As predicted by harmonic theory the outgoing inelastic spectrum of neutrons, scattered coherently by a single crystal, for a particular angle of scattering consists of a number of delta-function peaks superposed on a continuous background. The peaks correspond to one-phonon processes in which one phonon is absorbed or emitted by the neutron; the background corresponds to multi-phonon processes. When anharmonic forces are present the delta-function peaks are broadened into finite peaks and are shifted relative to those predicted in the harmonic approximation. These anharmonic effects are treated by means of many particle perturbation theory, in which the anharmonic part of the Hamiltonian is considered as the perturbation (phonon-phonon interaction). Use has been made of diagrams for representing the various matrix elements. Since the one-phonon peaks are considered as separate from the background without confining oneself to lowest order perturbation theory, the treatment is restricted to the case where the line width of a phonon state is small with respect to the energy of the phonon also for strong coupling between the phonons. In this connection the results obtained are expected to be valid only in the temperature range from absolute zero up to temperatures not much higher than the Debye temperature. For these temperatures the influence of two-phonon processes on the line shape may be neglected. An expansion for calculating the line shift and line width in powers of u/d and in terms of simple connected diagrams is obtained (u is the average atomic displacement, d is the smallest interatomic distance in the crystal). Formulae, which express the shift and width in the parameters of the lattice, are given valid to order (u/d2
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