The thermal diffuse reflexions from a single crystal result from an exchange of energy, as well as momentum, between the beam and the crystal. The theory is outlined for the case of neutron radiation, and some properties of the diffuse reflexions are derived. Diffuse streaks have been found in the Laue diffraction pattern of an iron single crystal placed in an unmonochromatized beam of neutrons from the Harwell pile. The 110 streak has been studied in detail, and its behaviour is compared with calculations from the Weinstock-Waller-Froman theory of inelastic neutron scattering. The theory is verified to within about 15%. Inelastic scattering cross-sections are given for neutrons incident at 20 degrees on the (110) plane; they are in the region of 10 millibarns.