The cumulant approach and the ratio method

Fornasini, P.

doi:10.1107/S1574870722005481

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International
Tables for
Crystallography
Volume I
X-ray absorption spectroscopy and related techniques
Edited by C. T. Chantler, F. Boscherini and B. Bunker

International Tables for Crystallography (2024). Vol. I. ch. 5.11, pp. 683-686
https://doi.org/10.1107/S1574870722005481

Chapter 5.11. The cumulant approach and the ratio method

Paolo Fornasini^a ^*

^aDepartment of Physics, University of Trento, 38123 Trento-Povo, Italy
Correspondence e-mail: [email protected]

For weak disorder, the distributions of interatomic distances sampled by extended X-ray absorption fine structure (EXAFS) can be parametrized in terms of their leading cumulants. For a sufficiently well isolated first coordination shell, accurate relative values of the cumulants with respect to a reference sample can be obtained by separate analysis of phases and amplitudes (the ratio method). The basics of both the cumulant approach and the ratio method are presented and their strengths and limitations are discussed.

Keywords: cumulant analysis; ratio method.

1. Distance distributions and cumulants

An EXAFS experiment samples a one-dimensional effective distribution of interatomic distances for each scattering path $[P(r,\lambda)=\rho(r)\exp[-2r/\lambda(k)]/r^{2}, \eqno (1)]$ where λ(k) is the photo-electron mean free path and ρ(r) is the real distribution of distances, the width and shape of which depend on atomic vibrations and sometimes also on structural disorder.

For disorder that is not too large, an effective distribution can effectively be characterized by the cumulant expansion method (see Appendix A ), whereby the EXAFS signal for one path is $[\chi(k)\propto{\textstyle\int\limits_{0}^{\infty}}P(r,\lambda)\exp(2ikr)\,{\rm d}r=\exp\left[\textstyle\sum\limits_{n=0}^{\infty}(2ik)^{n}C_{n}/n!\right], \eqno (2)]$ so that, for a single coordination shell, $[\eqalignno {\chi(k)& =S_{0}^{2}|f(k,\pi)|N\cr &\ \quad {\times}\ \exp\left[C_{0}-2C_{2}k^{2}+{{2} \over {3}}C_{4}k^{4}-{{4} \over {45}}C_{6}\,k^{6}+\cdots\right]\cr &\ \quad {\times}\ \sin\left[2C_{1}k-{{4} \over {3}}C_{3}k^{3}+{{4} \over {15}}C_{5}k^{5}+\cdots+ \varphi(k)\right]. & (3)}]$ Odd and even cumulants C_n separately determine the phase and the amplitude of the signal. C₁ and C₂ are the mean and the variance of the distribution, respectively; higher-order odd and even cumulants measure the asymmetric and symmetric deviations from the normal shape, respectively. In particular, positive or negative values of the third cumulant correspond to a long tail on the positive or the negative side of the distribution, respectively.

The cumulants $[C^{*}_{n}]$ of the real distribution ρ(r) can be recovered from the cumulants C_n of the effective distribution by the recursion formula (Vaccari et al., 2007 ) $[C_{n}^{*}\simeq C_{n}+2C_{n+1}(1/C_{1}+1/\lambda)\quad {\rm for} \quad n=1,2,3,\ldots \eqno (4)]$ The difference between $[C_{1}^{*}]$ and C₁ can be significant (Bunker, 1983 ; Freund et al., 1989 ), $[C_{1}^{*}=C_{1}+2C_{2}(1/C_{1}+1/\lambda)]$ , and the corresponding transformation is included in most data-analysis packages. For n ≥ 2, the difference is much smaller and is frequently neglected.

2. Cumulants and data analysis

The standard EXAFS analysis relies on the calculation of backscattering amplitudes, phase shifts and inelastic factors for all relevant single- and multiple-scattering paths. Coordination numbers and the leading cumulants (generally the first three for the first shell and the first two for the outer shells) are then evaluated by a nonlinear best fit of simulated EXAFS spectra to experimental spectra.

2.1. The ratio method

The ratio method (Bunker, 1983 ; Dalba et al., 1993 ) is an alternative approach for the analysis of the first shell when its contribution can be neatly singled out by Fourier filtering, for example when the leakage of contributions from longer paths is negligible (Schnohr et al., 2014 ). If the first coordination shell of the sample s only contains one atomic species, and if a reference m of known structure is available, the backscattering amplitudes, phase shifts and inelastic factors of which are the same as for the sample s, then the phases and the amplitudes of the first-shell EXAFS signal, $[\eqalignno {A(k) & =(S_{0}^{2}/k)N|f(k,\pi)| \exp(C_{0}-2k^{2}C_{2}+2 k^{4}C_{4}/3 \ldots), \cr \Phi(k) & = 2kC_{1} - 4k^{3}C_{3}/3\ldots + \varphi(k), &(5)}]$ of the sample s can be separately analysed by comparison with the reference m.

The separation of phase and amplitude is made possible by complex Fourier filtering. If the direct and inverse transforms were performed for −∞ < r < +∞, one would obtain $[{\rm Im} \,\tilde{\chi}(k)=0]$ from the inverse transform. Since the inverse transform is performed only for r > 0, both the real and the imaginary parts of $[\tilde{\chi}(k)]$ differ from zero. The real part corresponds to the filtered EXAFS signal (to within the artefact of the Fourier filtering procedure): $[{\rm Re} \,\tilde{\chi}(k) = \chi(k) = A(k) \sin\Phi(k).\eqno (6)]$ The real and imaginary parts can be combined to separately calculate the amplitude and phase of the filtered first-shell signal, $[\eqalignno {A(k) & = \{[{\rm Re}\,\tilde{\chi}(k)]^{2}+[{\rm Im}\, \tilde{\chi}(k)]^{2}\}^{1/2},\cr \Phi(k)& =\arctan[{\rm Re}\,\tilde{\chi}(k)/{\rm Im}\,\tilde{ \chi}(k)]. &(7)}]$

2.2. Phase analysis

If the phase shifts φ(k) of the sample s and reference m are identical, they can be removed by subtracting the total phase of the reference from that of the sample, $[\Phi^{s}(k)-\Phi^{m}(k)= 2k(C_{1}^{s}-C_{1}^{m})-4 k^{3}(C_{3}^{s}-C_{3}^{ m})/3\ +\ldots\eqno (8)]$ A polynomial fit allows one to obtain the differences between the first odd cumulants: $[\Delta C_{1}=C_{1}^{s}-C_{1}^{m}]$ , $[\Delta C_{3}=C_{3}^{s}-C_{3}^{m}]$ , …

If ΔΦ/2k is plotted against k², the intercept gives ΔC₁, the slope is proportional to ΔC₃ and deviations from linearity imply a non-negligible influence of ΔC₅ (Fig. 1 , left).

Figure 1

EXAFS at the Te K edge of CdTe: comparison of the phases (left) and amplitudes (right) of seven spectra measured at 300 K with a reference spectrum measured at 20 K. The dashed line in the left plot is the average best fit for all data in a central k range. Data from Abd el All et al. (2013 ).

2.3. Amplitude analysis

If |f(k, π)|, $[S_{0}^{2}]$ and λ(k) for the sample s and reference m are identical, they can be removed by dividing the sample amplitude by the reference amplitude. The logarithm of the amplitude ratio is $[\eqalignno {\ln{{A^{s}(k)} \over {A^{m}(k)}} & = \ln{{N^{s}} \over {N^{m}}}+(C_{0}^{s}-C_{0}^{m})-2k^{2}(C_{2}^{s}-C_{2}^{m}) \cr &\ \quad +\ 2k^{4}(C_{4}^{s}-C_{4}^{m})/3-\ldots &(9)}]$ A polynomial fit allows one to obtain the coordination number ratio N^s/N^m and the differences between the first even cumulants: $[\Delta C_{2}=C_{2}^{s}-C_{2}^{m}]$ , $[\Delta C_{4}=C_{4}^{s}-C_{4}^{m}]$ , …

If $[\ln[A^{s}(k)/A^{m}(k)]]$ is plotted against k², the intercept corresponds to $[\ln(N^{s}/N^{m})+(C_{0}^{s}-C_{0}^{m})]$ , the slope is proportional to ΔC₂ and deviations from linearity imply a non-negligible influence of ΔC₄ (Fig. 1 , right). The term $[C_{0}^{s}-C_{0}^{m} = -2(C_{1}^{s}-C_{1}^{m})/\lambda-2 (\ln C_{1}^{s}-\ln C_{1}^{m}) \eqno(10)]$ can be estimated from a rough knowledge of the interatomic distances $[C_{1}^{S}]$ and $[C_{1}^{m}]$ and mean free path λ; its neglect could lead to incorrect N^s/N^m ratios.

3. Interpretation of results

3.1. Relative and absolute values of cumulants

The ratio method avoids the theoretical input of backscattering amplitudes, phase shifts and inelastic factors, provided that these quantities can be considered to be identical for the sample s and reference m. Additionally, it cancels the artefacts of the Fourier filtering procedure and reduces curved-wave effects. On the other hand, it only gives ratios of coordination numbers and relative values of cumulants $[\Delta C_{n}=C_{n}^{s}-C_{n}^{m}]$ for well isolated first coordination shells. The method is particularly suited for temperature-dependent (Fornasini et al., 2004 ) or pressure-dependent (Freund et al., 1989 ; Fornasini et al., 2018 ) studies, where the low-T or low-p spectrum can be taken as a reference, or for the comparison of noncrystalline and crystalline phases of a system.

Relative values of the first cumulant are significant in themselves; they allow the direct measurement of the bond thermal expansion or of the bond compressibility by temperature- or pressure-dependent measurements, respectively.

Absolute values of the thermal contribution to the second cumulant (Debye–Waller exponent) in the harmonic approximation can be evaluated by fitting a correlated Debye model (Beni & Platzman, 1976 ; Sevillano et al., 1979 ), $[\sigma^{2}_{\rm D}(T)={3\hbar\over\omega_{\rm D}^{3}m}{\textstyle \int\limits_{0}^{\omega_{\rm D}}}\,{\rm d}\omega \,\omega\coth{\hbar\omega\over 2kT}\left[1-{{\omega_{\rm D}\sin(\omega Rq_{D}/\omega_{\rm D})} \over {\omega Rq_{\rm D}}}\right], \eqno(11)]$ or an Einstein model (Sevillano et al., 1979 ), $[\sigma^{2}_{\rm E}(T)=(\hbar/2\mu\omega_{\rm E})\coth(\omega_{\rm E}/2kT), \eqno (12)]$ to the temperature dependence of the experimental values. The comparison of experimental data with more refined quantum-perturbative approaches allows the evaluation of the absolute values of the thermal contribution to the third and fourth cumulants.

3.2. Ratio method versus nonlinear fitting

The temperature or pressure dependence of the first-shell cumulants obtained by the ratio method and by a nonlinear fit to simulated spectra should coincide: a comparison of the final results is thus an important cross-consistency check of the two procedures.

The detection of structural disorder effects by the ratio method is only possible in some cases, for example when an amorphous sample is compared with its crystalline counterpart or when a phase transition induces an abrupt variation of the temperature dependence with respect to Debye or Einstein behaviour. In general, the presence and the extent of structural contributions to the cumulants can only be estimated from the absolute values obtained by the nonlinear fit to simulated spectra.

The ratio method allows a quite effective elimination of the correlation between phase and amplitude parameters. Additionally, a simple inspection of plots such as those in Fig. 1 allows an immediate evaluation of the quality of the data and a visual estimation of the cumulant values.

Subtler technical differences have been discussed in Vaccari et al. (2007 ). The simulation codes can calculate the mean free path λ(k) as a function of k, but frequently take its effect into account through a factor exp[−2R/λ(k)], where R is an average distance; the effective distribution becomes $[\tilde{P}(r)=\rho(r)/r^{2}]$ and the correction to the first cumulant lacks the 1/λ term. The ratio method accounts for the instantaneous r-dependence of the whole effective distribution P(r, λ), but considers λ to be independent of k; the errors in phases and amplitudes due to this approximation cancel out to a good extent in the comparison of two files (Bunker, 1983 ).

In conclusion, the ratio method (when applicable) and the nonlinear fitting method should be considered as complementary procedures.

3.3. Reconstruction of the distance distribution

Once the absolute values of the leading cumulants have been determined, the characteristic function of the distance distribution can be evaluated by the first terms of its cumulant expansion. The distance distribution can in turn be reconstructed by a Fourier transformation of the characteristic function (Dalba et al., 1993 ).

4. Limitations and uncertainties

The strengths and limitations of the cumulant method have been highlighted by various authors. The convergence properties of the cumulant series for different distributions have been studied by Crozier et al. (1988 ), Dalba et al. (1993 ) and Yang et al. (1997 ). The inadequacy of the cumulant approach has been pointed out by Filipponi (2001 ) for systems affected by strong structural disorder such as liquids and by Mustre de Leon et al. (1992 ) for strongly anharmonic systems.

Only a limited number of polynomial coefficients $[\tilde{C}_{n}]$ are determined in EXAFS analyses. The agreement of the temperature dependence of the polynomial coefficients $[\tilde{C}_{n}]$ with theoretical expectations can be a self-consistent check of the convergence properties of the cumulant series, in order that the polynomial coefficients $[\tilde{C}_{n}]$ can be considered as good estimates of the cumulants C_n (Fornasini et al., 2004 ).

The assessment of the uncertainty of EXAFS results is far from trivial (Abd el All et al., 2013 ). A number of instrumental factors are not under the complete control of the experimenters and require suitable data-analysis procedures and a critical discussion of results for a sound a posteriori evaluation of uncertainties.

EXAFS measurements should be repeated a convenient number of times and the corresponding spectra should be separately analysed. The cumulants obtained from different spectra by the same procedure of analysis represent a restricted sample of a parent population due to short-term fluctuations; the corresponding uncertainty should be estimated from the standard deviation of the distribution of mean values, which decreases when the number of spectra increases.

Different choices of the analysis parameters for the same spectrum (such as the windows and weights of Fourier transforms and the fitting intervals) lead to different values of the cumulants, which cannot be considered as independent samples: increasing the number of fitting intervals cannot decrease the final uncertainty. The point is here a sound choice of the different Fourier parameters and of the different fitting intervals. In the ratio method, the largest fitting interval can be chosen as the interval where the phase differences or the logarithms of amplitudes ratios of different spectra exhibit reasonable agreement (Fig. 1 ). The visual choice is somewhat arbitrary, but generally leads to quite conservative estimates of uncertainty. The values of each cumulant obtained by varying the analysis parameters can be considered as sampling a uniform distribution between the maximum and minimum value, and the corresponding uncertainty can be evaluated as the standard deviation of the distribution.

A further contribution to the uncertainty evaluation is obtained by comparing independent measurements performed on the same system in different laboratories or on samples of different thicknesses and possibly at two different absorption edges.

Finally, external accuracy checks can rely on the comparison of experimental results with theoretical models.

APPENDIX A

Moments and cumulants of a distribution

The initial moments of a distribution Φ(r) are defined as (Cramér, 1966 ) $[\alpha_{n}=\langle r^{n}\rangle=(1/\alpha_{0}) \textstyle \int \limits_{-\infty}^{\infty} r^{n} \Phi(r)\, {\rm d}r, \eqno (13)]$ where $[\alpha_{0}=\textstyle\int\Phi(r)\,{\rm d}r]$ is the normalization integral and α₁ = m is the mean value of the variable r. The initial moment α_n only exists if the integral $[\textstyle \int|r^{n}|\Phi(r)\,{\rm d}r]$ exists; if the moment α of order n exists, all moments of order k < n also exist.

The central moments of the distribution Φ(r) are defined as $[\mu_{n}=\langle(r-\langle r\rangle)^{n}\rangle=(1/\alpha_{0})\textstyle \int \limits_{-\infty}^ {\infty}(r-\langle r\rangle)^{n}\Phi(r)\, {\rm d}r,\eqno(14)]$ so that μ₀ = 1, μ₁ = 0, $[\mu_{2}=\alpha_{2}-\alpha_{1}^{2}=\sigma^{2}]$ is the variance of the distribution, $[\mu_{3}=\alpha_{3}-3\alpha_{1}\alpha_{2}+2\alpha_{1}^{3}]$ measures the asymmetry (or skewness) and μ₄ measures the flatness (or kurtosis).

The characteristic function of a distribution Φ(r) is its Fourier transform $[\Psi(t) =\textstyle \int \limits_{-\infty}^{\infty}\Phi(r)\exp(itr)\, {\rm d}r = \alpha_{0} \langle \exp(itr)\rangle.\eqno (15)]$ There is a one-to-one correspondence between a distribution Φ(r) and its characteristic function Ψ(t).

If all moments α_n exist, the characteristic function can be expanded as a Maclaurin series around the origin t = 0, $[\Psi(t) = \alpha_{0}\langle \exp(itx)\rangle =\alpha_{0}\left[1+\textstyle \sum\limits_{n =1}^{\infty}(it)^{n}\alpha_{n}/n! \right].\eqno (16)]$

The second characteristic function is the logarithm of Ψ(t), $[\ln\Psi(t) =\ln|\Psi(t)| + i\theta(t).\eqno (17)]$ The second characteristic function can be expanded as a Maclaurin series around k = 0 as $[\ln\Psi(t) = \textstyle \sum\limits_{n=1}^{\infty}(it)^{n}C_{n}/n!, \eqno (18)]$ where the coefficients C_n are the cumulants or seminvariants of the distribution. The convergence interval of the cumulant series depends on the properties of the distribution Φ(r).

The first cumulant corresponds to the first initial moment C₁ = α₁ and the leading higher-order cumulants are simply connected to the central moments $[C_{2}=\mu_{2},\quad C_{3}=\mu_{3},\quad C_{4}=\mu_{4}-3\mu_{2}^{2}. \eqno (19)]$

For EXAFS t = 2k, so that $[\ln\Psi(2k) =\ln[\alpha_{0}\langle \exp(2ikr)\rangle] = C_{0}+\ln\langle \exp(2ikr)\rangle \eqno(20)]$ and C₀ = lnα₀ = 0 for the real distribution and C₀ ≠ 0 for the effective distribution of path lengths.

In the literature, the first cumulant has alternatively been defined as the average difference between the instantaneous distance r and the average distance of a reference spectrum (Stern et al., 1992 ) or even the minimum position of the effective potential (Yokoyama et al., 1996 ). These choices do not affect the cumulants of higher order but lead to different expressions for C₀.

References

Abd el All, N., Thiodjio Sendja, B., Grisenti, R., Rocca, F., Diop, D., Mathon, O., Pascarelli, S. & Fornasini, P. (2013). J. Synchrotron Rad. 20, 603–613.Google Scholar

Beni, G. & Platzman, P. M. (1976). Phys. Rev. B, 14, 1514–1518.Google Scholar

Bunker, G. (1983). Nucl. Instrum. Methods Phys. Res. 207, 437–444.Google Scholar

Cramér, H. (1966). Mathematical Methods of Statistics. Princeton University Press.Google Scholar

Crozier, E. D., Rehr, J. J. & Ingalls, R. (1988). X-ray Absorption, edited by D. C. Koningsberger & R. Prins, pp. 373–442. New York: John Wiley & Sons.Google Scholar

Dalba, G., Fornasini, P. & Rocca, F. (1993). Phys. Rev. B, 47, 8502–8514.Google Scholar

Filipponi, A. (2001). J. Phys. Condens. Matter, 13, R23–R60.Google Scholar

Fornasini, P., a Beccara, S., Dalba, G., Grisenti, R., Sanson, A., Vaccari, M. & Rocca, F. (2004). Phys. Rev. B, 70, 174301.Google Scholar

Fornasini, P., Grisenti, R., Irifune, T., Shinmei, T., Mathon, O., Pascarelli, S. & Rosa, A. D. (2018). J. Phys. Condens. Matter, 30, 245402.Google Scholar

Freund, J., Ingalls, R. & Crozier, E. D. (1989). Phys. Rev. B, 39, 12537–12547.Google Scholar

Mustre de Leon, J., Conradson, S. D., Batistić, I., Bishop, A. R., Raistrick, I. D., Aronson, M. C. & Garzon, F. H. (1992). Phys. Rev. B, 45, 2447–2457.Google Scholar

Schnohr, C. S., Araujo, L. J. & Ridgway, M. C. (2014). J. Phys. Soc. Jpn, 83, 094602.Google Scholar

Sevillano, E., Meuth, H. & Rehr, J. J. (1979). Phys. Rev. B, 20, 4908–4911.Google Scholar

Stern, E. A., Ma, Y., Hanske-Petitpierre, O. & Bouldin, C. E. (1992). Phys. Rev. B, 46, 687–694.Google Scholar

Vaccari, M., Grisenti, R., Fornasini, P., Rocca, F. & Sanson, A. (2007). Phys. Rev. B, 75, 184307.Google Scholar

Yang, D. S., Fazzini, D. R., Morrison, T. I., Tröger, L. & Bunker, G. (1997). J. Non-Cryst. Solids, 210, 275–286.Google Scholar

Yokoyama, T., Kobayashi, K., Ohta, T. & Ugawa, A. (1996). Phys. Rev. B, 53, 6111–6122.Google Scholar

International Tables for Crystallography (2024). Vol. I. ch. 5.11, pp. 683-686
https://doi.org/10.1107/S1574870722005481