 1999. doi:10.1107/S0909049598018147.