Scattering space

For a given structure, the values of S at which in-phase scattering occurs can be plotted these values make up the reciprocal lattice. The separation of the diffraction maxima is inversely proportional to the separation of the scatterers. In one dimension, the reciprocal lattice is a series of planes, perpendicular to the line of scatterers, spaced 2Jl/ apart. In two dimensions, the lattice is a 2D array of infinite rods perpendicular to the 2D plane. The rod spacings are equal to 2Jl/(atomic row spacings). In three dimensions, the lattice is a 3D lattice of points whose separation is inversely related to the separation of crystal planes. 

The smearing of the electron density due to thermal vibrations reduces the intensity of the diffracted beams, except in the forward S = 0 direction, for which all electrons scatter in phase, independent of their distribution. The reduction of the intensity of the Bragg peaks can be understood in terms of the diffraction pattern of a more diffuse electron distribution being more compact, due to the inverse relation between crystal and scattering space, discussed in chapter 1. 

It is noted that both the probability distribution of Eq. (2.16) and the temperature factor of Eq. (2.19) are Gaussian functions, but with inversely related mean-square deviations. Analogous to the relation between direct and reciprocal space, the Fourier transform of a diffuse atom is a compact function in scattering space, and vice versa. 

We note that Eq. (3.19) again illustrates the inverse relation between direct and scattering space, a contraction of charge density corresponding to an expansion in scattering space, and vice versa. Equation (3.19) implies that the /c-modified scattering factor can be obtained directly from the unperturbed IAM scattering factors tabulated in the literature. 

The size of scattering space is defined with a quantity like S 8 = 4 (sin0)/A... 

That is done in the penultimate section where we present some preliminary DWBA calculations of the resonances in the H+(X) -> H(X) addition reaction using a fit to an ab initio potential energy surface. A reduced-dimensionalty scattering space is derived based on a novel scattering path hamiltonian. 

Additionally, Porod (54) derived an integral over all scattering space, the so-called invariant, which is directly related to the mean square fluctuation of electron density. Irrespective of the geometrical features of the structure. 

Figure 2.1. On the definition of a scattering space V R is a receiver of. scattered light dil is a. solid angle element Vium is a luminous space K is a vessel with scattered medium, D are diaphragms...

The receiver measures the intensity 7 of light scattered on all the particles inside a scattering space V, which is cut by the receiver s solid angle from the luminous space Vjum (Garrabos < t al., 1978), see Figure 2.1... 

So, we conditionally divide the whole scattering space V into N equal elements of a volume V each. 

N being the number of molecules per unit volume. Considering fluctuations in the number of molecules in different elements of space v independent, we get for the intensity of scattering on all the scattered space V... 

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