Incoherent approximation for neutron up-scattering cross sections and its corrections for slow neutrons and low crystal temperatures

The incoherent approximation (IA) is often used for calculating the one-phonon inelastic neutron scattering cross section for arbitrary solids. It is valid for thermal neutrons but for slow neutrons it requires a correction, which is significant for isotopes that are strong coherent scatterers. In this article, we present the extension of the Placzek-Van Hove corrections for slow neutrons in the limit of low temperatures using the example of solid ortho-deuterium (sD2). Our approach yields realistic one-phonon up-scattering cross sections for sD2 and shows the IA to be a factor of 2-5 too high for ultracold neutron (UCN) up-scattering in sD2. Our calculations are compared with previously published Monte Carlo calculations of the one-phonon cross section based on the dynamic structure function S(q,ω) of polycrystalline ortho-deuterium and are found to be consistent with them. Furthermore, we provide the means for easily replicable calculations of the one-phonon up-scattering cross sections of solid ortho-deuterium for slow neutrons. These should from now on be used in calculations and simulations of UCN scattering in sD2.