approximately sinusoidal dependence on the interatomic distance R AB , an appropriately weighted Fourier transform yields peaks at approximately R AB . However, the
phase shifts from the absorber and scatterer are also k-dependent functions, and their
net effect on the Fourier transform is to shift peaks to shorter distances. The
nonlinearity of the phase shifts also skews the Fourier transform peaks and can
give rise to additional unphysical peaks.
An approximate solution to this problem is to include the phase shift functions in
the Fourier transform itself. If one transforms with respect to 2k + ϕ(k), where ϕ(k) is
the total phase shift, then the peaks for the A–B interaction will be approximately in
the proper position R AB . Of course, the correction is not perfect when there are
multiple types of neighbors, but in practice the resulting pictures are still approximately correct. Note in the Na 2 MoO 4 transform example that there are still additional non-physical peaks due to the finite range of the data being transformed. These
truncation effects are well-known to result when applying a Fourier transform to a
signal that has been terminated essentially by convolution with a square wave
envelope.
6.7 Interpretation of EXAFS
From the previous discussion, we see that all of the required physics for describing
EXAFS is basically understood, and it could be argued that structure determination
from EXAFS data should be straightforward. The defect in this assumption is that
structure determination from EXAFS is an inverse problem. The structure does not
fall out from the data; instead you usually have to follow an iterative procedure
involving the following: (a) assuming a structure, (b) calculating the EXAFS,
(c) calculating the disagreement between observed and calculated EXAFS,
(d) revising the proposed structure, (e) calculating the revised EXAFS, and
(f) calculating the new residual and iterating until convergence. As with EXAFS
extraction, a variety of software packages is available to accomplish these steps.
Iteration is required because of the nonlinear nature of the EXAFS equation, and
finding the optimum simulation is usually accomplished by a nonlinear least squares
analysis. To use Eq. 6.15, the nature of the backscatterer has to be assumed, and one
can then vary N, R, and σ to achieve the best match to the data. As part of the
analysis, you need expressions for the neighbor backscattering amplitude (Fig. 6.9)
and both absorber and scatterer phase shift functions (Fig. 6.8). Early on, these
functions were obtained empirically from experimental data [245] or interpolated
from tables of theoretically calculated values [225]. Nowadays, software packages
are available to calculate these functions as needed.
Writing about the things that can go wrong with EXAFS analysis is something of
a cottage industry. Indeed, I can think of no other technique that has more review
articles about pitfalls, limitations, or artefacts. Sadly, there is a long history of
EXAFS analyses gone bad.
6.7 Interpretation of EXAFS
157
phase shifts from the absorber and scatterer are also k-dependent functions, and their
net effect on the Fourier transform is to shift peaks to shorter distances. The
nonlinearity of the phase shifts also skews the Fourier transform peaks and can
give rise to additional unphysical peaks.
An approximate solution to this problem is to include the phase shift functions in
the Fourier transform itself. If one transforms with respect to 2k + ϕ(k), where ϕ(k) is
the total phase shift, then the peaks for the A–B interaction will be approximately in
the proper position R AB . Of course, the correction is not perfect when there are
multiple types of neighbors, but in practice the resulting pictures are still approximately correct. Note in the Na 2 MoO 4 transform example that there are still additional non-physical peaks due to the finite range of the data being transformed. These
truncation effects are well-known to result when applying a Fourier transform to a
signal that has been terminated essentially by convolution with a square wave
envelope.
6.7 Interpretation of EXAFS
From the previous discussion, we see that all of the required physics for describing
EXAFS is basically understood, and it could be argued that structure determination
from EXAFS data should be straightforward. The defect in this assumption is that
structure determination from EXAFS is an inverse problem. The structure does not
fall out from the data; instead you usually have to follow an iterative procedure
involving the following: (a) assuming a structure, (b) calculating the EXAFS,
(c) calculating the disagreement between observed and calculated EXAFS,
(d) revising the proposed structure, (e) calculating the revised EXAFS, and
(f) calculating the new residual and iterating until convergence. As with EXAFS
extraction, a variety of software packages is available to accomplish these steps.
Iteration is required because of the nonlinear nature of the EXAFS equation, and
finding the optimum simulation is usually accomplished by a nonlinear least squares
analysis. To use Eq. 6.15, the nature of the backscatterer has to be assumed, and one
can then vary N, R, and σ to achieve the best match to the data. As part of the
analysis, you need expressions for the neighbor backscattering amplitude (Fig. 6.9)
and both absorber and scatterer phase shift functions (Fig. 6.8). Early on, these
functions were obtained empirically from experimental data [245] or interpolated
from tables of theoretically calculated values [225]. Nowadays, software packages
are available to calculate these functions as needed.
Writing about the things that can go wrong with EXAFS analysis is something of
a cottage industry. Indeed, I can think of no other technique that has more review
articles about pitfalls, limitations, or artefacts. Sadly, there is a long history of
EXAFS analyses gone bad.
6.7 Interpretation of EXAFS
157