EXAFS spectra Fourier transforms

Figure 6.14 EXAFS and Fourier transform of rhodium metal, showing a) the measured EXAFS spectrum, b) the uncorrected Fourier Transform according to equation (6-10), c) the first Rh-Rh shell contribution being the inverse of the main peak in the Fourier Transform, and d) the phase- and amplitude-corrected Fourier Transform according to (6-11). The Fourier transform is a complex function, and hence the transforms give the magnitude of the transform (the positive and the negative curve are equivalent) as well as the imaginary part, which oscillates between the magnitude curves (from Martens (361).

Fig. 10.17. EXAFS and Fourier transforms of rhodium metal (a) measured EXAFS spectrum, (b) the uncorrected Fourier transform, (c) the inverse Fourier transform of the main contribution in (b), and (d) the phase and amplitude corrected Fourier transform [38],...

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 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.)...

Fig. 10 a Co K-edge XAS spectrum for CoP collected in transmission mode, showing the approximate regions where XANES and EXAFS features are observed and the assignment of dipolar and quadrupolar transitions, b EXAFS (x) vs. k curve, c Fourier transform of EXAFS... 

Fig. 16. Effect of soaking TS-1 with water on the XANES (a) and UV-Raman (b) spectra dried TS-1 (solid line) soaked TS-1 (dotted line). The inset in part (a) reports the U-weighted, phase-uncorrected Fourier transforms of the corresponding EXAFS spectrum [Reprinted from Ricchiardi et al. (41) with permission. Copyright (2001) American Chemical Society].

Figure 2. a) X-ray absorption spectrum near the Mo K-edge of the Co/Mo = 0.125 unsupported Co-Mo catalyst recorded in situ at room temperature b) normalized Mo EXAFS spectrum c) absolute magnitude of the Fourier transform d) fit of the first shell e) fit of the second shell. The solid line in d) and e) is the filtered EXAFS, and the dashed line is the least squares fit. 

The right panel of Figure 1.3 displays the radial function obtained by Fourier transformation of the -weighed background-subtracted EXAFS data from the solid heated to 420°C [31], This spectrum shows two major peaks, one at about 1.5 A associated with backscattering from O neighbors, and a second at 3 A related to the Nb-Mo pairs. The measured distances are consistent with a combination of niobium oxo species and heteropolymolybdate fragments, presumably the catalytically active phase. 

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... 

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

Figure 7. Fourier transform of the background-subtracted EXAFS spectrum for Cr3 53-montmorillonite.

Figure 6 (A) k1 weighted raw EXAFS spectra of Pt/K-Al203 after reduction at 673 K (solid line), after evacuation at 323 K (dotted line) and after evacuation at 473 K (dashed line). (B) Fourier transform (k2, 2.5 < k < 14 A 1) of the spectrum taken after reduction at 673 K (solid line) and best R-space fit (1.6 < R < 3.2 A, dashed line).

Fig. 10.18. Magnitude of the Fourier Transform of the Mo K edge EXAFS spectrum of carbon-supported, sulphided Co-Mo and Mo catalysts, along with EXAFS parameters (from...

From equation 5, it is apparent that each shell of scatterers will contribute a different frequency of oscillation to the overall EXAFS spectrum. A common method used to visualize these contributions is to calculate the Fourier transform (FT) of the EXAFS spectrum. The FT is a pseudoradial-distribution function of electron density around the absorber. Because of the phase shift [< ( )], all of the peaks in the FT are shifted, typically by ca. —0.4 A, from their true distances. The back-scattering amplitude, Debye-Waller factor, and mean free-path terms make it impossible to correlate the FT amplitude directly with coordination number. Finally, the limited k range of the data gives rise to so-called truncation ripples, which are spurious peaks appearing on the wings of the true peaks. For these reasons, FTs are never used for quantitative analysis of EXAFS spectra. They are useful, however, for visualizing the major components of an EXAFS spectrum. 

Figure 10. Fourier transforms of the EXAFS spectra for the OEC. Solid line indicates control, dark-adapted Sj spectrum dotted-dashed indicates 100 ilM NH2OH for 3, quenched by dilution and dashed line indicates 200 IJ.M H2Q, 3O quenched withferricyanide. Fourier transforms calculated over the range k = 3-11.5 A 1 using k3 weighted data.

In Figure 1 (a,b) the Fourier-Transformed (FT) EXAFS spectra of the Co/C and the Co-Mo/C catalyst together with the Co Sg reference compound are plotted. The absolute FT spectrum of Co Sg (Fig. 1 (a)) exhibits two peaks. The first peak is attributed to combined Co-S and Co-Co coordinations, the second one only to a Co-Co coordination (denoted Co-Co(2), to differentiate it from the Co-Co(1) coordination in the first peak). In bulk Co Sg 8/9 of the cobalt atoms are tetrahedrally coordinated and 1/9 are octahedrally coordinated by sulfurs. The Co-S coordination distances are in the range 2.13-2.39 A. The Co-Co coordination distance of Co-Co(1) is 2.50 A, for Co-Co(2) the distance is 3.51 A. From Figure 1 (a) it is apparent that in the catalyst spectra the first peak is shifted to lower r-values compared to that in Co Sg. This shift is larger for Co-Mo/C than for Co/C. It is furthermore clear that a Co-Co(2) coordination is also present in the Co/C catalyst, but not in the promoted catalyst. On the other hand, the latter catalyst shows an additional peak which is not present in COgSg and Co/C and, consequently, might be ascribed to Mo backscatterers. 

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