EXAFS phase-shift functions

For the osmium EXAFS, the first term in Eq. 4.10 represents the contribution of osmium backscattering atoms. In this term, the quantity /V, represents the number of nearest neighbor osmium atoms about an osmium absorber atom and R, represents the distance between the osmium atoms. The phase shift function 28, (/0 is that for an OsOs atomic pair. The quantity f,(/0 exp(—2/C 2tr,2) differs from the analogous quantity for pure metallic osmium by a factor exp(—2X 2Ao-,2), where Ao-,2 is the difference between the value of o-,2 for the OsOs pair in the osmium-copper catalyst and the value for the same pair in the pure metallic osmium. Note that the quantity F,(K) exp( —2/C2cr,2) for the pure metallic osmium is known from the analysis of EXAFS data on it, as indicated earlier. 

The approach adopted amounts to a trial and error procedure in which a series of values is chosen for OsCu and CuOs subject to the constraint of Eq. 4.12. For each set of trial phase shift functions, Eqs. 4.10 and 4.11 for the function Xi(XT, incorporating expressions of the form of Eq. 4.9 for the various x/MO terms, are fit to the corresponding functions derived from the osmium and copper EXAFS data on the osmium-copper catalyst. The fitting exercise yields values of various structural parameters, including the distance between an osmium atom and a copper atom (nearest neighbor atoms). For a given set of phase shift functions for OsCu and CuOs, limited only by the constraint of Eq. 4.12, this distance as derived from the osmium EXAFS will not in general be equal to the distance derived from the copper EXAFS. 

We adopt the additional criterion that the distance between nearest neighbor atoms of osmium and copper must have the same value when derived from either the osmium or copper EXAFS. The phase shift functions for OsCu and CuOs which yield this result are then taken as the correct pair. The functions which are shown for OsCu and CuOs in Figure 4.14 were determined in this manner. 

The distance values, as expected, are sensitive to the phase shift functions employed, and are different for the osmium and copper EXAFS, except for the set of phase shift functions corresponding to the point of intersection of the lines. The latter are the functions shown for OsCu and CuOs in Figure 4.14 and are characterized by a CuOs phase shift adjustment parameter A 0 approximately equal to —4 eV. The corresponding OsCu phase shift adjustment parameter A is approximately — 18 eV. 

It is interesting to note that the value of the 0 adjustment for CuOs is very close to the E0 adjustment ( — 3.3 eV) for OsOs required in the use of the Teo and Lee phase shift functions to fit the EXAFS results on pure metallic osmium. Similarly, the value of the E0 adjustment for OsCu is very close to the adjustment (—20.1 eV) for CuCu required to fit the EXAFS results on pure metallic copper. Thus, for the system of interest here, it appears that the adjustments to the theoretical phase shift functions are concerned primarily with the backscattering atom. 

In Figures 4.16 and 4.17 the uppermost fields (labeled a) illustrate the quality of fit of values of the function KnX](K), represented by the points, to the corresponding function (solid line) derived from the EXAFS data (32). The points were calculated for values of structural parameters corresponding to Af o = —4 eV in Figure 4.15. For the osmium EXAFS in Figure 4.16 the function fitted was K2x K), while for the copper EXAFS in Figure 4.17 it was K3x U0- The fits are excellent except at very low K values. The fits can be improved at the very low K values by modification of the details of the phase shift functions, but there is very little effect of such a modification on the values of the structural parameters obtained. 

We begin by considering the iridium EXAFS of a reference material such as metallic iridium or a catalyst containing pure iridium clusters. An EXAFS function for the iridium in the platinum-iridium catalyst is then generated from the function for the reference material by introducing adjustments for differences in interatomic distances, amplitude functions, and phase shifts. In making such adjustments, we are aided by the fact that the amplitude functions and phase shift functions of platinum are not very different from those of iridium, as shown in Figures 4.27 and 4.28. 

To obtain structural information on platinum-iridium clusters from EXAFS data, we concentrate primarily on the determination of interatomic distances. To obtain accurate values of interatomic distances, we need to have precise information on phase shifts. In this regard, we are fortunate that the phase shift functions of platinum and iridium are not very different. 

Similar considerations apply to the phase shifts of interest in the analysis of the iridium ZH, EXAFS for platinum-iridium catalysts. Therefore, for simplicity, the phase shift functions for PtPt and Irlr are used in the analysis of the EXAFS associated with the platinum and iridium edges, respectively. This simplifying assumption introduces an uncertainty of only about 0.001 A in the interatomic distances derived from the data. 

The experimental approach extracts the amphtude function Fj(A) and the total phase-shift function ij(A) from the spectrum of a standard sample of known structure which should be as similar as possible to the sample under investigation. When Nj and Fj are known, a modified backscattering amphtude function can be derived from the measured EXAFS Xjik) of the standard sample ... 

Similarly, the total phase-shift function 4>ij k) can be extracted. These functions can then be used to analyze the EXAFS of the compound under investigation. If a standard compound is chosen so that its electronic and chemical properties are close to that of the compound under investigation, then the influence of some usually unknown factors, such as, e.g., Sq or Aj(A), and of some simplifying assumptions made during the derivation of the EXAFS formula, such as, e.g., that of plane waves or that of a Gaussian distance distribution, is minimized. Instead of an absolute value of relative value Ao], describing the difference in devi-... 

Fig. 6. Backtransformed CoK edge EXAFS data of the first coordination shell of Co in CoAPO-20 (dashed line) fitted using backscattering amplitude and phase-shift functions determined on cobalt acetate hydrate (solid line). Two different sub-shells of oxygen neighbors are necessary in order to obtain a satisfactory fit. Their individual EXAFS functions are shown by dotted lines [42]...

Data were analysed using the EXAFS analysis program EXAI H [2]. The amplitude and phase shift functions for the Pt-Pt and Mo-O bonds were extracted from EXAFS spectra obtained from Pt foil and K2M0O4, respectively, at 298 K. The corresponding functions for the Pt-O, Pt-Mg and Mo-Mg bonds were theoretically calculated using the FEFF5 program [3]. 

The essence of analyzing an EXAFS spectrum is to recognize all sine contributions in x(k)- The obvious mathematical tool with which to achieve this is Fourier analysis. The argument of each sine contribution in Eq. (8) depends on k (which is known), on r (to be determined), and on the phase shift

characteristic property of the scattering atom in a certain environment, and is best derived from the EXAFS spectrum of a reference compound for which all distances are known. The EXAFS information becomes accessible, if we convert it into a radial distribution function, 0 (r), by means of Fourier transformation ... 

Figure 6.12 Left simulated EXAFS spectrum of a dimer such as Cu2, showing that the EXAFS signal is the product of a sine function and a backscattering amplitude F(k) divided by k, as expressed by (6-8) and (6-9). Note that F(k)/k remains visible as the envelope around the EXAFS signal y(k). Right the Cu EXAFS spectrum of a cluster such as Cu20 is the sum of a Cu-Cu and a Cu-O contribution. Fourier analysis is the mathematical tool to decompose the spectrum into the individual Cu-Cu and Cu-O contributions. Note the different backscattering properties of Cu and O, revealed in the envelope of the individual EXAFS contributions. For simplicity, phase shifts have been ignored in the simulations.

A straightforward Fourier transform of the EXAFS signal does not yield the true radial distribution function. First, the phase shift causes each coordination shell to peak at the incorrect distance second, due to the element-specific backscattering amplitude, the intensity may not be correct. The appropriate corrections can be made, however, when phase shift and amplitude functions are derived from reference samples or from theoretical calculations. The phase- and amplitude-corrected Fourier transform becomes ... 

In order to interpret an EXAFS spectrum quantitatively, the phase shifts for the absorber and backscatterer and the backscattering amplitude function must be known. Empirical phase shifts and amplitude functions can be obtained from studies of known structures which are chemically similar to that under investigati-... 

By Fourier transforming the EXAFS oscillations, a radial structure function is obtained (2U). The peaks in the Fourier transform correspond to the different coordination shells and the position of these peaks gives the absorber-scatterer distances, but shifted to lower values due to the effect of the phase shift. The height of the peaks is related to the coordination number and to thermal (Debye-Waller smearing), as well as static disorder, and for systems, which contain only one kind of atoms at a given distance, the Fourier transform method may give reliable information on the local environment. However, for more accurate determinations of the coordination number N and the bond distance R, a more sophisticated curve-fitting analysis is required. 

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