Phase-shift subtracted Fourier

Figure 4. Magnitudes of the phase-shift subtracted Fourier transform of calcite-rich bituminous coal (A), the Fust seam lignite (B), and calcium acetate (C).

Fig -8. Co XAS results for sorption to a-Al203 (A) background subtracted k3 Co(II) EXAFS spectra as a function of surface coverage, 7, (B) fast Fourier transformed radial structure functions of Co(II) EXAFS, uncorrected for phase shift. Uncorrected peaks at approximately 2600 and 5500 pm in the sorption samples are primarily due to Co-Co second shell and Co-Co fourth-shell interactions, respectively (after Hayes Katz, 1996). 

Figure 4. Fourier transforms (solid curves) [s(t) vs. r, A (before phase shift correction)] of the background-subtracted EXAFS spectra in Figure 3 and Fourier-filtering windows (dashed curves). Key A, cis-Pt(NHs) Cl2 B, Pt(2-A-6-MPR)t C, Pt(6-MPR)2 Z>, Pd(6-MPR)2 E, Pd(2-A-6-MPR)2 and F, Pd(Guo)2Cl2.

Ni K-edge k -weighted, background subtracted, EXAFS spectra (above) and their Fourier transforms (below) of (left) Ni(cyclam)-SAPO STA-6, as made, and (right) calcined to give Ni-SAPO STA-6, right. (The Fourier transform distances are not the atom-atom distances between shells, because of phase shifts in the scattering process.)... 

Fourier transformation of the EXAFS data permits it to be replotted as a function of distance (Fig. 9). This aids visualization of the data, since each peak in the Fourier transform in principle represents a shell of atoms. However, because a phase shift of about 0.4 A [1] appears in the Fourier-transformed data, it is not possible to use transformed data to accurately determine M-L distances. Nonetheless, it is possible to use the transformed data to make an initial guess as to the radial distribution function of atoms surrounding the metal. The Fourier transform can also be used to check the background subtraction procedure and the noise level of the spectrum. The presence of peaks of significant intensity at very low R (A) suggests that errors were made in the subtraction process peaks at high R result from especially noisy data. 

Equation 14 consists of an unmodulated part with amplitude 1 - U2, the basic frequencies and cop with amplitudes kJl, and the combination frequencies < and w+ with amplitudes k 4, and inverted phase. To compute the frequency-domain spectram, first the unmodulated part is subtracted, as it gives a dominant peak at zero frequency for the usual case of small k values. A cosine Fourier transform (FT) of the time trace results in a spectrum that contains the two nuclear frequencies, w and cop, with positive intensity, and their sum and difference frequencies, a>+ and m, with negative intensity. If the initial part of the time-domain trace is missing, then the spectrum can be severely distorted by frequency-dependent phase shifts and it may be best to FT the time-domain trace and compute the magnitude spectrum. 

The usual procedure of the evaluation of an EXAFS data of the XA spectrum involves first the background subtraction and the evaluation of % (Figure 1.4c). Then one has to separate the contribution of the various coordination shells to visualize their data. This is done by a Fourier transform to the distance space. Figure 1.4d clearly shows four well-developed shells of crystalline Ni metal, R to R, which are characteristic of a cubic face-centered metal. This result still needs a correction by the phase shift <1> and the evaluation of Ni and o. This occurs for specific specimens by comparison with EXAFS data of well-characterized standards like nickel metal in this case or by calculation of the EXAFS results with a program like FEFF [12] or data analysis packages from the Internet with reasonable assumptions [13]. For this purpose, the % data of each shell are separated and... 

Refinement also may be accomplished in real space by calculating difference Fourier syntheses using as coefficients A F = F i,s — Fcaic and phases caic derived from the trial structure. In general, if an atom has been incorrectly placed near its true location, a negative peak will appear at its assigned position and a positive peak will appear at its proper location. That is, in the difference Fourier we have subtracted electron density from where the atom isn t, and not subtracted density from where it is. The atom is then shifted by altering x, y, z, improved atomic parameters are included in a new round of structure factor calculations, and another difference Fourier computed. The process is repeated until the map is devoid of significant features. 

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