Fourier transform of the density correlation function

In the classical limit the probability 11/ of initial state / represents essentially the time Fourier transform of the density correlation function. It contains information about the particle density fluctuations in the scattering system. 

Classical scattering theory shows that the coherent scattering from a one-component system is proportional to the Fourier transform of the density correlation function, C(r), of the particles in the system (19,21,22) ... 

Thus, the angular dependence of the scattering intensity of a binary mixture can be analyzed as the Fourier transform of the density correlation function of any of the scattering particles scaled by the square of the difference in scattering lengths of the two components. 

Since experiments for Kr have been performed at small angle neutron scattering for some low density states, we present the results of the Fourier transform of the direct correlation function, c(q) = (S(q) — 1 )/p.S (7/), rather than those of the structure factor S(q). Figure 20 shows the curves of c q). As it can be seen, the theoretical results, obtained by HMSA+WCA and MD with the AS plus AT potentials, are in excellent agreement with the experimental data [12]. While the AT contribution is included by means of an effective pair potential in the SCIET, it is used under its original form owing to Eqs.(l 19) and... 

Equation 14.29 defines the density correlation function C(r), where p(f) is the density of material at position r, and the brackets represent an ensemble average. In Equation 14.30, A is a normalization constant, D is the fractal dimension of the object, and d is the spatial dimension. Also in Equation 14.30 are the limits of scale invariance, a at the smaller scale defined by the primary or monomeric particle size, and at the larger end of the scale h(rl ) is the cutoff function that governs how the density autocorrelation function (not the density itself) is terminated at the perimeter of the aggregate near the length scale As the structure factor of scattered radiation is the Fourier transform of the density autocorrelation function. Equation 14.30 is important in the development below. 

Cross-correlation and spectral analysis have proven invaluable tools for quantifying the frequency dependent characteristics of the human subject. The cross-spectral density function, or cross-spectrum Sxyif), can be obtained from the random target x t) and random response y t) by taking the Fourier transform of the cross-correlation function Vxyir), that is, Sxyif) = Ffr yfr), or in the frequency domain via Sxy if) = X(/) y(/), or by a nonparametric system identification approach (e.g., spa.m in Matlab ). The cross-spectrum provides estimates of the relative amphtude (i.e., gain) and phase-lag at each frequency. Gain, phase, and remnant frequency response curves provide objective measures of pursuit... 

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... 

An IR spectrum reflects the Fourier transform of the molecular dipole moment. The absorption coefficient, a(P) measured by IR spectroscopy is given by Eq. (6), where the infrared spectral density is the Fourier transform of the time-correlation function for the dipole moment [11] ... 

The measured spin relaxation parameters (longitudinal and transverse relaxation rates, Ri and P2> and heteronuclear steady-state NOE) are directly related to power spectral densities (SD). These spectral densities, J(w), are related via Fourier transformation with the corresponding correlation functions of reorientional motion. In the case of the backbone amide 15N nucleus, where the major sources of relaxation are dipolar interaction with directly bonded H and 15N CSA, the standard equations read [21] ... 

A similar approach, also based on the Kubo-Tomita theory (103), has been proposed in a series of papers by Sharp and co-workers (109-114), summarized nicely in a recent review (14). Briefly, Sharp also expressed the PRE in terms of a power density function (or spectral density) of the dipolar interaction taken at the nuclear Larmor frequency. The power density was related to the Fourier-Laplace transform of the time correlation functions (14) ... 

According to standard NMR theory, the spin-lattice relaxation is proportional to the spectral density of the relevant spin Hamiltonian fluctuations at the transition frequencies coi. The spectral density is given by the Fourier transform of the auto-correlation fimction of the single particle fluctuations. For an exponentially decaying auto-correlation function with auto-correlation time Tc, the well-known formula for the spectral density reads as ... 

Derive an expression for the Fourier transform of the density-density correlation function and hence the scattering from a system of dilute, and hence noninteracting polymers. The chains are described by the position R s) of monomer s along the contour. One chain end has s = 0 and the other has 5 = A, for a finite-sized chain of N monomers. The density is related to the position vector by... 

The Fourier transform of this temporal correlation function (Oberhettinger, 1973) is the spectral density accurate up to the constant (see Elquation 2.3-26) and represents the Loreiitzian... 

The subscripts 1 and 2 refer to one substance dissolved in another substance (including one solid sample mixed with another solid sample) coh refers to the coherent scattering, and V is the partial molar volume. The scattering law 5 (Q) here is the Fourier transform of the density fluctuation correlation function [instead of G(r, f)] of the scattering centers ... 

The above theoretical framework provides a technique to compute the scattering functions of ordered phases. The experimentally observed scattering intensities correspond to the Fourier transform of the density-density correlation function [35],... 

Spontaneous thermal fluctuations of the density, p r,t), the momentum density, g(r,t), and the energy density, e(r,t), are dynamically coupled, and an analysis of their dynamic correlations in the limit of small wave numbers and frequencies can be used to measure a fluid s transport coefficients. In particular, because it is easily measured in dynamic light scattering. X-ray, and neutron scattering experiments, the Fourier transform of the density-density correlation function - the dynamics structure factor - is one of the most widely used vehicles for probing the dynamic and transport properties of liquids [56]. 

The qualitative difference between low-density and high-density rotational relaxation is clearly reflected in the Fourier transform of the normalized angular momentum correlation function ... 

In the experimental geometry used in this study, the distribution of scattered intensity measured by the detector is the two dimensional Fourier transform of the cross section of the electron density correlation function with a plane perpendicular to the extrusion direction (11). 

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