Thermal and Elastic Properties of Crystals at Low Temperatures

Abstract

It is shown that to a first approximation (first-order perturbation theory in the quartic terms and second order in the cubic terms) the thermal free energy of an anharmonic crystal at low temperatures is given by an effective harmonic frequency distribution. The limiting form of this distribution at low frequencies can be calculated from the velocity of long elastic waves just as for a harmonic crystal, which implies that the Debye temperature ${{\ensuremath{\Theta}}_{0}}^{c}$ calculated from the heat capacity at low temperatures is equal to ${{\ensuremath{\Theta}}_{0}}^{\mathrm{el}}$ calculated from the elastic constants at low temperatures. Discrepancies between ${{\ensuremath{\Theta}}_{0}}^{c}$ and ${{\ensuremath{\Theta}}_{0}}^{\mathrm{el}}$ found in earlier theoretical work are discussed, and it is suggested that they are mainly due to a specific approximation employed in deriving ${{\ensuremath{\Theta}}_{0}}^{\mathrm{el}}$.