Correlation functions static structure factor

The decay of the structural correlations measured by the static structure factor can be studied by dynamic scattering techniques. From the simulations, the decay of structural correlations is determined most directly by calculating the coherent intermediate scattering function, which differs from Eq. [1] by a time shift in one of the particle positions as defined in Eq. [2] ... 

Fig. 4.1 a Typical time evolution of a given correlation function in a glass-forming system for different temperatures (T >T2>...>T ), b Molecular dynamics simulation results [105] for the time decay of different correlation functions in polyisoprene at 363 K normalized dynamic structure factor at the first static structure factor maximum solid thick line)y intermediate incoherent scattering function of the hydrogens solid thin line), dipole-dipole correlation function dashed line) and second order orientational correlation function of three different C-H bonds measurable by NMR dashed-dotted lines)... 

There is an important case which is intermediate between small bounded systems and macroscopic fully extended systems, namely the description of the surface region of a macroscopic metal. The correlation functions which describe density fluctuations in the surface region are extremely anisotropic and of long range, very unlike their counterparts in the bulk, and the thermodynamic limit must be taken with sufficient care. Consider the static structure factor for a large system of N particles contained within a volume Q,... 

For the discussion of the properties of the static structure factors, it is often more convenient to write the scattering functions in terms of a space correlation function y(r)4wr2dr. 

To calculate Rpp(t), the two-particle direct correlation function c 2(q) is required which is obtained from the nearly analytical expression given by Baus and Colot for a 2-D system [177]. The static structure factor S(q) has been calculated from the two-particle direct correlation function through the well-known Omstein-Zemike relation [21]. 

Detailed high-frequency (terahertz) dynamical studies of glasses have been probed by inelastic X-ray scattering (IXS) [139], The advantage of this technique is that with reliable measurements it allows determination of the so-called nonergodicity parameter f(q, T) as a function of wavevector q this quantity is defined by the long time limit of the density-density correlation function F(q, t) divided by the static structure factor [15],... 

The liquid-like order present in the pair correlation function manifests itself as a peak in the static structure factor (S(q)). The scaling of the position qm of this maximum with the density has attracted much attention in the literature [40, 51-53]. Scaling arguments suggest [35, 42, 49, 51] that qm obeys the relation qm p1/3 for dilute solutions and qm pv/(3v 1) for semidilute solutions. Here v is the scaling exponent for the end-to-end distance, i.e., RE hT. The overlap threshold concentration is estimated as p N1 3v. As a conse-... 

The static structure of three-dimensional colloidal suspensions is usually determined experimentally, not by measuring directly g(r) in real space, but by measuring the static structure factor S(k) in the reciprocal space, which is the Fourier transform of the local particle-concentration correlation function. The radial distribution function is directly related to the Fourier transform of S(k), as it is explained below. Let us consider a system of N particles in a volume V. The local particle concentration p(r) at the position r is given by... 

The function P(k) = b(ko/2)2, called the form factor, accounts for interference effects arising from the shape of the scattering particles, and the static structure factor S(k) accounts for interference effects due to spatial particle correlation. 

Being of fourth order in the particle operators, the static structure factor provides a measure of second order correlation. Defining the normalized second order correlation function as... 

Figure 2.15 Scheme of the pair potential U(r), the pair correlation function g(r) and the static structure factor 5(g). 

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... 

FiGURE 15.3. (a) The static structure factors for sulfonate groups and water molecules, Ssiq) and S q), and (b) the SO3H-SO3H and S03H-water pair correlation functions, gssif) and gsw(r), for different partial densities of water, p , at T = 300 K. Adapted from Ref. [53]. 

Liquids (at least the ones concerned with here) are isotropic, and therefore the correlation function depends only on the magnitude of r (r), and the structure factor (static and dynamic) depends only on the magnitude of Q (Q). The relations between the static structure factor and the radially symmetric pair distribution function (also known as the radial distribution function, rdf) then can be expressed as (compare Eq. (29.21))... 

Equation 2.64 illustrates that the static structure factor, and hence the scattering pattern obtained in the light-scattering experiments, is the Fourier transform (see Appendix A2) of the autocorrelation function of the local segment density. 5(k) indicates which wave vector components are present in the correlation function. 

In the first two chapters, we learned about thermodynamics (free energy, osmotic pressure, chemical potential, phase diagram) of polymer solutions at equilibrium and static properties (radius of gyration, static structure factor, density correlation function) of dissolved polymer chains. This chapter is about dynamics of polymer solutions. Polymer solutions are not a dead world. Solvent molecules and polymer chains are constantly and vigorously moving to change their positions and shapes. Thermal energy canses these motions in a microscopic world. 

The static structure factor 5 (k) for this correlation function is simple ... 

Figure 4.15. The reciprocal of the static structure factor S(k) plotted as a function of k. The slope of the plot is equal to the square of the correlation length...

The static structure factor can be Fourier transformed into real space to give the pair correlation function, g(r), which is proportional to the probability of finding an atom at a position r relative to a reference atom at the origin. Due to the isotropic nature of liquids and glasses it is only necessary to consider the distance from the reference atom hereafter the vector dependence becomes a magnitude dependence. Therefore the pair correlation function, g(r), is expressed in terms of S(q) ... 

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 ... 

The study of the structures within the GFH picture can be accomplished in a way closely related to that of the PI case [35]. The reason lies in the fact that every actual particle is described by a proper thermal packet Eq. (58) about its centroid. As a matter of fact, can be taken as an approximation to the P oo limit of the particle thermal quantum spread in a PI treatment. The formal operations to derive the GFH structures utilize the functional derivatives technique and can be found, together with the related formulas, in works by the author (see, e.g.. Refs. 35, 131, and 132). One obtains the three static structure factors, namely, instantaneous total continuous linear response and centroids together with their associated radial correlation functions. It is... 

Apphcation of the standard functional procedure involving an external field acting on the SVP centroids leads to the static structure factor for centroids cM SVP which takes the classical forms contained in Eq. (66), but involving the SVP pair radial correlation function between centroids The absence of... 

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