The lattice dynamics of an anharmonic crystal

The theory of the physical properties of an anharmonic crystal is discussed by using the thermodynamic Green's functions for the phonons. A perturbation procedure is developed to obtain the Green's functions and it is shown that for some purposes a quasi-harmonic approximation is useful, in which the frequencies of the normal modes are those determined by infra-red or neutron spectrometry. The thermodynamic, elastic, dielectric and scattering properties of an anharmonic crystal are discussed in terms of the Green's functions, and detailed expressions are given for the more important contributions. Detailed numerical calculations are presented of the thermal expansion, dielectric properties and shapes of some of the inelastically scattered neutron groups, for sodium iodide and potassium bromide. The calculations, which give reasonable agreement with experiment, show that even at quite low temperatures, the lifetimes of some of the normal modes can be quite short. By using the quasi-harmonic approximation it is shown that the large temperature dependence of the normal modes in a ferroelectric crystal can be treated adequately.