Extended radial distribution function

For molecules the radial distribution function can be extended with orientational degrees of freedom to characterize the angular distribution. 

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. 

Alben and Boutron suggest that the peak in the X-ray and neutron scattering functions at 1.7 A-1 is indicative of an anisotropic layer structure extending over at least 15 A in Polk type continuous random network models. To show this better Fig. 52 displays the radial distribution function of the Alben-Boutron modified... 

With reference to the minima of the radial distribution function D r), SCF analyses [61] using the near-Hartree-Fock wavefunctions of dementi [64] indicate that the numbers of electrons found in the inner shell extending up to the minimum of D r) amount to = 2.054 e (Be), 2.131 (C), 2.186 (O), 2.199 (F) and 2.205 electron (Ne). The results of Smith et al. [65] bearing on the boundaries in position space that enclose the exact number given by the Aufbau principle support the idea of physical shells compatible with that principle. The maxima of D r), on the other hand, also appear to be topological features indicative of shells, their positions correlate well with the shell radii from the Bohr-Schrodinger theory of an atom... 

Figure 2.1 Radial distribution functions for an NaCl melt at 1148 K, as found by neutron diffraction. Concentric, mutually exclusive shells of anions and cations can clearly be seen extending to large distances relative to the sizes of the ions. (From Edwards, F.G., Enderby, J.E., Howe, R.A., and Page, D.I., /. Phps. C Solid State Phys., 8, 3483-3490, 1975. With permission.)...

Since the Fourier transformation of equation (2.2) yields only a radial distribution function about the absorber, we note that information obtained from EXAFS is limited to an average, one-dimensional representation of structure. Furthermore, in order that the transform be comparatively free of ripples, the data should extend to at least... 

In order to extend the analytical equations to a fractal lattice, we will need the radial distribution function rdf(r) of the Sierpinski gasket, rdf(r) dr being the average number of sites with distance between r and r + dr from a given site. For fractal lattices one has... 

Schlesinger and Marton (15) studied the nucleation and growth of electrolessly deposited thin nickel (Ni-P) films. These studies were later extended and complemented by the studies performed by Cortijo and Schlesinger (19, 20) on radial distribution functions (RDFs). RDF curves were derived from electron diffraction data obtained from similar types of films as well as electrolessly deposited copper ones. Those studies, taken together, have elucidated the process of crystallization in the electroless deposition of thin metal films. 

This mechanism is in agreement with the mechanism proposed by others (Belin et al., 1989 and 1995 Martin et al., 1986a, Willermet et al., 1992) using extended X-ray absorption fine structure spectroscopy (EXAFS) and infrared spectroscopy. When ZDDP is present in the lubricant formulation, the radial distribution function (RDF) indicates that crystalline iron oxide diffuses into the polyphosphate network material. 

The MFA [1] introduces the perturbation due to the solvent effect in an averaged way. Specifically, the quantity that is introduced into the solute molecular Hamiltonian is the averaged value of the potential generated by the solvent in the volume occupied by the solute. In the past, this approximation has mainly been used with very simplified descriptions of the solvent, such as those provided by the dielectric continuum [2] or Langevin dipole models [3], A more detailed description of the solvent has been used by Ten-no et al. [4], who describe the solvent through atom-atom radial distribution functions obtained via an extended version of the interaction site method. Less attention has been paid, however, to the use of the MFA in conjunction with simulation calculations of liquids, although its theoretical bases are well known [5]. In this respect, we would refer to the papers of Sese and co-workers [6], where the solvent radial distribution functions obtained from MD [7] calculations and its perturbation are introduced a posteriori into the molecular Hamiltonian. 

Liquids, Gases and Disordered Solids. Liquids, disordered solids, gases, and single crystals can diffract X rays. For liquids and disordered solids, where there is no long-range order, and the short-range order extends from 0 to maybe 1 or 2 nm, the diffraction consists of very broad maxima in the intensity function 7(s), where s is the scattering vector defined in Eq. (11.23.2). The one-dimensional Fourier transform of I(s) is the radial distribution function R(r) ... 

Hereafter, the expressions of h, c and p are matrices. In Equation (1) the notation f ), denotes the derivatives of quantity / with respect to a under constant value of b. The integral equation of the number density derivatives of the radial distribution function is obtained in a similar way. The differential form of the closure relation is obtained in a similar way. In what follows a rather extended description is given for calculation of the heat capacity. Detailed procedure for calculation of these thermodynamic quantities will be published elsewhere. ) The functions S, and calculated from... 

In order to have accurate informations from the radial distribution function (RDF), it is necessary to have a good counting statistics and to extend the measurement as far as possible in the k space (k = 4ji sin 0/A). This leads of course to time-consuming data collections. The improvement of linear detector permits to shorten the collection time anyhow, with a classical X-ray tube and Mo Ka wavelength, the data collection takes a few hours to go up to 17 A. The synchrotron beam allows faster experiments and the possibility to work in the hard X-ray part of the spectrum extending considerably the k region available. A realistic possibility for the future ESRF machine seems to be k , = 100 (2). 

Finally, we note that it is also possible to extend the methods of this chapter to study other aspects of the hard sphere problem. For example, by essentially the same approach one can treat the radial distribution function g(r). (Like the equation of state, this can be expanded in powers of the density, and then the r-dependent coefficients expanded in terms of r-dependent Mayer or Ree-Hoover integrals.) There are again several options, depending on whether one uses dimensional scaling in isolation or in conjunction with an in-... 

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