The One- and Two-Body Densities of Crystalline Matter and Bragg and Diffuse Scattering of Neutrons and X-Rays

Abstract

This work presents a theoretical treatment of the Bragg and diffuse scattering of an incident double-beam of neutrons or X-rays from crystalline matter. The incoming double-beam is a linear superposition of two plane waves with the same energy. It is shown that the total Bragg plus diffuse scattering differential cross section is a sum of terms stemming from the scattering of the two individual plane waves in the incoming double-beam and of terms originating in the interference of the scattering of the two superimposed incident plane waves. The leading-order dominant Bragg peaks in the total differential cross section are size-extensive and depend on the lattice Fourier transform of the one-body density of the scattering material. It is proven that the Bragg peaks are a consequence of the long-range behavior of the two-body density for large interparticle distances. The diffuse background scattering contributions in the total differential cross section for the scattering of a double-beam depend on the lattice Fourier transform of the one-body density and, more importantly, on the continuous Fourier transform of the (discrete) lattice Fourier transform of the two-body density of the scatterer, which latter transform is a function of the relative position vector of the two particles. One of the most significant conclusions from these findings is that the full two-body density of a crystalline material is experimentally accessible by means of the diffuse background scattering of a double-beam of neutrons or X-rays in which the difference of the two wave number vectors in the incoming two-beam is a reciprocal lattice vector of the scattering crystal. This paper deals also with the crystallographic symmetries in the Bragg and the diffuse scattering structure functions. Based on results of previous research, it is found that both scattering structure functions follow exactly the same crystallographic symmetry patterns as the lattice Fourier transform functions of the two-body density do. The formal theory of Bragg and diffuse scattering of a neutron or X-ray two-beam derived here may be readily related to previous and ongoing exact numerical Monte Carlo calculations of the spatial microstructure of crystals, in which the local one- and two-body densities are computed by means of exact group-theoretical Fourier path integral Monte Carlo simulations.