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 arbitrary solids. It valid thermal neutrons but slow it requires a correction, which significant isotopes that are strong coherent scatterers. In this article, we present extension of Placzek--Van Hove corrections in limit low temperatures using example solid \emph{ortho}-deuterium (sD$_2$). Our approach yields realistic up-scattering sections sD$_2$ and shows IA to be factor 2 5 too high ultracold (UCN) sD$_2$. calculations compared with previously published Monte Carlo based on dynamic structure function $S(q,\omega)$ polycrystalline found consistent them. Furthermore, provide means easily replicable neutrons. These should from now simulations UCN