EXAFS amplitude functions

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-... 

In the present study we have used the phase and amplitude functions of absorber-scatterer pairs in known model compounds to fit the EXAFS of the catalysts. By use of Fourier filtering, the contribution from a single coordination shell is isolated and the resulting filtered EXAFS is then non-linear least squares fitted as described in Ref. (19, 20). 

In order to obtain data with reduced temperature smearing, experiments were also carried out at 77 K. However, such experiments could not be carried out in. situ and the catalysts were thus exposed to air before the measurements. EXAFS data of three catalysts with Co/Mo atomic ratios of 0.0., 0.25, and 0.50 were obtained. The results show many similarities with the data recorded in situ and were fitted in a similar fashion using phase and amplitude functions of the well-crystallized model compound M0S2 recorded at 77 K. The results, which are given in Table III, show that the bond lengths for the first and second coordination shell are the same for all the catalysts and identical to the values obtained for the catalyst recorded in situ (Table II). The coordination numbers for both shells appear, however, to be somewhat smaller. Although coordination numbers determined by EXAFS cannot be expected to be determined with an accuracy better than + 20, the observed reduction... 

The Co/Mo = 0.125 catalyst has all the cobalt atoms present as Co-Mo-S and, therefore, the EXAFS studies of this catalyst can give information about the molybdenum atoms in the Co-Mo-S structure. The Fourier transform (Figure 2c) of the Mo EXAFS of the above catalyst shows the presence of two distinct backscatterer peaks. A fit of the Fourier filtered EXAFS data using the phase and amplitude functions obtained for well-crystallized MoS2 shows (Table II) that the Mo-S and Mo-Mo bond lengths in the catalyst are identical (within 0.01 A) to those present in MoS2 (R =... 

The first stage in EXAFS data analysis is the removal of the background absorption and the extraction of the EXAFS and its normalization to that for a unit metal atom. An inadequate removal of the background can result in a number of deficiencies in the EXAFS data, which may lead to inaccurate determination of the amplitude function and/or distortions of the low R contribution. The EXAFS may be plotted as a function of k. Often it is convenient to weight the data by... 

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 back-scattering amplitude, the intensity may not be correct. Third, coordination numbers of distant shells will be too low mainly because of the term 1/r in the amplitude (10.10) and also because of the small inelastic mean free path of the photoelectron. The appropriate corrections can be made, however, when phase shift and amplitude functions are derived from reference samples or from theoretical calculations. Figure 11.17 illustrates the effect of phase and amplitude correction on the EXAFS of a Rh foil [38]. Note that unless the sample is that of a single element, N is a fractional coordination number, i.e. the product of the real coordination number and the concentration of the element involved. Also, the EXAFS information is an average over the entire sample. As a consequence, meaningful data on supported catalysts are only obtained when the particles have a monodisperse size distribution. 

The EXAFS analysis followed standard procedures as described in ref. [11]. The weighted spectra were Fourier transformed within the limits k=4 to k=16. EXAFS of the first Rh-Rh shell was fitted using phase shift and amplitude functions obtained from a Rh foil under the assumption... 

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. 

The amplitude functions in Figure 4.27 were determined from EXAFS data on pure metallic platinum and iridium reference materials (48). The amplitude functions are modified by multiplication by the factor (A 2,/A/,), where and A/, are the nearest neighbor distance and coordination number, respectively. Both the shape and magnitude of the amplitude functions are very similar. 

Figure 4.27 Modified amplitude functions (for the first coordination shell of atoms) for the EXAFS associated with the L (ll absorption edges of pure metallic platinum and iridium at 100°K (48). (Reprinted with permission from the American Institute of Physics.)...

Because of the similarity rn the amplitude functions of platinum and iridium, we do not separate the backscattering contributions of platinum and iridium atoms in the analysis of EXAFS data on platinum-iridium clusters or alloys. In our quantitative treatment of EXAFS arising from nearest neighbor atoms of platinum and iridium, the EXAFS function of Eq. 4.3 consists of only one term, as will be seen in the following discussion. 

Returning to our discussion of the contribution of the function /,(A0 to the extended fine structure associated with the platinum Z,M edge of a platinum-iridium catalyst, we generate an EXAFS function for the iridium in the catalyst using the amplitude function AR(Klr) for an iridium reference material, either pure iridium or an iridium reference catalyst (48). The superscript R signifies that the amplitude function is for the reference material while the subscript Ir on K denotes that the wave vector is defined for iridium EXAFS. 

The quantity D is a scaling factor related to coordination numbers, and Adifference between the value of cr2 characteristic of the platinum L m EXAFS of the platinum-iridium catalyst and the corresponding value of cr2 for a platinum reference material (48). The reference material is pure metallic platinum or a platinum catalyst. The quantity A (K) is the amplitude function of the platinum reference material. It is emphasized that the parameter A different from the analogous parameter in Eq. 4.14. Also, the values of the interatomic distance R in Eqs. 4.14 and 4.17 will not in general be equivalent. 

Figure 23 shows a contour map of St as a function of / and jurt. For a thick sample, even a small fraction of beam skimming by can affect the EXAFS amplitude. Similarly, if there are harmonics in the beam, they go through the sample with less absorption than the fundamental, leading to the same effect as if there were a hole (Stern and Kim 1981). While the sample may not have holes in it, it is common to encounter particles smaller than the beam. Suppose one is dealing with a primary mineral such as magnetite ( 04),... 

Figure 23. Reduction of the EXAFS amplitude as a function of the sample s hole fraction and the absorption edge jump. Percentages correspond to the ratio of the amplitude of the measured EXAFS wiggle to the actual signal.

The theoretical form of the EXAFS as described by Eq. (11) is a sum of damped sinusoidal functions, with frequencies related to the distance of the absorber atom from the backscattering atoms, and an amplitude function which contains information about the number of backscatterers at that distance. This structural information can be best extracted by the Fourier transform technique, which converts data from k or momentum space into R or distance space. The following Fourier transformation of... 

In both backtransformed EXAFS functions, it can be seen that the envelope of the EXAFS does not drop continuously with k, as was observed with backscat-terers of small atomic number (as oxygen, see Fig.4j, k). Here, we have elements with higher atomic numbers as backscatterers, where Ramsauer-Townsend resonances introduce non-monotonic behavior into the backscattering amplitude functions (see Sect 3.1). 

---