SUMMIT, Energy Transfer Diffusion Cross-Sections, Crystalline Moderator, Phonon Expansion

1 - Nature of physical problem solved: The program evaluates the differential energy-transfer cross section for scattering by a crystalline moderator, utilizing the so-called phonon expansion. The scattering kernel for a 1-phonon change in energy is added to that for a 2-phonon energy exchange, and so on. This program has been used to determine scattering matrices for beryllium, graphite, and oxygen. Σ(E(0) to E)/Σ((0)=(((M+1)/M)2) * √(E/E(0)) * 1/2 the integral from -1 to 1 of Σ((E(0) to E,cos(θ)) D(Cos(θ)) where E(0) and E are the initial and final energies, θ is the angle of scattering, sigma(0) the free-atom cross section, and M the ratio of the mass of the scattering nucleus to that of the neutron. 2 - Method of solution: The general method used to evaluate the scattering kernel involves two different expansions. The one that is to be used for given initial and final energies and for a given angle of scattering is determined by the magnitude of the free-atom recoil energy, (K*K)/2M. If (K*K)/2M is smaller than some quantity which is controlled by input, we use the phonon expansion. Each of the first N(PHO) terms in this expansion, where N(PHO) is an input number, is computed by numerically performing the convolution integral which determines the N-phonon cross section. The terms of order N greater than N(PHO) are approximated by means of the central-limit theorem. For larger values of (K*K)/2M, we use the short-collision-time approximation for the low-frequency modes, making the phonon expansion only for the high-frequency modes. In this case, the central-limit theorem is used to approximate the contributions from the high-frequency modes for all values of N