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Martin J. Fay, Andrew Proctor, Douglas P. Hothnann, and David M. Hercules Department of Chemistry University of Pinsburgh Pittsburgh, PA 15260
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The past several years have witnessed the emergence of Extended X-ray Absorption Fine Structure (EXAFS) spectroscopy from an esoteric technique to a widely available structural spectroscopic tool. Several in-depth review articles and books are available on the subject ( I d ) ,hut some aspects of theory or data reduction may be heyond the scope or need of the average chemist. In this article we will try to reduce the mystique surrounding EXAFS and clarify the terminology and concepts in the EXAFS literature. We will concentrate primarily on tbeory and data analysis and conclude with a few applications.
are ejected with kinetic energy, E, ap. proximately equal to ( E A - E ). Understanding the EXAFS phenomenon requiresthat we consider these phutnemitted electrons in their alternate guise, as waves. The EXAFS oscillations in an ab-
sorption spectrum arise from the backscatteringoflheemil~dphotoelectron waves from atom A off neighboring atoms 9. Figure I illustrates this phennmenon in two dimensions. The circles associated with each atnm pair (A, B) represent wave tronts of outgoing
INSTRUMENTATION
EXAFS and the X-ray absorption
spectrum EXAFS provides structural information about atoms (9)in the local threedimensionar environment surrounding a central atom (A) (Figure 1). This information is obtained by analyzing the oscillatory fine structure that occurs beyond the edge of the X-ray absorption spectrum of the central atom A. In Figure 2a, the Ni K-edge spectrum obtained from NiO is shown as an example of a typical X-ray absorption spectrum. The sudden large increase in ahsorption or “edge” occurs where the energy of the incident X-rays (EA”)is equal to the threshold energy (Eo) necessary to photoeject the Ni 1s electron. (EOfor the K-edge of Ni is approximately 8333 eV). At energies greater than Eo the resultant photoelectrons 003-2700/88/A360-1225/a0 1.50/0 @ 1988 American Chemical Society
Flgure 1. Schematlc diagram of the backscattering process. ANALYTICAL CHEMISTRY, VOL. 60. NO. 21, NOVEMBER 1, 1988
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(from A) and backscattered (from B) waves a t intervals of the photoelectron wavelength X,. The backscattered photoelectron waves can interfere with the outgoing photoelectron waves constructively or destructively, depending on the photoelectron wavelength X, (and so the incident X-ray energy) and the nearest neighbor distance between A and B, RAB. The interference produces a corresponding modulation in the absorption cross section, which results in oscillations in the X-ray absorption spectrum. Backscattering atoms B need not have a different atomic number than the absorbing atoms A. For example, in a metal where A andB are identicalatoms,anabsorbing atom will also be a backscattering atom. Extraction of structural information from EXAFS is a multistep procedure that involves considerable data manipulation. The first step in the data analysis is extraction of the EXAFS fraction of the X-ray absorption spectrum, which is termed x. Figure 2b shows the extracted and weighted EXAFS spectrum, k 3 x ( k ) , derived from the NiO absorption spectrum in Figure 2a. The energy scale (eV) used in the original absorption spectrum has been converted to the photoelectron wave vector scale and x ( k ) has been weighted by k3 to offset the rapidly decaying EXAFS oscillations. (These data manipulations will be discussed in more detail below.) It is clear that the EXAFS oscillations in NiO extend well beyond the energy (-1000 eV) or k range actually measured. Chi, x ( k ) , is the sum of individual sinusoidal components arising from the backscattering by different coordination shells. The frequencies of these components depend primarily on the nearest neighbor distance, RAB,of the coordination shells. Fourier transformation of k3x(k) will produce a frequency spectrum called a Pseudo Radial Distribution Function (PRDF), as illustrated in Figure 2c. The peaks in the PRDF correspond to individual sinusoidal frequepcies or, more significantly, to radial coordination shells. The radial positions are shifted slightly (to lower R ) from the actual nearest neighbor distances and thus are only approximate or pseudo distances. The PRDF provides an informative picture of the local environment surrounding the central atom, hut quantitative structural information is not generally obtained from the PRDF itself. To obtain specific structural information, an individual peak (or coordination shell) in the PRDF is backtransformed into k space producing a single shell &(k). This procedure is commonly termed “Fourier filtering.” Figure 2d shows the backtransformed or Fourier filtered k 3 x ( k ) from the first 1228A
coordination shell (Ni-0) of NiO. The hacktransform window is indicated by the bars surrounding the first shell peak in Figure 2c. The single shell k3x(k)is then analyzed to obtain structural information such as nearest neighbor distances, coordination numbers, Debye-Waller factors (indicative of the degree of vibrational and static disorder), and types of nearest neighbors. EXAFS is a valuable technique he-----e its short-range nature allows
structural information to be obtained from amorphous and highly dispersed materials that lack sufficient longrange order to be structurally characterized by traditional X-ray diffraction methods. A few examples of the wide range of materials studied by EXAFS are heterogeneous catalysts (6, 7), amorphous materials such glasses (S), and metal sites in bioinorganic materials (9). Another feature of EXAFS that greatly enhances its potential as an analytical tool is the ability to e x a r
Flgure 2. Major stages of EXAFS analysis of h (a) NI K 4 g e X-ray absorpth spec” ol NO, (b) exbaded @x(M specr” ol NiO, (c) PRDF ol NiO. and (d) backlranalomad Px(M from llrst cmrdlnsHon shell Ot NiO (bednansfam window Dhow” in F@re 2c).
ANALYTICAL CHEMISTRY, VOL. 60, NO. 21, NOVEMBER 1, IS88
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Figure 3. Basic EXAFS experimental setup.
ine the local structure of individual elements simply by tuning the incident Xrays to the desired edge energy. Experimental/instmmentai aspects ol EXAFS
Although it is possible to conduct EXAFS experiments in house using the Bremsstrahlung radiation from a rotating anode X-ray source, it is preferable to use synchrotron radiation hecause of its broad, smooth spectral range and high intensity (10).Our endeavors in EXAFS have been carried out a t Beam Line X18B at the National Synchrotron Light Source a t Brookhaven National Laboratory (Upton, NY). The basic experimental setup for an EXAFS experiment is illustrated in Figure 3. Polychromatic X-rays (white light) from the synchrotron storage ring (or a rotating anode source) are first energy-selected by a crystal monochromator. Generally, curved crystal monochromators are used with laboratory spectrometers to increase the flux of highly divergent X-rays from the anode source (11, 12); flat crystal monochromators are used at synchrotron facilities because the natural collimation of synchrotron radiation reduces divergence (13). The monoenergetic X-rays (-1 eV fwhm) then pass through a detector that measures their incident intensity, Io. Thereafter, the radiation impinges on the sample, and the intensity of the transmitted (I,)or the fluorescent (Idflux is measured. Ionization chambers are,the most common form of detector, but solid-state and scintillation detectors are also used, particularly for fluorescence detection (14). The X-ray absorption coefficients of a sample of thickness x for transmission and fluorescence modes, rtx and pfx, are given hy rtx = log, (I&) rfx = If&
(la) (Ih)
Figure 4a shows the Co K-edge (1s) transmission spectrum, and Figure 4h shows the fluorescence spectrum of cos04. The transmission spectrum is superimposed on a background that decreases with increasing incident energy, as described by the Victoreen for1228A
* ANALYTICAL
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Figure 4. Cobalt K-edge X-ray absorption spectra. la) Tran8mIssi.m measurememof &SO,. (b) flwre6CBnCBmeeSurement of Cos04,(c) exbacled x(k) from transmlsslon measurement of Cos0,. and Id) extracted x(k) from fluoressencemeawement of COA
mula (15). The fluorescence spectrum shows an increase in signal (and hackground) primarily because of the increasing penetration depth of the Xrays with increasing incident energy, which produces a concomitant increase in X-ray fluorescence.Variations in detector response and the use of filters (16) (in fluorescence detection) can also influence spectral backgrounds. Figures 4c and 4d show the corresponding x(k)’s obtained for Co304to illustrate that essentially identical results are obtained from transmission (Figure 4c) and fluorescence (Figure 4d) measurements under ideal conditions. This is important, because fluorescence-derived EXAFS data are critical for the analysis of dilute materials, especially when transmission data deteriorate because of high residual ahsorption by the sample matrix. EXAFS measurements can be ob-
CHEMISTRY, VOL. 60. NO. 21, NOVEMBER i,1988
tained under ambient conditions because the high-energy X-rays used to examine most elements pass through air unattenuated. When low-energy Xrays are used to measure the EXAFS of lighter elements, the loss of intensity from ahsorDtion hv air becomes sienificant, and experiments must be carried out under vacuum or in a low-ahsorhing gas such as helium. It is also common (and preferable) to measure spectra a t a low temperature to reduce atomic vibrations that tend to reduce the EXAFS amplitude. Surface EXAFS measurements also have been made by measuring the yield of secondary and Auger electrons (17,18). This requires that the experiment be performed under high vacuum. Theory of EXAFS The “EXAFS spectrum”: x(E). The theoretical description of EXAFS con-
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siders only the oscillatory portion of the absorption spectrum attributable to the backscattering process. This normalized fractional portion of the absorption spectrum, termed x ( E ) (Figure 2b), is defined as
x ( E ) = M E ) - i@)I/fio(E)
(2)
where @ ( E )is the measured absorption at a given X-ray wavelength (Figure 2a) and po(E) is the “smooth” absorption that would be observed a t that same wavelength if no EXAFS structure were present (an isolated atom). Except in the case of a monatomic gas, po(E) cannot be measured experimentally because of the physical impossibility of isolating the sample atoms in space; therefore,po(E) is approximated by a “smooth line,” which is determined numerically. I t is convenient to convert x ( E ) from the energy scale (eV) to the photoelectron wave vector or k scale (A-’); k is related to the photoelectron kinetic energy, E = (Ehv - Eo), by k = [(8r2m/h2)E]’”
(3)
where m is the rest mass of the electron and h is Planck’s constant. For E in eV, k 2 0.51(E)’”A-‘. The conversion of %(E)to x ( k ) using Equation 3 requires that Eo be known. This is a slight problem; usually it is impossihle to know exactly where EOoccurs in the absorption spectrum. Fortunately, a small error ie not catastrophic, and an arhitrary choice of EOnear the edge is adequate because changes can be accommodated during data analysis. The EXAFS equation The “EXAFS equation” is a theoretical expression that describes x ( k ) in terms of structural parameters and thereby allows structural information to be derived from the experimental x ( k ) . Strictly speaking, the following form of the EXAFS equation is valid only for K edges; however, it adequately describes Lm edges within experimental error (19). The general equation fof describing the weighted x ( k )is shown in Equation 4 (see box). This equation is in essence
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ANALYTICAL CHEMISTRY, VOL.
Flgure 5. First coordination shell of Co m x(M and A(M; (b) h%hl and PAW;(cl o s o i I ~ l wpart, ain[Zkr, + #g(k)]; and (d) amplitude terms (S,(k)
(8)
= 1.0, N = amstam): h o k a c a m n i n g a t W i i , F/(W Debye-Waiier amtibution, exp(-Zok21 ine!astic M mean mX, pem mnbibdon. ey, [-2r/lukll: l/kdependence in F i ~ v 5a; e and k2 depenjenoe in F l a w 5b.
BO, NO. 21. NOVEMBER 1, 1988
the sum of modulated sine waves of varying frequency (from each backscattering coordination shell). By combining the numerous terms in Equation 4, one can more clearly see the general form of the EXAFS equation.
where A,(k) is the total backscattering amplitude of the j t h coordination shell and *,J(k) is the corresponding total phase. The general behavior of the terms in Equations 4 and 5 are illustrated in Figure 5. Figures 5a and 5b show the unweighted and k3-weighted x ( k ) from the first coordination shell in Co metal ( R = 2.50 A, N = 12) with their total backscattering amplitudes A,(k) and k3AJ(k)as dashed lines. Figure 5c shows the sinusoidal part of x ( k ) , and Figure 5d illustrates the general behavior of the various terms that describe A,(k) (S,(k) = 1.0 and N, = constant). No attempt is made to show the absolute magnitude of the individual curves in Figure 5d because we wish only to highlight their respective shapes. The k* factor in Figure 5d represents the kn-1 factor in Equations 4 and 5, which reflects the intrinsic Ilk dependence of x ( k ) . Therefore, the k-I and k2 terms represent the kn-1 weighting factors for n = 0 (no Weighting) and n = 3. The kn-I weighting factor roughly compensates for the combined attenuation effects of AJ(k)at high k and helps to prevent the first-shell backscatterers from dominating higher order ones
leads to a phase change given by ( 2 r X 2rjIAd. which reduces to Zkr, ( k = 2r/
u. The second contribution to the total
phase, @ij(k),arises from interaction of the backscattering photoelectron with the potentials of the scattering pair. This results in additional phase changes as the photoelectron wave is emitted from A, backscattered off B, and when it finally arrives back a t the absorber atom A. Although &(k) generally accounts for only 10%of the total phase, it is still necessary that &j(k) be known before the nearest neighbor distance can be determined with an accuracy better than f0.1 A (29, 30). An experimental value of $ij(k) can be determined from standards with absorberlscatterer pairs similar to the material of interest, or theoretically calculated $vj(k)'s can be used (19,31). The experimental or theoretical &j(k) is subtracted from the total phase of the unknown to obtain accurate nearest neighbor distances from the 2krj
portion of *ij(k). The shift of the peaks in the PRDF away from their true radial values (see above) is the result of the influence of the additional &,(k) phase term to 2krj. Amplitude Backscattering amplitude: Fj(k). The backscattering amplitude function, Fj(k), is purely a function of the scatterer atom type and not the absorber atom. Figure 6 shows theoretical backscattering amplitudes calculated by Teo and Lee (19)for Rh, Co, S, and 0. Several distinct trends can be observed in the behavior of Fj(k) as a function of atomic number (Z). The amplitude of Fj(k) a t high k generally increases with increasing atomic number, and the position of the amplitude maximum moves to higher k as the atomic number of the backscatterer increases. It can be seen in Figure 5 that the overall shape of the EXAFS envelope is controlled by the shape of Fj(k),partic-
(19).
The full EXAFS equation (Equation 4) is based on a short-range, single-scattering theory developed in the early 1970s by Sayers, Stem, and Lytle (20, 21). Thia was followed by several other important theoretical contributions (22-25). The succeea of this formulation for providing accurate structural information from known materials (26-28) indicated the wtential of EXAFS for structural characterization and spurred its rapid growth. The EXAFS theory will be examined in terms of the individual components in Equation 4. phrwe
The total phase (see Equations 4 and 5) of the backscattered photoelectron wave at the central atom, U,,(k),is considered in two parts. The first part arises as a result of the photoelectron wave traveling a distance 2rJ from the absorber to the backscatterer and then back to the absorber atom. The number of wavelengths to which this distance corresponds is Zr,/X,, where & is the photoelectron wavelength corresponding to its kinetic energy, E. This
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ularly when the k" weighting scheme is employed. This influence of Fj(k) on the characteristic shape of the total backscattering amplitude (Aj(k)) can be used as a qualitative aid in identifying different types of nearest neighbor atoms (29,32,33). However, this method is limited to backscatterers with significantly different atomic numbers (e.& different rows of the periodic table) because of the modifying nature of the remaining terms in Aj(k) and errors induced by signal to noise. In addition to the elements shown in Figure 6, backscattering amplitudes have been calculated for elements from C to Pb. These calculated backscattering amplitudes (and @ij(k)'s)are available in tabulated (19) and parameterized (34) forms and are widely used in EXAFS data analysis. A new set of theoretical backscattering amplitudes and phases was recently calculated using a curved-wave formalism in an attempt m to improve the accuracy of the theoretical function (35).The previous calculations employed the plane-wave approximation (19). The remaining terms. The other terms that constitute Aj(k) account for the various physical effects that attenuate the backscattering amplitude, Fj(k).The reduction factor in Equation 4, Si(k), accounts for losses in Fj(k) caused by multiple excitation of the absorber from processes such as shake-up and shake-off. Studies on the effect of multiple excitations have shown that they can reduce Fj(k) by as much as a factor of two (36).Si(k) is difficult to estimate, and it is generally ignored when standard compounds similar to the experimental unknown are used as references for analysis. This assumes that Si(k) is similar for both the standard and the unknown and that they effectively cancel in the data analysis. FILTRATION Inelastic scattering of the photoelecInline and leetyp0 filterst0 pmtect tron also results in a decrease in A;(k). instrumen16 by removing hard particle This effect is approximated by exp wntaminatkmfromIluidiiws ( - 2 r j / A ( k ) ) , where A ( k ) is the mchoiceolsintersdandwire mesh photoelectron mean free path. Because slement~fmm0.5to440microns A(k) increases as k increases, attenuacompact designs tion is greater at low k (37)(see Figure mail meld construction 5d). Again, it is difficult to accurately approximate the effect of inelastic ~%
