XAFS Spectroscopy of Liquid and Amorphous Systems: Presentation and Verification of a Newly Developed Program Package

XAFS Spectroscopy of Liquid and Amorphous Systems: Presentation and Verification of a Newly Developed Program Package

T E J A S E B A S T I A N E R T E L , * H E L M U T B E R T A G N O L L I , S A B I N E H ~ C K M A N N , U W E K O L B , and D I E T M A R P E T E R Institut [Er Physikalische Chemie der Universit6t WErzburg, Marcusstr. 9-11, D-8700 WErzburg, Germany

A program package for XAFS data analysis, especially of liquid and amorphous samples, has been developed. For the first time a consequent error propagation is presented for all functions to be calculated in the course of the data analysis. The structural investigation of the Grignard compound CH3MgBr in diethyl ether is taken as an example for the various steps of the data analysis.

Index Headings: X-ray absorption; Data analysis; Structure; Liquids; Amorphous systems; Grignard compounds.

I N T R O D U C T I O N

Only 11% of the literature (up to September 1991) has dealt with the structural investigation of liquid or amorphous systems. The fraction of publications is re- duced to 2 %, if we consider only liquid systems. In view of the fact that most reactions in organic, inorganic, and biochemistry occur in the liquid state, this circumstance is rather astonishing. Furthermore, interest in the structure of amorphous materials has grown over the last decade with regard to the development of new ceramics. An explanation for this situation may be the complicated nature and intricacies of the transmission or fluorescence measurements and data evaluation. In this paper, we present a data evaluation procedure specially developed for the analysis of liquid or amorphous systems. In view of the great variety of existing program packages, one might ask why a new one has been developed. Initially, we used several of the available program packages to evaluate the data for liquid samples. In the course of these data evaluations it was found that in the case of solid crystalline materials the program packages yielded satisfactory results, but in the case of noncrystalline systems several shortcomings were detected, which neces- sitated the development of a new data evaluation, which is particularly well suited for liquid and amorphous systems. Of course, our program package can also be applied to crystalline materials, and it should be noted that it was tested for its applicability to standard systems such as metal foils.

BASIC THEORETICAL ASPECTS

X-ray Absorption Fine Structure (XAFS) is based upon the absorption of x-rays in a particular absorbing atom and the oscillatory variation of the x-ray absorption coefficient tL as a function of the photon energy E beyond an absorption edge. The modulation of #(E) provides

Received 25 August 1991; revision received 29 October 1991. * Author to whom correspondence should be sent.

information about the kind, number, and distance of neighbors surrounding the absorbing atom. In a transmission or fluorescence experiment, u(E) is calculated by

t t ( E ) t r ~ s d = ln(Io/It,ans) (1)

tt(E)nu d = Inu/Io (2)

where d is the sample thickness, and I0, It . . . . and I~u are the intensities of the incident, the transmitted, or the fluorescence beams, respectively. In order to relate the modulation of the absorption rate in XAFS x (E) (cf. Eq. 3) to structural parameters, #(E) must be expressed in terms of the photoelectron wave vector k (cf. Eq. 4):

it(E) - tt0(E) x(E) = ( 3 )

fro(E)

k = Vr(2m/~2)(E - Eo). (4)

This transformation of ×(E) in E space yields x(k) in k space

2 2 _ 2 k 2 a j 2 x(k) = ~_~ (NJkr j )S O (k)Fj(k)e J [ -2(r~- A)/X(k)]

e sin[2krj + Oj(k)] (5)

where So2(k) is the damping factor resulting from many- body effects at the central atom. Fi(k) is the backscat- tering amplitude from each of the Nj neighboring atoms with a Debye-Waller factor % Inelastic losses in the scattering process are expressed by the exponent -2(r~ - A)A(k), }, being the electron mean free path due to the finite core hole lifetime and interactions with the valence electrons. The parameter A is a core radius of the central atom that eliminates double counting of inelastic pro- cesses in the core region. The nearest-neighbor distance appears to be a reasonable approximation for A. ~j(k) is the total phase shift experienced by the photoelectron. Fourier transform of weighted x(k) over a finite range between kmi. and km~ yields a modified distribution function in r space

fk km~ F(r) = 1/V/-~ W(k)k~x(k)ei2krdk, (6) rain

W(k) being a window function. Back-transformation into k space of one or more maxima of the Fourier transform provides the contribution of one or more backscatterers to these separated co-ordination shells. Finally, a curve- fitting routine yields the distance r between absorbing atom and backscatterer, the accompanying co-ordination number N, and the Debye-Waller factor a. 1,2

690 Volume 46, Number 4, 1992 0003-7028/92/4604-069052.00/0 APPLIED SPECTROSCOPY © 1992 Society for Applied Spectroscopy

EXPERIMENTAL

The Investigated System. Since the Grignard reagents RMgX (with R = alkyl, aryl and X = C1, Br, I) were first synthesized in 1900 by Victor Grignard, ~ their impor- tance and applicability as precursors in synthetic organic and inorganic chemistry have increased over the decades. Grignard reagents can be successfully employed, for example, in the synthesis of alcohols, aldehydes, ketones, carboxylic acids, esters, amines, and mercaptanes. Al- though the Grignard reagents have been used for over ninety years, considerable confusion exists as to their structure in solution. It has often been interpreted in terms of the Schlenk equilibrium 4

2 RMgX = R2Mg + MgX2 ~- R2Mg'MgX2 ~- (RMgX)2

or equilibria including dimeric and oligomeric species. 5 Almost all methods of physical chemistry have been used to elucidate the possible equilibria and species contrib- uting to it. It is known from molecular-weight studies that alkyl magnesium chlorides are monomeric in 2 M THF solution and dimeric in ethereal solution at concentrations down to 0.05 M. 6,7 Alkyl and aryl magnesium bromides and iodides are monomeric in THF as well as in ethereal solution at concentrations up to 0.05 M. 6,7,s The degree of aggregation in ethereal solution depends on concentration rather than on the organic group. In THF, however, Grignard compounds in general are monomeric, even at high concentrat ions. 7 Tracer experiments 9,1° provided evidence that the structure of the dimer is either a symmetric (RMgX)2 or an unsym- metric R2Mg" MgX2. 1H-NMR studies H of various alkyl Grignard compounds in diethyl ether suggest that R2Mg- MgX2 is the predominant species. THF solutions of aryl magnesium compounds showed a dependence of the equilibrium constant on the type of aryl group. Fur- thermore, the contribution of the R2Mg species increases with the co-ordinating ability of the solvent22 Infrared spectroscopic studies, however, showed that the type of the alkyl or aryl group, the halogen atom, and the solvent have an effect on the structure of species present in Gri- gnard solution, and therefore no generalizations can be made. Conclusions may only be transferred from one system to another with restrictions23

Although x-ray diffraction studies have been performed on ethyl and phenyl magnesium bromide diether- ates in the solid state, ~4,~5 little information is available on structural features such as the intra- and interatomic distances of the Grignard compounds in solution. Ex- tended x-ray absorption fine structure (EXAFS) and large-angle x-ray scattering (LAXS) were used to determine the structure of magnesium bromide and iodide in diethyl ether and THF. The structure of methyl, ethyl, and phenyl magnesium bromide in diethyl ether in the concentration range of 0.1 to 1.0 M was also investigated with these two spectroscopic methods. 16,17

We decided to perform XAFS measurements on a 3 M solution of methyl magnesium bromide in diethyl ether, in order to test the experimental setup and the sample cell with this highly reactive as well as moisture- and air- sensitive compound. Since in a recent study ~7 the existence of a second shell could not be observed, we inves-

tigated a more highly concentrated solution (factor 3), thereby increasing the probability of detecting a second or even higher shell, which is prerequisite for a reasonable structural proposal.

Experimental Details. The XAFS experiment was carried out at the Br K-edge at 13,474.0 eV at the beam line R(~MO II at the Hamburger Synchrotronstrahlungslabor (HASYLAB) at DESY, Hamburg, at 20°C, with the use of a Si(311) crystal monochromator under ambient conditions (5.4 GeV, beam current 30 mA). A sample cell suitable for measuring air- and moisture-sensitive liquid compounds, adjustable with high precision between 35,000.0 and 0.0 #m, with a step of 5 ttm, was used.t Data were collected in transmission mode with ion chambers.

The first ion chamber, monitoring Io, with a length of 15 cm was continuously flushed with a mixture of 7 % argon and 93 % nitrogen. The second and third ion chambers, monitoring I1 and I2, respectively, with a length of 30 cm were flushed with 100% argon. The fill of the ion chambers was chosen in such a way that about 10% of the incident x-rays were absorbed in the first and second ion chambers and about 80% in the sample itself. TM En- ergy calibration was monitored with a 30-ttm thick anhydrous MgBr2 sample as reference. The difference between the pre-edge absorption coefficient and the post- edge absorption coefficient, multiplied by the sample thickness, should be about 1.5. We calculated a sample thickness of ~396 gm for a 3 M solution of CH3MgBr in diethyl ether, and adjusted it to 480 ttm.

DATA REDUCTION

The program package for the data reduction procedure was designed in such a way as to guarantee compatibility with any type of computer and operating system without change of the Fortran source code.

Conversion of the Experimental Data (Program EXRD). After calculation of the total absorption coefficient (cf. Eq. I and 2), the energy threshold E o is determined. The usual procedure involves differentiating the data with respect to E and assigning the first maximum to Eo. This method fails if the absorption edge has pre-peaks or shoulders (see for example Fig. 1B). A more convenient method is the convolution of the data with a series of increasingly broader Gaussian functions. The convolution is calculated by means of the convolution theorem and the Fast Fourier Transform. 19,2° Figure 1A shows the effect of convolving the experimental data of MgBr2 at the Br K-edge at 13,474.0 eV (used as reference data) with three Gaussian functions with FWHMs of 16, 32, and 48 eV. The intersection point closest to the middle of the edge was assumed to be the threshold energy Eo ref. The deviation of Eo ~°f from the value given in the literature, AEo ref = 0.072 eV, is then subtracted from the experimental data for CH3MgBr in diethyl ether. Figure 1B shows an example (crystalline CuBr2 measured at the Br K-edge) where the determination of Eo with the derivative method fails owing to the shape of the absorption edge. The pre-edge absorption in the range from 13,000.0

t For detailed information about the sample holder, please contact the authors.

APPLIED SPECTROSCOPY 691

3.5 1.550

A L O

2.5

1.5

0.5

- 0 . 5 13.300

!

i

i i' i i

i i / i

/ • i

i

13,412 13.525 13.637 13.750 Energy [keY]

7.5

B

5.5 "- . - - . . . . . .

3.5

-0 .5 , ' ~ I ' ' ' I ~ I ' ' ' 13.425 13.450 13.475 13.500 13.525

Energy [keV]

FIG. 1. Effect of convolving the absorp t ion edge of the exper imen ta l and Victoreen-corrected da ta (A) of crystal l ine MgBr~ (solid line) and (B) of crystal l ine CuBr2 (solid line), wi th Gauss ian func t ions of F W H M 16 eV (dashed line), 32 eV (dot ted line), and 48 eV (dashed-do t ted line).

to 13,450.0 eV is then fitted with a Victoreen-type polynomial

~v(E) = a E -3 + bE -4 (7)

where a and b are fit parameters and E is the x-ray energy. The pre-edge fit is then extrapolated beyond the edge and subtracted from the total absorption to give the corrected absorption coefficient. In practice, the corrected absorption coefficient still contains other background contributions, such as higher beam harmonics and elastic scattering, etc. 1

In addition, the post-edge experimental absorption data are also convoluted with three Gaussian functions of F W H M 16, 32, and 48 eV. The common intersection points of all damped spectra (shown in Fig. 2) with the experimental data are called nodal points and can be used to determine the background (see below), because it is assumed that the nodal points define the shape of the background. It should be noted that all the other intersection points of the damped spectra are not nodal points. Their inclusion in the determination of the background causes only negligible error, since they are po- sitioned symmetrically with respect to the background29

1.475

w~ 1.400

1.325 - j ,

1.250 '

13.480

k _ / . - - : - ~ .oao, po~.t~

i J

s i

b i i I i , i I i i i j i i i

13.535 13.590 13.645 13.700 Energy [keV]

FIG. 2. Inflect ion po in t s de t e rmined by convolving the exper imen ta l and Victoreen-corrected da ta of CH~MgBr in d ie thyl e the r (solid line) wi th Gauss ian func t ions of F W H M 16 eV (dashed line), 32 eV (dot ted line), and 48 eV (dashed-do t t ed line).

Background Removal and Normalization (Program EXBR). The interference function above the edge is defined as

x ( E ) = I Z v ' e ( E ) - #° (E) (8) uo(E)

where/~0(E) is the absorption coefficient of an isolated atom. Since/~o(E) is generally not known, it is assumed that the smooth part of the Victoreen and edge-corrected /~v.e(E) without wiggles or oscillations approximates/~0(E). A modified smoothing-spline algorithm, based on the IMSL subroutine ICSSV, proved to be very useful for determining the background ~o(E) of spectra of liquids. 21 The degree of smoothing is determined by the condition

N

~_~ [(C(x,) - Yi) /Wi] 2 < SM (9) i = l

where C(x~) is the smoothing-spline approximation to a set of noisy data Yi, W~ is the weight assigned to point i, and SM is a user-specified smoothing parameter. C is a natural cubic spline with knots at all data points, i = 1, 2 . . . N. The two extremes of the approximation are SM = 100, resulting in a straight line, and SM = 0, when the spline follows the data accurately. 22

When evaluating Victoreen and edge-corrected data of liquids, one has to pay attention to several facts. Up to about 75 eV above the edge, the shape of the spectrum demands a flexible smoothing spline. On the other hand, a high flexibility in the range up to about 300 eV above the edge increases the risk of removing or reducing part of the few XAFS oscillations. The high noise level caused by the static disorder of liquids is an additional problem.

These problems are successfully dealt with by impos- ing three conditions included in this program package. First, the set of data points may be weighted by factors W~ that decrease with increasing i. After application of this procedure, the smoothing spline function follows the experimental data more closely near the edge. For spectra of metal foils or other solid materials, the use of a weighting function is not necessary. Second, the smoothing spline is forced to go through the nodal points. They are calculated by convolving the data with increasingly

692 Volume 46, Number 4, 1992

1.550 , 6.00

a b s o r p t i o n c o e f f i c i e n t p ( E ]

1.462 _0" ~

1 . 2 0 0 ~ i ~ I ' ~ i I ~ i ~ I I I

15.480 13.610 1 ̀5.740 15.870 14.000 Energy [keV]

~-~ 1.,575 :&

-1.50

b

-9.00

-16.50

i

-24.00 13.480

r

i

i - - , - - ~ c - ~ - - =

i i

, !

B

i i I i i i I i i i I i ~ i

1,5.610 13.740 15.870 14.000 Energy [keV]

9.00 --

4.50

~o rn 0.00

-4.50

-9.00

C

.00 3.75 6.50 9.25 12.00 k [Angstroem -1]

5.00

3.75

E

f 2.5o < "L"

D

m

0 . 0 0 ' ' I ' ~ ' I ~ '

0.00 1.25 2.50 3.75 5.00 r+~ [Angstroem]

Fro. 3. The effect of the smoothing-factors 14.9 (dashed line), 2.2 (solid line), and 0.05 (dashed-dotted line): (A) on the shape of #o(E), (B) on the derivative of #o(E), (C) on the ×(k) function, (D) on the pseudo-radial distribution functions, for CH3MgBr in diethyl ether.

broader Gaussian functions, as described above. Third, the smoothing spline may be applied to a convolved and therefore damped t~v,e(E) spectrum to reduce the influence of spurious elements, such as over- or undershoots, on the shape of ttv.e(E).

Subsequently, a decision has to be made as to which of the calculated smoothing splines represents the ade- quate background absorption. Several empirical and mathematical criteria were found to be helpful in as- sessing the shape of the background. The most simple criterion is monitoring the shape of the background and its derivative. An additional test is provided by the Fou- rier transformation of the XAFS data weighted by k ~. Cook and Sayers 23 suggested the use of three parameters obtained from the Fourier transform as criteria for choosing smoothing parameters.

CS = ( H , - HN)/HM (10)

where HR is the average value of the transform magnitude between 0 and 0.25/~, H M is the maximum value of the transform magnitude between 1 and 5/~, and HN is the average value of the transform magnitude between 9 and 10 ~. H R is a measure of any remaining low-fre- quency (i.e., low r) component in the EXAFS data. HM is a reference value against which HR and HN can be judged. The criteria for choosing a good smoothing parameter are either HR - HN ~ 0.05 HM, i f H N <- 0.1 HM;

or HR > 0.1 HM, if HN > 0.1 H M. The latter criterion should be applied to noisy data.

The removal of the background from the data for CH3MgBr in diethyl ether was very difficult. The smoothing spline was determined from the experimental data convolved with a Gaussian function with an FWHM of 24 eV by applying a linear weighting function and a smoothing factor of 2.2 in the range from 13,480.8 to 13,998.8 eV. Note that the smoothing spline is determined from the convolved curve, but applied to the experimental spectrum. Figure 3 illustrates the effect of background correction in the described manner with the smoothing factors 14.9, 2,2, and 0.05 on the shape of tt0(E), the derivative of tt0(E), x (k ) , and the pseudo-radial distribution function for CH3MgBr in diethyl ether.

Two major difficulties arose when we corrected the background of several spectra of liquid systems with EXCURV88 from the Daresbury Laboratory and the program package developed at HASYLAB (DESY, Ham- burg). Neither a single polynomial fit over the whole data range in E space (realized in the program package at HASYLAB) nor cubic splines defined over a series of intervals (realized in EXCURV88) helped remove a spurious and remarkably pronounced low-distance peak in the 0-1 A region. The use of higher orders of the polynomial splines combined with a larger number of sections more or less removes the low-distance peak, but damps


2.0 ~ , 2 .50

- 0 . 5

-5.5

-8.0

FIG. 4.

. I N ( r ) l 2 ( e x t r a p o l a t e d )

' ' i " i

i i 1 4 - - ] S ( r ) l 2 ( deduced ) i i t i i i

I c ( r ] r 2 ( m e a s u r e d )

0.0 2.5 5.0 7.5 10.0 r/rm~ , [10 ~ %1

The power spectral density of the x(k) function for CH3MgBr in diethyl ether (solid line) and the deduced signal (dashed line) and noise (dashed-dotted line).

1.25

E o o -g

o.oo <~ Q-

- 1 . 2 5

- 2 . 5 0

2.50 4.50 6.50 8.50 k [Angstroem -1]

10.50

FIG. 5. k3x(k) function for CH3MgBr in diethyl ether after background correction and normalization {solid line), the filtered and deconvoluted function (dashed line), and the calculated error bars.

the first shell peak in the radial distribution function, and therefore causes problems in the determination of N and a. In the case of cubic splines defined over a series of intervals, the choice of limits is somewhat arbitrary. Comparison of the results of different background corrections of the program packages mentioned above is almost meaningless due to that arbitrariness.

Data were normalized with the spline (cf. Eq. 8) and compared with the results of a normalization according to Heitler, 24 who derived a theoretical derivation of tt0' (k)

tt0'(k) = J[1 - 8/3(h2/2mEo)]k 2 (11)

where J is the jump at the edge. The results from both methods were identical within the experimental margin of error.

Deconvolution of the x(k) Function (Program EXDC). In order to determine the co-ordination number N more accurately, the program EXDC calculates the deconvolution of the x(k) function with the monochromator resolution as the response function g(k). This procedure compensates for the decrease of the x(k) amplitude caused by the limited and energy-dependent resolution of the monochromator. The FWHM of the Si(311) monochromator rocking curve at the Br K-edge at 13,474.0 eV was determined in the usual way from the first derivative of the absorption edge with respect to E. The obtained value of 3.0 eV is in good agreement with the theoretically calculated 2.6 eV. 25

The deconvolution is generally quite sensitive to noise in the input data and to the accuracy to which the response function g(k) is known. Therefore, reasonable trials may sometimes yield meaningless results. In this case the additional use of optimal Wiener filtering with a Fourier transform helps to solve the problem. The equation of the optimal filter is given by 2°

IS(r) l 2 • (r) = (12)

IS(r) 12 + [N(r) 12

where IS(r)l 2 and IN(r)l 2 are the deduced signal and the noise in r space, respectively. The two of these add up to I C(r) 12, the power spectral density of the measured signal.

IS(r)l 2 + IN(r)l 2 = IC(r)l 2. (13)

To determine the optimal filter from Eq. 12, IS(r) [ 2 and IN(r) 12 must be estimated separately; in the plot of the power spectral density [C(r)[ 2 of the x(k) function a smooth curve is drawn through the noise spectrum and extrapolated into the region dominated by the signal to give IN(r)]2. Since IN(r)12 is known, IS(r)]2 can be deduced from Eq. 13 (see Fig. 4). Because the optimal filter results from a minimization problem, the quality of the results obtained by optimal filtering differs from the true optimum by an amount that is second order in the precision to which the optimal filter is determined. 2° In other words, the separation of the measured signal into signal and noise components can be done by inspecting a crude plot of the power spectral density. With the optimal filter O(r), the deconvoluted signal Xdec(k) in k space is then given by a Fourier transformation

C(r) ~(r) -~ Xdec(k) (14)

G(r) where G(r) is the resolution function in r space. 2° Figure 5 shows the k3x(k) function and the Wiener-filtered function, deconvolved with a Gaussian function of FWHM 3.0 eV. Figure 6 demonstrates the effect of the deconvolution on the radial distribution function. Deconvo- lution of the absorption spectrum #v,e(E) instead of x(k) and an analogous data reduction procedure would also be possible. We tested both methods, and the procedures yielded the same results. The shape of the absorption edge, however, makes it more difficult to determine the optimal Wiener filter. Therefore, we decided to apply the process of deconvolution to background-corrected and normalized x(k) functions.

Note that the following steps of the data interpretation (i.e., Fourier transformation, Fourier filtering, and curve- fitting) should be performed with both the experimental and the deconvolved x(k) functions, because of the un- certainties in the determination of the monochromator resolution and the Wiener filter. We tested the effect of the deconvolution on the structural parameters (r, N, a, AE o) and found that only the co-ordination numbers N are affected. Additionally, the deconvolution has a re-


3.50

2.62 I E

~ 1.75 <~

0.88

0.00

t ~

0,00 1.25 2.50 3.75 5,00 r + a [Angs[ roem]

Radial distribution function for CH3MgBr in diethyl ether FiG. 6. (solid line), the Fourier transformed deconvoluted function (dashed line), and the random error bars.

markable effect on the co-ordination numbers of higher shells. We discovered an increase in N of 7.2% for the first shell and 13.7 % for the second shell of CH3MgBr in diethyl ether. For a comparison with model compounds the corrections mentioned above must of course be applied to the experimental data for the model compound, too. Deconvolution becomes more important when the resolution of the monochromator crystal decreases with increasing energy. At the Br K-absorption edge the resolution of the Si(311) crystal employed was 3.0 eV, in contrast with a value of 7.0 eV at the Zr K-edge at 17,998.0 eV.

Fourier Transformation and Fourier Filtering (Program EXFT). The next step in the data reduction procedure is the Fourier transformation of the k"×(k) function and the Fourier filtering of a relevant region in r space. Square, Welch, Parzen, Hanning, and Square/Hanning windows can be applied, and the spectrum can be weighted by k% with n = 0, 1, 2, 3. A direct comparison of the influence of the different window functions and the applied weighting scheme to the radial distribution functions makes it possible to detect artificial peaks and sidelobe effects. Thus, the suitable r range for the subsequent Fourier filtering can be selected quite easily.

On the forward transform (k to r space) the data are multiplied by the k-grid spacing and divided by V% Likewise, back-transformation involves multiplying the data by the r-grid spacing and dividing them by V~. This is done to eliminate any dependence of the scale of the ordinate axis on the number of points used in the transform and the grid spacing. The dimension of the transform of the knx(k) data then is (length)-("+1). 26 Back- transformation will be performed with a Square and a Hanning/Square window. The small differences at the endpoints of the two filtered k~x'(k) functions indicate termination-of-series effects. An appropriate k range for the following curve-fitting procedure can therefore be chosen.

Additionally, back-transformation into k space provides the amplitude and phase functions of the filtered k"x ' (k) functions. The amplitude and phase function (after subtraction of an approximated 2kr term) of the selected k range may now be compared with those tabu-

lated by McKale 27 or model amplitudes and phases. In the case of a single shell, the shape of the amplitude and phase function may sometimes be directly assigned to a certain type of backscatterer.

In our data evaluation we selected a range from 2.55 to 10.45/~-1 of k 3 weighted x(k) data for CH3MgBr in diethyl ether for the forward transform with a Square window. Square windows in the range from 1.56 to 2.74 /~, 2.74 to 3.34/~, and 1.56 to 3.34/~ were chosen for the inverse transform. Figure 6 shows the radial distribution function of CH3MgBr in diethyl ether.

Estimation of Errors. Fitting theoretical functions Xtheo(k) to normalized and background-corrected or to Fourier filtered functions Xe.,(k) involves minimizing X 2 according to the advice published by the International Workshop on Standards and Criteria in XAFS: 26

X 2 = Npt~[1/(Npt~ - v)]

{k?[Xe, ,p(k , ) -- Xtheo(h i ) ] } 2 (15) I

2 i=1 iTerror,i

Npt s -- 2 A k A r / T r (16)

where N is the number of data points; Ak -- kmax - kmi, is the k space range of the data; and Ar = rm~ - rml . is the width of the window in r space, v in Eq. 15 is the number of fit parameters, and a ...... , is the standard deviation for each data point and includes statistical as well as nonstatistical errors in the data:

IT . . . . . . i = iTstatistical, , ~ - iT . . . . tatistical,/" (17)

Normally, iT ...... i is not exactly known and is therefore approximated by ae ..... i = 1. In order to determine iT ...... , exactly, we included an estimation of errors in each program module. We assumed that the measured intensities I (E) obey a Poisson distribution with a standard deviation oF °

IT, = x / i ~ , ) . (18)

With Eq. 18 the standard deviation of #(E) for a transmission or a fluorescence experiment can be calculated from Eq. 1 and 2 by Gaussian error propagation

°'it . . . . . . i = k/(1/Io(Ei)) + (1~It,ms(E,)) (19)

IT,No,~ = N/(I,,(E~)/Io(Ei) 2) + (I~u(E,)2/Io(E,)D • (20)

These IT,., values represent the statistical error and do not take account of nonstatistical errors. The errors may partially compensate for each other, and the estimated error could possibly be smaller. This approximation, however, provides a good estimation. The error on the accumulated spectra O'totald must be calculated with the following equation:

O'total, j = (l/n) ( 1 ~ 2 ITs = [1/(n) ''5] 2 ITs (21) j=l j=l

where ITj is the error on the jth spectrum, and n the number of spectra. Equation 21 shows that an accu- mulation of four or ten spectra reduces the error by about 50 % and 68 %, respectively. In each of the following data reduction steps this error is input into the Gaussian error propagation. The error on Victoreen and edge-corrected data is calculated by


O'ttv, e, i = V a t t , i 2 -}- O-vic,i 2 ~ o'u, i (22)

where O-vic. i is calculated by

0"vic, i = ~¢/Ei-%-2a -F Ei-8o-2 b. (23)

Here ao and ab are the errors on the Victoreen-fit parameters in E space. Because a,.~ >> a~i~,i, a,i¢,i may be neglected, and the approximation in Eq. 22 is valid. The error on the x(k) function is given by

aX, i = X,/ (1/#o(Ei)2)ty ..... i 2 (24)

provided that a,~.~.~ >> a,o,~ because a,0 . cannot be accurately calculated.

Figure 5 also shows the k 3 weighted error bars of the k3x(k) function. The error bars are shifted on the ordinate axis to negative values, for the sake of clarity. These error bars seem to be very large. At higher k-values, however, they provide a very good description of the possible deviation of one data point from its "true" value. The calculated error function is used as a ...... i (cf. Eq. 15), and so was taken into account in the least-squares refinement: the iterative procedure compensates for error-matched deviations of the theoretical functions from the experimental data.

Subsequent Gaussian error propagation (cf. the following equations) for the Fourier transformation and Fourier filtering yields an unexpected result. The errors on the imaginary part, a~(r), the real part, aRe(r), and the Modulo function, aMod(r) , a r e calculated by

~-' /N)] aim (r) : ~//[/=~1 sin (Trir a×,i= (25)

N--1 ] ~ / [ [ i ~ 1 cosOrir/N) a×,i~ (26) aR~(r) =

\ / ( R e / ~ / R e 2 + Im2)aRe(r) 2

aMod(r) = V + (Im/k/Re 2 + Im2)aim(r) 2 (27)

where N is the number of points used in the Fourier transform. An analogous calculation was carried out on the inverse transform. The errors on the Fourier transform and the Fourier filtered spectra calculated by Eqs. 25 to 27 are constant. This is a consequence of the contribution from each point (and each error) in k space to a point in r space, and vice versa. This, however, does not reflect reality. In order to tackle this problem, we added a random error to the original ×(k) function

Xrandom(k) = x ( k ) "4- aia×,i (28)

where ax, i is the calculated error on the original x(k) function, and ai is a random number between - 1 and 1. The random error on the Fourier transform is now given by the difference between the Fourier transform of ×,~dom(k) and the Fourier transform of the original x(k):

a~ndo~ = IF~ndom(r) -- F(r)]. (29)

The error on the Fourier filtered x'(k) function is calculated analogously. Figure 6 shows the random error on the Fourier transform.

Curve Fitting (Program EXCF). For interpreting the k"x(k) function and the Fourier filtered k"×'(k) function we developed the program EXCF. This program is based on the Levenberg-Marquard method. 2° We modified this iteration algorithm in such a way that reasonable bound- ary conditions may be assigned to all varied parameters (r, N, a, AEo). This feature has been included to avoid meaningless results and waste of computing time. The optimization can be performed with McKale, FEFF, and model amplitudes and phases. 27,28 An exceptional feature is that any combination of theoretical and model amplitudes and phases for multi-shell fits is possible. The basic formula is given by Eq. 5. With McKale phases and amplitudes, k (k) is approximated by the universal curves for elements, inorganic and organic compounds, and ad- sorbed gases 1

k(k) = (1/rl)[(~/k) 4 + k n] (30)

where ~ and ~ can be refined to best-fit the XAFS data. 29 So=(k) (cf. Eq. 5) can be taken account of by a linear factor =0.7 and 0.8. 2 When using McKale or model amplitude and phase functions, one should allow Eo to vary, since this variation preserves the otherwise less reliable phase transferability. The program EXCF provides the possibility of considering AEo by

ki = k/k 2 - 0.2625(AEo) (31)

where k is the experimental wave vector, with the Eo chosen in the data reduction process, and ki is the refined wave vector with E'o (and AEo = Eo - E'o). Such an iteration procedure will take up much more computing time, because each change of E'o will entail an explicit calculation of all k-dependent terms by interpolation. But this "slow iteration" proved to yield more reasonable results for AEo in comparison to the commonly used approximation of Eq. 32, where, as often as not, very high shifts of about _+25 eV or more may occur. Accord- ing to our tests, however, Eq. 32 is only valid for AE o _-< 18 eVI. In this region the functions calculated according to Eq. 32 give nearly the same results as the exactly calculated ones (cf. Eq. 31).

x(k) = ~_~ (NJkrj2)So2(k)Fj(k) e-2k2~i~et-2(rj-~)/~(h)] J

sin[2krj + 4~j(k) - 0.2625rjAEo/k]. (32)

It should further be noted that with the approximation (cf. Eq. 32) a good-quality fit in the low-k region is easy to achieve, because the k-dependence of the amplitude is neglected. When one is using the exactly calculated functions it is difficult to compensate for the deviations from the experimental k"x(k) function, which are caused by the shortcomings of the theoretical amplitude and phase functions of McKale 27 in this region. The reason for this is that, in the exact calculation of all k-dependent terms, the fit-parameters h(k) and a have negligible influence in the low-k region. A k-dependent So would help to solve this problem, but its value is not exactly known. The only possibility of achieving a satisfactory agreement in the k-region below 5 /~-1 is the use of either theoretical amplitude and phase functions from FEFF, 28 together with an exact calculation of h(k) and S02(k) , o r empirical ones, i.e., model amplitude and phase func-


2.50 3.50

1.25

E

o~ 0.00

-1 .25

-2 .50 2.50 4.50 6.50 8.50 10.50

k [Angstroem -1]

FIG. 7. k3x(k) f u n c t i o n (so l id l ine) , t h e c o n t r i b u t i o n f r o m t h e s e c o n d a n d t h i r d she l l t o t h e k3x(k) f u n c t i o n ( d a s h e d l ine) , a n d t h e t h e o r e t - i ca l ly c a l c u l a t e d f u n c t i o n o b t a i n e d f r o m F E F F (see T a b l e I, i n d e x b) (do t s ) w h i c h b e s t f i ts t h e e x p e r i m e n t a l d a t a .

tions. When one is using FEFF amplitudes and phases (FEFF V3.11, March 28th 1990), FEFF-calculated So2(k) and k(k) are used in the least-squares refinement, and E0 is not allowed to vary. 28 In addition, the "slow iteration" makes it much easier to distinguish between different types of backscatterers.

RESULTS AND DISCUSSION

The results--i.e., the absorber-backscatterer distance r[/~], the correspondin~ co-ordination number N, the Debye-Waller factor a[A], and the shift of the threshold energy AEo[eV ] together with the standard errors, obtained with the amplitude and phase functions of McKale and FEFF- -a re shown in Table I. The table summarizes the different single- and multiple-shell fit results, which were obtained by fitting the three different Fourier filtered k3×'(k) functions (see above). Figure 7 shows the theoretically calculated function (FEFF) representing the best multiple-shell fit and the contribution from the second and third shell to the M×(k) function. In Fig. 8 the Fourier transformed best fit is plotted together with the radial distribution function.

In addition to the calculations with theoretical amplitude and phase functions, we also tried MgBr2 as a model compound. However, it was impossible to isolate the Br-Mg model amplitude and phase from the first peak in the radial distribution function, because oxygen backscatterers contribute to this peak. MgBr2 is extreme- ly moisture sensitive and, despite considerable efforts, we could not remove the water completely. So we decided to use NaBr as a model compound for reasons of phase

2.62 _

~ 1.75 <

S

0.88

• /

0.00 ' ' I f ' i I ' ' ' I ~ '

0.00 1.25 2.50 3.75 5.00 r+c~ [Angstroem]

Fro . 8. R a d i a l d i s t r i b u t i o n f u n c t i o n ( so l id l ine) a n d t h e F o u r i e r t r a n s - f o r m e d b e s t f i t (do t s ) .

transferability. With this model compound we determined the structural parameters of the Br-Mg pair in the usual way. 1 The obtained co-ordination number N now allows determination of the degree of association. This yielded a degree of dimerization of 60 _+ 7 % in a 3 M ethereal solution. This was deduced from the co- ordination number of 1.60 __+ 0.07. The Mg-Br distance increases from 2.475/~ to 2.556/~ when the model phase and amplitude functions are used. The co-ordination number of the Br-Br pair could not be used to determine the degree of dimerization, because the contribution from this pair to the experimental k3x (k) function is too small, and therefore a normalization with the model compound Br2 seemed to be useless. But even this small contribution from a Br-Br pair confirms the existence of dimeric species. The ratio of the co-ordination numbers for the Mg-Br and Br-Br pairs suggests the existence of dimeric species with the deduced degree of dimerization.

Most of the complexes of magnesium in solution have octahedral or tetrahedral geometries, and the co-ordination sphere is completed by solvent molecules2 ° As- suming only the existence of these species and restricting the discussion to monomers and dimers, we can deduce the Br-Mg-X bond angle (X = Br, O), i.e., the type of co-ordination, from the obtained distances. From the Br- Mg and Br-Br distance of 2.475 _ 0.001/~ and 3.500 _+ 0.005/~ (3.495 + 0.005/~), respectively, an angle of about 90 ° was calculated; this implies an octahedral co-ordination in the dimeric species. Assuming an average Mg- O distance of 2.10/~,14 from the obtained Br-O distance of 3.680 + 0.015/~, a Br-Mg-O angle of about 107 ° can be calculated, which indicates the existence of tetrahe- drally co-ordinated monomeric species. Octahedrally co-

T A B L E I. Structural parameters for CH3MgBr in diethyl ether obtained with amplitude and phase functions calculated by McKale (index a) and F E F F ( i n d e x b); the co-ordination number N was corrected for monochromator resolution.

r [A] N a [A] AEo[eV ]

M a g n e s i u m 2 .475 + 0 .001 o 0 .71 + 0 .02 o 0 .0680 _+ 0 . 0 0 1 0 ~ 3 .42 _ 0 .015 a 2 .475 _+ 0 . 0 0 P 0 .71 ___ 0 .02 b 0 . 0 5 8 4 _+ 0 .0015 b . - .

B r o m i n e 3 .495 + 0 .005 ° 0 .23 _ 0 .05 ° 0 . 0 8 6 6 + 0 .0175 a 0.01 _ 0 .005 a 3 .500 _+ 0 .005 b 0 .23 + 0 .05 ~ 0 . 0 8 5 8 + 0 .0175 b . . .

O x y g e n 3 .680 + 0 .015 ° 0 .74 _+ 0 .10 ~ 0 .0747 ___ 0 . 0 1 5 0 ~ - 0 . 0 1 _+ 0 .005 ° 3 .680 ___ 0 .015 b 0 .74 _+ 0 .10 b 0 . 0 7 4 0 +_ 0 . 0 1 5 0 b - . .


ordinated monomers would show a much shorter Br-O distance of about 3.25/~. A detailed structural investigation of the Grignard compounds CH3MgBr, CH3CH2MgBr, and C6HsMgBr in diethyl ether and CH2 = CH2MgBr in tetrahydrofuran with XAFS and other spectroscopic methods is beyond the scope of this paper and will be summarized in a second paper.

An EXAFS data analysis and structural interpretation of CH3MgBr in diethyl ether is presented. This example was chosen because it features most of the difficulties which are typical of EXAFS spectra of noncrystalline systems. Further examples indicating the utility of our program package, such as structural studies of aqueous RbBr solutions, amorphous precursors for the sol-gel processing of lead zirconate titanate, and some organo- metallic complexes, are published in Refs. 31-34. In each case we also tested the other available program packages. The results were very similar for EXAFS spectra of crystalline samples, but differed, sometimes considerably, for spectra without pronounced EXAFS oscillations. For example, the manner in which the background is determined seemed to be critical and somewhat arbitrary. Our intention was to reduce that arbitrariness and restrict the user-specified options as much as possible.

ACKNOWLEDGMENTS

We wish to thank the SFB 347 (Selektive Reaktionen Metall-akti- vierter Molekfile) for generous financial support, as well as HASYLAB at DESY, Hamburg, for the provision of synchrotron radiation and the data evaluation program. The Daresbury Laboratory is gratefully acknowledged for the provision of EXCURV86 and EXCURV88. J. J. Rehr is gratefully acknowledged for the provision of FEFF V3.11. The authors T. S. Ertel and H. Bertagnolli wish to thank the Stipendien- fonds des Verbands der Chemischen Industrie for generous financial support.

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Documents

XAFS Spectroscopy of Liquid and Amorphous Systems: Presentation and Verification of a Newly Developed Program Package