Distribution function pair

We conclude this section with some remarks on pair distribution functions that have been used in multiple scattering problems. For the hole-correction 

Passing to spherical coordinates, using the spherical wave expansion of the plane waves exp (jp-r) and assuming spherical symmetry for H, i.e., H p) = H p), we obtain 

The Ornstein-Zernike equation with Percus-Yevick approximation has a closed-form solution for the case of a hard-sphere potential. In the Fourier transform domain, the solution to the Ornstein-Zernike equation is 

the Fourier transform Hip) can be calculated readily while the total correlation function is obtained from (2.192). 

Amorphous polymers, which do not have any crystalline reflections, are not amenable to structure analysis by the methods described in Sections 2.4.2 and 2.4.3. In such instances, the stmcture can be assessed using atomic pair distribution function (PDF) [27-29]. PDF, g r), is the probability of finding two atoms separated by a distance r. It can be obtained by a numerical Fourier transform of the measured 

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Between the limits of small and large r, the pair distribution function g(r) of a monatomic fluid is detemrined by the direct interaction between the two particles, and by the indirect interaction between the same two particles tlirough other particles. At low densities, it is only the direct interaction that operates through the Boltzmaim distribution and... 

Toby B FI and Egami T 1992 Accuracy of pair distribution function analysis applied to crystalline and noncrystalline materials Aota Crystaiiogr.k 48 336-46... 

The description of the atomic distribution in noncrystalline materials employs a distribution function, (r), which corresponds to the probability of finding another atom at a distance r from the origin atom taken as the point r = 0. In a system having an average number density p = N/V, the probability of finding another atom at a distance r from an origin atom corresponds to Pq ( ). Whereas the information given by (r), which is called the pair distribution function, is only one-dimensional, it is quantitative information on the noncrystalline systems and as such is one of the most important pieces of information in the study of noncrystalline materials. The interatomic distances cannot be smaller than the atomic core diameters, so = 0. 

The pair correlation functions can be expressed directly in terms of the computed coefficients from Eq. (61) in particular, the number-number pair distribution function gN ir) and the number-number structure factor SNN k). Thus,... 

FIG. 4 Pair distribution functions (the main part of the figure) and... 

On the other hand, the connected pair distribution function for a given matrix configuration p (ri,r2 ... 

Let us proceed with the description of the results from theory and simulation. First, consider the case of a narrow barrier, w = 0.5, and discuss the pair distribution functions (pdfs) of fluid species with respect to a matrix particle, gfm r). This pdf has been a main focus of previous statistical mechanical investigations of simple fluids in contact with an individual permeable barrier via integral equations and density functional methodology [49-52]. 

We conclude, from the results given above, that both the ROZ-PY and ROZ-HNC theories are sufficiently successful for the description of the pair distribution functions of fluid particles in different disordered matrices. It seems that at a low adsorbed density the PY closure is preferable, whereas... 

The simulations are repeated several times, starting from different matrix configurations. We have found that about 10 rephcas of the matrix usually assure good statistics for the determination of the local fluid density. However, the evaluation of the nonuniform pair distribution functions requires much longer runs at least 100 matrix replicas are needed to calculate the pair correlation functions for particles parallel to the pore walls. However, even as many as 500 replicas do not ensure the convergence of the simulation results for perpendicular configurations. 

In order to get the pair distribution functions gjj, which satisfy the symmetry constraints, so-called minor iterations must be converged. When we use gij(r >r j) of (fl) in (2), we obtain the minor equations regarding the rotation operator R of the angle tt/2 ... 

Although we work with the pair distribution functions, what we are to solve are essentially the point probability functions fj(rj), i=A and B. 

We have solved the set of integral equations on the pair distribution functions by discretizing them. This is equivalent to allowing the atomic displacements to finite number of points. When they are discretized, to solve them is a straightforward application of the exsisting CVM. 

In conclusion, we have presented a new formulation of the CVM which allows continuous atomic displacement from lattice point and applied the scheme to the calculations of the phase diagrams of binary alloy systems. For treating 3D systems, the memory space can be reduced by storing only point distribution function f(r), but not the pair distribution function g(r,r ). Therefore, continuous CVM scheme can be applicable for the calculations of phase diagrams of 3D alloy systems [6,7], with the use of the standard type of computers. 

Now let us add the possibility of collisions. Before we proceed, we make the following two assumptions (1) only binary collisions occur, i.e. we rule out situations in which three or more hard-spheres simultaneously come together (which is a physically reasonable assumption provided that the gas is sufficiently dilute), and (2) Boltzman s Stosszahlansatz, or his molecular chaos assumption that the motion of the hard-spheres is effectively pairwise uncorrelated i.e. that the pair-distribution function is the product of individual distribution functions ... 

We turn now to the orientational correlations which are of particular relevance for liquid crystals that is involving the orientations of the molecules with each other, with the vector joining them and with the director [17, 28]. In principal they can be characterised by a pair distribution function but in view of the large number of orientational coordinates the representation of the multi-dimensional distribution can be rather difficult. An alternative is to use distance dependent orientational correlation coefficients which are related to the coefficients in an expansion of the distribution function in an appropriate basis set [17, 28]. 

Figure 5. The Fourier transformed signal AS[r, i] of I2/CCI4. The pump-probe delay times are I = 200 ps, 1 ns, and 1 ps. The green bars indicate the bond lengths of iodine in the X and A/A states. The blue bars show the positions of the first two intermolecular peaks in the pair distribution function gci-ci- (See color insert.)...

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