Scattering of Neutrons by an Anharmonic Crystal

The one-phonon differential scattering cross section for the coherent scattering of thermal neutrons by an anharmonic Bravais crystal is obtained correct to the lowest nonvanishing order in the anharmonic force constants. Cubic and quartic anharmonic terms are retained in the crystal's Hamiltonian. It is found that the $\delta$-function peaks in the energy distribution of the scattered neutrons for a fixed momentum transfer (which occur at the unperturbed phonon energies) in the harmonic approximation are broadened and their positions are shifted in an anharmonic crystal. Some numerical results for the magnitudes of the phonon widths and shifts are obtained for a simple model of a face-centered cubic crystal.