Correlation functions Van Hove

Polymer Melts Coherent Scattering and van Hove Correlation Functions. Part I Dynamics in the p-Relaxation Regime. 

The correlation functions G are the same functions which Oppenheimer and Bloom introduced in their theory of nuclear spin relaxation [304], The function has been called the time-dependent pair distribution function it should not, however, be confused with the van Hove correlation function which is often referred to by the same name. The name time-dependent intermolecular correlation function (TDICF) has, therefore, been proposed for G [71]. 

We shall now discuss three Van Hove correlation functions 18,72 obtained from our CO simulations. These functions are defined as follows ... 

Simulations can be used to evaluate several other rather informative correlation functions such as van Hove correlation function given below. 

The van Hove correlation function and the dynamic structure factor The following section is mainly based on the textbook by Egelstaff [93] and in parts on other standard texts. Again, we use the monoatomic liquid as example. In Section 2.5.2.1, only the spatial correlation of the different positions of the atoms in a liquid were discussed. 

Figure 2.18 Schematic explanation of the meaning of the van Hove correlation function. Here, O indicates the origin, (a) At t = 0, there is a scattering centre at the position indicated by the full circle. The dashed circle indicates a position where there may or may not be such a scattering centre, (b) At time t, there is a scattering centre at the position marked by the full circle, while again there may be or may not be a scattering centre at the position of the dashed circle.

The van Hove correlation function can be easily related to quasi- and inelastic scattering experiments. In the following this relation will be derived for the case of neutron scattering. 

This equation reveals the unique possibility to directly measure the self-part of the van Hove correlation function with neutrons. Since light scattering is also used in the present book the equivalent relationship for light will be briefly mentioned here too. For light scattering only small values of the wave number are significant. In this limit it is possible... 

For H in metals the measured QENS intensity, after the necessary ra v data corrections (background subtraction, detector efficiency, etc.) is proportional to the incoherent scattering function S , (Q, co) which can be written as the two-fold Fourier transform (in space and time) of the single-particle, space-time van Hove correlation function, Gg(r,t),... 

Figure 8.2 Fourier transform relationships among the van Hove correlation function G (r, 0, the intermediate scattering function F(q, f), and the dynamic structure factor S(q, co).

Figure 8.3 Qualitative behavior of the van Hove correlation functions. The full curve is Gd(r, t) and the broken curve is Gs(r, t). (After van Hove.5)...

The self part Gs(r,f) of the van Hove correlation function represents the probability that a proton, which was at the origin at time 0, will be at position r at time t. For a particle undergoing a translational diffusion with diffusion coefficient D, Gs(r,0 therefore obeys the Fickian diffusion equation... 

Gibb s free energy of mixing. 6.1.1.2 van Hove correlation function. 8.1.1 [8.14]... 

However, it should be emphasized that rather drastic superposition approximations have to be made to obtain this result these assumptions are not explicit in the original paper. From this point Hills shows that can be expressed in terms of the van Hove correlation function S(k,Uij) ... 

For data fitting, it is easier to work with the scattering function than with the double-differential cross-section since there are no experimental parameters. During a normal QENS experiment, one measures in Q-o) space motions, which occur in real space and in time. The scattering fimctions are the four-dimensional Fourier transforms of the van Hove correlation function G(r,t) [10]. 

Fig. 3 Schematic view of Van Hove correlation functions. See text for details...

We also need to remark that the Van Hove correlation function Gdir, t) is used in the analysis of inelastic neutron scattering from liquids and is defined by ... 

A detailed description of the time evolution of spatial correlations in liquids requires the introduction of a time-dependent generalization of the radial distribution function. It is the van Hove correlation function [24] which retains the microscopic nature of the system and yet are tractable within the current development in the statistical mechanical theory of liquids. 

The relevant dynamical variable which defines the van Hove correlation function is the local number density of particles... 

The SSSV theory is a generalization of the Smoluchowski-Vlasov theory of Calef and Wolynes [41] based on the interaction-site model, and its application has shown that the theory predicts some of the essential features of van Hove correlation functions of water [46]. However, the SSSV theory in its current form is valid only in the diffusion regime, and the non-Markovian effects in the memory function, which are important in the dense-liquid dynamics, cannot be properly taken into account. 

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