Partial pair correlation function

Packing effects 94 Pair correlation function 60 Pair potential, effective 243 Partial structure factor 220 Partition function 25, 27, 30, 65, 85 Pentamer 78... 

Figure. 3 (a) Partial pair correlation function.s gij(B.) in liquid K-Sb alloys, (b) Total, partial, and local electronic densities of states in liquid Ko.soSbo.so- Cf. text. 

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. 

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. 

Here, H and C are symmetric matrices whose elements are the partial total hap(r) and direct cap,(r) pair correlation functions a,ft = A,B) W is the matrix of intramolecular correlation functions wap r) that characterize the conformation of a macromolecule and its sequence distribution and p is the average number density of units in the system. Equation 17 is complemented by the closure relation corresponding to the so-called molecular Percus-... 

The total G(rz) can be expressed as a neutron-weighted sum (dependent on the scattering lengths of the constituent atoms) of the partial pair correlation functions 8ifrz) according to... 

Using the method of isotope substitution, it is possible to obtain partial pair correlation functions experimentally. Two isotopically different samples were prepared at c = 0.1 M, one where all the hydrogen in the butylammonium chains was ordinary II and one where they were entirely deuterated, to make use of the large difference in scattering... 

How would pair correlation functions and partial structural factors be written for a binary molten salt, e.g., sodium chloride The corresponding correlation function would be given by... 

Determine and explain the terms radial distribution function, pair correlation function, and partial structural factors. 

Postorino, P., Ricci, M. A., and Soper, A. K., Water above its boiling point Study of the temperature and density dependence of the partial pair correlation functions. I. Neutron diffraction experiment, J. Chem. Phys. 101,4123-4-132 (1994). 

Figure 2 Pair correlation function d r) for (a) confined D2O from fully hydrated Vycor (27°C) (b) confined D2O from partially hydrated Vycor (35°C) as compared with (c) bulk water (27°C) [38],...

There have been a number of modeling efforts that employ the concept of clustering in supercritical fluid solutions. Debenedetti (22) has used a fluctuation analysis to estimate what might be described as a cluster size or aggregation number from the solute infinite dilution partial molar volumes. These calculations indicate the possible formation of very large clusters in the region of highest solvent compressibility, which is near the critical point. Recently, Lee and coworkers have calculated pair correlation functions of solutes in supercritical fluid solutions ( ). Their results are also consistent with the cluster theory. 

Figure 5 Comparison of the partial pair correlation functions for the initial (dashed lines) and final (solid lines) stages at 120 K. In both cases, systems of unit cells were applied.

If the individual partial structure factors Sy k ) can be determined, one can calculate the individual partial pair correlation function gij(r) on the basis of a Fourier transformation ... 

The fundamental problem in classical equilibrium statistical mechanics is to evaluate the partition function. Once this is done, we can calculate all the thermodynamic quantities, as these are typically first and second partial derivatives of the partition function. Except for very simple model systems, this is an unsolved problem. In the theory of gases and liquids, the partition function is rarely mentioned. The reason for this is that the evaluation of the partition function can be replaced by the evaluation of the grand canonical correlation functions. Using this approach, and the assumption that the potential energy of the system can be written as a sum of pair potentials, the evaluation of the partition function is equivalent to the calculation of... 

Here m refers to the discretized fe-space variable k = mdk, where m = 0,1,... A, while i and j refer to the discretized distance already introduced. We notice that, in the general matrix case, y couples to all other through the partial derivative dy Jdc only, as the Fourier transform, its inverse, and the closure all relate correlation functions for the same pair of sites. Two of the required partial derivatives can be calculated from the discrete Fourier transform and its discrete inverse. These are... 

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