Phase-shift functions

The discrepancy between the two values is a result of the phase-shift function, which is included in Equation 7.3. Phase shifts are dependent on the absorber atom and can be calculated by ah initio methods. The model building for phase-shift files include a model of the most abundant atomic neighbor for a specific atom. 

Phase shifts determined in this manner for OsOs and CuCu were employed to be consistent with the use of adjusted theoretical phase shifts for the osmium-copper atomic pair, which will be considered subsequently in the analysis of data on silica-supported osmium-copper clusters. For the osmium-copper pair, two phase-shift functions are necessary, depending on which of the atoms is the absorber atom and which is the backscattering atom. The two situations are distinguished by using the designation OsCu for the... 

Figure 4.14 Phase shift functions for the various possible atomic pairs in the osmium-copper system (32). (For the osmium-copper pair, the phase shift function is designated OsCu when Os is the absorber atom and CuOs when the absorber atom is Cu.) (Reprinted with permission from the American Institute of Physics.)...

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. 

The values of the osmium-osmium and copper-copper distances were insensitive to the phase shift functions employed for CuOs and OsCu over the range of A values for the CuOs phase shift adjustment parameter shown in Figure 4.15. The ranges of variation found for the osmium-osmium and copper-copper distances were, respectively, only 0.014 A (2.674 to 2.688 A) and 0.005 A (2.549 to 2.554 A). In both cases, but especially for the osmium-osmium pair, the distances appear to be smaller than the corresponding distances in metallic osmium and copper, which are 2.705 and 2.556 A, respectively. As indicated earlier, the value of 2.705 A for metallic osmium is the average of the interatomic distance (2.735 A) in a hexagonal layer and the distance of closest approach (2.675 A) between two atoms in adjacent... 

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. 

In Figure 4.28 phase shift functions are shown for the various possible combinations of absorber and backscattering atoms in the platinum-iridium system (48). For the platinum-iridium pair there are two functions, since there are two different combinations of absorber and backscattering atoms. The two functions are distinguished by using the designation Ptlr when Pt is the absorber atom and IrPt when Ir is the absorber atom. 

The phase shift functions for PtPt and Irlr were determined from data on... 

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 partial wave sum is now reduced to a sum over few pole contributions in the complex plane of /.. The contribution of a single pole to the phase shift function and the deflection function can be obtained from the parameterization (55). Fig. 10 illustrates the result. (/) is essentially a pulse centred at / = Re (Xp — ) with the depth 2/Im Xp and the width 2 Im Xp. Now one proceeds as follows. Starting with N poles, which are placed on a small circle centred at 7.p in the complex /-plane, the number of these poles (N) and the real and imaginary part of the central pole (/p) are derived from semiclassical quantities. The rainbow angle is given by 9r = 2N/lm Xp,... 

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-... 

The experimental way to derive the backscattering ampHtude and the phase-shift functions is very intuitive. Its appHcation is described in some detail in Sect. 4. 

In this section we describe the process of data analysis using the intuitive method of determining the backscattering ampHtude and the phase-shift functions experimentally and then to use these functions to determine the structural parameters of the sample under investigation. Differences to data analysis using theoretical functions will be explained. 

ZnGa204 was chosen as a standard substance to determine experimental backscattering ampHtude and phase-shift functions. In this compound with the normal spinel structure, galHum is coordinated octahedraUy by oxygen atoms at a Ga-0 distance of 1.99 A. 

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]...

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