Radial distribution functions analysis

The majority of the various studies on Bi2Ss used thioacetamide as the source of sulphur. In the first of these studies [20], very broad peaks were observed in the XRD spectrum and the film was classified as amorphous. (A radial distribution function analysis of these films allowed a structure to be proposed [29]). Much sharper XRD peaks were obtained after a mild annealing at 150°C (for 6 hr). The broad peaks in the as-deposited film seem, for the most part, to have different values of 20 from those of the annealed film. 

Maron, J., M.J. Winokur, and B.R. Mattes. 1995. Processing-induces changes in the local-structure of amorphous polyaniline by radial-distribution-function analysis of X-ray scattering data. Macromolecules 28 4475. 

For the former, i.e., diffraction, structural refinement techniques that wholly address both the crystalline and diffuse portions of the scattering profiles would be a tremendous advance. The latter component will entail heavier reliance on a variety of powerful probes of local structure including nuclear magnetic resonance, scanning tunneling microscopy, anomalous scattering methods, and radial distribution function analysis [21 j. As the number of chemical/structural architectures available continues to increase, determining the fundamental nature of the structure-property relationships will become an ever more important issue. 

The simplest way computationally of obtaining a sedimentation coefficient distribution is from time derivative analysis of the evolving concentration distribution profile across the cell [40,41]. The time derivative at each radial position r is d c r,t)/co /dt)r where cq is the initial loading concentration. Assuming that a sufficiently small time integral of scans are chosen so that Ac r t)/At= dc r t)ldt the apparent weight fraction distribution function g (s) n.b. sometimes written as (s ) can be calculated... 

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 ... 

The power of X-ray methods can be extended to investigate the local structure on a scale of a few angstroms by means of the analysis of the fine structure and the radial distribution function. 

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 model-free approach to analysis of DEER data in the absence of orientation selection was proposed based on shell factorization.22 The decay curves are simulated as the products of orientationally averaged thin shells of interacting electrons. The dipolar time-evolution data can be separated into a linear contribution and a non-linear contribution from background. The linear contribution can be converted to a radial distribution function for spin-spin interaction. 

Finally, an important if tentative conclusion emerges concerning the symmetry of distribution functions, which could be relevant to the general theory of polarization and to current ideas about liquid structure. Theoretical analysis in terms of radially symmetric models does not necessarily tell us about the structure of a system, because the short-range forces, by their very nature, have a directional character which is not lost in any averaging process. Is it this averaged directional force which is responsible for structure in liquids and solutions ... 

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