EXAFS radial distribution function

FIGURE 2.33 Co edge EXAFS radial distribution function for Co (CO) 4. 

Figure 17 Ni(100)-O2. EXAFS radial distribution function (Adapted from Japan J. Appl. Phys., 1978, 17 (Suppl. 2), 217 and/. Vac. Sci. Tech., 1979, 16, 37)...

Several qualitative conclusions are readily accessible from the EXAFS radial distribution functions gathered in Fig. 28. 

Figure 4.1-11 The EXAFS data and pseudo-radial distribution functions of Co(ll) in (a) basic and (b) acidic chloroaluminate ionic liquid. Reproduced from reference 46 with permission.

Figure 4.1-13 Comparison of the experimental without (—) and with (—) triphenylphosphine at (solid line) and fitted (dashed line) (a) EXAFS 80 °C and in the presence of triphenylphosphine and (b) pseudo-radial distribution functions and reagents at 50 °C for 20 min (—). Repro-...

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.)...

Examination of the EXAFS formulation in wave vector form reveals that it consists of a sum of sinusoids with phase and amplitude. Sayers et al32 were the first to recognize the fact that a Fourier transform of the EXAFS from wave vector space (k or direct space) to frequency space (r) yields a function that is qualitatively similar to a radial distribution function and is given by ... 

Data reduction of EXAFS spectra was performed using WinXAS [14], The normalized spectra were analyzed over the Arrange of 2.5 to 10 A1. A square-weighted degree 7 spline was used to remove the background of the x(k) function. Finally, the data in -space were converted to R-space using a Bessel window to obtain the radial distribution function. 

The EXAFS function becomes understandable if we look at the Fourier transform of x(k), which resembles a radial distribution function ... 

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 ... 

The presence of local cation ordering in Mg2Ga and MgsGa - CO3 LDHs noted in Sect. 3.3.1 has been confirmed by means of both EXAFS and by calculation of the electron radial distribution function from the Fourier transform of the diffracted X-ray intensity. In each case the gallium was found to have six magnesium ions and no galhum ions as next-nearest neighbors [39]. 

Fig. 2.3 Basic structural units and Fe-Fe distances (in nm) for hematite, goethite, akaganeite and lepidocrocite and their associated radial distribution functions as obtained from EXAFS spectra. The first peak in the radial distribution...

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 2.24 EXAFS data for (a) (NH4 )2MnBr4 (upper) extracted EXAFS data (lower) the radial distribution function, solid line experimental, dotted line calculated. 

Since the Fourier transformation of equation (2.2) yields only a radial distribution function about the absorber, we note that information obtained from EXAFS is limited to an average, one-dimensional representation of structure. Furthermore, in order that the transform be comparatively free of ripples, the data should extend to at least... 

In principle, EXAFS information may be obtained for most or all of the elements in a catalyst. Thus, for multicomponent samples, the characterization of local surroundings for all (or almost all) the elements may be obtained. However, we stress that the radial distribution function cannot be transformed into a unique three-dimensional structure. Therefore, the EXAFS technique is not ideal for providing such information and the data representing materials consisting of several different phases may often be too difficult to analyze meaningfully. 

Figure 1 shows Fourier transforms of EXAFS spectra of a few samples prepared. The radial distribution functions of these samples are different from that of nickel oxide or cobalt oxide [7]. All the Fourier transforms showed two peaks at similar distances (phase uncorrected) the peak between 1 and 2 A is ascribed to the M-0 bond (M divalent cation) and the peak between 2 and 3 A is ascribed to the M-O-M and M-O-Si bonds. The similar radial distribution functions in Figure 1 indicate that the local structures of X-ray absorbing atoms (Ni, Co, and Zn) are similar. No other bonds derived from metal oxides (nickel, cobalt and zinc oxides) were observed in the EXAFS Fourier transforms of the samples calcined at 873 K, which suggests that the divalent cations are incorporated in the octahedral lattice. 

The EXAFS spectroscopy results strongly confirm the existence of local order in the mineral part of lead isooctane reverse micelles in dodecane and reveal quantitative information concerning the first and second coordination shells. The radial distribution functions (RDFs) exhibit peaks at around 0.19 nm, corresponding to the first shell of oxygen atoms and at around 0.35 nm corresponding to the shell of lead. Analytical transmission electron microscopy (ATEM) indicates the size of the mineral core of the micelles (1-1.5 nm) and the discoid shape of the particles when the micelles aggregate (Mansot et al. 1994). 

This mechanism is in agreement with the mechanism proposed by others (Belin et al., 1989 and 1995 Martin et al., 1986a, Willermet et al., 1992) using extended X-ray absorption fine structure spectroscopy (EXAFS) and infrared spectroscopy. When ZDDP is present in the lubricant formulation, the radial distribution function (RDF) indicates that crystalline iron oxide diffuses into the polyphosphate network material. 

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