Radial distribution function dimensions

The first satisfactory definition of crystal radius was given by Tosi (1964) In an ideal ionic crystal where every valence electron is supposed to remain localised on its parent ion, to each ion it can be associated a limit at which the wave function vanishes. The radial extension of the ion along the connection with its first neighbour can be considered as a measure of its dimension in the crystal (crystal radius). This concept is clearly displayed in figure 1.7A, in which the radial electron density distribution curves are shown for Na and Cl ions in NaCl. The nucleus of Cl is located at the origin on the abscissa axis and the nucleus of Na is positioned at the interionic distance experimentally observed for neighboring ions in NaCl. The superimposed radial density functions define an electron density minimum that limits the dimensions or crystal radii of the two ions. We also note that the radial distribution functions for the two ions in the crystal (continuous lines) are not identical to the radial distribution functions for the free ions (dashed lines). 

Inspection of Eq. 7 reveals that the molecular interference function, s(x), can be derived from the ratio of the total cross-section to the fitted IAM function, when the first square bracketed factor has been accounted for. A widely used model of the liquid state assumes that the molecules in liquids and amorphous materials may be described by a hard-sphere (HS) radial distribution function (RDF). This correctly predicts the exclusion property of the intermolecular force at intermolecular separations below some critical dimension, identified with the sphere diameter in the HS model. The packing fraction, 17, is proportional for a monatomic species to the bulk density, p. The variation of r(x) on 17 is reproduced in Fig. 14, taken from the work of Pavlyukhin [29],... 

A complication arising from the extension of the theory to flexible macromolecules is that in general, the intermolecular and intramolecular radial distribution functions depend on each other.In modeling the bulk of a one-phase polymer melt, however, the situation resolves itself because the excluded volume effect is insignificant under these conditions the polymer chains assume unperturbed dimensions (see also the section on Monte Carlo simulations by Corradini, as described originally in Ref. 99). One may therefore calculate the structure of the unperturbed single chain and employ the result as input to the PRISM theory to calculate the intermolecular correlation functions in the melt. 

Equation (o) is illustrated in Figure 6-15. (Note that co[r) has dimensions of reciprocal length.) The maximum in the curve corresponds to the most probable end-to-end distance, and it can be found easily by differentiating equation (o). The result, or the most probable value of r, is Mb or (2nl 2/3)Y The mean square end-to-end distance is given by the second moment of the radial distribution function ... 

This gives rise to a uniformly decreasing intensity with 9 which depends upon the characteristic dimension a. This approach may be used, for example, to describe the X-ray scattering from an atom where the electron density varies with radius by an amount described by quantum mechanics. Thus the scattering depends upon the radial distribution function of electronic density. The agreement between the predicted variation of scattered intensity with that which is measured serves as a test of the correctness of the quantum mechanical description. 

It is difficult to trace the origin of these discrepancies between the computed and experimental radial distribution function. It is possible that the pair potential does not produce enough preference for the tetrahedral geometry (hence, the coordination number and the location of the second peak do not show the characteristic values as in water), or that the numerical procedure was not run with a sufficient number of particles and configurations. [Sixty four particles in three dimensions corresponds to four particles in one dimension, which is quite a small number. Furthermore, the number of configurations, of the order of 10 , is quite small to ensure convergence. See also the discussion in Section 6.11.] Some preliminary recent Monte Carlo results were reported by Popkie et ah (1973). 

For a detailed characterization and understanding of different dimeric arrangements in the crystal, the radial distribution function g(r) is calculated, considering a super cell of dimension 3 x 3 x 3 A and maintaining a cutoff distance of 15 A. 

FIGURE 8.8 Radial distribution function g r) for the octathio[8]circulene crystal in supercells of dimension 3 x 3 x 3 A. Note that a Lorentzian broadening (F) of 0.06 is used for smoothening the crystalline 5-functions at the peak positions (dotted histogram). 

Fig. 9.4 Radial distribution functions obtained from the liquid state simulations in a three- and b two-dimensions at 1000 K above the respective isocoordinate (red lines) and isochoric (black lines) crystal to liquid phase transitions. The abscissa length-scale is normalized by the position of the first peak, ri. The length-scales corresponding to ideal tetrahedral and trigonal C-C-C bond angles are indicated as dashed vertical lines...

There are two dimensions, which are the main system criteria. They are the order parameter P2 and the radial distribution function center of the ehain mass g(r). The parameter... 

The detailed approach would require a meticulous summation of all molecule-molecule interactions. Consider the situation, shown schematically in Figure 3, of a deep layer of adsorbate, j molecules thick. The particular molecule shown in the ith layer experiences a potential given by the sum of interactions with the semi-infinite solid and the interactions with other adsorbate molecules, infinite in two dimensions, but finite in the third. The last was approximated in Equation 16 by treating the adsorbed film as normal liquid in state. The detailed analysis would involve complete potential functions, radial distributions, and structural information, and would then lead to some total expression for the free energy of an adsorbed film as a function of its thickness. From the corresponding partial molar free energy, a relationship to kT In P °/P could in principle be obtained. 

As for the one-dimensional case, the function L makes features emerge from the electron density that p itself does not clearly show. What then does the function L reveal for the spherical electron density of a free atom Because of the spherical symmetry, it suffices to focus on the radial dimension alone. Figure 7.2a shows the relief map of p(r) in a plane through the nucleus of the argon atom. Figure 7.2b shows the relief map of L(r) for the same plane, and Figure 7.2c the corresponding contour map. Since the electron density distribution is... 

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