Fourier filtering

Figures Fourier transform (soiid curve), Osir ) versus r (A, without phase-shift correction), of the Mo K-edge EXAFS of Figure 5 for moiybdenum metal foii. The Fourier filtering window (dashed curve) is applied over the region -1.5-4.0 A to isolate the two nearest Mo-Mo peaks.

Figure Fourier-filtered Mo K-edge EXAFS, versus k (A ) (soiid curve), for...

Figure 5.49. (a) STM image (unfiltered) of the initially sodium-contaminated Pt(l 1 l)-(2x2)-0 adlattice (b) corresponding Fourier transform spectrum (c) Fourier-filtered STM image of the overlapping Pt(l 1 l)-(2x2)-0 and Pt(111)-(12x12)-Na adlayers (bias Ut = 80 mV, tunelling current I, = 10 nA, total scan size 319 A).78 Reprinted with permission from Elsevier Science. 

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... 

Figure 22. Fourier-filtered data for a lead upd monolayer on a silver (111) electrode at two applied potentials. Solid curve, -1.0 V dashed curve, -0.53 V. (From Ref. 88, with permission.)...

Figure 25. EXAFS data for K3[Fe(CN)6] (A) k2-weighted EXAFS (B) Fourier transform of (A) showing Fe—C and Fe— N peaks (C) Fourier-filtered back-transformation of the Fe—C peak. (From Ref. 97, with permission.)...

Figure 3A. Phase uncorrected radial distribution functions (solid line) and fourier filter window (dashed line) for bulk [Ru(v-bpy)3]+2.

In the present study we have used the phase and amplitude functions of absorber-scatterer pairs in known model compounds to fit the EXAFS of the catalysts. By use of Fourier filtering, the contribution from a single coordination shell is isolated and the resulting filtered EXAFS is then non-linear least squares fitted as described in Ref. (19, 20). 

Table II. Bond lengths and coordinations numbers obtained by fitting the Fourier filtered Mo EXAFS of the Co-Mo unsupported catalyst recorded in situ at room temperature. ...

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 =... 

Figure 12.5 [18] Structure of the monolayer of S05-3 arborescent graft (dendri-graft) polystyrenes cast on mica as prepared (A) and Fourier filtered (B)...

Figure 7.19 Ni K-edge X-ray absorption spectra for as-isoiated (solid line) and NADH- and H2-reduced (dashed line) samples of the hydrogenase from R. eutropha HI6. (a) Edge region (b) Fourier-filtered EXAFS (backtransform window = I.I-2.6 A). Reprinted with permission from Gu, eta/. (1996) and the American Chemical Society.

Anyway, due to the limited precision of the EDP - intensity measurement, especially for the (100) reflection, which is too close to the central beam - this argument is not very strong and cannot reject the 1/3 substitution at all. To confirm the hypotheses of the proposed 2/3 substitution HREM observation in [001] and [021] zones were performed in parallel with multislice calculations, based on the 2/3 substitution. These projections are used because the fourfold modulation due to the Mo atoms is well expressed in them. The observed images are enhanced by a Fourier filter with Bragg mask to eliminate the noise frequencies. 

Key words nanocrystalline materials, Z1O2, Z1O2 -Y2O3, HRTEM, IP, Fourier filtering... 

Figure 3. Continued. C) Back-transform (Fourier filter) of data. Upper trace corresponds...

DeTitta, G. T., Weeks, C. M., Thuman, P, Miller, R. and Hauptman, H. A. (1994). Structure solution by minimal-function phase refinement and Fourier filtering. 

Running mean Running median Running polynomial Fourier filtering... 

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