Correlation Function and Structure Factor

Physical observables that can be measured in both the path-integral representation and the SSE representation include, next to the energy and the specific heat, any expectation value or correlation function that is diagonal in the basis set i). This includes the uniform or staggered magnetization in the z direction, the equal time correlation functions and structure factor of the... 

These conclusions have been strengthened by an analysis of suitable correlation functions and structure factors [99]. These results show (Fig. 31) that a cylindrical bottle brush is a quasi-lD object and, as expected for any kind of ID system, from basic principles of statistical thermodynamics, statistical fluctuations destroy any kind of long-range order in one dimension [108]. Thus, for instance, in the lamellar structure there cannot be a strict periodicity of local composition along the z-axis, rather there are fluctuations in the size of the A-rich and B-rich domains as one proceeds along the z-axis, these fluctuations are expected to add up in a random fashion. However, in the molecular dynamics simulations of Erukhimovich et al. [99] no attempt could be made to study such effects quantitatively because the backbone contour length L was not very large in comparison with the domain size of an A-rich (or B-rich, respectively) domain. 

Relationship Between the Correlation Function and Structure Factor The statistical average in the definition of the structure factor 5 (k) in Eq. 2.60 is taken with respect to the pair distribution. With Eq. 2.62, Eq. 2.60 is rewritten to... 

Finally, we comment on the difference between the self part and the full density autocorrelation function. The full density autocorreration function and the dynamical structure factor ire experimentally measured, while in the present MD simulation only the self pairt was studied. However, the difference between both correlation functions (dynamical structure factors) is considered to be rather small except that additional modes associated with sound modes appear in the full density autocorrelation. We have previously computed the full density autocorrelation via MD simulations for the same model as the present one, and found that the general behavior of the a relaxation was little changed. General trends of the relaxation are nearly the same for both full correlation and self part. In addition, from a point of numerical calculations, the self pMt is more easily obtained than the full autocorrelation the statistics of the data obtained from MD simulatons is much higher for the self part than for the full autocorrelation. 

Another method, Reverse Monte Carlo (RMC) [33], is effectively a modification of the Metropolis Monte Carlo [34] (MMC) method instead of specifying a pair potential and minimising the model s configurational energy, RMC requires one or more experimental pair correlation functions (or structure factors) and minimises the squared difference between the experimental G(r) and the model G(r). The RMC algorithm proceeds as follows ... 

The Fourier transform of the pair correlation function, the structure factor, can be measured experimentally by X-ray, neutron, or light scattering techniques [37,38,43,47]. Moreover, in the simple and often... 

The value of these approaches to the problem is that they demonstrate the possibility of evaluating the memory function through the intermolecular correlation functions and structural dynamic factor of the system of interacting Brownian particles. Theoretical evaluations of the memory function are based on some approximations which, apparently, do not allow the calculation of quantities for the limiting case c M, but they are good in the cross-over region from Rouse to entanglement dynamics. 

Typical results for a semiconducting liquid are illustrated in figure Al.3.29 where the experunental pair correlation and structure factors for silicon are presented. The radial distribution function shows a sharp first peak followed by oscillations. The structure in the radial distribution fiinction reflects some local ordering. The nature and degree of this order depends on the chemical nature of the liquid state. For example, semiconductor liquids are especially interesting in this sense as they are believed to retain covalent bonding characteristics even in the melt. 

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. 

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. 

Here, the matrices H( ), C q), and W( ) contain the functions h q), c q), and w q) which are the Fourier transformations of the corresponding correlation functions with wave vector q. Having these functions, one can find the partial static structure factors, Saffl), which are the Fourier transformed density-density fluctuation correlation functions and are proportional to the scattering intensities observable in experiments. They are defined as... 

If go(r), g CrX and g (r) are known exactly, then all three routes should yield the same pressure. Since liquid state integral equation theories are approximate descriptions of pair correlation functions, and not of the effective Hamiltonian or partition function, it is well known that they are thermodynamically inconsistent [5]. This is understandable since each route is sensitive to different parts of the radial distribution function. In particular, g(r) in polymer fluids is controlled at large distance by the correlation hole which scales with the radius of gyration or /N. Thus it is perhaps surprising that the hard core equation-of-state computed from PRISM theory was recently found by Yethiraj et aL [38,39] to become more thermodynamically inconsistent as N increases from the diatomic to polyethylene. The uncertainty in the pressure is manifested in Fig. 7 where the insert shows the equation-of-state of polyethylene computed [38] from PRISM theory for hard core interactions between sites. In this calculation, the hard core diameter d was fixed at 3.90 A in order to maintain agreement with the experimental structure factor in Fig. 5. 

So, one can consider the parameters r, Xi and Me to be equivalent. One of these parameters is introduced in each theory of polymer dynamics. Note that the correlation time is expressed through the two-particle correlation function and the dynamic structure factor of the system of the interacting Brownian particles in many-chain theories [54, 91]. 

Using the Omstein-Zemike relationship between the direct correlation function and the static structure factor, the gradient terms for the two different limits are given by ... 

An important complication arises here because the intermolecular structure factor as introduced in Eq. (2.13) has now become a function of the form factor P(q), i.e., the distribution of polyions depends on their mutual orientation and their shape and vice versa. It is only in the case of spherical polyions that S(q) and P(q) are separable by the use of center-of-mass coordinates. For rod-like polyions the mutual orientation and the spatial distribution are correlated, and for flexible polyions the chain conformation and the spatial distribution of chains depend on each other. Assuming weak interactions, several approximations were introduced to separate form- and structure factor. However, for strong, long-range electrostatic interactions intra- and intermolecular correlations cannot yet be properly separated [28]. This is an important limitation to all current theories except for Monte Carlo simulations. 

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