Fourier transform of the density

One of the main points in the papers by Mermin and his collaborators [9 - 18] is the insistence on the primacy of reciprocal space. The properties of the Fourier transform of the density rather than the density itself determine those properties which are of importance for "generalized" crystallography. As pointed out by Mermin that point view was stressed in a paper by Bienenstock and Ewald already in 1962 [26]. 

For a uniform charge distribution within a spherical atom the Fourier transform of the density has been shown (equation 5.6) to be of the form sin a/a, for a wave of phase a in momentum space. From the Bragg equation (Figure 2.9), A = 2dsin0, it follows that electrons at a distance d = A/2sin0 apart, scatter in phase, i.e. with phase difference 27T. At a separation r the relative phase shift a, is given by ... 

The one-dimensional Fourier transform of the density function becomes (u =... 

Systems in the collinear eZe configuration which have tori would be the antiproton-proton-antiproton (p-p-p) system, the positronium negative ion (Pr- e-e-e)), which corresponds to the case of Z= 1, = 1, and If these systems have bound states, we can see the effect of our finding in the Fourier transform of the density of states for the spectrum. For a positronium negative ion, the EBK quantization was done [34]. Stable antisymmetric orbits were obtained and were quantized to explain some part of the energy spectrum. As hyperbolic systems, H and He have been already analyzed in Refs. 11 and 17, respectively. Thus, Li+ is the next candidate. We might see the effect of the intermittency for this system in quantum defect as shown for helium [14]. 

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. 

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. 

The double differential scattering cross section, according to (8.15), is proportional to the space-time Fourier transform of G(r,f). This is analogous to the fact, discussed in Section 1.5.2, that in the static approximation the intensity I(q) (or the differential scattering cross section ds/dQ) is given by the spatial Fourier transform of the density-density autocorrelation function defined in (1.79). In the special case of t = 0, G(r,0) denotes the probability of finding a particle at r when there is already a particle at position 0. G(r,0) is therefore related to the pair distribution function g(r) discussed in Section 4.1.1, as in... 

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... 

Characteristic function. The characteristic function of continuous distributions is the expected value of the complex exponential function e" (where i = — 1). In other words, it is the Fourier transform of the density function / ... 

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. 

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 ... 

For a periodic solid, the Fourier transforms of the density and the potentials provide a particularly convenient platform for evaluating the various terms of the total energy. We follow here the original formulation of the total energy in terms of the plane waves exp(iG r), defined by the reciprocal-space lattice vectors G, as derived by Dun, Zunger and Cohen [66]. The Fourier transforms of the density and potentials are given by... 

Before closing this section let us go back and discuss in more detail the quantity 4>(Qz), the Fourier transform of the density gradient, in the case of a... 

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]. 

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