Thermal Expansion and Other Anharmonic Properties of Crystals

Abstract

The temperature dependences of the thermodynamic functions, as derived from lattice dynamics, are examined for the limit of low temperature and also for high temperatures (those above a high characteristic temperature). Particular attention is given to the effects of anharmonic terms in the lattice potential energy. Detailed calculations are reported for central-potential models for fcc, bcc, and hcp lattices. In particular, the normal-mode frequencies and Gr\"uneisen parameters were calculated for a large number of points in the Brillouin zone as a function of volume; and the specific heat, compressibility, thermal-expansion coefficient, and macroscopic Gr\"uneisen parameter were calculated as functions of temperature and volume. At fixed volume the isothermal compressibility shows little temperature dependence and the explicit anharmonic contribution is small; at zero pressure the compressibility increases with increasing temperature and the explicit anharmonic contribution is again small. The thermal-expansion coefficient exhibits similar behavior at high temperatures. The anharmonic specific heat is proportional to temperature at high temperatures, and also depends strongly on the volume. The effective Debye temperatures and the macroscopic Gr\"uneisen parameters exhibit a wide variety of temperature and volume dependences. Approximations are developed for quantities which determine the behavior of thermodynamic functions at low and high temperatures, and approximate relations between several anharmonic properties are found. These approximations are tested by comparison with accurate calculations for the central-potential models.