Scattering of phonons from a substitutional impurity in a body-centred cubic lattice

A model of a body-centred cubic lattice is considered, in which the polarization of phonons is taken into account in the description of scattering of phonons from a substitutional impurity involving changes in mass and force constants. In the limit of long waves, the total cross section is found to correspond to a Rayleigh type of scattering with resonance possibilities. It is shown that the cross section depends upon the direction of incidence and the state of polarization of incident phonons. This behaviour persists even in the limit of long waves and is manifested through the specific partial wave contributions to the total scattering cross section. The dispersion law for this model, which does not go over to the Debye model in the long wavelength limit, is held responsible for this behaviour. The case of an isotopic impurity is treated in detail and a directional asymmetry is predicted for the scattering process. The scalar model studied earlier is shown to be a special case of the model under consideration, with a specific choice for the force constants. The relevant Green functions are discussed in an appendix.