Difference pair correlation function

For all the above reasons we have defined g(C) without reference to any hypothetical, independent-site system. One simply extracts both 1(C) and all from the experimental data, and then constructs the quantity g(C). When the sites are identical in a weak sense, i.e., all k = k, some of the correlations for a given / might differ. For example, four identical subunits arranged in a square will have only one intrinsic binding constant k, but two different pair correlation functions. For this particular example we have four nearest-neighbor pair correlations g (2), and two second-nearest-neighbor pair correlations gJJ)- The average correlation for this case is... 

FIGURE 8.4 Two difference pair-correlation functions AG(rz). The dotted line shows AG(rz)(H), obtained from the difference between the vermiculite and its H-salt solution. The full line shows AG(rz)(D), obtained from the difference between the vermiculites in D- and H-salt solutions. 

FIGURE 13.5 Difference pair correlation functions AG(rz) between the atomic pair correlation functions of samples with deuterated and hydrogenous PEO. In the case of the 0.1 M salt concentration (solid line), the butylammonium chains were deuterated in both samples, whereas for the 0.03 M salt concentration (dashed line), both samples contained hydrogenous counterions. The volume fraction of PEO was 4% in all cases. 

The generalization to multicomponent systems is quite straightforward. Instead of one pair correlation g(X, X"), we shall have pair correlation functions for each pair of species a/ . For instance, if A and B are spherical particles, then we have three different pair correlation functions g,, (R), gAB(R) = 8ba(R) and gBB(R)- We shall describe these in more detail in section 2.9. 

In other words, the lEPA is accurate up to second order of perturbation theory based on the Hartree-Fock (HF) model as zeroth-order approximation. The exact third-order contribution involves—besides the terms which are properly included—the interaction between different pair correlation functions... 

However, before proceeding with the description of simulation data, we would like to comment the theoretical background. Similarly to the previous example, in order to obtain the pair correlation function of matrix spheres we solve the common Ornstein-Zernike equation complemented by the PY closure. Next, we would like to consider the adsorption of a hard sphere fluid in a microporous environment provided by a disordered matrix of permeable species. The fluid to be adsorbed is considered at density pj = pj-Of. The equilibrium between an adsorbed fluid and its bulk counterpart (i.e., in the absence of the matrix) occurs at constant chemical potential. However, in the theoretical procedure we need to choose the value for the fluid density first, and calculate the chemical potential afterwards. The ROZ equations, (22) and (23), are applied to decribe the fluid-matrix and fluid-fluid correlations. These correlations are considered by using the PY closure, such that the ROZ equations take the Madden-Glandt form as in the previous example. The structural properties in terms of the pair correlation functions (the fluid-matrix function is of special interest for models with permeabihty) cannot represent the only issue to investigate. Moreover, to perform comparisons of the structure under different conditions we need to calculate the adsorption isotherms pf jSpf). The chemical potential of a... 

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. 

The correlation functions of the partly quenched system satisfy a set of replica Ornstein-Zernike equations (21)-(23). Each of them is a 2 x 2 matrix equation for the model in question. As in previous studies of ionic systems (see, e.g.. Refs. 69, 70), we denote the long-range terms of the pair correlation functions in ROZ equations by qij. Here we apply a linearized theory and assume that the long-range terms of the direct correlation functions are equal to the Coulomb potentials which are given by Eqs. (53)-(55). This assumption represents the mean spherical approximation for the model in question. Most importantly, (r) = 0 as mentioned before, the particles from different replicas do not interact. However, q]f r) 7 0 these functions describe screening effects of the ion-ion interactions between ions from different replicas mediated by the presence of charged obstacles, i.e., via the matrix. The functions q j (r) need to be obtained to apply them for proper renormalization of the ROZ equations for systems made of nonpoint ions. 

The prerequisite for an experimental test of a molecular model by quasi-elastic neutron scattering is the calculation of the dynamic structure factors resulting from it. As outlined in Section 2 two different correlation functions may be determined by means of neutron scattering. In the case of coherent scattering, all partial waves emanating from different scattering centers are capable of interference the Fourier transform of the pair-correlation function is measured Eq. (4a). In contrast, incoherent scattering, where the interferences from partial waves of different scatterers are destructive, measures the self-correlation function [Eq. (4b)]. 

The pair-correlation function for the segmental dynamics of a chain is observed if some protonated chains are dissolved in a deuterated matrix. The scattering experiment then observes the result of the interfering partial waves originating from the different monomers of the same chain. The lower part of Fig. 4 displays results of the pair-correlation function on a PDMS melt (Mw = 1.5 x 105, Mw/Mw = 1.1) containing 12% protonated polymers of the same molecular weight. Again, the data are plotted versus the Rouse variable. 

The result for the like and unlike partners definition can be obtained by very similar arguments and involves all three pair correlation functions. The various definitions and results can equally be applied to defects which are not ionic by merely substituting the words different kind for opposite charge and either kind for either charge in the definitions. 

A reasonable approximation for the pair correlation function of the j8-process may be obtained in the following way. We assume that the inelastic scattering is related to imcorrelated jumps of the different atoms. Then all interferences for the inelastic process are destructive and the inelastic form factor should be identical to that of the self-correlation function, given by Eq. 4.24. On... 

Three doubly spin-labelled [2]catenanes with different sizes were studied by 4-pulse DEER.52 The experimental distribution of interspin distances was compared with a theoretical pair-correlation function computed based on geometrical constraints. In chloroform solution the medium and large catenanes were close to fully expanded, but in glassy o-terphenyl they were partially collapsed. For the smaller catenane there was a higher population of shorter interspin distances, which was attributed to interactions between unsaturated sections of the molecule. 

In the last region K = 0 for T -C c/(p2/imp) we come back to the strong pinning case, discussed in section 3.3 before, and calculate the pair correlation function exactly. Taking into account that the hi s are independent on different lattice sites, i.e., hihj oc the (discrete) phase correlation function is given... 

Chapter 8 provides a unified view of the different kinetic problems in condensed phases on the basis of the lattice-gas model. This approach extends the famous Eyring s theory of absolute reaction rates to a wide range of elementary stages including adsorption, desorption, catalytic reactions, diffusion, surface and bulk reconstruction, etc., taking into consideration the non-ideal behavior of the medium. The Master equation is used to generate the kinetic equations for local concentrations and pair correlation functions. The many-particle problem and closing procedure for kinetic equations are discussed. Application to various surface and gas-solid interface processes is also considered. 

The auxiliary equations for pair correlation functions also differ from previous ones in the very same respect ... 

In a solution containing n atomic species, p, q, etc., the number of different pair interactions is n(n + l)/2. The pair correlation function, gm(r), measures the probability of finding an atom q at a distance r... 

The PRISM (Polymer-Reference-Interaction-Site model) theory is an extension of the Ornstein-Zernike equation to molecular systems [20-22]. It connects the total correlation function h(r)=g(r) 1, where g(r) is the pair correlation function, with the direct correlation function c(r) and intramolecular correlation functions (co r)). For a primitive model of a polyelectrolyte solution with polymer chains and counterions only, there are three different relevant correlation functions the monomer-monomer, the counterion-counterion, and the monomer-counterion correlation function [23, 24]. Neglecting chain end effects and considering all monomers as equivalent, we obtain the following three PRISM equations for a homogeneous and isotropic system in Fourier space ... 

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