Reduced pair correlation function

Fig. 2 Reduced pair-correlation functions for amorphous (a) CUgyTi33, (b) cusoTiso and (c) CuasTtes alloys.

Far from the surface, the theory reduces to the PY theory for the bulk pair correlation functions. As we have noted above, the PY theory for bulk pair correlation functions does not provide an adequate description of the thermodynamic properties of the bulk fluid. To eliminate this deficiency, a more sophisticated approximation, e.g., the SSEMSA, should be used. 

Much more detailed information about the microscopic structure of water at interfaces is provided by the pair correlation function which gives the joint probability of finding an atom of type/r at a position ri, and an atom of type v at a position T2, relative to the probability one would expect from a uniform (ideal gas) distribution. In a bulk homogeneous liquid, gfn, is a function of the radial distance ri2 = Iri - T2I only, but at the interface one must also specify the location zi, zj of the two atoms relative to the surface. We expect the water pair correlation function to give us information about the water structure near the metal, as influenced both by the interaction potential and the surface corrugation, and to reduce to the bulk correlation Inunction when both zi and Z2 are far enough from the surface. 

In this nonvariational approach for the first term represents the potential of the exchange-correlation hole which has long range — 1/r asymptotics. We recognize the previously introduced splitup into the screening and screening response part of Eq. (69). As discussed in the section on the atomic shell structure the correct properties of the atomic sheU structure in v arise from a steplike behavior of the functional derivative of the pair-correlation function. However the WDA pair-correlation function does not exhibit this step structure in atoms and decays too smoothly [94]. A related deficiency is that the intershell contributions to E c are overestimated. Both deficiencies arise from the fact that it is very difficult to represent the atomic shell structure in terms of the smooth function p. Substantial improvement can be obtained however from a WDA scheme dependent on atomic shell densities [92,93]. In this way the overestimated intershell contributions are much reduced. Although this orbital-depen-... 

Figure 8. Pair correlation function g(r) at T — 1.5, for five reduces densities p — 0.4, 0.6, 0.7, 0.8, and 0.9, calculated with the ODS scheme, compared to MC simulation data [36], Taken from Ref. [80],...

Fig. 3. Pair correlation functions as functions of reduced distance r. The b.c.c. arrangement transforms 10 f.c.c. ai 220 MC moves when ihe Lcnnard-Jones potential is employed. If at this stage the interaction is changed to cesium potential, (here is a change back to the b.c.c. at 450 MC moves. (From Yashonalh and Rao (7).)...

Reduced radial distribution function, or oxygen-oxygen pair correlation function g0Q(r), for H20(l) at 300 K X rays, solid line [61,62] X rays dashed line [63] neutron, dot-dashed line [64] neutron, gray line [65]. The peaks indicate a first (nearest)-neighbor, a second, and a third 0-0 distance at approx 2.9 A, 4.3 A, and 6.5 A, respectively. 

The local structure in liquids can be measured by X-ray diffraction and described by either a radial distribution function or the pair correlation function. In particular, the oxygen-oxygen pair correlation function or reduced radial distribution function for water, goo(r) Fig. 11.74, can be obtained from... 

Figure 1 shows the predicted spatial pair correlation functions, yg. reduced separation distance, r = for a typical, dilute supercritical solution using the LJ parameters given... 

There have been many analytic and numerical studies of the structure that solids induce in an adjacent fluid. Early studies focussed on layering in planes parallel to a flat solid surface. The sharp cutoff in fluid density at the wall induces density modulations with a period set by oscillations in the pair correlation function for the bulk fluid [169 173]. An initial fluid layer forms at the preferred wall fluid spacing. Additional fluid molecules tend to lie in a second layer, at the preferred fluid fluid spacing. This layer induces a third, and so on. The pair correlation function usually decays over a few molecular diameters, except near a critical point or in other special cases. Simulations of simple spherical fluids show on the order of 5 clear layers [174 176], while the number is typically reduced to 3 or less for chain or branched molecules that have several competing length scales [177 180]. 

Several approaches to estimation of the internal energy have been described. These involve assuming specific relationships between the pair correlation functions gjj and the form of the interaction energy uy. The simplest of these is based on the van der Waals treatment of fluids and its application of the law of corresponding states. Examination of typical radial distribution functions for mixtures such as those shown in fig. 2.16 reveals that the maximum in each distribution function g r) occurs close to the diameter a describing the distance of closest approach for the two molecules involved. Thus, it is better to describe the radial distribution function in terms of the reduced distance r/a instead of the distance r. This conclusion leads to the assumption that... 

In this book, we shall only be interested in homogeneous and isotropic fluids. In such a case, there is a redundancy in specifying the full configuration of the pair of particles by 12 coordinates (X, X"). It is clear that for any configuration of the pair X, X", the correlation g(X, X ) is invariant to translation and rotation of the pair as a unit, keeping the relative configuration of one particle toward the other fixed. Therefore, we can reduce to six the number of independent variables necessary for the full description of the pair correlation function. For instance, we may choose the location of one particle at the origin of the coordinate system, R = 0, and fix its orientation, say, at (j) = O = t// = 0. Hence, the pair correlation function is a function only of the six variables X" = R", S2". 

The difference between the two results (3.113) and (3.114) arises from the finite number of particles in the T,V,N system. Even when there are no interactions, U Rn) = 0, there is still correlation between the particles. The density at any point in the system is p(R) = N/V. The conditional density at R given a particle at any other point R is not p(R) = N/Vbut (AT- 1 )/V. Fixing one particle at some point has an effect on the density at any other point merely because the number of particles was reduced from N to N— 1. Such an effect does not exist if we open the system, in which case the pair correlation function g0(R) is unity everywhere for an ideal gas. 

Several comments are in order as to the form (36) associated with G,(/ ), and these comments can apply also to the higher-order terms in the expansion of X(R) as well. The factor f occurring in G2(R) is our expansion parameter in the present problem, so that the higher-order terms in the expansion of X(R) contain higher powers of f. In the reduced form (36) for G2(R), the k dependence (therefore, T and p dependence) occur only in the second factor 0 as a proportionality constant. A detailed investigation we made shows that this linear oecurrence of i is a common feature associated with all terms involved in the expansion. Therefore this term can be factored out of the expansion and used to reduce the cell-pair correlation function t (a ) which is only a function of the reduced distance x, and its corresponding expansion has the following form ... 

The measurements were performed in the same manner as iho-c with the organomagnesium iodides. The reduced intensity and the total atom pair correlation functions nrc shown in Figures 10.19. The interpretation of the total atom pair correlation function followed the outline given in the previous section. The peak at 2.55 A was assigned to a Mg—Br distance and the shoulder at 3.6 A to a Br-Br distance. From the area of this shoulder—the limits of the integration are not given—the number of... 

Chiew and Glandt46 modeled the structure of a dispersion of identical spheres as an equilibrium hard sphere fluid. They used pair correlation functions to compute the contributions of pairs of spheres to the effective thermal conductivity of the dispersions. Their results, derived for arbitrary values of the conductivity of the two phases, reduce to the following equation for the case of gas bubbles in electrolyte ... 

X-ray and neutron scattering experiments yield direct information on the atomic pair-correlation functions, gj/fj, fj,). by the help of the scattering cross-section do/dQ 241,242) gj. reduced intensity I both of which depend on the scattering angle 6, which is expressed as the scattering vector k = [4n sin (0/2)]/X. X is the wavelength of the incident radiation. The intensity of the radiation scattered by each atom i depends on its scattering factor fj. 

Thus the problem of calculating the thermodynamic properties and pair correlation function of a low-density gas has been reduced to that of evaluating the integrals corresponding to the graphs with certain specified numbers of field points. There is an extensive literature that makes use of this virial expansion. 

First, we recall that the averaging process carried out over the full pair correlation function g(R, 22) has reduced the very... 

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