On the neglecting of higher-order cumulants in EXAFS data analysis

Sanson, A.

doi:10.1107/S0909049509037716

research papers

JOURNAL OF
SYNCHROTRON
RADIATION

ISSN: 1600-5775

Volume 16| Part 6| November 2009| Pages 864-868

doi:10.1107/S0909049509037716

On the neglecting of higher-order cumulants in EXAFS data analysis

A. Sanson ^a ^*

^aUniversità degli Studi di Verona, Facoltà di Scienze, Dip. di Informatica, Strada le Grazie 15, I-37134 Verona, Italy
^*Correspondence e-mail: andrea.sanson@univr.it

(Received 16 July 2009; accepted 17 September 2009; online 3 October 2009)

The cumulant expansion is one of the most powerful and useful methods for EXAFS data analysis, in which the higher-order cumulants allow to consider deviations from a simple Gaussian distribution. In this work, analytical expressions have been derived to show the effects of neglecting higher-order cumulants in EXAFS analysis by the ratio method. The errors in the best-fitting procedure owing to the omission of the higher-order cumulants, as well as of the coordination number, can be determined.

Keywords: EXAFS; cumulant analysis of EXAFS.

1. Introduction

The cumulant method is a model-independent technique based on the expansion of extended X-ray absorption fine-structure (EXAFS) amplitudes and phases as a series of cumulants of the interatomic distance distribution (Teo, 1986 ). In the EXAFS analysis, the contributions of the different coordination shells are singled out by Fourier filtering and separately analyzed. The Fourier filtering process allows the phase Φ(k) and amplitude A(k) of the single shell to be separated. The difference between the EXAFS phases and the logarithm of the amplitude ratio can be written through the ratio method as (Bunker, 1983 ; Fornasini et al., 2001 )

$[\Phi_{\rm{s}}(k)-\Phi_{\rm{r}}(k) = 2k\Delta C_{1}-{{4}\over{3}}k^{3}\Delta C_{3}+{{4}\over{15}}k^{5}\Delta C_{5}+\ldots,\eqno(1)]$

$[\ln{{A_{\rm{s}}(k)}\over{A_{\rm{r}}(k)}} = \ln N-2k^{2}\Delta C_{2}+{{2}\over{3}}k^{4}\Delta C_{4}-{{4}\over{45}}k^{6}\Delta C_{6}+\ldots,\eqno(2)]$

where k is the photoelectron wavevector, N = N_s/N_r is the coordination number ratio, ΔC_n indicates the cumulant difference C_n^s-C_n^r, and the subscripts s and r refer to the sample and reference, respectively. The cumulant method allows the characterization of the first coordination shell in terms of parameters which describe the distance distributions: the first cumulant C₁ is the mean value, C₂ is the variance, C₃ measures the distribution asymmetry and C₄ measures its flatness. For a Gaussian distribution the cumulants C_n are zero for n > 2.

Anharmonicity effects on EXAFS were detected quite early (Eisenberger & Brown, 1979 ). After the first pioneering studies on AgI (Boyce et al., 1981 ) and CuBr (Tranquada & Ingalls, 1983 ), it has been shown that anharmonicity cannot be neglected even in systems like germanium (Dalba et al., 1995 ) or copper (a Beccara et al., 2003 ), where the third cumulant has been taken into account in the analysis to obtain accurate values of the first cumulant; even more so in the case of systems affected by structural disorder, such as liquids, glasses, molten salts and alloys, where the higher-order cumulants must be taken into account when fitting the EXAFS data (Wei et al., 2000 ; Sanson et al., 2008 ; Swilem et al., 2005 ). In strongly disordered systems, where the convergence of the cumulant series is in principle questionable (Filipponi, 2001 ), the cumulants can sometimes be considered to parameterize only the short-range component of the whole distance distribution, as tested in α-AgI (Boyce et al., 1977 , 1981) and more recently in silver molybdate glasses (Sanson, Rocca, Dalba et al., 2007 ).

The importance of including higher-order cumulants in EXAFS analysis has been recognized in many works (Yokoyama et al., 1997 ; Soldo et al., 1998 ; Bus et al., 2006 ; Vaccari et al., 2007 ; Ahmed et al., 2009 ). Other groups recognized the asymmetry in the distance distribution, but did not use the cumulants beyond the second order (Diaz-Moreno et al., 1997 ; Berlier et al., 2002 ; Katsikini et al., 2008 ; Chu et al., 2009 ), with the consequence that the resulting errors in the fit parameters may have drastic effects on the EXAFS structural parameters. In some specific cases, the errors owing to the use of a Gaussian pair distribution have been estimated (Mobilio & Incoccia, 1984 ; Wei et al., 2000), with the result that it produces a significant error for the distance and coordination number. At present, a general treatment of this problem is still lacking.

In this work, for the first time, analytical expressions have been derived to determine the errors in the best-fitting procedure owing to the neglect of the higher-order cumulants (up to the sixth order). The paper is organized as follows: the procedure to derive these expressions is briefly described in §2; the results are reported in §3 and discussed in §4; §5 is dedicated to conclusions.

2. Procedure

Let us consider the best fit of the phases difference, assuming that it is sufficient to truncate equation (1) at the third order (ΔC₃) to have a good fit. In order to evaluate the resulting error on the relative first cumulant (i.e. on the bond distance variation) owing to the neglect of the third cumulant, we can solve the following equation with respect to $[\Delta C_{1}^{\,\prime}]$ ,

$[{{\partial}\over{\partial\Delta C_{1}^{\,\prime}}}\int\limits_{k_{\rm{m}}}^{k_{\rm{M}}}\left[\left(2k\Delta C_{1}-{{4}\over{3}}k^{3}\Delta C_{3}\right)-2k\Delta C_{1}^{\,\prime}\right]^{2}\,{\rm{d}}k=0,\eqno(3)]$

which corresponds to minimize the fitting difference between 2kΔC₁ − (4/3)k³ΔC₃ and $[2k\Delta C_{1}^{\,\prime}]$ ; k_m and k_M are the minimum and maximum values of the fitting interval, respectively. Expanding (3) we obtain

$[\left[\,{{8}\over{3}}k^{3}\Big(\Delta C_{1}^{\,\prime}-\Delta C_{1}\Big)+{{16}\over{15}}k^{5}\Delta C_{3}\right]^{k_{\rm{M}}}_{k_{\rm{m}}} = 0,\eqno(4)]$

and so

$[\Delta C_{1}^{\,\prime} = \Delta C_{1}-\bigg({{2}\over{5}}{{k_{\rm{M}}^{5}-k_{\rm{m}}^{5}}\over{k_{\rm{M}}^{3}-k_{\rm{m}}^{3}}}\bigg)\Delta C_{3}.\eqno(5)]$

As a result, from (5) it can be observed that for ΔC₃ > 0 the neglect of the third cumulant gives an underestimation of the relative first cumulant. On the contrary, for ΔC₃ < 0 the relative first cumulant is overestimated. More important, equation (5) allows the error on ΔC₁ to be quantitatively estimated. For example, by (5), the neglect of a third cumulant ΔC₃ ≃ 0.0005 Å³ in the fitting interval k = 2–10 Å⁻¹ (i.e. k_m = 2 and k_M = 10 Å⁻¹) gives an underestimation of ΔC₁ of about 0.020 Å.

As a second example, let us consider the best fit of the amplitudes ratio, assuming that it is sufficient to truncate (2) at the fourth order (ΔC₄) to have a good fit. To evaluate the resulting error on the coordination number and on the second cumulant owing to the neglecting of the fourth cumulant, we can solve the following system of equations with respect to N′ and $[\Delta C_{2}^{\,\prime}]$ ,

$[\eqalignno{{{\partial}\over{\partial\ln N^{\,\prime}}}\int\limits_{k_{\rm{m}}}^{k_{\rm{M}}}\Big[&\Big(\ln N-2k^{2}\Delta C_{2}+{{2}\over{3}}k^{4}\Delta C_{4}\Big)\cr&-\left(\ln N^{\,\prime}-2k^{2}\Delta C_{2}^{\,\prime}\right)\Big]^{2}\,{\rm{d}}k=0,&(6)}]$

$[\eqalignno{{{\partial}\over{\partial\Delta C_{2}^{\,\prime}}}\int\limits_{k_{\rm{m}}}^{k_{\rm{M}}}\Big[&(\ln N-2k^{2}\Delta C_{2}+{{2}\over{3}}k^{4}\Delta C_{4}\Big)\cr&-\left(\ln N^{\,\prime}-2k^{2}\Delta C_{2}^{\,\prime}\right)\Big]^{2}\,{\rm{d}}k=0,&(7)}]$

whose solutions are

$[\ln N' = \ln N-\bigg({{2}\over{35}}{{4k_{\rm{M}}^{6}+16k_{\rm{M}}^{5}k_{\rm{m}}+40k_{\rm{M}}^{4}k_{\rm{m}}^{2}+55k_{\rm{M}}^{3}k_{\rm{m}}^{3}\,+}\over{4k_{\rm{M}}^{2}+7k_{\rm{M}}k_{\rm{m}}+4k_{\rm{m}}^2}}]$

$[{{+\,40k_{\rm{M}}^{2}k_{\rm{m}}^{4}+16k_{\rm{M}}k_{\rm{m}}^{5}+4k_{\rm{m}}^{6}}\over{}}\bigg)\Delta C_{4}\eqno(8)]$

and

$[\Delta C_{2}^{\,\prime} = \Delta C_{2}-\bigg({{1}\over{7}}{{8k_{\rm{M}}^{4}+17k_{\rm{M}}^{3}k_{\rm{m}}+20k_{\rm{M}}^{2}k_{\rm{m}}^{2}\,+}\over{4k_{\rm{M}}^{2}+7k_{\rm{M}}k_{\rm{m}}+4k_{\rm{m}}^2}}]$

$[{{+\,17k_{\rm{M}}k_{\rm{m}}^{3}+8k_{\rm{m}}^4}\over{}}\bigg)\Delta C_{4}.\eqno(9)]$

As a result, from (8)–(9) it can be seen that the neglect of the fourth cumulant gives an underestimation/overestimation of the relative second cumulant and of the coordination number ratio, according to the sign of ΔC₄. These errors, which depend on the fitting interval (k_m–k_M), can be quantitatively estimated by (8) and (9). For example, the neglect of the fourth cumulant ΔC₄ ≃ 0.0001 Å⁴ in the fitting interval 2–10 Å⁻¹ gives an underestimation of ΔC₂ of about 0.0032 Å² and on the logarithm of N of about 0.096.

3. Results

Following the procedure described in the previous section, the errors on the cumulants analysis have been derived for different cases. The results are listed below.

3.1. Phases difference

3.1.1. Neglect of the third and fifth cumulant

Neglecting both the third and fifth cumulant, the relative first cumulant results as follows,

$[\Delta C_{1}^{\,\prime} = \Delta C_{1}-\bigg({{2}\over{5}}{{k_{\rm{M}}^{5}-k_{\rm{m}}^{5}}\over{k_{\rm{M}}^{3}-k_{\rm{m}}^{3}}}\bigg)\Delta C_{3}+\bigg({{2}\over{35}}{{k_{\rm{M}}^{7}-k_{\rm{m}}^{7}}\over{k_{\rm{M}}^{3}-k_{\rm{m}}^{3}}}\bigg)\Delta C_{5}.\eqno(10)]$

When the fifth cumulant is negligible (ΔC₅ = 0), (10) reduces to (5).

3.1.2. Neglect of the fifth cumulant

With the neglect of the fifth cumulant, the first and third cumulant become

$[\Delta C_{1}^{\,\prime} = \Delta C_{1}-\bigg({{2}\over{63}}{{4k_{\rm{M}}^{10}+16k_{\rm{M}}^{9}k_{\rm{m}}+40k_{\rm{M}}^{8}k_{\rm{m}}^{2}+80k_{\rm{M}}^{7}k_{\rm{m}}^{3}\,+}\over{4k_{\rm{M}}^{6}+16k_{\rm{M}}^{5}k_{\rm{m}}+40k_{\rm{M}}^{4}k_{\rm{m}}^{2}\,+}}]$

$[{{+\,140k_{\rm{M}}^{6}k_{\rm{m}}^{4}+175k_{\rm{M}}^{5}k_{\rm{m}}^{5}+140k_{\rm{M}}^{4}k_{\rm{m}}^{6}+80k_{\rm{M}}^{3}k_{\rm{m}}^{7} \,+}\over{+55k_{\rm{M}}^{3}k_{\rm{m}}^{3}+40k_{\rm{M}}^{2}k_{\rm{m}}^{4}+16k_{\rm{M}}k_{\rm{m}}^{5}+4k_{\rm{m}}^{6}}}]$

$[{{+\,40k_{\rm{M}}^{2}k_{\rm{m}}^{8}+16k_{\rm{M}}k_{\rm{m}}^{9}+4k_{\rm{m}}^{10}}\over{}}\bigg)\Delta C_{5}\eqno(11)]$

and

$[\Delta C_{3}^{\,\prime} = \Delta C_{3}-\bigg({{1}\over{9}}{{8k_{\rm{M}}^{8}+32k_{\rm{M}}^{7}k_{\rm{m}}+80k_{\rm{M}}^{6}k_{\rm{m}}^{2}+125k_{\rm{M}}^{5}k_{\rm{m}}^{3}\,+}\over{4k_{\rm{M}}^{6}+16k_{\rm{M}}^{5}k_{\rm{m}}+40k_{\rm{M}}^{4}k_{\rm{m}}^{2}\,+}}]$

$[{{+\,140k_{\rm{M}}^{4}k_{\rm{m}}^{4}+125k_{\rm{M}}^{3}k_{\rm{m}}^{5}+80k_{\rm{M}}^{2}k_{\rm{m}}^{6}+32k_{\rm{M}}k_{\rm{m}}^{7} +8k_{\rm{m}}^{8}}\over{+\,55k_{\rm{M}}^{3}k_{\rm{m}}^{3}+40k_{\rm{M}}^{2}k_{\rm{m}}^{4}+16k_{\rm{M}}k_{\rm{m}}^{5}+4k_{\rm{m}}^{6}}}\bigg)\Delta C_{5},\eqno(12)]$

respectively. As a result, both the relative first and third cumulant are underestimated when ΔC₅ > 0 and overestimated when ΔC₅ < 0.

3.2. Amplitudes ratio

3.2.1. Neglect of the coordination number, fourth and sixth cumulant

In the case that the coordination number, fourth and sixth cumulant are neglected, the second cumulant becomes

$[\eqalignno{\Delta C_{2}^{\,\prime} ={}& \Delta C_{2}-\bigg({{5}\over{6}}{{k_{\rm{M}}^{3}-k_{\rm{m}}^{3}}\over{k_{\rm{M}}^{5}-k_{\rm{m}}^{5}}}\bigg)_{\vphantom{\Big|}}\ln N-\bigg({{5}\over{21}}{{k_{\rm{M}}^{7}-k_{\rm{m}}^{7}}\over{k_{\rm{M}}^{5}-k_{\rm{m}}^{5}}}\bigg)\Delta C_{4}\cr&+\bigg({{2}\over{81}}{{k_{\rm{M}}^{9}-k_{\rm{m}}^{9}}\over{k_{\rm{M}}^{5}-k_{\rm{m}}^{5}}}\bigg)\Delta C_{6}.&(13)}]$

3.2.2. Neglect of the fourth and sixth cumulant

Neglecting both the fourth and sixth cumulant, the coordination number and the second cumulant become

$[\ln N^{\,\prime} = \ln N- \bigg({{2}\over{35}}{{4k_{\rm{M}}^{6}+16k_{\rm{M}}^{5}k_{\rm{m}}+40k_{\rm{M}}^{4}k_{\rm{m}}^{2}+55k_{\rm{M}}^{3}k_{\rm{m}}^{3}\,+}\over{4k_{\rm{M}}^{2}+7k_{\rm{M}}k_{\rm{m}}+4k_{\rm{m}}^{2}}}]$

$[{{+\,40k_{\rm{M}}^{2}k_{\rm{m}}^{4}+16k_{\rm{M}}k_{\rm{m}}^{5}+4k_{\rm{m}}^{6}}\over{}}\bigg)\Delta C_{4}]$

$[-\,\bigg({{4}\over{945}}{{8k_{\rm{M}}^{8}+32k_{\rm{M}}^{7}k_{\rm{m}}+80k_{\rm{M}}^{6}k_{\rm{m}}^{2}+125k_{\rm{M}}^{5}k_{\rm{m}}^{3}+140k_{\rm{M}}^{4}k_{\rm{m}}^{4}\,+}\over{4k_{\rm{M}}^{2}+7k_{\rm{M}}k_{\rm{m}}+4k_{\rm{m}}^{2}}}]$

$[{{+\,125k_{\rm{M}}^{3}k_{\rm{m}}^{5}+80k_{\rm{M}}^{2}k_{\rm{m}}^{6}+32k_{\rm{M}}k_{\rm{m}}^{7}+8k_{\rm{m}}^{8}}\over{}}\bigg)\Delta C_{6},\eqno(14)]$

$[{{+\,17k_{\rm{M}}k_{\rm{m}}^{3}+8k_{\rm{m}}^4}\over{}}\bigg)\Delta C_{4}]$

$[+\,\bigg({{2}\over{63}}{{4k_{\rm{M}}^{6}+9k_{\rm{M}}^{5}k_{\rm{m}}+12k_{\rm{M}}^{4}k_{\rm{m}}^{2}+13k_{\rm{M}}^{3}k_{\rm{m}}^{3}+12k_{\rm{M}}^{2}k_{\rm{m}}^{4}\,+}\over{4k_{\rm{M}}^{2}+7k_{\rm{M}}k_{\rm{m}}+4k_{\rm{m}}^{2}}}]$

$[{{+\,9k_{\rm{M}}k_{\rm{m}}^{5}+4k_{\rm{m}}^{6}}\over{}}\bigg)\Delta C_{6}.\eqno(15)]$

When the sixth cumulant is negligible (ΔC₆ = 0), equations (14) and (15) reduce to (8) and (9), respectively.

3.2.3. Neglect of the sixth cumulant

When the sixth cumulant is neglected, the coordination number, the second and the fourth cumulant change as

$[\ln N^{\,\prime} = \ln N- \bigg({{4}\over{2079}}{{64k_{\rm{M}}^{12}+576k_{\rm{M}}^{11}k_{\rm{m}}+2880k_{\rm{M}}^{10}k_{\rm{m}}^{2}\,+}\over{64k_{\rm{M}}^{6}+351k_{\rm{M}}^{5}k_{\rm{m}}+855k_{\rm{M}}^{4}k_{\rm{m}}^{2}\,+}}]$

$[{{+\,9335k_{\rm{M}}^{9}k_{\rm{m}}^{3}+20655k_{\rm{M}}^{8}k_{\rm{m}}^{4}+32535k_{\rm{M}}^{7}k_{\rm{m}}^{5}+37695k_{\rm{M}}^{6}k_{\rm{m}}^{6} \,+}\over{+\,1135k_{\rm{M}}^{3}k_{\rm{m}}^{3}+855k_{\rm{M}}^{2}k_{\rm{m}}^{4}+351k_{\rm{M}}k_{\rm{m}}^{5}+64k_{\rm{m}}^{6}}}]$

$[{{+\,32535k_{\rm{M}}^{5}k_{\rm{m}}^{7}+20655k_{\rm{M}}^{4}k_{\rm{m}}^{8}+9335k_{\rm{M}}^{3}k_{\rm{m}}^{9}+2880k_{\rm{M}}^{2}k_{\rm{m}}^{10}\,+}\over{}}]$

$[{{+\,576k_{\rm{M}}k_{\rm{m}}^{11}+64k_{\rm{m}}^{12}}\over{}}\bigg)\Delta C_{6},\eqno(16)]$

$[\Delta C_{2}^{\,\prime} = \Delta C_{2}- \bigg({{2}\over{99}}{{64k_{\rm{M}}^{10}+401k_{\rm{M}}^{9}k_{\rm{m}}+1305k_{\rm{M}}^{8}k_{\rm{m}}^{2}\,+}\over{64k_{\rm{M}}^{6}+351k_{\rm{M}}^{5}k_{\rm{m}}+855k_{\rm{M}}^{4}k_{\rm{m}}^{2}\,+}}]$

$[{{+\,2895k_{\rm{M}}^{7}k_{\rm{m}}^{3}+4695k_{\rm{M}}^{6}k_{\rm{m}}^{4}+5535k_{\rm{M}}^{5}k_{\rm{m}}^{5}+4695k_{\rm{M}}^{4}k_{\rm{m}}^{6} \,+}\over{+\,1135k_{\rm{M}}^{3}k_{\rm{m}}^{3}+855k_{\rm{M}}^{2}k_{\rm{m}}^{4}+351k_{\rm{M}}k_{\rm{m}}^{5}+64k_{\rm{m}}^{6}}}]$

$[{{+\,2895k_{\rm{M}}^{3}k_{\rm{m}}^{7}+1305k_{\rm{M}}^{2}k_{\rm{m}}^{8}+401k_{\rm{M}}k_{\rm{m}}^{9}+64k_{\rm{m}}^{12}}\over{}}\bigg)\Delta C_{6},\eqno(17)]$

$[\Delta C_{4}^{\,\prime} = \Delta C_{4}- \bigg({{2}\over{11}}{{64k_{\rm{M}}^{8}+366k_{\rm{M}}^{7}k_{\rm{m}}+990k_{\rm{M}}^{6}k_{\rm{m}}^{2}\,+}\over{64k_{\rm{M}}^{6}+351k_{\rm{M}}^{5}k_{\rm{m}}+855k_{\rm{M}}^{4}k_{\rm{m}}^{2}\,+}}]$

$[{{+\,1655k_{\rm{M}}^{5}k_{\rm{m}}^{3}+1935k_{\rm{M}}^{4}k_{\rm{m}}^{4}+1655k_{\rm{M}}^{3}k_{\rm{m}}^{5}+990k_{\rm{M}}^{2}k_{\rm{m}}^{6}\,+ }\over{+\,1135k_{\rm{M}}^{3}k_{\rm{m}}^{3}+855k_{\rm{M}}^{2}k_{\rm{m}}^{4}+351k_{\rm{M}}k_{\rm{m}}^{5}+64k_{\rm{m}}^{6}}}]$

$[{{+\,366k_{\rm{M}}k_{\rm{m}}^{7}+64k_{\rm{m}}^{8}}\over{}}\bigg)\Delta C_{6}.\eqno(18)]$

Accordingly, they are underestimated when ΔC₆ > 0 and overestimated when ΔC₆ < 0.

3.2.4. Neglect of the coordination number

Let us consider the case that (2) truncated at the fourth order (ΔC₄) gives a good fit. If the variation of the coordination number is neglected, the second and the fourth cumulant result as follows,

$[\Delta C_{2}^{\,\prime} = \Delta C_{2}- \bigg({{7}\over{6}}{{8k_{\rm{M}}^{8}+32k_{\rm{M}}^{7}k_{\rm{m}}+80k_{\rm{M}}^{6}k_{\rm{m}}^{2}\,+}\over{4k_{\rm{M}}^{10}+16k_{\rm{M}}^{9}k_{\rm{m}}+40k_{\rm{M}}^{8}k_{\rm{m}}^{2}\,+}}]$

$[{{+\,125k_{\rm{M}}^{5}k_{\rm{m}}^{3}+140k_{\rm{M}}^{4}k_{\rm{m}}^{4}+125k_{\rm{M}}^{3}k_{\rm{m}}^{5}+80k_{\rm{M}}^{2}k_{\rm{m}}^{6} \,+}\over{+\,80k_{\rm{M}}^{7}k_{\rm{m}}^{3}+140k_{\rm{M}}^{6}k_{\rm{m}}^{4}+175k_{\rm{M}}^{5}k_{\rm{m}}^{5}+140k_{\rm{M}}^{4}k_{\rm{m}}^{6}+80k_{\rm{M}}^{3}k_{\rm{m}}^{7}\,+}}]$

$[{{+\,32k_{\rm{M}}k_{\rm{m}}^{7}+8k_{\rm{m}}^{8}}\over{+\,40k_{\rm{M}}^{2}k_{\rm{m}}^{8}+16k_{\rm{M}}k_{\rm{m}}^{9}+4k_{\rm{m}}^{10}}}\bigg)\ln N,\eqno(19)]$

$[\Delta C_{4}^{\,\prime} = \Delta C_{4}- \bigg({{63}\over{10}}{{4k_{\rm{M}}^{6}+16k_{\rm{M}}^{5}k_{\rm{m}}\,+}\over{4k_{\rm{M}}^{10}+16k_{\rm{M}}^{9}k_{\rm{m}}+40k_{\rm{M}}^{8}k_{\rm{m}}^{2}\,+}}]$

$[{{+\,40k_{\rm{M}}^{4}k_{\rm{m}}^{2}+55k_{\rm{M}}^{3}k_{\rm{m}}^{3}+40k_{\rm{M}}^{2}k_{\rm{m}}^{4}\,+ }\over{+\,80k_{\rm{M}}^{7}k_{\rm{m}}^{3}+140k_{\rm{M}}^{6}k_{\rm{m}}^{4}+175k_{\rm{M}}^{5}k_{\rm{m}}^{5}+140k_{\rm{M}}^{4}k_{\rm{m}}^{6}+80k_{\rm{M}}^{3}k_{\rm{m}}^{7}\,+}}]$

$[{{+\,16k_{\rm{M}}k_{\rm{m}}^{5}+4k_{\rm{m}}^{6}}\over{+\,40k_{\rm{M}}^{2}k_{\rm{m}}^{8}+16k_{\rm{M}}k_{\rm{m}}^{9}+4k_{\rm{m}}^{10}}}\bigg)\ln N.\eqno(20)]$

Equation (19) shows the correlation between coordination number and EXAFS Debye–Waller factor. As expected, a decrease (or increase) of the coordination number, if neglected, leads to an increase (or decrease) of the second cumulant, according to the amplitude of the EXAFS signal. This is particularly important in EXAFS studies of glasses or disordered systems, where coordination number and Debye–Waller factor play a key role (Kuzmin et al., 2006 ; Sanson, Rocca, Dalba et al., 2007; Sanson, Rocca, Fornasini et al., 2007 ).

4. Discussion

Let us test the equations calculated in the previous sections through an experimental example. To this aim, let us consider the phases difference and the logarithm of the amplitudes ratio of the EXAFS signals measured in silver molybdate glasses at room temperature against the same glass at 25 K used as reference (Sanson, Rocca, Dalba et al., 2007).

Fig. 1 shows the difference of the phases and the corresponding best fits in the range k = 2.5–12 Å⁻¹. The fits were performed (a) using only the first cumulant (ΔC₁), (b) including the third cumulant (ΔC₁ + ΔC₃) and (c) including the fifth cumulant (ΔC₁ + ΔC₃ + ΔC₅). The corresponding fitting results are reported in the first part of Table 1. It can be observed that the third cumulant is essential to obtain accurate relative values of the first cumulant. In this example, the discrepancy on ΔC₁ between fit (a) and fit (b) [or fit (c)] is about 0.02 Å. The discrepancy on ΔC₃ between fit (b) and fit (c) (although less important) is about 10⁻⁴ Å³. These discrepancies can be directly estimated by (5), (10) and (11)–(12) (depending on the fit case) with k_m = 2.5 and k_M = 12 Å⁻¹. The results are listed in the second part of Table 1. It can be seen that the agreement between predicted values (i.e. second part of Table 1) and best-fit values (i.e. first part of Table 1) is excellent.

Table 1
Fit of the phases difference

First part: relative cumulants obtained from the fits. Second part: relative cumulants predicted from the equations calculated in §2 and §3.

	ΔC₁ (Å)	ΔC₃ (Å³)	ΔC₅ (Å⁵)
Resulting cumulants by fit
(a) ΔC₁	−0.0397	–	–
(b) ΔC₁ + ΔC₃	−0.0201	3.37 × 10⁻⁴	–
(c) ΔC₁ + ΔC₃ + ΔC₅	−0.0179	4.42 × 10⁻⁴	3.24 × 10⁻⁶

Estimated cumulants of fit
(a) by (5) with values of fit (b)	−0.0397	–	–
(a) by (10) with values of fit (c)	−0.0397	–	–
(b) by (11)–(12) with values of fit (c)	−0.0201	3.37 × 10⁻⁴	–

Figure 1
Example of phases difference (black solid line) fitted with ΔC₁ (dashed line), ΔC₁ + ΔC₃ (solid line) and ΔC₁ + ΔC₃ + ΔC₅ (dash-dotted line). The results are listed in the first part of Table 1

and compared with the values predicted from equations of §2

and §3

Analogously, Fig. 2 shows the logarithm of the amplitude ratios and the corresponding best fits in the same interval k = 2.5–12 Å⁻¹. The fits were performed (a) using only the second cumulant (ΔC₂), (b) including the fourth cumulant (ΔC₂ + ΔC₄), (c) only including the coordination number and the second cumulant (N + ΔC₂) and (d) including coordination number, second and fourth cumulants (N + ΔC₂ + ΔC₄). For simplicity, the best fits that include the sixth cumulant are not reported, but the reliability of the corresponding equations (14)–(18) is assured anyway.

Figure 2
Example of logarithm of amplitudes ratio (black solid line) fitted with ΔC₂ (dotted line), ΔC₂ + ΔC₄ (dashed line), N + ΔC₂ (dash-dotted line) and N + ΔC₂ + ΔC₄ (solid line). The results are listed in the first part of Table 2

and compared with the values predicted from equations of §2

and §3

The best-fitting results are listed in the first part of Table 2. It can be seen, for example, that the changes of the coordination number, when neglected, drastically affect the values of the second cumulant, as well as of the fourth cumulant. The discrepancy on ΔC₂ between fit (b) and fit (d) is almost 0.004 Å², and about 7 × 10⁻⁵ Å⁴ on ΔC₄. The cumulant differences among the fits can be directly estimated from the equations of §2 and §3. The results are listed in the second part of Table 2. The agreement with the best-fit values (i.e. with the first part of Table 2) confirms the goodness of the analytical expressions derived in this paper.

Table 2
Fit of the logarithm of amplitude ratio

First part: relative cumulants obtained from the fits. Second part: relative cumulants predicted from the equations calculated in §2 and §3.

	N	ΔC₂(Å²)	ΔC₄ (Å⁴)
Resulting cumulants by fit
(a) ΔC₂	–	4.98 × 10⁻³	–
(b) ΔC₂ + ΔC₄	–	7.20 × 10⁻³	6.48 × 10⁻⁵
(c) ln N + ΔC₂	0.795	3.66 × 10⁻³	–
(d) ln N + ΔC₂ + ΔC₄	0.787	3.45 × 10⁻³	−4.76 × 10⁻⁶

Estimated cumulants of fit
(a) by (13) with values of fit (b)	–	4.98 × 10⁻³	–
(a) by (13) with values of fit (c)	–	4.98 × 10⁻³	–
(a) by (13) with values of fit (d)	–	4.99 × 10⁻³	–
(b) by (19)–(20) with values of fit (d)	–	7.20 × 10⁻³	6.45 × 10⁻⁵
(b) by (19)–(20) with values of fit (c)	–	7.25 × 10⁻³	6.63 × 10⁻⁵
(c) by (8)–(9) with values of fit (d)	0.795	3.67 × 10⁻³	–

Before the conclusions, let us make a final observation. The experimental data cannot be fitted using an unrestricted number of fitting parameters, otherwise the fit becomes better but the essential parameters (i.e. distance, Debye–Waller factor, coordination number) can give worse results. However, on the other side, the main higher-order cumulants cannot be neglected in many cases, but it is necessary to find a good balance.

5. Conclusions

In this work, analytical expressions have been derived to determine the errors in the EXAFS analysis, by the ratio method, owing to the neglect of the higher-order cumulants. The reliability of the present results has been tested on experimental data. The importance of the higher-order cumulants to obtain accurate values of the lower-order cumulants, i.e. bond distance, coordination number and Debye–Waller factor, is demonstrated.

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Volume 16| Part 6| November 2009| Pages 864-868

doi:10.1107/S0909049509037716