Isotropic pair distribution function

For the rototranslational spectra, within the framework of the isotropic interaction approximation, the expressions for the zeroth and first moments, Eqs. 6.13 and 6.16, are exact provided the quantal pair distribution function (Eq. 5.36) is used [314]. A similar expression for the binary second translational moment has been reported [291],... 

Pseudo-affine model, the deformation process of polymers in cold drawing is very different from that in the rubbery state. Elements of the structure, such as crystallites, may retain their identity during deformation. In this case, a rather simple deformation scheme [12] can be used to calculate the orientation distribution function. The material is assumed to consist of transversely isotropic units whose symmetry axes rotate on stretching in the same way as lines joining pairs of points in the bulk material. The model is similar to the affine model but ignores changes in length of the units that would be required. The second moment of the orientation function is simply shown to be ... 

The isotropic nature of a liquid implies that any structure factor, S(k), obtained from a scattering experiment (typically X-ray or neutron) on that liquid will contain no angular dependence (of the molecules). Thus, the Fourier transform of any S(k) will yield a radial distribution function. Recently developed techniques of isotopic substitution [5-7] have been utilized in neutron diffraction experiments in order to extract site-site partial structure factors, and hence site-site radial distribution functions, gap(r). Unfortunately, because g p(r) represents integrals (convolutions) over the full pair distribution function, even a complete set of site-site radial distribution functions can not be used to reconstruct unambiguously the full molecular pair distribution function [2]. However, it should be mentioned at... 

Quantum sum formulas based on exact pair distribution functions (obtained in the isotropic potential approximation) are also known for n = 0,1,2, and 3 [318,319] we mention also unpublished work by J. D. Poll. Levine has given detailed estimates based on classical moments, assessing the bound dimer contributions [302]. 

Hence, the pair distribution function for an isotropic ideal gas is given by... 

The second equality holds for a homogeneous and isotropic fluid. A related quantity is the locational pair correlation function, defined in terms of the locational pair distribution function, i.e.,... 

Liquids (at least the ones concerned with here) are isotropic, and therefore the correlation function depends only on the magnitude of r (r), and the structure factor (static and dynamic) depends only on the magnitude of Q (Q). The relations between the static structure factor and the radially symmetric pair distribution function (also known as the radial distribution function, rdf) then can be expressed as (compare Eq. (29.21))... 

The isotropic approximation to elastic motion makes the treatment remarkably simple by converting the evaluation of the solute distribution function p(r) to a trivial multiplication of independent terms describing the pair interaction between the solute molecule and the polymer atoms localized at their average positions . A set of mean positions describes the structure of the... 

The interpolation between the low and high density limits, which is inherent to this variational approach, leads in a very natural way to the scaled particle theory for the structure and thermodynamics of isotropic fluids of hard particles. This unifies, for the first time the Percus Yevick theory, which is based on diagram expansions, and the scaled particle theory of Reiss, Frisch and Lebowitz, and, at the same time yields the analytical expressions of the dcf conformal to those of the hard spheres. It provides an unified derivation of the most comprehensive analytic description available of the hard sphere thermodynamics and pair distribution functions as given by the Percus Yevick and scaled particle theories, and yields simple explicit expressions for the higher direct direct correlation functions of the uniform fluid. 

If we consider many ion pairs which have an isotropic distribution function g(r,0) given as... 

A perturbation approach leads to an analytical expression for the pair-distribution function of weakly interacting short rods and is in good agreement with MC-simulations [38]. A more general treatment was recently proposed by Maeda [39] for charged rods in the isotropic regime. 

Figure 11.3 The pair distribution function determined by Stokesian dynamics simulation for hard spheres at ( ) = 0.45 and Pe = 1000 shown (a) in the full shear plane and (b) in spherical average as a function of pair separation. In (b), a comparison is made to the radial dependence of the isotropic equilibrium structure, which was shown in a planar view in Figure 11.2. Shear flow is as in Figure 11.2, left to right and increasing velocity up the page.

Isotropic means that the direction is not important. We can then write for the pair distribution function... 

Let a unit volume of an isotropic medium comprise Vvib/2 of such pairs (nonrigid dipoles). We shall calculate the generated complex susceptibility x by using the high-frequency approximation for which it is assumed that at the instant just after a strong collision the velocities and position coordinates are given by the Boltzmann distribution (marked by the subscript B). Then, in view of Eq. (3.5) in GT1, the complex susceptibility x is proportional to the spectral function L ... 

Fig. 4.11 a Pair correlation function g(r) of an ideal crystal and b the corresponding intensity distribution I(q) obtained by Fourier transform (FT), c For isotropic liquids the pair correlation function g(r) is characterized by an exponentially decaying wave function which result in d a Lorentzian intensity distribution I(q) in the scattering image... 

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