Radial distribution function local

Typical results for a semiconducting liquid are illustrated in figure Al.3.29 where the experunental pair correlation and structure factors for silicon are presented. The radial distribution function shows a sharp first peak followed by oscillations. The structure in the radial distribution fiinction reflects some local ordering. The nature and degree of this order depends on the chemical nature of the liquid state. For example, semiconductor liquids are especially interesting in this sense as they are believed to retain covalent bonding characteristics even in the melt. 

A combination of physicochemical, topological, and geometric information is used to encode the environment of a proton, The geometric information is based on (local) proton radial distribution function (RDF) descriptors and characterizes the 3D environment of the proton. Counterpropagation neural networks established the relationship between protons and their h NMR chemical shifts (for details of neural networks, see Section 9,5). Four different types of protons were... 

When the system contains more than one component it is important to be able to explore the distribution of the different components both locally and at long range. One way in which this can be achieved is to evaluate the distribution function for the different species. For example in a binary mixture of components A and B there are four radial distribution functions, g (r), g (r), g (r) and g (r) which are independent under certain conditions. More importantly they would, with the usual definition, be concentration dependent even in the absence of correlations between the particles. It is convenient to remove this concentration dependence by normalising the distribution function via the concentrations of the components [26]. Thus the radial distribution function of g (r) which gives the probability of finding a molecule of type B given one of type A at the origin is obtained from... 

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. 

We also examined the fold statistics in this Ciooo system. The distribution of the inter-stem vectors connecting stems linked by the loops, and their radial distribution function again indicated that about 60-70% of the folds are short loops connecting the nearest or the second and third nearest stems, though the crystallization did not complete. The presence of local order in the under cooled melt in the present Ciooo system is also examined through the same local order P(r) parameter, the degree of bond orientation as a function of position r, but again we did not detect any appreciable order in the undercooled melt. 

The presence of local cation ordering in Mg2Ga and MgsGa - CO3 LDHs noted in Sect. 3.3.1 has been confirmed by means of both EXAFS and by calculation of the electron radial distribution function from the Fourier transform of the diffracted X-ray intensity. In each case the gallium was found to have six magnesium ions and no galhum ions as next-nearest neighbors [39]. 

The average local electrostatic potential V(r)/p(r), introduced by Pohtzer [57], led Sen and coworkers [58] to conjecture that the global maximum in V(r)/p(r) defines the location of the core-valence separation in ground-state atoms. Using this criterion, one finds N values [Eq. (3.1)] of 2.065 and 2.112 e for carbon and neon, respectively, and 10.073 e for argon, which are reasonable estimates in light of what we know about the electronic shell structure. Politzer [57] also made the significant observation that V(r)/p(r) has a maximum any time the radial distribution function D(r) = Avr pir) is found to have a minimum. 

In principle, EXAFS information may be obtained for most or all of the elements in a catalyst. Thus, for multicomponent samples, the characterization of local surroundings for all (or almost all) the elements may be obtained. However, we stress that the radial distribution function cannot be transformed into a unique three-dimensional structure. Therefore, the EXAFS technique is not ideal for providing such information and the data representing materials consisting of several different phases may often be too difficult to analyze meaningfully. 

Fig. 5. VET rate constants of benzene in scC02 as a function of reduced density (filled circles). The solid line represents calculations of the local density at the position of the first maximum of the radial distribution function around an attractive solute in a Lennard-Jones fluid (see Fig. 7 and text for details). Experimental conditions pred = 2.1 (500bar, 318K), prei = 1.6 (150 bar, 318K), pred= 1.2 (lOObar, 318K), pred= 0.7 (lOObar, 328K).

Figure 1 shows Fourier transforms of EXAFS spectra of a few samples prepared. The radial distribution functions of these samples are different from that of nickel oxide or cobalt oxide [7]. All the Fourier transforms showed two peaks at similar distances (phase uncorrected) the peak between 1 and 2 A is ascribed to the M-0 bond (M divalent cation) and the peak between 2 and 3 A is ascribed to the M-O-M and M-O-Si bonds. The similar radial distribution functions in Figure 1 indicate that the local structures of X-ray absorbing atoms (Ni, Co, and Zn) are similar. No other bonds derived from metal oxides (nickel, cobalt and zinc oxides) were observed in the EXAFS Fourier transforms of the samples calcined at 873 K, which suggests that the divalent cations are incorporated in the octahedral lattice. 

Despite the formation of clathrate-like clusters and complete 512 cages during these simulations, the increased ordering observed from the radial distribution functions and local phase assignments resulted in the authors concluding that their simulation results are consistent with a local order model of nucleation, and therefore do not support the labile cluster model. 

For a gas at equilibrium, i.e., no mean deformation, there is a spatial homogeneity and thus g(ri,r2) depends only on the separation distance r = ri — r2. Then g = go(r) is termed the radial distribution function, which may be interpreted as the ratio of the local number density at a distance r from the central particle to the bulk number density. For a system of identical spheres, the radial distribution function go O ) at contact (i.e., r = dp) can be expressed in terms of the volume fraction of solids ap as... 

The EXAFS spectroscopy results strongly confirm the existence of local order in the mineral part of lead isooctane reverse micelles in dodecane and reveal quantitative information concerning the first and second coordination shells. The radial distribution functions (RDFs) exhibit peaks at around 0.19 nm, corresponding to the first shell of oxygen atoms and at around 0.35 nm corresponding to the shell of lead. Analytical transmission electron microscopy (ATEM) indicates the size of the mineral core of the micelles (1-1.5 nm) and the discoid shape of the particles when the micelles aggregate (Mansot et al. 1994). 

The static structure of three-dimensional colloidal suspensions is usually determined experimentally, not by measuring directly g(r) in real space, but by measuring the static structure factor S(k) in the reciprocal space, which is the Fourier transform of the local particle-concentration correlation function. The radial distribution function is directly related to the Fourier transform of S(k), as it is explained below. Let us consider a system of N particles in a volume V. The local particle concentration p(r) at the position r is given by... 

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