- Email: [email protected]
EXAFS multiple-scattering data-analysis: GNXAS methodology and applications
ELSEVIER
Physica B 208&209 (1995) 125 128
EXAFS multiple-scattering data-analysis: GNXAS methodology and applications A. Di Cicco Dipartimento di Matematica e Fisica, Universit& di Camerino, Via Madonna delle Carceri, 62032 Camerino, Italy
Abstract
A method for multiple-scattering EXAFS data analysis, denominated GNXAS, is described. Reference to previous work is given. Accuracy of the structural results is discussed. An application to crystalline iron is briefly presented.
In recent times, the need to develop improved EXAFS data-analysis methods has been largely recognized. Earlier plane-wave single-scattering theories, and empirical methodologies based on the use of model compounds, are being gradually replaced by ab initio multiple-scattering (MS) calculations. These advances are related to the development, in the last 20 years, of both an efficient theory and convenient algorithms to calculate the structural contribution to the X-ray absorption cross-section. The inclusion of MS signals has been found to be necessary, both in the low-energy and high-energy regimes, for EXAFS structural analysis. Important MS effects have been evidenced not only in crystalline and molecular systems [1-51, but also in disordered matter [6, 71. The need for accounting correctly of MS contributions has stimulated the development of sophisticated packages for EXAFS data analysis. In this contribution we briefly describe the GNXAS method, originally presented in 1991 [8,91, which has been already applied to a large variety of systems. Applications included diatomic [10] and polyatomic gas-phase molecules [11,121, carbonyl clusters [13], a-Si [6], high-To cuprates [14], aqueous solutions [15,161, elemental and molecular liquids [7, 17, 18], and biological systems [19]. A full description of the theory and several examples of data analysis will
be given in a forthcoming paper [201. Generally speaking, a very good agreement with structural data determined by other techniques has been found. Typical statistical errors are often found well below 0.01 A for first-shell average bond distance, while absolute values have been found to be reliable within 0.02,g, [201. The method is based on a comparison between the experimental signal and a theoretical model one'attempting to optimize the relevant structural parameteFwalues. The comparison is performed on the absorption coefficient (ctoxp(E)) instead of the structural g(k) signal. The reason is that EXAFS background can hardly be defined exactly and, as a consequence, it is refined in conjunction with a structural contribution on the raw data. The model absorption signal is 0¢mod(E)= g0(1 + Xmod(E-- Eo)) + 0~bks(E) + C%x¢(E), (1) where gbks(E)is a smooth polynomial spline accounting for underlying absorption channels, and ctoxc(E)accounts for possible multi-electron excitation channels, ctox¢(E) simulates in an empirical way steplike or slope changes occurring at precise energies above the main excitation edge. Examples of the importance of the ctexc(E)contribution have been widely discussed in the literature (see Ref. [21] and the references therein).
0921-4526/95/$09.50 © 1995 Elsevier Science B.V. All rights reserved SSDI 0 9 2 1 - 4 5 2 6 ( 9 4 ) 0 0 6 4 7 - 4
126
A. Di Cicco/Physica B 208&209 (1995) 125 128
The structural EXAFS term x(k)mod is an oscillating function of the wave vector k ~ , ~ - - E0 (E0 is the threshold energy) and is calculated according to the following steps: (a) calculation of a model potential for the final state photoelectron (muffin-tin approximation), (b) the phase shifts calculation, (c) the signal calculation, and (d) the configurational average. The basic theory related to the steps (a) and (b) is described in detail elsewhere (see Ref. [22] for example). The central step (c) in the calculation of g(k)rnod, in the framework of the MS theory within the muffin-tin approximation [22], is the inversion of a matrix (I + GT) containing phase shifts (matrix T) and propagators (matrix G) which describe the final state photo-electron in a selected cluster of atoms (see. for example, Refs. [23, 20] and the references therein). A power series expansion of the (I + GT)- 1 matrix is usually convergent in the high-energy limit and the first non-trivial term g2, representing the single-scattering, has been widely used to analyze EXAFS spectra. However, there are practical difficulties in applying the multiple-scattering series expansion mainly because of the great number of contributions (MS pathways) to be considered to achieve convergence. A different approach consists in calculating directly the total contribution to the cross-section due to couples, triplets, and quadruplets of atoms. Calculation of the total signals due to a few atoms are very fast also at high energies by using the continued-fraction algorithm [23]. Proper n-body 71"~signals are defined by subtracting loworder terms from n-atoms total calculations [20]. The use of the 7t"l signals allows us to establish quite simple relationships between structure and signals in terms of the main two-atom and three-atom configurations. Another important aspect of the theory is how the thermal (configurational) average of the signals has to be done. An appropriate theory has to be applied for calculating the averages of the n-body signals [24, 20] and is conceptually incorrect to apply different effective DebyeWaller-like factors for each n-body configuration. The signals (Zm or 7) to be averaged are oscillating functions where amplitudes A(k, R) and phases if(k, R) are smooth functions of k and of the geometrical parameters R. Suitable expressions, based on a Taylor expansion of A(k, R) and tk(k, R), have been derived for the average of the n-body configurations using Gaussian and nearly Gaussian distribution functions and have been implemented into the GNXAS programs [24, 9, 20]. As anticipated, the GNXAS programs provide a fit of the raw experimental data incorporating all the abovementioned algorithms. Calculation of the ),t"~functions is performed by using a muffin-tin potential and advanced models for the energy-dependent exchange-correlation self-energy.
The comparison between ~modand ~expis evaluated by means of a square residual function of the type R({)o})
i=~1
[~Xexp(Ei) - ~Xmod(Ei, }q, )~2. . . . . )~p)]2 0"2
(2)
Following very standard statistical procedures for non-linear fitting problems it is possible to perform a rigorous statistical evaluation of the results [20]. In particular, (i) the optimal values for the p structural parameters are {2} = { ~ corresponding to the minimum of the residual function; (ii) the statistical ~2-test can be performed to check whether the actual value of R({J.}) is only due to residual noise or it contains unexplained physical information; (iii) the statistical significance of the inclusion of different structural details, such as further n-body contributions or a splitting of a shell into two, can be analyzed using the F-test; (iv) the correlated statistical error in the parameters can be evaluated looking at confidence intervals defined by the equation R({2}) < Rmi n + C, with C depending on the confidence level of the analysis. Similar procedures based on statistical tests are indeed commonly applied to the EXAFS data-analysis problem [25]. Numerous GNXAS applications to molecular cases have been published so far. Here we briefly present an application to a crystalline case: the BCC iron. Full details are given elsewhere [26]. The main frequency content of the EXAFS signal is confined within 5 ,~ (Fig. 1, panel c). The five two-atom and four three-atom configurations contributing with 7 signals in this frequency region have been considered in the g(k)mod calculation. A pictorial view of the two-atom and three-atom configurations relevant to the BCC structure is contained in Ref. [27], Fig. 2. Angles and relative distances have been kept fixed, only the cell side a, the variances of distances and angles, and the correlation parameters of the covariance matrices have been floated [9, 20]. The total number of structural parameters is 14. Other two non-structural parameters, required to compare the model and experimental signals are Eo and So2. Eo accounts for the energy difference between the theoretical and experimental energy scales while S~ is an overall amplitude factor accounting for many-body effects and inaccurate jump normalization. In the upper panel (a) of Fig. 1 the proper best-fit two-atom and three-atom signals are shown from the top to the bottom. The first two, ~(2) and 7(,I.2), are the proper MS two-atom signals associated with the first and the fourth shell of neighbors, respectively. The third and fourth 7 signals are related to the first triangular configuration distribution (angle of 70.53 degrees). The 7t22)signal is the proper two-atom signal related to second
127
,4. Di Cicco /Physica B 208&209 (1995) 125 128 I/
(z)4 (a)
C
/
v
'
I ....
I'
''
0.0050 v
a
V
I ....
v
~
7'(a) t (==)
v
.....
%
~
0.0045
7(a)= ...~..~, 7(a)3~ 7(=)5
I
70) 4
0.0040
2.5
....
0.75 ,,I
....
I ....
I ....
I ....
I ....
101.,I . . . . I ? , , , I , : , , I , , , , I z.5
5~o
//
g
I ....
o
I .... a
7.6
12.5
I
6
I ....
0.0
I.,.
0.95
Fig. 2. Fitting of the Fe K-edge EXAFS spectrum: a 2 - So2 contour map. The inner elliptically shaped curves represent the 95% confidence interval.
~5.o
"X
I .... 4 R (1)
I ....
0.85
~o~
. . . . t,
to.o
A
I,
I ....
0.8
.... B
Fig. 1. Proper two-atom and three-atom multiple-scattering contributions to EXAFS of crystalline iron (a). Comparison between calculated (solid) and experimental (dots) crystalline iron k2z(k)(b). Fourier transform (FT) of experimental (dots) and calculated (solid) signals (c).
neighbors while the y~3)represents the proper three-atom signal. The successive two y signals are associated with the second three-body configuration (angle of 109.47 degrees). The seventh y signal is the proper three-atom contribution due the third triangular configuration (angle of 90 degrees). The last two curves are related to the collinear configurations involving first and fifth neighbors. As expected, the proper three-atom contribution (Tt,3)) is much stronger than the fifth-shell two-atom one (y~2)) due to the well-known focusing effect. In the center panel (b) the total best-fit MS signal and the experimental structural term k2g(k) are shown. The agreement is excellent in the whole energy range and the unexplained features are mainly due to contributions located in the region above 5.A in R-space (see Fig. 1, panel (c)). All of the best-fit structural parameter values and their estimated statistical errors are given elsewhere. In this contribution, due to the limited space, we focus on the results of the main parameters: the cell side a and the
first-shell bond variance a~. We found a = 2.855 (5)/{, in agreement within 0.01 ,~ with the value 2.8662 (5) A measured by diffraction. The slight shortening of the cell side a can be mainly attributed to systematic errors both in the theoretical calculation and in the experimental energy scale. The bond variance a 2 = 4.7 (6) × 10- 3 ~=2 is also in agreement with the calculated one tr2 = 5.06 x 10-3/~ 2 [28]. The multi-parameter correlation matrix has been calculated showing large correlation coefficients between some parameters. For example, both a and Eo and So2 and a 2 are highly correlated ( ~ 0.88), as expected, mainly contributing to increase the errors in a and in try, respectively. The Eo energy parameter was found to coincide within 0.5 eV with the energy position of the second peak of the derivative (around 7120 eV) of the experimental spectrum (used to calculate the k scale of the figure). The S g factor has been found to be around 0.85. The contour plots associated with the S2-tr 2 two-dimensional sections is shown in Fig. 2. The correlated statistical errors at the 95% confidence level, which depend on the noise level and on the energy extension of the measurement, are found quite low in the present case: + 0.005 A for a and 0.6 x 10 - 3 ~2 for a 2.
References
[1] P.A. Lee and J.B. Pendry, Phys. Rev. B 11 (1975) 2795. [2] M. Benfatto, C.R. Natoli, A. Bianconi, J. Garcia, A. Marcelli, M. Fanfoni and I. Davoli, Phys. Rev. B 34 (1986) 5774.
128
A. Di Cicco/Physica B 208&209 (1995) 125-128
[3] S.J. Gurman, N. Binsted and I. Ross, J. Phys. C 19 (1986) 1845. [4] A. Bianconi, A. Di Cicco, N.V. Pavel, M. Benfatto, A. Marcelli, C.R. Natoli, P. Pianetta and J. Woicik, Phys. Rev. B 36 (1987) 6426. [5] M.F. Ruiz-Lrpez, M. Loos, J. Goulon, M. Benfatto and C.R. Natoli, Chem. Phys. 121 (1988) 419. [6] A. Filipponi, A. Di Cicco, M. Benfatto and C.R. Natoli, Europhys. Lett. 13 (1990) 319. I-7] A. Di Cicco and A. Filipponi, Europhys. Lett. 27 (1994) 407. [8] The acronym GNXAS derives from 9n (the n-body distribution functions) and XAS (X-ray absorption spectroscopy). [9] A. Filipponi, A. Di Cicco, T.A. Tyson and C.R. Natoli, Solid State Commun. 78 (1991) 265. [10] P. D'Angelo, A. Di Cicco, A. Filipponi and N.V. Pavel, Phys. Rev. A 47 (1993) 2055. [11] A. Di Cicco, S. Stizza, A. Filipponi, F. Boscherini and S. Mobilio, J. Phys. B 25 (1992) 2309. [12] E. Burattini, P. D'Angelo, A. Di Cicco, A. Filipponi and N.V. Pavel, J. Phys. Chem. 97 (1993) 5486. [13] A. Filipponi, A. Di Cicco, R. Zanoni, M. Bellatreccia, V. Sessa, C. Dossi and R. Psaro, Chem. Phys. Lett. 184 (1991) 485. [14] A. Di Cicco and M. Berrettoni, Phys. Lett. A 176 (1993) 375. [15] P. D'Angelo, A. Di Nola, A. Filipponi, N.V. Pavel and D. Roccatano, J. Chem. Phys. 100 (1994) 985. [16] A. Filipponi, P. D'Angelo, N.V. Pavel and A. Di Cicco, Chem. Phys. Lett. 225 (1994) 150.
1,17] L. Ottaviano, A. Filipponi, A. Di Cicco, S. Santucci and P. Picozzi, J. of Non-Cryst. Sol. 156-158 (1993) 112. [18] A. Filipponi, L. Ottaviano, M. Passacantando, P. Picozzi and S. Santucci, Phys. Rev. E 48 (1993) 4575. 1-19] E. Nordlander, S.C. Lee, W. Cen, Z.Y. Wu, C.R. Natoli, A. Di Cicco, A. Filipponi, B. Hedman, K.O. Hodgson and R.J. Holm, J. Am. Chem. Soc. 115 (1993) 5549; S.D. Conradson, B.K. Burgess, W.E. Newton, A. Di Cicco, A. Filipponi, J. Wu, C.R. Natoli, B. Hedman and K.O. Hodgson, Proc. Natl. Acad. Sci. USA 91 (1994) 1290; H.I. Liu, A. Filipponi, N. Gavini, B.K. Burgess, B. Hedman, A. Di Cicco, C.R. Natoli and K.O. Hodgson, J. Am. Chem. Soc. 116 (1994) 248; T.E. Westre, A. Di Cicco, A. Filipponi, C.R. Natoli, B. Hedman, E.I. Solomon and K.O. Hodgson, J. Am. Chem. Soc. 116 (1994) 6757. [20] A. Filipponi, A. Di Cicco and C.R. Natoli, submitted. [21] A. Fillipponi, Physica B 208&209 (1995) 29. [22] C.R. Natoli and M. Benfatto, J. Phys. (Paris) Colloq. 47 (1986) C8-11; T.A. Tyson, K.O. Hodgson, C.R. Natoli and M. Benfatto, Phys. Rev. B 46 (1986) 5997. [23] A. Filipponi, J. Phys.: Condensed Matter 3 (1991) 6489. [24] M. Benfatto, C.R. Natoli and A. Filipponi, Phys. Rev. B 40 (1989) 9626. 1,25] R.W. Joyner, K.J. Martin and P. Meehan, J. Phys. C 20 (1987) 4005. [26] A. Di Cicco, M. Berrettoni, S. Stizza, E. Bonetti and G. Cocco, Phys. Rev. B 50 (1994) 12386. [27] A. Filipponi, L. Lozzi, M. Passacantando, P. Picozzi, S. Santucci and M. Diociaiuti, Phys. Rev. B 47 (1993) 8494. 1-28] E. Sevillano, H. Meuth and J.J. Rehr, Phys. Rev. B 20 (1979) 4908.
Physica B 208&209 (1995) 125 128
EXAFS multiple-scattering data-analysis: GNXAS methodology and applications A. Di Cicco Dipartimento di Matematica e Fisica, Universit& di Camerino, Via Madonna delle Carceri, 62032 Camerino, Italy
Abstract
A method for multiple-scattering EXAFS data analysis, denominated GNXAS, is described. Reference to previous work is given. Accuracy of the structural results is discussed. An application to crystalline iron is briefly presented.
In recent times, the need to develop improved EXAFS data-analysis methods has been largely recognized. Earlier plane-wave single-scattering theories, and empirical methodologies based on the use of model compounds, are being gradually replaced by ab initio multiple-scattering (MS) calculations. These advances are related to the development, in the last 20 years, of both an efficient theory and convenient algorithms to calculate the structural contribution to the X-ray absorption cross-section. The inclusion of MS signals has been found to be necessary, both in the low-energy and high-energy regimes, for EXAFS structural analysis. Important MS effects have been evidenced not only in crystalline and molecular systems [1-51, but also in disordered matter [6, 71. The need for accounting correctly of MS contributions has stimulated the development of sophisticated packages for EXAFS data analysis. In this contribution we briefly describe the GNXAS method, originally presented in 1991 [8,91, which has been already applied to a large variety of systems. Applications included diatomic [10] and polyatomic gas-phase molecules [11,121, carbonyl clusters [13], a-Si [6], high-To cuprates [14], aqueous solutions [15,161, elemental and molecular liquids [7, 17, 18], and biological systems [19]. A full description of the theory and several examples of data analysis will
be given in a forthcoming paper [201. Generally speaking, a very good agreement with structural data determined by other techniques has been found. Typical statistical errors are often found well below 0.01 A for first-shell average bond distance, while absolute values have been found to be reliable within 0.02,g, [201. The method is based on a comparison between the experimental signal and a theoretical model one'attempting to optimize the relevant structural parameteFwalues. The comparison is performed on the absorption coefficient (ctoxp(E)) instead of the structural g(k) signal. The reason is that EXAFS background can hardly be defined exactly and, as a consequence, it is refined in conjunction with a structural contribution on the raw data. The model absorption signal is 0¢mod(E)= g0(1 + Xmod(E-- Eo)) + 0~bks(E) + C%x¢(E), (1) where gbks(E)is a smooth polynomial spline accounting for underlying absorption channels, and ctoxc(E)accounts for possible multi-electron excitation channels, ctox¢(E) simulates in an empirical way steplike or slope changes occurring at precise energies above the main excitation edge. Examples of the importance of the ctexc(E)contribution have been widely discussed in the literature (see Ref. [21] and the references therein).
0921-4526/95/$09.50 © 1995 Elsevier Science B.V. All rights reserved SSDI 0 9 2 1 - 4 5 2 6 ( 9 4 ) 0 0 6 4 7 - 4
126
A. Di Cicco/Physica B 208&209 (1995) 125 128
The structural EXAFS term x(k)mod is an oscillating function of the wave vector k ~ , ~ - - E0 (E0 is the threshold energy) and is calculated according to the following steps: (a) calculation of a model potential for the final state photoelectron (muffin-tin approximation), (b) the phase shifts calculation, (c) the signal calculation, and (d) the configurational average. The basic theory related to the steps (a) and (b) is described in detail elsewhere (see Ref. [22] for example). The central step (c) in the calculation of g(k)rnod, in the framework of the MS theory within the muffin-tin approximation [22], is the inversion of a matrix (I + GT) containing phase shifts (matrix T) and propagators (matrix G) which describe the final state photo-electron in a selected cluster of atoms (see. for example, Refs. [23, 20] and the references therein). A power series expansion of the (I + GT)- 1 matrix is usually convergent in the high-energy limit and the first non-trivial term g2, representing the single-scattering, has been widely used to analyze EXAFS spectra. However, there are practical difficulties in applying the multiple-scattering series expansion mainly because of the great number of contributions (MS pathways) to be considered to achieve convergence. A different approach consists in calculating directly the total contribution to the cross-section due to couples, triplets, and quadruplets of atoms. Calculation of the total signals due to a few atoms are very fast also at high energies by using the continued-fraction algorithm [23]. Proper n-body 71"~signals are defined by subtracting loworder terms from n-atoms total calculations [20]. The use of the 7t"l signals allows us to establish quite simple relationships between structure and signals in terms of the main two-atom and three-atom configurations. Another important aspect of the theory is how the thermal (configurational) average of the signals has to be done. An appropriate theory has to be applied for calculating the averages of the n-body signals [24, 20] and is conceptually incorrect to apply different effective DebyeWaller-like factors for each n-body configuration. The signals (Zm or 7) to be averaged are oscillating functions where amplitudes A(k, R) and phases if(k, R) are smooth functions of k and of the geometrical parameters R. Suitable expressions, based on a Taylor expansion of A(k, R) and tk(k, R), have been derived for the average of the n-body configurations using Gaussian and nearly Gaussian distribution functions and have been implemented into the GNXAS programs [24, 9, 20]. As anticipated, the GNXAS programs provide a fit of the raw experimental data incorporating all the abovementioned algorithms. Calculation of the ),t"~functions is performed by using a muffin-tin potential and advanced models for the energy-dependent exchange-correlation self-energy.
The comparison between ~modand ~expis evaluated by means of a square residual function of the type R({)o})
i=~1
[~Xexp(Ei) - ~Xmod(Ei, }q, )~2. . . . . )~p)]2 0"2
(2)
Following very standard statistical procedures for non-linear fitting problems it is possible to perform a rigorous statistical evaluation of the results [20]. In particular, (i) the optimal values for the p structural parameters are {2} = { ~ corresponding to the minimum of the residual function; (ii) the statistical ~2-test can be performed to check whether the actual value of R({J.}) is only due to residual noise or it contains unexplained physical information; (iii) the statistical significance of the inclusion of different structural details, such as further n-body contributions or a splitting of a shell into two, can be analyzed using the F-test; (iv) the correlated statistical error in the parameters can be evaluated looking at confidence intervals defined by the equation R({2}) < Rmi n + C, with C depending on the confidence level of the analysis. Similar procedures based on statistical tests are indeed commonly applied to the EXAFS data-analysis problem [25]. Numerous GNXAS applications to molecular cases have been published so far. Here we briefly present an application to a crystalline case: the BCC iron. Full details are given elsewhere [26]. The main frequency content of the EXAFS signal is confined within 5 ,~ (Fig. 1, panel c). The five two-atom and four three-atom configurations contributing with 7 signals in this frequency region have been considered in the g(k)mod calculation. A pictorial view of the two-atom and three-atom configurations relevant to the BCC structure is contained in Ref. [27], Fig. 2. Angles and relative distances have been kept fixed, only the cell side a, the variances of distances and angles, and the correlation parameters of the covariance matrices have been floated [9, 20]. The total number of structural parameters is 14. Other two non-structural parameters, required to compare the model and experimental signals are Eo and So2. Eo accounts for the energy difference between the theoretical and experimental energy scales while S~ is an overall amplitude factor accounting for many-body effects and inaccurate jump normalization. In the upper panel (a) of Fig. 1 the proper best-fit two-atom and three-atom signals are shown from the top to the bottom. The first two, ~(2) and 7(,I.2), are the proper MS two-atom signals associated with the first and the fourth shell of neighbors, respectively. The third and fourth 7 signals are related to the first triangular configuration distribution (angle of 70.53 degrees). The 7t22)signal is the proper two-atom signal related to second
127
,4. Di Cicco /Physica B 208&209 (1995) 125 128 I/
(z)4 (a)
C
/
v
'
I ....
I'
''
0.0050 v
a
V
I ....
v
~
7'(a) t (==)
v
.....
%
~
0.0045
7(a)= ...~..~, 7(a)3~ 7(=)5
I
70) 4
0.0040
2.5
....
0.75 ,,I
....
I ....
I ....
I ....
I ....
101.,I . . . . I ? , , , I , : , , I , , , , I z.5
5~o
//
g
I ....
o
I .... a
7.6
12.5
I
6
I ....
0.0
I.,.
0.95
Fig. 2. Fitting of the Fe K-edge EXAFS spectrum: a 2 - So2 contour map. The inner elliptically shaped curves represent the 95% confidence interval.
~5.o
"X
I .... 4 R (1)
I ....
0.85
~o~
. . . . t,
to.o
A
I,
I ....
0.8
.... B
Fig. 1. Proper two-atom and three-atom multiple-scattering contributions to EXAFS of crystalline iron (a). Comparison between calculated (solid) and experimental (dots) crystalline iron k2z(k)(b). Fourier transform (FT) of experimental (dots) and calculated (solid) signals (c).
neighbors while the y~3)represents the proper three-atom signal. The successive two y signals are associated with the second three-body configuration (angle of 109.47 degrees). The seventh y signal is the proper three-atom contribution due the third triangular configuration (angle of 90 degrees). The last two curves are related to the collinear configurations involving first and fifth neighbors. As expected, the proper three-atom contribution (Tt,3)) is much stronger than the fifth-shell two-atom one (y~2)) due to the well-known focusing effect. In the center panel (b) the total best-fit MS signal and the experimental structural term k2g(k) are shown. The agreement is excellent in the whole energy range and the unexplained features are mainly due to contributions located in the region above 5.A in R-space (see Fig. 1, panel (c)). All of the best-fit structural parameter values and their estimated statistical errors are given elsewhere. In this contribution, due to the limited space, we focus on the results of the main parameters: the cell side a and the
first-shell bond variance a~. We found a = 2.855 (5)/{, in agreement within 0.01 ,~ with the value 2.8662 (5) A measured by diffraction. The slight shortening of the cell side a can be mainly attributed to systematic errors both in the theoretical calculation and in the experimental energy scale. The bond variance a 2 = 4.7 (6) × 10- 3 ~=2 is also in agreement with the calculated one tr2 = 5.06 x 10-3/~ 2 [28]. The multi-parameter correlation matrix has been calculated showing large correlation coefficients between some parameters. For example, both a and Eo and So2 and a 2 are highly correlated ( ~ 0.88), as expected, mainly contributing to increase the errors in a and in try, respectively. The Eo energy parameter was found to coincide within 0.5 eV with the energy position of the second peak of the derivative (around 7120 eV) of the experimental spectrum (used to calculate the k scale of the figure). The S g factor has been found to be around 0.85. The contour plots associated with the S2-tr 2 two-dimensional sections is shown in Fig. 2. The correlated statistical errors at the 95% confidence level, which depend on the noise level and on the energy extension of the measurement, are found quite low in the present case: + 0.005 A for a and 0.6 x 10 - 3 ~2 for a 2.
References
[1] P.A. Lee and J.B. Pendry, Phys. Rev. B 11 (1975) 2795. [2] M. Benfatto, C.R. Natoli, A. Bianconi, J. Garcia, A. Marcelli, M. Fanfoni and I. Davoli, Phys. Rev. B 34 (1986) 5774.
128
A. Di Cicco/Physica B 208&209 (1995) 125-128
[3] S.J. Gurman, N. Binsted and I. Ross, J. Phys. C 19 (1986) 1845. [4] A. Bianconi, A. Di Cicco, N.V. Pavel, M. Benfatto, A. Marcelli, C.R. Natoli, P. Pianetta and J. Woicik, Phys. Rev. B 36 (1987) 6426. [5] M.F. Ruiz-Lrpez, M. Loos, J. Goulon, M. Benfatto and C.R. Natoli, Chem. Phys. 121 (1988) 419. [6] A. Filipponi, A. Di Cicco, M. Benfatto and C.R. Natoli, Europhys. Lett. 13 (1990) 319. I-7] A. Di Cicco and A. Filipponi, Europhys. Lett. 27 (1994) 407. [8] The acronym GNXAS derives from 9n (the n-body distribution functions) and XAS (X-ray absorption spectroscopy). [9] A. Filipponi, A. Di Cicco, T.A. Tyson and C.R. Natoli, Solid State Commun. 78 (1991) 265. [10] P. D'Angelo, A. Di Cicco, A. Filipponi and N.V. Pavel, Phys. Rev. A 47 (1993) 2055. [11] A. Di Cicco, S. Stizza, A. Filipponi, F. Boscherini and S. Mobilio, J. Phys. B 25 (1992) 2309. [12] E. Burattini, P. D'Angelo, A. Di Cicco, A. Filipponi and N.V. Pavel, J. Phys. Chem. 97 (1993) 5486. [13] A. Filipponi, A. Di Cicco, R. Zanoni, M. Bellatreccia, V. Sessa, C. Dossi and R. Psaro, Chem. Phys. Lett. 184 (1991) 485. [14] A. Di Cicco and M. Berrettoni, Phys. Lett. A 176 (1993) 375. [15] P. D'Angelo, A. Di Nola, A. Filipponi, N.V. Pavel and D. Roccatano, J. Chem. Phys. 100 (1994) 985. [16] A. Filipponi, P. D'Angelo, N.V. Pavel and A. Di Cicco, Chem. Phys. Lett. 225 (1994) 150.
1,17] L. Ottaviano, A. Filipponi, A. Di Cicco, S. Santucci and P. Picozzi, J. of Non-Cryst. Sol. 156-158 (1993) 112. [18] A. Filipponi, L. Ottaviano, M. Passacantando, P. Picozzi and S. Santucci, Phys. Rev. E 48 (1993) 4575. 1-19] E. Nordlander, S.C. Lee, W. Cen, Z.Y. Wu, C.R. Natoli, A. Di Cicco, A. Filipponi, B. Hedman, K.O. Hodgson and R.J. Holm, J. Am. Chem. Soc. 115 (1993) 5549; S.D. Conradson, B.K. Burgess, W.E. Newton, A. Di Cicco, A. Filipponi, J. Wu, C.R. Natoli, B. Hedman and K.O. Hodgson, Proc. Natl. Acad. Sci. USA 91 (1994) 1290; H.I. Liu, A. Filipponi, N. Gavini, B.K. Burgess, B. Hedman, A. Di Cicco, C.R. Natoli and K.O. Hodgson, J. Am. Chem. Soc. 116 (1994) 248; T.E. Westre, A. Di Cicco, A. Filipponi, C.R. Natoli, B. Hedman, E.I. Solomon and K.O. Hodgson, J. Am. Chem. Soc. 116 (1994) 6757. [20] A. Filipponi, A. Di Cicco and C.R. Natoli, submitted. [21] A. Fillipponi, Physica B 208&209 (1995) 29. [22] C.R. Natoli and M. Benfatto, J. Phys. (Paris) Colloq. 47 (1986) C8-11; T.A. Tyson, K.O. Hodgson, C.R. Natoli and M. Benfatto, Phys. Rev. B 46 (1986) 5997. [23] A. Filipponi, J. Phys.: Condensed Matter 3 (1991) 6489. [24] M. Benfatto, C.R. Natoli and A. Filipponi, Phys. Rev. B 40 (1989) 9626. 1,25] R.W. Joyner, K.J. Martin and P. Meehan, J. Phys. C 20 (1987) 4005. [26] A. Di Cicco, M. Berrettoni, S. Stizza, E. Bonetti and G. Cocco, Phys. Rev. B 50 (1994) 12386. [27] A. Filipponi, L. Lozzi, M. Passacantando, P. Picozzi, S. Santucci and M. Diociaiuti, Phys. Rev. B 47 (1993) 8494. 1-28] E. Sevillano, H. Meuth and J.J. Rehr, Phys. Rev. B 20 (1979) 4908.