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On the determination of interatomic potential anharmonicities from EXAFS measurements
Volume 157, number 4,5
PHYSICS LETTERSA
29 July 1991
On the determination of interatomic potential anharmonicities from EXAFS measurements J. F r e u n d FachbereichPhysik, Universit&Paderborn, W-4790 Paderborn, Germany Received 21 May 1991; accepted for publication 28 May 1991 Communicated by A. Lagendijk
EXAFS data sets of Cu, a-Ge, NaBr, KBr and RbCI are analyzed by fitting with and without parameters pertaining to interatomic potential anharmonicities. The resulting goodness-of-fitparameters, X2, are subjected to an F-test showing that incorporation into the fitting of anharmonic parameters is not justified.
With applications of (extended) X-ray-absorption fine structure, (E)XAFS, in solid state physics, chemistry and biology growing rapidly, the interest in obtaining quantitative information about interatomic potential anharmonicities has risen. In general, the first (second, ...) perturbation of the harmonic potential can be related to the third (fourth, ...) cumulant o f the nearest-neighbor distance. Since the cumulants enter the equation for single-scattering EXAFS and can be determined easily from standard data analysis techniques (log-ratio and parameter fitting) [1,2], one is inclined to assume that EXAFS does, in fact, contain anharmonic information. A state-of-the-art report from 1988 with numerous references is given in ref. [3]. After a short review of the relation between nearest-neighbor distance cumulants and interatomic potentials, the goodness-of-fit parameter, X~, for a large number of Cu, A-Ge, NaBr, KBr and RbCI data sets at room temperature and pressure is calculated. An F-test is then used to show that X~ does not change in a statistically significant way when the third or the fourth cumulant is included as an extra fit parameter. This leaves two physically meaningful fit parameters: the mean and the variance of the nearestneighbor distance. The single-scattering single-shell EXAFS formula including cumulants up to fourth order is given by 256
z ( k ) = A ( k ) sin[ ~ ( k ) ] ,
(la)
with the amplitude
A(k)_
NF(k,n)f , ~ expt-2R/2(k)]
× e x p ( - 2 6 2 k 2 ) exp( ~o'(4~k 4 )
(lb)
and the phase ~U(k) = 2k~-R- 2a2 [ I.-
-40"(3)k3+ tJS(k) .
(lc)
N is the coordination number, F(k, 7t) the back-scatter amplitude, f a yet undetermined amplitude correction factor, R the distance to the shell, 2 ( k ) the electron mean free path, a 2 the second cumulant, o 4 3 ) the third cumulant, 0-(47 the fourth cumulant and • (k) the combined central atom and back-scatter phase shift, k is the electron wave-vector. Since it depends on an arbitrary choice of zero, it must be related to the real wave-vector k' (in A - ~) by
k=x/k'2-AE/3.81 ,
(ld)
where AE (in eV) is the shift between the assumed and the real zero of the free electron states. The second, third and fourth cumulants are expressed in terms of the first, second, third and fourth Elsevier Science Publishers B.V. (North-Holland)
Volume 157, number 4,5
PHYSICS LETTERS A
moments,/q, /-~2,/-~3and ]-~4,respectively by a2 =/~2 -/12 ,
(2a)
0"(3) ~-~,U3- 3fl2,/21 +2/z 3 ,
(2b)
0"(4) = 114- 4 f 1 3 f l l - 3 t z ~ + 1 2 1 z 2 g ~ - 6 / z
4.
(2c)
The nth moment of the distance between two atoms at positions r~ and r2 in a crystal consisting of N atoms is defined as f...f lr2 - r l I% - v¢,~ ......u)/kBr drl ... drN lt~= f...f e -v<'' ......u ) / ~ r drl ...drN ,
(3)
where U is the potential energy of the crystal. The integrations are over all 3N dimensions of configuration space. The only assumption entering this ansatz is the classical approximation (summation replaced by integration) which is generally justified above about half of the Debye temperature. In principle, eq. (3) allows determination of harmonic and anharmonic terms of interatomic potentials from nearest-neighbor distance cumulants. In practice, given potentials must be inserted into the variable U and the resulting cumulants must then be compared with those measured from the EXAFS [ 2 ]. If more assumptions are introduced eq. (3) can be solved analytically [ 1 ]. Note at this point that a perturbation of the harmonic potential does not describe every imaginable interatomic potential. One can think of a symmetric double=well potential, e.g., that would defy the perturbative approach. But even for single-well potentials problems may arise, since the cumulant expansion is an expansion about k = O and will diverge at sufficiently large k-values. The data analysis technique used here is parameter fitting: while N must be known, F ( k , n), 2(k) and q~(k) are taken from the FEFF calculations (version 3.23) of Rehr, Albers and Mustre de Leon [4]. This leaves f R, a 2, a t3), a t4) and AE as fit parameters. The criterion for the goodness of the fit is the parameter X~ defined by [ 5,6 ] 1 Nina (~data - - x p t ) 2
~, \ X2= 7, ~=
a~
--
/
,
(4)
where v designates the number of degrees of freedom, the summation is over all Ni,d independent data points, Xi stands for the single-shell EXAFS interfer-
29 July 1991
ence function and ai for the error of data point number i. v is given by v=Ni,d -Npar,
(5)
where Npar is the number of fit parameters. The number of independent data points is given by [7 ] 2 Ak Ar
(6)
Nin d = - - ,
where Ak is the k-space range used for the Fourier forward transform from k- to r-space and Ar is the r-space range used for the Fourier back transform from r- to k-space. It is more convenient to sum over the arbitrary number of points, N, of the Fourier forward transform or the Fourier back transform. This can be done if the sum is multiplied by a correction factor, Ni,d/ N, to give Xz~ =
N Nina - Npa~ ,= , \
-a-~
-/ .
(7)
This is eq. (2) of the "Report of the International Workshop on Standards and criteria in X-ray absorption spectroscopy" [ 8 ]. In general, one does not use the EXAFS interference function, x ( k ) , but x ( k ) weighted with k X,where x is a number typically between 1 and 3. Then the final formula for the goodness-of-fit parameter becomes Ni,d Xz~ =
1 ~ (Zdi'tak-i~-%cltki':~ ~ N N i . a - N p ~ r ~=, \ a, kl ~ -] .
(8)
It is a common experience that Z~ decreases with an increasing number of parameters and thus an increasing flexibility of the fit function, but even an increase ofxZ~ is possible due to the term Ni,,a-Npar in the denominator of eq. (8). The problem is whether the change ofx~ is statistically significant or not when an extra fit parameter is included. According to standard statistical analysis procedures (e.g. ref. [ 5 ] ) the F-test is the appropriate statistical means to check the significance of an extra fit parameter. The F-statistic for out problem, i.e. the comparison of a four-parameter fit with a five-parameter fit, is given by Fx=
2 N ind - - 4 )--XNind-z /~Nind--4( 5( Nind - 5 ) 2
(9)
Nind ~ 5
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Volume 157. number 4.5
PHYSICS LETTERS A
The F-statistic thus defined follows an F-distribution with one degree of freedom for the numerator and Ni,d-- 5 degrees of freedom for the denominator and is tabulated (e.g. refs. [5,9]). The F-distribution table in ref. [5] shows the probability, P, that a given value of F, or any value larger than that, occurs by change in an experiment. Some tables, like in ref. [9], show the complement, l - P , i.e. the probability that a given value o f F , or any value larger than that, does not occur by change. In other words, I - P is a measure for the confidence that a given value of F d o e s not occur by change in an experiment. ai should contain random, or statistical, and nonstatistical contributions of the error. The random error can be estimated by fitting a polynomial or a Victoreen function into the pre-edge data region since (fine) structure is not expected in that part of the XAFS spectrum. The error thus determined must be divided by the step height to turn it into an error of x ( k ) . Non-statistical errors are largely due to inhomogeneities of the sample (pinhole effect) and a nonnegligible contribution of higher harmonics in the beam due to insufficient detuning of the monochromator crystal. These errors are hard to estimate quantitatively. However, while a wrong choice for 0.i has an influence on Z2,, it cancels in eq. (9). Nonstatistical errors are indirectly taken into account in this study, because we have a large number of data sets from different experiments, in part because some of the materials investigated were used as pressure calibrants for our high pressure EXAFS experiments of materials such as perovskites, high-Tc superconductors, etc. Table 1 presents some of the input and output of the parameter fits. The error 0.i in the second column is a dimensionless quantity like z ( k ) . It shows considerable variation between the experiments and should be used as a guide to interpret the results. The m i n i m u m and m a x i m u m cutoffs in k-space (krange), as well as the m i n i m u m and m a x i m u m cutoffs in r-space (r-range) in the two columns to follow determine the number o f independent data points, Nmd. The k-range is largely determined by the noise in the data, as expressed by ai in the second column. Therefore, a certain negative correlation between Ak and 0., can be observed. The next columns contain the goodness-of-fit parameters, Z 2, as given by eq. (8), for three cases: ( 1 ) only four parameters 258
29 July 1991
are fitted, namely f, R, AE and a 2, (2) 0.(3) is included as the fifth fit parameter, and (3) 0.(4) is included as the fifth fit parameter. The last columns, finally, give the F-statistics for the comparison between cases ( 1 ) and (2), as well as for the comparison between cases ( 1 ) and (3). The probabilities, P, pertaining to the F-values were looked up in the tables mentioned above. The cases marked with asterisks ( , ) and number signs (~;) are those with probabilities of occurrence of 25% or less and 10% or less, respectively. Let us take Cu as an example. We made 10 independent experiments, therefore we should expect to find three , ' s and two # ' s in both columns combined, if no anharmonic contributions existed in Cu, i.e. if no improvement were to be expected by including a fifth fit parameter. In fact we find one • and two #'s, i.e. even two , ' s less than expected for a perfectly harmonic material. If incorporation of 0.(3) or 0.(4) into the parameter fitting o f Cu yielded a significant improvement of the fit, we would have to find #'s, or at least .'s, attached to many or all of the numbers of the pertaining column. For RbCI, which was also repeated ten times, we find seven . ' s in the o-(4) column, 5.5 more than expected for a harmonic material. However there is no ~, i.e. one less than expected. For all five materials combined we should find ten . ' s and seven # ' s under the harmonic assumption, and we do find ten , ' s and two #'s. Again, if one of the materials were anharmonic, we would expect to have # ' s attached to most numbers of the respective column. It is also interesting to note that in three cases 0.(3) as the fifth fit parameter seems to give a slightly better fit, while in nine other cases it is O"(4). We must therefore conclude that in all materials investigated there is no indication whatsoever that incorporation of 0.(3) o r 0.(4) into the parameter fitting procedure is justified on statistical grounds. We must conclude that nearest-neighbor EXAFS data of Cu, a-Ge, NaBr, KBr and RbCI do not contain anharmonic information. Let us now turn to the information content o f our nearest-neighbor EXAFS data. After elimination o f 0.(3) and 0.(4~ we have four fit parameters left: f R, 0.2 and zXE.f a n d zXE are parameters that have to be included into the fitting routine but do not contain structural information. So we are left with the mean
Volume 157, number 4,5
PHYSICS LETTERS A
29 July 1991
Table 1 Input and output for the parameter fits of a large number o f C u , a-Ge, NaBr, KBr and RbCI data sets. The column labelled "error" is the statistical error of the EXAFS interference function, z ( k ) , as determined from the standard deviation of data points in the pre-edge region. The data sets were Fourier forward transformed and back transformed after application of Gaussian windows with 10% values as given under "k-range" and "r-range", respectively. The goodness-of-fit parameters, as defined by eq. (8), are given for the three cases where a 2, a 2 and tr t3), as well as a 2 and tr t4~ are the parameters fitted, along with f, R and AE. The last column contains the values of the F-statistic for the comparisons between four-parameter and five-parameter fits, where the fifth parameter is tr t3J and tr t4j, respectively. Asterisks ( , ) and number signs ( # ) indicate F-values with probabilities of occurrence less than 2 5% and less than 10%, respectively, if a purely harmonic interatomic potential is assumed. There is no column with a large number o f # ' s or even .'s. This shows that a fifth fit parameter, tr t3~ or tr ~4~, respectively, does not improve the fit; i.e. the data sets do not contain anharmonic information. No.
Cu
Error
k-range
r-range
(A-~)
(A)
Goodness-of-fit parameter
F-statistic
2nd cumulant
2nd and 3rd cumulants
2nd and 4th cumulants
2nd versus 2nd and 3rd cumulants
2nd versus 2nd and 4th cumulants
1 2 3 4 5 6 7 8 9 10
0.0013 0.0010 0.0007 0.0020 0.0015 0.0001 0.0022 0.0022 0.0040 0.0020
2.0-12.0 2.0-12.0 2.0-15.0 2.0-14.0 2.0-14.0 2.0-15.0 2.0-16.0 2.0-14.0 2.5-12.5 2.0-15.0
1.69-2.79 1.60-2.79 1.75-2.67 1.72-2.70 1.75-2.67 1.75-2.67 1.81-2.61 1.69-2.73 1.66-2.70 1.81-2.67
0.0449 0.2146 0.0283 0.0078 0.0228 3.8898 0.0013 0.0125 0.0101 0.0139
0.0613 0.2648 0.0325 0.0094 0.0168 1.3671 0.0019 0.0164 0.0106 0.0158
0.0284 0.0759 0.0222 0.0107 0.0325 5.0562 0.0019 0.0151 0.0154 0.0111
0.20 0.32 0.53 0.41 2.08 7.67# 0.01 0.06 0.88 0.63
2.74, 7.53# 1.99 0.06 0.10 0.17 0.01 0.32 0.10 1.79
a-Ge
1 2 3
0.0024 0.0005 0.0015
2.0-10.6 2.0-16.0 2.0-14.0
1.47-2.73 1.69-2.49 1.60-2.55
0.0099 0.0671 0.0075
0.0147 0.0828 0.0088
0.0077 0.0869 0.0075
0.05 0.41 0.52
1.83 0.29 1.00
NaBr
1 2 3 4 5 6 7
0.0003 0.0009 0.0006 0.0020 0.0006 0.0007 0.0007
2.02.02.02.02.02.02.0-
8.4 8.0 7.0 7.0 7.0 8.4 8.0
1.63-3.01 1.56-3.04 1.47-3.13 1.50-3.22 1.50-3.13 1.56-2.98 1.50-3.07
0.3058 0.0091 0.1502 0.0203 0.1170 0.0537 0.0561
0,3288 0.0148 0.5641 0.0395 0.7822 0.0133 0.0129
0.7952 0.0235 0.6198 0.0112 0.3726 0.0825 0.1115
0.89 0.36 0.06 0.28 0.00 6.42 7.69*
0.00 0.00 0.03 2.20 0.18 0.38 0.01
KBr
1 2 3
0.0040 0.0018 0.0018
2.0- 9.0 2.0- 8.0 2.0- 8.0
1.93-3.34 1.93-3.34 1.96-3.31
0.0005 0.0017 0.0027
0.0008 0.0059 0.0197
0.0007 0.0059 0.0164
0.14 0.01 0.00
0.35 0.01 0.03
RbCI
1 2 3 4 5 6 7 8 9 10
0.0009 0.0007 0.0015 0.0010 0.0005 0.0008 0.0016 0.0008 0.0006 0.0004
2.0-10.8 2.0- 9.0 2.0- 8.0 2.0- 7.6 2.0-10.0 2.0- 9.0 2.0- 9.2 2.0- 8.0 2.0- 8.8 2.0- 8.8
1.81-3.16 1.81-3.31 1.53-3.47 1.50-3.59 1.87-3.34 1.69-3.47 1.81-3.37 1.50-3.50 1.69-3.41 1.72-3.44
0.0079 0.0257 0.0215 0.0859 0.0349 0.0843 0.0069 0.1579 0.1019 0.2322
0.0057 0.0368 0.0305 0.1212 0.0294 0.0852 0.0092 0.2173 0.1333 0.2903
0.0078 0.0227 0.0152 0.0495 0.0418 0.0550 0.0047 0.0786 0.0494 0.0942
2.38* 0.19 0.00 0.00 1.65 0.96 0.21 0.01 0.19 0.31
1.05 1.35 2.41. 3.54* 0.42 3.09* 2.47* 4.67* 4.66* 6.05*
259
Volume 157. number 4,5
PHYSICS LETTERS A
and the variance o f the nearest-neighbor distance, R and a 2, respectively. In principle, these two pieces o f information, along with the known c o o r d i n a t i o n number, can be used to construct a t h r e e - p a r a m e t e r probability density function for the length o f the nearest-neighbor distance, i.e. a nearest-neighbor radial distribution function. However, with only three parameters at hand, the nearest-neighbor radial distribution function thus constructed must be a simple function, such as a p a r a b o l a or a Gaussian or a Lorentzian or any other function that is d e t e r m i n e d completely by the location a n d height o f the maxim u m and its width. All attempts to find more sophisticated distributions, such as a s y m m e t r i c or double-well distributions, must be ruled out a priori when nearest-neighbor E X A F S d a t a o f Cu, a-Ge, NaBr, KBr or RbCI are investigated. At this point we cannot draw a general conclusion about the suitability o f E X A F S for the d e t e r m i n a tion o f interatomic potential anharmonicities, double-well potentials and the like. However, we want to point out that with our choice o f materials we have covered a wide range: metallic ( C u ) , covalent ( a - G e ) and ionic ones (NaBr, KBr, R b C I ) ; hard (Cu, a - G e ) and soft ones (NaBr, KBr, RbC1). Therefore, if more than two pieces o f structural information are claimed to be the content o f a nearest-neighbor E X A F S data set o f some other material, it is i n c u m b e n t upon the data analyst to prove quantitatively that the extra information is indeed contained in the data. I would like to thank R. Ingalls, J.E. W h i t m o r e , J.M. T r a n q u a d a and B. Houser, my former Univer-
260
29 July 1991
sity o f Washington group, and E.D. Crozier, N. AIberding, A.J. Seary, K.R. Bauchspiess and D. Jiang, our collaborators from Simon Fraser University, for having done the experiments with us at SSRL. Special thanks go to E.D. Crozier for numerous contributions to i m p r o v e the E X A F S data analysis. This work was supported by the Deutsche Forschungsgemeinschaft, the U.S. D e p a r t m e n t o f Energy and the Natural Sciences and Engineering Research Council o f Canada. SSRL is s u p p o r t e d by the U.S. D e p a r t m e n t o f Energy (Office o f Basic Energy Sciences) and the National Institutes o f Health (Biotechnology Research Program, Division o f Research Resources).
References [1] J. Freund, R. lngalls and E.D. Crozier, Phys. Rev. B 39 (1989) 12537. [ 2 ] J. Freund, R. Ingalls and E.D. Crozier, Phys. Rev. B ( 1991 ), in press. [3] E.D. Crozier, J.J. Rehr and R. lngalls, in: X-ray absorption, eds. D.C. Koningsberger and R. Prins (Wiley, New York, 1988) p. 373. [4] J.J. Rehr, R.C. Albers and J. Mustre de Leon, Physica B 158 (1989) 417. [5 ] Ph.R. Bevington, Data reduction and error analysis for the physical sciences (McGraw-Hill, New York, 1969 ). [6]W.H. Press, B.P. Flannery, S.A. Teukolsky and W.T. Venerling, Numerical recipes (Cambridge Univ. Press, Cambridge, 1986). [7 ] P.A. Lee, P.H. Citrin, P. Eisenberger and B.M. Kincaid, Rev. Mod. Phys. 53 (1981) 769. [8] F.W. Lytle, D.E. Sayers and E.A. Stem, Physica B 158 (1989) 701. [9] W.J. Dixon and F.J. Massey Jr., Introduction to statistical analysis, 3rd Ed. (McGraw-Hill, New York, 1969).
PHYSICS LETTERSA
29 July 1991
On the determination of interatomic potential anharmonicities from EXAFS measurements J. F r e u n d FachbereichPhysik, Universit&Paderborn, W-4790 Paderborn, Germany Received 21 May 1991; accepted for publication 28 May 1991 Communicated by A. Lagendijk
EXAFS data sets of Cu, a-Ge, NaBr, KBr and RbCI are analyzed by fitting with and without parameters pertaining to interatomic potential anharmonicities. The resulting goodness-of-fitparameters, X2, are subjected to an F-test showing that incorporation into the fitting of anharmonic parameters is not justified.
With applications of (extended) X-ray-absorption fine structure, (E)XAFS, in solid state physics, chemistry and biology growing rapidly, the interest in obtaining quantitative information about interatomic potential anharmonicities has risen. In general, the first (second, ...) perturbation of the harmonic potential can be related to the third (fourth, ...) cumulant o f the nearest-neighbor distance. Since the cumulants enter the equation for single-scattering EXAFS and can be determined easily from standard data analysis techniques (log-ratio and parameter fitting) [1,2], one is inclined to assume that EXAFS does, in fact, contain anharmonic information. A state-of-the-art report from 1988 with numerous references is given in ref. [3]. After a short review of the relation between nearest-neighbor distance cumulants and interatomic potentials, the goodness-of-fit parameter, X~, for a large number of Cu, A-Ge, NaBr, KBr and RbCI data sets at room temperature and pressure is calculated. An F-test is then used to show that X~ does not change in a statistically significant way when the third or the fourth cumulant is included as an extra fit parameter. This leaves two physically meaningful fit parameters: the mean and the variance of the nearestneighbor distance. The single-scattering single-shell EXAFS formula including cumulants up to fourth order is given by 256
z ( k ) = A ( k ) sin[ ~ ( k ) ] ,
(la)
with the amplitude
A(k)_
NF(k,n)f , ~ expt-2R/2(k)]
× e x p ( - 2 6 2 k 2 ) exp( ~o'(4~k 4 )
(lb)
and the phase ~U(k) = 2k~-R- 2a2 [ I.-
-40"(3)k3+ tJS(k) .
(lc)
N is the coordination number, F(k, 7t) the back-scatter amplitude, f a yet undetermined amplitude correction factor, R the distance to the shell, 2 ( k ) the electron mean free path, a 2 the second cumulant, o 4 3 ) the third cumulant, 0-(47 the fourth cumulant and • (k) the combined central atom and back-scatter phase shift, k is the electron wave-vector. Since it depends on an arbitrary choice of zero, it must be related to the real wave-vector k' (in A - ~) by
k=x/k'2-AE/3.81 ,
(ld)
where AE (in eV) is the shift between the assumed and the real zero of the free electron states. The second, third and fourth cumulants are expressed in terms of the first, second, third and fourth Elsevier Science Publishers B.V. (North-Holland)
Volume 157, number 4,5
PHYSICS LETTERS A
moments,/q, /-~2,/-~3and ]-~4,respectively by a2 =/~2 -/12 ,
(2a)
0"(3) ~-~,U3- 3fl2,/21 +2/z 3 ,
(2b)
0"(4) = 114- 4 f 1 3 f l l - 3 t z ~ + 1 2 1 z 2 g ~ - 6 / z
4.
(2c)
The nth moment of the distance between two atoms at positions r~ and r2 in a crystal consisting of N atoms is defined as f...f lr2 - r l I% - v¢,~ ......u)/kBr drl ... drN lt~= f...f e -v<'' ......u ) / ~ r drl ...drN ,
(3)
where U is the potential energy of the crystal. The integrations are over all 3N dimensions of configuration space. The only assumption entering this ansatz is the classical approximation (summation replaced by integration) which is generally justified above about half of the Debye temperature. In principle, eq. (3) allows determination of harmonic and anharmonic terms of interatomic potentials from nearest-neighbor distance cumulants. In practice, given potentials must be inserted into the variable U and the resulting cumulants must then be compared with those measured from the EXAFS [ 2 ]. If more assumptions are introduced eq. (3) can be solved analytically [ 1 ]. Note at this point that a perturbation of the harmonic potential does not describe every imaginable interatomic potential. One can think of a symmetric double=well potential, e.g., that would defy the perturbative approach. But even for single-well potentials problems may arise, since the cumulant expansion is an expansion about k = O and will diverge at sufficiently large k-values. The data analysis technique used here is parameter fitting: while N must be known, F ( k , n), 2(k) and q~(k) are taken from the FEFF calculations (version 3.23) of Rehr, Albers and Mustre de Leon [4]. This leaves f R, a 2, a t3), a t4) and AE as fit parameters. The criterion for the goodness of the fit is the parameter X~ defined by [ 5,6 ] 1 Nina (~data - - x p t ) 2
~, \ X2= 7, ~=
a~
--
/
,
(4)
where v designates the number of degrees of freedom, the summation is over all Ni,d independent data points, Xi stands for the single-shell EXAFS interfer-
29 July 1991
ence function and ai for the error of data point number i. v is given by v=Ni,d -Npar,
(5)
where Npar is the number of fit parameters. The number of independent data points is given by [7 ] 2 Ak Ar
(6)
Nin d = - - ,
where Ak is the k-space range used for the Fourier forward transform from k- to r-space and Ar is the r-space range used for the Fourier back transform from r- to k-space. It is more convenient to sum over the arbitrary number of points, N, of the Fourier forward transform or the Fourier back transform. This can be done if the sum is multiplied by a correction factor, Ni,d/ N, to give Xz~ =
N Nina - Npa~ ,= , \
-a-~
-/ .
(7)
This is eq. (2) of the "Report of the International Workshop on Standards and criteria in X-ray absorption spectroscopy" [ 8 ]. In general, one does not use the EXAFS interference function, x ( k ) , but x ( k ) weighted with k X,where x is a number typically between 1 and 3. Then the final formula for the goodness-of-fit parameter becomes Ni,d Xz~ =
1 ~ (Zdi'tak-i~-%cltki':~ ~ N N i . a - N p ~ r ~=, \ a, kl ~ -] .
(8)
It is a common experience that Z~ decreases with an increasing number of parameters and thus an increasing flexibility of the fit function, but even an increase ofxZ~ is possible due to the term Ni,,a-Npar in the denominator of eq. (8). The problem is whether the change ofx~ is statistically significant or not when an extra fit parameter is included. According to standard statistical analysis procedures (e.g. ref. [ 5 ] ) the F-test is the appropriate statistical means to check the significance of an extra fit parameter. The F-statistic for out problem, i.e. the comparison of a four-parameter fit with a five-parameter fit, is given by Fx=
2 N ind - - 4 )--XNind-z /~Nind--4( 5( Nind - 5 ) 2
(9)
Nind ~ 5
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The F-statistic thus defined follows an F-distribution with one degree of freedom for the numerator and Ni,d-- 5 degrees of freedom for the denominator and is tabulated (e.g. refs. [5,9]). The F-distribution table in ref. [5] shows the probability, P, that a given value of F, or any value larger than that, occurs by change in an experiment. Some tables, like in ref. [9], show the complement, l - P , i.e. the probability that a given value o f F , or any value larger than that, does not occur by change. In other words, I - P is a measure for the confidence that a given value of F d o e s not occur by change in an experiment. ai should contain random, or statistical, and nonstatistical contributions of the error. The random error can be estimated by fitting a polynomial or a Victoreen function into the pre-edge data region since (fine) structure is not expected in that part of the XAFS spectrum. The error thus determined must be divided by the step height to turn it into an error of x ( k ) . Non-statistical errors are largely due to inhomogeneities of the sample (pinhole effect) and a nonnegligible contribution of higher harmonics in the beam due to insufficient detuning of the monochromator crystal. These errors are hard to estimate quantitatively. However, while a wrong choice for 0.i has an influence on Z2,, it cancels in eq. (9). Nonstatistical errors are indirectly taken into account in this study, because we have a large number of data sets from different experiments, in part because some of the materials investigated were used as pressure calibrants for our high pressure EXAFS experiments of materials such as perovskites, high-Tc superconductors, etc. Table 1 presents some of the input and output of the parameter fits. The error 0.i in the second column is a dimensionless quantity like z ( k ) . It shows considerable variation between the experiments and should be used as a guide to interpret the results. The m i n i m u m and m a x i m u m cutoffs in k-space (krange), as well as the m i n i m u m and m a x i m u m cutoffs in r-space (r-range) in the two columns to follow determine the number o f independent data points, Nmd. The k-range is largely determined by the noise in the data, as expressed by ai in the second column. Therefore, a certain negative correlation between Ak and 0., can be observed. The next columns contain the goodness-of-fit parameters, Z 2, as given by eq. (8), for three cases: ( 1 ) only four parameters 258
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are fitted, namely f, R, AE and a 2, (2) 0.(3) is included as the fifth fit parameter, and (3) 0.(4) is included as the fifth fit parameter. The last columns, finally, give the F-statistics for the comparison between cases ( 1 ) and (2), as well as for the comparison between cases ( 1 ) and (3). The probabilities, P, pertaining to the F-values were looked up in the tables mentioned above. The cases marked with asterisks ( , ) and number signs (~;) are those with probabilities of occurrence of 25% or less and 10% or less, respectively. Let us take Cu as an example. We made 10 independent experiments, therefore we should expect to find three , ' s and two # ' s in both columns combined, if no anharmonic contributions existed in Cu, i.e. if no improvement were to be expected by including a fifth fit parameter. In fact we find one • and two #'s, i.e. even two , ' s less than expected for a perfectly harmonic material. If incorporation of 0.(3) or 0.(4) into the parameter fitting o f Cu yielded a significant improvement of the fit, we would have to find #'s, or at least .'s, attached to many or all of the numbers of the pertaining column. For RbCI, which was also repeated ten times, we find seven . ' s in the o-(4) column, 5.5 more than expected for a harmonic material. However there is no ~, i.e. one less than expected. For all five materials combined we should find ten . ' s and seven # ' s under the harmonic assumption, and we do find ten , ' s and two #'s. Again, if one of the materials were anharmonic, we would expect to have # ' s attached to most numbers of the respective column. It is also interesting to note that in three cases 0.(3) as the fifth fit parameter seems to give a slightly better fit, while in nine other cases it is O"(4). We must therefore conclude that in all materials investigated there is no indication whatsoever that incorporation of 0.(3) o r 0.(4) into the parameter fitting procedure is justified on statistical grounds. We must conclude that nearest-neighbor EXAFS data of Cu, a-Ge, NaBr, KBr and RbCI do not contain anharmonic information. Let us now turn to the information content o f our nearest-neighbor EXAFS data. After elimination o f 0.(3) and 0.(4~ we have four fit parameters left: f R, 0.2 and zXE.f a n d zXE are parameters that have to be included into the fitting routine but do not contain structural information. So we are left with the mean
Volume 157, number 4,5
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Table 1 Input and output for the parameter fits of a large number o f C u , a-Ge, NaBr, KBr and RbCI data sets. The column labelled "error" is the statistical error of the EXAFS interference function, z ( k ) , as determined from the standard deviation of data points in the pre-edge region. The data sets were Fourier forward transformed and back transformed after application of Gaussian windows with 10% values as given under "k-range" and "r-range", respectively. The goodness-of-fit parameters, as defined by eq. (8), are given for the three cases where a 2, a 2 and tr t3), as well as a 2 and tr t4~ are the parameters fitted, along with f, R and AE. The last column contains the values of the F-statistic for the comparisons between four-parameter and five-parameter fits, where the fifth parameter is tr t3J and tr t4j, respectively. Asterisks ( , ) and number signs ( # ) indicate F-values with probabilities of occurrence less than 2 5% and less than 10%, respectively, if a purely harmonic interatomic potential is assumed. There is no column with a large number o f # ' s or even .'s. This shows that a fifth fit parameter, tr t3~ or tr ~4~, respectively, does not improve the fit; i.e. the data sets do not contain anharmonic information. No.
Cu
Error
k-range
r-range
(A-~)
(A)
Goodness-of-fit parameter
F-statistic
2nd cumulant
2nd and 3rd cumulants
2nd and 4th cumulants
2nd versus 2nd and 3rd cumulants
2nd versus 2nd and 4th cumulants
1 2 3 4 5 6 7 8 9 10
0.0013 0.0010 0.0007 0.0020 0.0015 0.0001 0.0022 0.0022 0.0040 0.0020
2.0-12.0 2.0-12.0 2.0-15.0 2.0-14.0 2.0-14.0 2.0-15.0 2.0-16.0 2.0-14.0 2.5-12.5 2.0-15.0
1.69-2.79 1.60-2.79 1.75-2.67 1.72-2.70 1.75-2.67 1.75-2.67 1.81-2.61 1.69-2.73 1.66-2.70 1.81-2.67
0.0449 0.2146 0.0283 0.0078 0.0228 3.8898 0.0013 0.0125 0.0101 0.0139
0.0613 0.2648 0.0325 0.0094 0.0168 1.3671 0.0019 0.0164 0.0106 0.0158
0.0284 0.0759 0.0222 0.0107 0.0325 5.0562 0.0019 0.0151 0.0154 0.0111
0.20 0.32 0.53 0.41 2.08 7.67# 0.01 0.06 0.88 0.63
2.74, 7.53# 1.99 0.06 0.10 0.17 0.01 0.32 0.10 1.79
a-Ge
1 2 3
0.0024 0.0005 0.0015
2.0-10.6 2.0-16.0 2.0-14.0
1.47-2.73 1.69-2.49 1.60-2.55
0.0099 0.0671 0.0075
0.0147 0.0828 0.0088
0.0077 0.0869 0.0075
0.05 0.41 0.52
1.83 0.29 1.00
NaBr
1 2 3 4 5 6 7
0.0003 0.0009 0.0006 0.0020 0.0006 0.0007 0.0007
2.02.02.02.02.02.02.0-
8.4 8.0 7.0 7.0 7.0 8.4 8.0
1.63-3.01 1.56-3.04 1.47-3.13 1.50-3.22 1.50-3.13 1.56-2.98 1.50-3.07
0.3058 0.0091 0.1502 0.0203 0.1170 0.0537 0.0561
0,3288 0.0148 0.5641 0.0395 0.7822 0.0133 0.0129
0.7952 0.0235 0.6198 0.0112 0.3726 0.0825 0.1115
0.89 0.36 0.06 0.28 0.00 6.42 7.69*
0.00 0.00 0.03 2.20 0.18 0.38 0.01
KBr
1 2 3
0.0040 0.0018 0.0018
2.0- 9.0 2.0- 8.0 2.0- 8.0
1.93-3.34 1.93-3.34 1.96-3.31
0.0005 0.0017 0.0027
0.0008 0.0059 0.0197
0.0007 0.0059 0.0164
0.14 0.01 0.00
0.35 0.01 0.03
RbCI
1 2 3 4 5 6 7 8 9 10
0.0009 0.0007 0.0015 0.0010 0.0005 0.0008 0.0016 0.0008 0.0006 0.0004
2.0-10.8 2.0- 9.0 2.0- 8.0 2.0- 7.6 2.0-10.0 2.0- 9.0 2.0- 9.2 2.0- 8.0 2.0- 8.8 2.0- 8.8
1.81-3.16 1.81-3.31 1.53-3.47 1.50-3.59 1.87-3.34 1.69-3.47 1.81-3.37 1.50-3.50 1.69-3.41 1.72-3.44
0.0079 0.0257 0.0215 0.0859 0.0349 0.0843 0.0069 0.1579 0.1019 0.2322
0.0057 0.0368 0.0305 0.1212 0.0294 0.0852 0.0092 0.2173 0.1333 0.2903
0.0078 0.0227 0.0152 0.0495 0.0418 0.0550 0.0047 0.0786 0.0494 0.0942
2.38* 0.19 0.00 0.00 1.65 0.96 0.21 0.01 0.19 0.31
1.05 1.35 2.41. 3.54* 0.42 3.09* 2.47* 4.67* 4.66* 6.05*
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and the variance o f the nearest-neighbor distance, R and a 2, respectively. In principle, these two pieces o f information, along with the known c o o r d i n a t i o n number, can be used to construct a t h r e e - p a r a m e t e r probability density function for the length o f the nearest-neighbor distance, i.e. a nearest-neighbor radial distribution function. However, with only three parameters at hand, the nearest-neighbor radial distribution function thus constructed must be a simple function, such as a p a r a b o l a or a Gaussian or a Lorentzian or any other function that is d e t e r m i n e d completely by the location a n d height o f the maxim u m and its width. All attempts to find more sophisticated distributions, such as a s y m m e t r i c or double-well distributions, must be ruled out a priori when nearest-neighbor E X A F S d a t a o f Cu, a-Ge, NaBr, KBr or RbCI are investigated. At this point we cannot draw a general conclusion about the suitability o f E X A F S for the d e t e r m i n a tion o f interatomic potential anharmonicities, double-well potentials and the like. However, we want to point out that with our choice o f materials we have covered a wide range: metallic ( C u ) , covalent ( a - G e ) and ionic ones (NaBr, KBr, R b C I ) ; hard (Cu, a - G e ) and soft ones (NaBr, KBr, RbC1). Therefore, if more than two pieces o f structural information are claimed to be the content o f a nearest-neighbor E X A F S data set o f some other material, it is i n c u m b e n t upon the data analyst to prove quantitatively that the extra information is indeed contained in the data. I would like to thank R. Ingalls, J.E. W h i t m o r e , J.M. T r a n q u a d a and B. Houser, my former Univer-
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sity o f Washington group, and E.D. Crozier, N. AIberding, A.J. Seary, K.R. Bauchspiess and D. Jiang, our collaborators from Simon Fraser University, for having done the experiments with us at SSRL. Special thanks go to E.D. Crozier for numerous contributions to i m p r o v e the E X A F S data analysis. This work was supported by the Deutsche Forschungsgemeinschaft, the U.S. D e p a r t m e n t o f Energy and the Natural Sciences and Engineering Research Council o f Canada. SSRL is s u p p o r t e d by the U.S. D e p a r t m e n t o f Energy (Office o f Basic Energy Sciences) and the National Institutes o f Health (Biotechnology Research Program, Division o f Research Resources).
References [1] J. Freund, R. lngalls and E.D. Crozier, Phys. Rev. B 39 (1989) 12537. [ 2 ] J. Freund, R. Ingalls and E.D. Crozier, Phys. Rev. B ( 1991 ), in press. [3] E.D. Crozier, J.J. Rehr and R. lngalls, in: X-ray absorption, eds. D.C. Koningsberger and R. Prins (Wiley, New York, 1988) p. 373. [4] J.J. Rehr, R.C. Albers and J. Mustre de Leon, Physica B 158 (1989) 417. [5 ] Ph.R. Bevington, Data reduction and error analysis for the physical sciences (McGraw-Hill, New York, 1969 ). [6]W.H. Press, B.P. Flannery, S.A. Teukolsky and W.T. Venerling, Numerical recipes (Cambridge Univ. Press, Cambridge, 1986). [7 ] P.A. Lee, P.H. Citrin, P. Eisenberger and B.M. Kincaid, Rev. Mod. Phys. 53 (1981) 769. [8] F.W. Lytle, D.E. Sayers and E.A. Stem, Physica B 158 (1989) 701. [9] W.J. Dixon and F.J. Massey Jr., Introduction to statistical analysis, 3rd Ed. (McGraw-Hill, New York, 1969).