Interaction of Ultra-Cold Neutrons with Condensed Matter

General theory of neutron scattering (elastic and inelastic) is presented. It is applicable for the whole domain of slow neutrons and includes as limiting cases existing theories for thermal and cold neutrons and for elastic scattering of UCN. New expression for inelastic scattering cross section for UCN is proposed. It differs from the usually used by proper account of re-scattering processes. Evidence for small heating and cooling of UCN is given.