Fitting EXAFS data Fourier transforms

Fig. 28. EXAFS data [Fourier transform of Rh K-edge k x k) shell, oscillation curve fitting] of (a) [RhjeO),J-NaY, (b) [Rhjj-NaY, and (c) CO + [RhJ-NaY.

Figure 9. Data reduction and data analysis in EXAFS spectroscopy. (A) EXAFS spectrum x(k) versus k after background removal. (B) The solid curve is the weighted EXAFS spectrum k3x(k) versus k (after multiplying (k) by k3). The dashed curve represents an attempt to fit the data with a two-distance model by the curve-fitting (CF) technique. (C) Fourier transformation (FT) of the weighted EXAFS spectrum in momentum (k) space into the radial distribution function p3(r ) versus r in distance space. The dashed curve is the window function used to filter the major peak in Fourier filtering (FF). (D) Fourier-filtered EXAFS spectrum k3x (k) versus k (solid curve) of the major peak in (C) after back-transforming into k space. The dashed curve attempts to fit the filtered data with a single-distance model. (From Ref. 25, with permission.)...

Such a function exhibits peaks (Fig. 9C) that correspond to interatomic distances but are shifted to smaller values (recall the distance correction mentioned above). This finding was a major breakthrough in the analysis of EXAFS data since it allowed ready visualization. However, because of the shift to shorter distances and the effects of truncation, such an approach is generally not employed for accurate distance determination. This approach, however, allows for the use of Fourier filtering techniques which make possible the isolation of individual coordination shells (the dashed line in Fig. 9C represents a Fourier filtering window that isolates the first coordination shell). After Fourier filtering, the data is back-transformed to k space (Fig. 9D), where it is fitted for amplitude and phase. The basic principle behind the curve-fitting analysis is to employ a parameterized function that will model the... 

The data collected are subjected to Fourier transformation yielding a peak at the frequency of each sine wave component in the EXAFS. The sine wave frequencies are proportional to the absorber-scatterer (a-s) distance /7IS. Each peak in the display represents a particular shell of atoms. To answer the question of how many of what kind of atom, one must do curve fitting. This requires a reliance on chemical intuition, experience, and adherence to reasonable chemical bond distances expected for the molecule under study. In practice, two methods are used to determine what the back-scattered EXAFS data for a given system should look like. The first, an empirical method, compares the unknown system to known models the second, a theoretical method, calculates the expected behavior of the a-s pair. The empirical method depends on having information on a suitable model, whereas the theoretical method is dependent on having good wave function descriptions of both absorber and scatterer. 

Fourier transformation of Cu EXAFS data gathered on the Cu(MPG) complex reveals two separate peaks representing shells at distances of 1.9 and 2.3 A. When tested for Ns (coordination number), metal-ligand distance (R as), and Debye-Waller parameter difference (Aa2as) followed by comparison to known model compounds, results show that the presence of both a Cu-(N, O) and Cu-S shell is necessary to obtain an adequate fit to the EXAFS data. Therefore it was concluded that a Cu-S bond is present in the compound. 

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 effects of particle size on the EXAFS region of the XAS spectra are reflected in the coordination numbers obtained in the fits to the EXAFS data. Figure 15 shows the EXAFS or xi ) data and corresponding Fourier transforms for a Pt foil, a Pt02... 

The effect of the applied potential on the XANES region of the XAS spectra for Pt/C catalysts has been briefly introduced above and is related to both the adsorption of H at negative potentials and the formation of the oxide at more positive potentials. The adsorption of H and the formation of oxides are also apparent in the EXAFS and corresponding Fourier transforms, as seen in the work by Herron et al. shown in Figure 15. As the potential is increased from 0.1 to 1.2 V vs SCE, the amplitude of the peak in the Fourier transform at 2.8 A decreases and that at 1.8 A increases. The effect on the EXAFS, (A), data is less easily observed the amplitude of the oscillations at A > 8 A decreases as the potential is increased, with the greatest change seen between 0.8 and 1.0 V. The results of fitting these data are shown in Table 2. Note that a value for the inner potential... 

Figure 27. weighted Pt L3 EXAFS (a and c) and the corresponding Fourier transforms (b and d) for (a and b) a poorly mixed PtRu/C alloy electrode and (c and d) a well mixed PtRu/C alloy electrode at 0.05 V vs RffE in 1 mol dm H2SO4 experimental data (solid line) and fits (dotted line). (Reproduced with permission from ref 87. Copyright 2002 S. Maniguet.)... 

Figure 31. Ru K EXAFS data (insets) and correspmding Fourier transforms for Ru tSej particles on a Sn02 F support in (a) nitrogen and (b) oxygen saturated 0.5 i ol dm H2SO4 experimental data (thin lines) and fits hicK lines).(Reproduced with permission from ref 149. Copyright 2000 Elsevier Sequoia S.A., Lausanne.)...

Fig. 10. Fourier transform of Pt Lj-edge SEXAFS data for 0.8 ML Pt/Si(l 11)7x7 (dots) and of the simulated EXAFS for a Pt atom in the sixfold interstitial site within the top Si(lll) double layer (solid line). The derived chemisorption model is represented in the insets. The first neighbour distance is Pt—Si = 2.48 + 0.03 A, and the Pt coordination number is 6 + 1 Si neighbours. The distances beyond the first neighbour peak are fitted up to the fifth Pt—Si distance, and indicate the deformation of the Si cage, as indicated in the modot, due to the presence of the host Pt atom...

The EXAFS function is obtained from the X-ray absorption spectrum by subtracting the absorption due to the free atom. A Fourier transform of the EXAFS data gives a radial distribution function which shows the distribution of the neighbouring atoms as a function of internuclear distance from the absorbing atom. Shells of neighbours, known as coordination shells, surround the absorbing atom. Finally, the radial distribution function is fitted to a series of trial structural models until a structure which best fits the... 

The second stage of improvement in the EXAFS data is achieved through Fourier filtering (Eccles, 1978). Here, a particular peak in (R), corresponding to the ith scattering shell, is transformed back to k-space and the transformed function is fitted to a parameterized expression for EXAFS (Eccles, 1978 Cramer 1978)... 

Figure 6.17 EXAFS data of a reduced Pt/AEO catalyst. Full lines are measured data dotted lines represent fits. Left magnitude of a -weighted Fourier transform of the range 1,9

---