Structural modeling pair distribution function

It is also possible to prepare crystalline electrides in which a trapped electron acts in effect as the anion. The bnUc of the excess electron density in electrides resides in the X-ray empty cavities and in the intercoimecting chaimels. Stmctures of electri-dides [Li(2,l,l-crypt)]+ e [K(2,2,2-crypt)]+ e , [Rb(2,2,2-crypt)]+ e, [Cs(18-crown-6)2]+ e, [Cs(15-crown-5)2]" e and mixed-sandwich electride [Cs(18-crown-6)(15-crown-5)+e ]6 18-crown-6 are known. Silica-zeolites with pore diameters of vA have been used to prepare silica-based electrides. The potassium species contains weakly bound electron pairs which appear to be delocalized, whereas the cesium species have optical and magnetic properties indicative of electron locahzation in cavities with little interaction between the electrons or between them and the cation. The structural model of the stable cesium electride synthesized by intercalating cesium in zeohte ITQ-4 has been coirfirmed by the atomic pair distribution function (PDF) analysis. The synthetic methods, structures, spectroscopic properties, and magnetic behavior of some electrides have been reviewed. Theoretical study on structural and electronic properties of inorganic electrides has also been addressed recently. ... 

The driving force behind the rapid development of powder diffraction methods over the past 10 years is the increasing need for structural characterization of materials that are only available as powders. Examples are zeolite catalysts, magnets, metal hydrides, ceramics, battery and fuel cell electrodes, piezo- and ferroelectrics, and more recently pharmaceuticals and organic and molecular materials as well as biominerals. The emergence of nanoscience as an interdisciplinary research area will further increase the need for powder diffraction, pair-distribution function (PDF) analysis of powder diffraction pattern allows the refinement of structural models regardless of the crystalline quality of the sample and is therefore a very powerful structural characterization tool for nanomaterials and disordered complex materials. 

In addition to the repulsive part of the potential given by Eq. (4), a short-range attraction between the macroions may also be present. This attraction is due to the van der Waals forces [17,18], and can be modelled in different ways. The OCF model can be solved for the macroion-macroion pair-distribution function and thermodynamic properties using various statistical-mechanical theories. One of the most popular is the mean spherical approximation (MSA) [40], The OCF model can be applied to the analysis of small-angle scattering data, where the results are obtained in terms of the macroion-macroion structure factor [35], The same approach can also be applied to thermodynamic properties Kalyuzhnyi and coworkers [41] analyzed Donnan pressure measurements for various globular proteins using a modification of this model which permits the protein molecules to form dimers (see Sec. 7). 

Nj and Rj are the most important structural data that can be determined in an EXAFS analysis. Another parameter that characterizes the local structine aroimd the absorbing atom is the mean square displacement aj that siunmarizes the deviations of individual interatomic distances from the mean distance Rj of this neighboring shell. These deviations can be caused by vibrations or by structural disorder. The simple correction term exp [ 2k c ] is valid only in the case that the distribution of interatomic distances can be described by a Gaussian function, i.e., when a vibration or a pair distribution function is pmely harmonic. For the correct description of non-Gaussian pair distribution functions or of anhar-monic vibrations, different special models have been developed which lead to more complicated formulae [15-18]. This term, exp [-2k cj], is similar to the Debye-Waller factor correction used in X-ray diffraction however, the term as used here relates to deviations from a mean interatomic distance, whereas the Debye-Waller factor of X-ray diffraction describes deviations from a mean atomic position. 

The task of extending the pair distribution function based on theoretical considerations has been addressed many times (Verlet 1968 Galam and Hansen 1976 Jolly, Freasier, and Bearman 1976 Ceperley and Chester 1977 Dixon and Hutchinson 1977 Foiles, Ashcroft, and Reatto 1984). Often the goal has been to study the correlation functions themselves or to calculate structure factors, not to obtain properties. Here we will emphasize applications aimed toward representing thermodynamic properties of molecular fluids that do not have conformational variations. While many publications have been confined to atomic model fluids, such as LJ particles, we focus here on applications for real molecular systems and their mixtures. 

Fig. 1.1 The Faber-Ziman Sap k) a, /3 = M, X) and Bhatia-Thornton Su k) (/, / = N, C) partial structure factors for liquid and glassy ZnCl2. The points with vertical (black) error bars are the measured functions in (a) and (c) for the liquid at 332(5) °C [ 16] and in (b) and (d) for the glass at 25(1) °C [15, 16]. The solid (red) curves are the Fourier backtransforms of the corresponding partial pair-distribution functions after the unphysical oscillations at r-values smaller than the distance of closest approach between the centres of two atoms are set to the calculated Unlit at r = 0. The broken (green) curves in (a) are from the polarisable ion model of Sharma and Wilson [63] for the Uquid at 327 °C...

Fig. 1.2 The Faber-Ziman partial structure factors Sap(k) and partial pair-distribution functions gafi (r) (a, /S = M, X) as calculated for models using two different values for the anion polarisability ax [61]. The curves in (a) and (b) correspond to a rigid ion model (RIM) with ax = 0, while the curves in (c) and (d) correspond to a polarisable ion model (PIM) with ax = 20 au. The introduction of anion polarisability leads to the appearance of an FSDP in S mm (k) at kpsop 1.2 A and to an alignment of the principal peaks in aU three Safi(k) functions atkpp 2 A. The alignment of the principal peaks in (c) arises from in-phase large-r oscillations in the ga r) functions shown in (d)...

Comparing structural information from X-ray and neutron diffraction provides a very valuable way to validate MD simulation results of glasses. In some simple systems, the partial pair distribution function or partial structure factors of all atom pairs can be determined experimentally and they provide excellent validations for simulated structures. However, as the composition becomes more complicated and more elements included, larger number of pair contributions will complicate the comparison and the validation becomes more and more difficult in multicomponent glass systems. For example, for binary oxides, e.g. sodium silicate, there are six partial pair distribution functions, but for a four component systems, for example the bioactive glass composition, there are a total of fifteen partials contributions. The overlap between partial contributions makes it very challenging to assign the peaks and to determine the quality of comparison and hence the validation of the simulated structure models. 

There have been a number of reports discussing where a dopant sits in the Ge2Sb2Tc5 host network [8], It is therefore interesting to see how these dopants (Cu, Ag, Au) adapt to the local structure. The atomic structure of our models is studied through a set of pair correlation functions. A pair correlation function is a position distribution function based on the probability of finding atoms at some distance r from a central atom. Following [9], we tersely develop the expressions for correlation functions and present this below. A general expression for the pair distribution function [9] is ... 

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