Line broadening Fourier methods

When applied to the XRD patterns of Fig. 4.5, average diameters of 4.2 and 2.5 nm are found for the catalysts with 2.4 and 1.1 wt% Pd, respectively. X-ray line broadening provides a quick but not always reliable estimate of the particle size. Better procedures to determine particle sizes from X-ray diffraction are based on line-profile analysis with Fourier transform methods. 

Chemisorption, transmission electron microscopy, and XRD line broadening do not necessarily result in the same calculated dispersion for a given catalyst. Chemisorption may be biased toward a lower average crystallite size and line broadening toward a higher size. In fact, line broadening and chemisorption methods are not directly comparable unless Fourier analysis is applied to the X-ray data. Chemisorption and transmission electron microscope results are directly comparable. 

As suggested by Equation (13) and the related discussion in Section 13.2.3, Fourier analysis is a natural choice for treating line profiles that makes it possible to separate the various sources of line broadening in a much better and more general way than using IB methods. 

The use of certain apodization functions improves the frequency resolution we obtain in our Fourier-transformed spectrum, but caution should be exercised when employing this technique. The use of negative line broadening and shifted Gaussian or squared sine bells (with the maximum to the right of the start of the FID) can be used to resolve a small peak that formerly appeared as the shoulder of a larger peak, but supervisors and reviewers frown upon the excessive application of these methods the starting NMR spectroscopist would do well to exercise restraint in this area. 

Mu-substituted hexadienyl radicals are easily observed by high-field tS Rotation and their reaction with benzoquinone has been studied. The method is principally the same as in the Mu relaxation kinetics but, since Mu-substituted tmlicals are measured, the disappearance rate of MuO H is determined from the line broadening of the Fourier transformation power spectrum. The obtained value is kjj = 2.8 x 10 M- s- . It is important to note tlmt the reaction rate constant of Mu-substituted radicals may not differ much from that of the H-counterparts. The isotope effect should be small since the mass of Mu- and H-radicals are not much different, and since the reaction center is arated from the site of Mu- and H-addition. 

Applications to Biological Samples. - Methods of distance measurements were compared for four doubly spin-labelled derivatives of human carbonic anhydrase.53 The distances between the spin labels were obtained from continuous wave spectra by analysis of the relative intensity of the half-field transition, Fourier deconvolution of the line-shape broadening, and computer simulation of line-shape changes. For variants with interspin distances greater than 18 A, the DEER method also was used. For each variant, at least two methods were applicable and reasonable agreement between distances obtained by different methods was obtained. The useful distance ranges for the techniques employed at X-band with natural isotope abundance spin labels were estimated to be half-field transition (5-10 A), line-shape simulation (up to 15 A), Fourier deconvolution (8 - 20 A), and four-pulse DEER (> 18 A).53... 

Fig. 3.4.1 Homogeneously and inhomogeneously broadened lines, (a) Echo train generated by repeated refocussing of the FID (CPMG method, cf. Fig 2.2.10(b)). (b) The Fourier transform of the slowly decaying echo envelope is the homogeneously broadened line, (c) The Fourier transform of the fast decaying echo is the inhomogeneously broadened line.

Powder X-ray diffraction (XRD) patterns of the catalysts separated from the reaction mixture were measured using a Shimadzu VD-1 diffractometer with CuKa radiation. The mean crystallite size (D. ) of iron metal in a catalyst was calculated from the half-maximum breadth of the (110) peak of a-Fe after correction for instrumental broadening (ref. 10). The crystallite size distributions (CSD) of iron in some catalysts were obtained according to the Fourier transform method (Stokes method) for X-ray line profile analysis (ref. 11). 

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