Radial Gaussian distribution function

The radial Gaussian distribution function. The radial Gaussian distribution function in the interpenetrational domain gives for spheres... 

Besides the coefficient of radial spreading, to characterize the liquid spreading, the distribution width is introduced [142]. This value is connected with a modified Gaussian distribution function and can be obtained also experimentally. 

W(Xy yy z)y W r) Density and radial distribution functions for the end-to-end coordinates of a polymer chain (usually Gaussian functions). 

History. Wilke [129] considers the case that different orders of a reflection are observed and that the orientation distribution can be analytically described by a Gaussian on the orientation sphere. He shows how the apparent increase of the integral breadth with the order of the reflection can be used to separate misorientation effects from size effects. Ruland [30-34] generalizes this concept. He considers various analytical orientation distribution functions [9,84,124] and deduces that the method can be used if only a single reflection is sufficiently extended in radial direction, as is frequently the case with the streak-shaped reflections of the anisotropic... 

These results point towards a universality in the particle radial distribution function from r = 0 to 2R. This is shown in Fig. 8 where the first contact peaks, which are self-similar, are collapsed for all packing fractions onto the master Gaussian curve ... 

Fig. 8 Rescaled radial distribution function g(r) for monodisperse packings versus the modified radial distance r. Data for different volume fractions collapse on a single curve, which is well approximated by a standard Gaussian function...

Clarke has examined the thermodynamic equation of state and the specific heat for a Lennard-Jones liquid cooled through 7 at zero pressure. He found that drops with decreasing temperature near where the selfdiffusion becomes very small. Wendt and Abraham have found that the ratio of the values of the radial distribution function at the first peak and first valley shows behavior on cooling much like that observed for the volume of real glasses (Fig. 6), with a clearly defined 7. Stillinger and Weber have studied a Gaussian core model and find a self-diffusion constant that drops essentially to zero at a finite temperature. They also find that the ratio of the first peak to the first valley in the radial distribution function showed behavior similar to that found by Wendt and Abraham" for Lennard-Jones liquids. However, the first such evidence for a nonequilibrium (i.e. kinetic) nature of the transition in a numerical simulation was obtained by Gordon et al., who observed breakaways in the equation of state and the entropy of a hard-sphere fluid similar to those in real materials. 

Figure 15. Radial distribution function T r) of the low-density a-C (p = 2.20 g/cm ) obtained from TBMD simulation (solid curve) compared with the neutron scattering data of Ref. 64 (dotted curve). The simulation result has been broadened by A Gaussian function with a width of 0.085 A. (From Ref. 62.)...

Fig. 9.7. The distance dependence of the nomalized segment density distribution function for 1, an exponential function, 2, a radial Gaussian function and 3, a constant segment density function (after Smitham and Napper, 1979).

Figure 1. Radial distribution functions in direct space and in reciprocal space, for D = 100, D — 1000 and D = 10000. When plotted in terms of the scaled coordinates r and k, the distribution functions for high values of D are sharply peahed at r = 1 and k = 1 and they can be closely approximated by Gaussians (equations (115) and (116)). The direct- and reciprocal-space curves for i = 100 can be resolved, but for D = 1000 and D = 10000 they are indistinguishable.

Fig. 27. First maximum in the radial distribution functions of the amorphous alloys Gdo.3sFeo.64 and Gdo.jgCoo gj. The curves and the Gaussian fits were drawn by Cargill III (1975) using his experimental data.

This overlap is shown in Table 3.1. The optimum fit occurs for a = 0.270950 and is shown in Fig. 3.2a. The corresponding radial distribution functions (4itr i/r) ) are compared in Fig. 3.2b. Notice the different behavior near the origin and the more rapid fall-off of the Gaussian function at large r. The overlaps S of (3.217) can be maximized for the STO-2G and STO-3G cases also and, if one does so, the optimum fits are as follows ... 

If the atoms in a molecule were rigidly held at certain distances, then the radial distribution function would consist of a series of lines, corresponding to those distances, with intensities that would depend on the atomic numbers of the pair of atoms and the distance between them. But, of course, the atoms vibrate. Thus, instead of a line, one obtains a Gaussian function where the area under the curve depends upon the variables mentioned. 

As discussed in Chap. 4, under equilibrium conditions the function fi becomes a gaussian or maxwellian distribution function for the molecular velocities, and g becomes an equilibrium radial distribution function, i.e.. 

Fig. 1.28. Comparisons among the rotational isomeric (RIS) radial distribution functions at 413 K for polyethylene (o) and PDMS ( ) chains having n — 20 skeletal bonds, and the Gaussian approximation ( — ) to the distribution for PDMS [45]. The RIS curves represent cubic-spline fits to the discrete Monte Carlo data, for 80 000 chains, and each curve is normalized with respect to an area of unity (with / being the skeletal bond length).

The mathematical description of the model is out of the scope of this paper. Briefly, in this model, each reactant beam density is fitted to gaussian radial and temporal distribution functions, the spread in relative translational energy is neglected and the densities are assumed to be constant within the probed volume, which is smaller than the reaction zone. These assumptions result in a simple analytic expression of the overlap integral. Calculations are carried out for each rovibrational state of the outcoming molecule and for extreme velocity vector orientations, i.e, forwards and backwards. An example of the correction function, F, obtained for the A1 + O2 reaction at = 0.49 eV is displayed on Fig. 1, together with the... 

Figure 6. Predicted interchain radial distribution function for a hard-core polyethylene melt described by three single-chain models atomistic RIS at 430 K, overlapping (lid = 0.5) SFC model with appropriately chosen aspect ratio and site number density (see text), and the Gaussian thread model (shifted horizontally to align the hard core diameter with the value of rld = l).

The electron diffraction structure analysis can accordingly be characterized as if it were the determination of the frequencies and the damping of the components of a sum of sine functions. As the molecular intensity is Fourier-transformed, another important but more descriptive function is attained, which bears, somewhat unfortunately, the name of radial distribution function, and which is indeed connected with the probability distribution function of the intramolecular inter-nuclear distances. It is a Gaussian-like distribution that... 

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