Techniques of data analysis

Creagh, D. C.

doi:10.1107/97809553602060000592

International
Tables for
Crystallography
Volume C
Mathematical, physical and chemical tables
Edited by E. Prince

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International Tables for Crystallography (2006). Vol. C. ch. 4.2, pp. 217-218

Section 4.2.3.4.1.2. Techniques of data analysis

D. C. Creagh^b

4.2.3.4.1.2. Techniques of data analysis

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Three assumptions must be made if XAFS data are to be used to provide accurate structural and chemical information:

(i) XAFS occurs through the interaction of waves singly scattered by neighbouring atoms;
(ii) the amplitude function of the atoms is insensitive to the type of chemical bond (the postulate of transferability), which implies that one can use the same amplitude function for a given atom in problems involving compounds of that atom, whatever the nature of its neighbours or the nature of the bond; and
(iii) the phase function can be transferred for each pair of absorber–back-scatterer atoms.

Of these three assumptions, (ii) is of the most questionable validity. See, for example, Stern, Bunker & Heald (1981).

It is usual, when analysing XAFS data, to search the literature for, or make sufficient measurements of, $[\mu_{l0}]$ remote from the absorption edge to produce a curve of $[\mu_{l0}(E)]$ versus E that can be extrapolated to the position of the edge. From equation (4.2.3.8), it is possible to produce a curve of $[\chi(E)]$ versus E from which the variation of $[\chi(k)]$ with k can be deduced using equation (4.2.3.9).

It is also customary to multiply $[\chi(k)]$ by some power of k to compensate for the damping of the XAFS amplitudes with increasing k. The power chosen is somewhat arbitrary but [k^3] is a commonly used weighting function.

Two different techniques may be used to analyse the new data set, the Fourier-transform technique or the curve-fitting technique.

In the Fourier-transform technique (FF), the Fourier transform of the $[k^n\chi(k)]$ is determined for that region of momentum space from the smallest, [k_1] , to the largest, [k_2] , wavevectors of the photoelectron, yielding the radical distribution function $[\rho_n(r')]$ in coordinate [(r')] space. $[\rho_n(r')={ 1\over(2\pi)^{1/2}}\int\limits^{k_2}_{k_1}k^n\chi(k)\exp(i2kr'){\, \rm d}k. \eqno (4.2.3.13)]$

The Fourier spectrum contains peaks indicating that the nearest-neighbour, next-nearest-neighbour, etc. distances will differ from the true spacings by between 0.2 to 0.5 Å depending on the elements involved. These position shifts are determined for model systems and then transferred to the unknown systems to predict interatomic spacings. Fig. 4.2.3.4 illustrates the various steps in the Fourier-transform analysis of XAFS data.

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Steps in the reduction of data from an XAFS experiment using the Fourier transform technique: (a) after the removal of background χ(k) versus k; (b) after multiplication by a weighting function (in this case k³); (c) after Fourier transformation to determine r′.

The technique works best for systems having well separated peaks. Its primary weakness as a technique lies in the fact that the phase functions are not linear functions of k, and the spacing shift will depend on [E_0] , the other factors including the weighting of data before the Fourier transforms are made, the range of k space transformed, and the Debye–Waller factors of the system.

In the curve-fitting technique (CF), least-squares refinement is used to fit the spectra in k space using some structural model for the system. Such techniques, however, can only indicate which of several possible choices is more likely to be correct, and do not prove that that structure is the correct structure.

It is possible to combine the FF and CF techniques to simplify the data analysis. Also, for data containing single-scatter peaks, the phase and amplitude components can be separated and analysed separately using either theory or model compounds (Stern, Sayers & Lytle, 1975).

Each XAFS data set depends on two sets of strongly correlated variables: $[\{F(k),\sigma,\rho,N\}]$ and $[\{\varphi(k),E_0,r\}]$ . The elements of each set are not independent of one another. To determine N and σ, one must know F(k) well. To determine r, $[\varphi(k)]$ must be known accurately.

Attempts have been made by Teo & Lee (1979) to calculate F(k) and $[\varphi(k)]$ from first principles using an electron–atom scattering model. Parametrized versions have been given by Teo, Lee, Simons, Eisenberger & Kincaid (1977) and Lee et al. (1981). Claimed accuracies for r, σ, and N in XAFS determinations are 0.5, 10, and 20%, respectively.

Acceptable methods for data analysis must conform to a number of basic criteria to have any validity. Amongst these are the following:

(i) the data analysis must not give rise to systematic error in the sense that it must provide unbiased estimates of parameters;
(ii) the assumed (hypothetical) model must be able to describe the data adequately;
(iii) the number of parameters used to describe the best fit of data must not exceed the number of independent data points;
(iv) where multiple solutions exist, supplementary information or assumptions used to resolve the ambiguity must conform to the philosophy of choice of the model structure.

The techniques for estimation of the parameters must always be given, including all known sources of uncertainty.

A complete list of criteria for the correct analysis and presentation of XAFS data is given in the reports of the International Workshops on Standards and Criteria in XAFS (Lytle, Sayers & Stern, 1989; Bunker, Hasnain & Sayers, 1990).

References

Bunker, G., Hasnain, S. S. & Sayers, D. E. (1990). Report of the International Workshops on Standards and Criteria in XAFS. In X-ray absorption spectroscopy, edited by S. S. Hasnain. London: Ellis Horwood.Google Scholar

Lee, P. A., Citrin, P. H., Eisenberger, P. & Kincaid, B. M. (1981). Extended X-ray absorption fine structure – its strengths and limitations as a structural tool. Rev. Mod. Phys. 53, 769–787.Google Scholar

Lytle, F. W., Sayers, D. E. & Stern, E. A. (1989). Report of the International Workshops on Standards and Criteria in XAFS. In X-ray absorption spectroscopy. Physica (Utrecht), B158, 701–722.Google Scholar

Stern, E. A., Bunker, B. & Heald, A. (1981). Understanding the causes of non-transferability of EXAFS amplitude. EXAFS spectroscopy: techniques and applications, edited by B. K. Teo & D. C. Joy, pp. 59–79. New York: Plenum.Google Scholar

Stern, E. A., Sayers, D. E. & Lytle, F. W. (1975). Extended X-ray absorption fine structure technique. III. Determination of physical parameters. Phys. Rev. B, 11, 4836–4846.Google Scholar

Teo, B. K. & Lee, P. A. (1979). Ab initio calculations of amplitude and phase functions for extended X-ray absorption fine structure spectroscopy. J. Am. Chem. Soc. 101, 2815–2832.Google Scholar

Teo, B. K., Lee, P. A., Simons, A. L., Eisenberger, P. & Kincaid, B. M. (1977). EXAFS. Approximation, parameterization and chemical transferability of amplitude function. J. Am. Chem. Soc. 99, 3854–3856.Google Scholar

International Tables for Crystallography (2006). Vol. C. ch. 4.2, pp. 217-218