Fourier transform distance distributions

The Fourier transform of the EXAFS of Figure 5 is shown in Figure 6 as the solid curve It has two large peaks at 2.38 and 2.78 A as well as two small ones at 4.04 and 4.77 A. In this example, each peak is due to Mo—Mo backscattering. The peak positions are in excellent correspondence with the crystallographically determined radial distribution for molybdenum metal foil (bcc)— with Mo—Mo interatomic distances of2.725, 3.147, 4.450, and 5.218 A, respectively. The Fourier transform peaks are phase shifted by -0.39 A from the true distances. 

Where, /(k) is the sum over N back-scattering atoms i, where fi is the scattering amplitude term characteristic of the atom, cT is the Debye-Waller factor associated with the vibration of the atoms, r is the distance from the absorbing atom, X is the mean free path of the photoelectron, and is the phase shift of the spherical wave as it scatters from the back-scattering atoms. By talcing the Fourier transform of the amplitude of the fine structure (that is, X( )> real-space radial distribution function of the back-scattering atoms around the absorbing atom is produced. 

Surface forces measurement is a unique tool for surface characterization. It can directly monitor the distance (D) dependence of surface properties, which is difficult to obtain by other techniques. One of the simplest examples is the case of the electric double-layer force. The repulsion observed between charged surfaces describes the counterion distribution in the vicinity of surfaces and is known as the electric double-layer force (repulsion). In a similar manner, we should be able to study various, more complex surface phenomena and obtain new insight into them. Indeed, based on observation by surface forces measurement and Fourier transform infrared (FTIR) spectroscopy, we have found the formation of a novel molecular architecture, an alcohol macrocluster, at the solid-liquid interface. 

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.)...

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 ... 

With this equation in place we now may evaluate the statistical average for any distance k= h-j along the chain, where h and j denote the position of chain segments. Fourier transforming r n ) (Eq. 3.3) we obtain a probability distribution in Fourier space ... 

At the beginning the interatomic distances were determined by measuring the positions of the maxima and minima on the interference pattern. Soon, however, a more direct method was proposed by Pauling and Brockway for determining the interatomic distances. They obtained a so-called radial distribution by Fourier-transforming the estimated intensity data. The radial distribution is related to the probability distribution of interatomic distances. The position of maximum on the radial distribution gives the interatomic distance, while its halfwidth provides information on the associated vibrational amplitude. 

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 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. 

Equation (D.2) shows Fourier transform of the pair distance distribution W (rn) for a path with its one end at r = 0 (the root) and the other at r (n-th generation). The particle scattering factor... 

X-ray scattering from coal was the subject of several early studies which led to the postulation that coal contains aromatic layers about 20 to 30 A in diameter, aligned parallel to near-neighbors at distances of about 3.5 A (Hirsch, 1954). Small-angle x-ray scattering, which permits characterization of the open and closed porosity of coal, has shown a wide size distribution and the radius of gyration appears to be insufficient to describe the pore size. Application of the Fourier transform technique indicated that some coals have a mesoporosity with a mean radius of 80 to 100 A (Guet, 1990). 

For a uniform charge distribution within a spherical atom the Fourier transform of the density has been shown (equation 5.6) to be of the form sin a/a, for a wave of phase a in momentum space. From the Bragg equation (Figure 2.9), A = 2dsin0, it follows that electrons at a distance d = A/2sin0 apart, scatter in phase, i.e. with phase difference 27T. At a separation r the relative phase shift a, is given by ... 

Take the position of one atom as origin and denote the atomic density, in a given direction, at a distance r from the centre by the number of atoms between r and dr, p(r). The volume of the corresponding spherical shell is 47rr2dr and the number of atoms in the shell, 47rr2p(r)dr, defines the radial distribution function. The intensity of scattered radiation from the sample is given by the Fourier transform... 

The EXAFS function becomes understandable if we look at the Fourier Transform of %(k), which resembles a radial distribution function (the probability of finding an atom at a distance r from the absorbing atom) ... 

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. 

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