Pair-density-function analysis

Egami (2001) has developed pair-density-function analysis of pulsed neutron-diffraction data to determine the structure in a time interval that is short relative to a bond-length fluctuation. This method provides a direct observation of the pattern of bond-length fluctuations. Several other measurements have been shown to provide indirect signatures of the presence of bond-length fluctuations (Goodenough and Zhou, 2001). For example,... 

Atomic pair-density function analysis of X-ray diffraction patterns is used to resolve the local stmcture of the catalysts from 0.15 to 2 nanometers (nm). 

Pair-density function analysis of X-ray diffraction patterns shows that active iron oxide (FeOx) catalysts are disordered at the nanoscale by the platinum and also have micropores, while inactive catalysts do not share these features. 

Although the catalysts are termed amorphous, they can have nanoscale order in their atomic stmctures that can have a significant bearing on their activity and transport properties. These nanoscale structural properties are not easily discerned with conventional materials science tools, but can be elucidated using an emerging technique atomic pair-density function analysis of X-ray diffraction (PDF-XRD). ... 

In addition, many of the ferroelectric solids are mixed ions systems, or alloys, for which local disorder influences the properties. The effect of disorder is most pronounced in the relaxor ferroelectrics, which show glassy ferroelectric behavior with diffuse phase transition [1]. In this chapter we focus on the effect of local disorder on the ferroelectric solids including the relaxor ferroelectrics. As the means of studying the local structure and dynamics we rely mainly on neutron scattering methods coupled with the real-space pair-density function (PDF) analysis. 

The population balance equation so obtained is not closed because the right-hand side involves a fresh unknown in the pair density function /2-Even without a detailed consideration, it should be obvious that an equation for /2 would involve and so on. Thus, an infinite hierarchy of equations is obtained, and unless some form of closure approximation is made the population density cannot be obtained. In population balance analysis, one makes the approximation... 

Analysis of Local Atomic Structure by the Pulsed Neutron Atomic Pair-Density Function (PDF) Method. 117... 

Rosenfield H, Barton R Jr. Pair-density function of nano-scale morphology in oriented polymer fibers Application to nomex aramid. In GRfrich JV, Cev Noyan I, Jenkins R, Hnang TC, Snyder RL, Smith DK, Zaitz MA, Predecki PK, editors. Advances in X-Ray Analysis. Springer 1998. p 523-533. 

Also in 2-substituted ethanesulphonates,35 the 33S chemical shift has a reverse substituent effect and correlates with both Taft substituent constants and the chemical shift of the carboxylic carbon in related carboxylic acids. It seems that the substituent effect does not depend on conformation, but prevailingly on intramolecular electronic effects. Density functional theory (DFT) calculations of 33S nuclear shielding constants and natural bond orbital (NBO) analysis made it possible to conclude that substituents cause a variation in the polarization of the S-C and S-O bonds and of the oxygen lone pairs of the C — S03 moiety. This affects the electron density in the surroundings of the sulphur nucleus and consequently the expansion or contraction of 3p sulphur orbitals. 

Analysis of the radial pair distribution function for the electron centroid and solvent center-of-mass computed at different densities reveals some very interesting features. At high densities, the essentially localized electron is surrounded by the solvent resembling the solvation of a classical anion such as Cr or Br. At low densities, however, the electron is sufficiently extended (delocalized) such that its wavefunction tunnels through several neighboring water or ammonia molecules (Figure 16-9). 

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 main source of experimental structural results are diffraction experiments, either of X-rays or neutrons, as their wavelength is of the right order of magnitude to probe typical interparticle separations of condensed phase system at moderate to high density. It is worth recalling, however, that pair correlation functions are not a direct experimental data. Rather, raw intensity values as a function of diffraction angle require careful data analysis to extract a structure function that is a combination of Fourier transforms of pair correlation functions. 

Modern theory of associative fluids is based on the combination of the activity and density expansions for the description of the equilibrium properties. The activity expansions are used to describe the clusterization effects caused by the strongly attractive part of the interparticle interactions. The density expansions are used to treat the contributions of the conventional nonassociative part of interactions. The diagram analysis of these expansions for pair distribution functions leads to the so-called multidensity integral equation approach in the theory of associative fluids. The AMSA theory represents the two-density version of the traditional MSA theory [4, 5] and will be used here for the treatment of ion association in the ionic fluids. 

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