Theory of the temperature dependence of second-order elastic constants in cubic materials

Abstract

The temperature dependence of the second-order elastic constants of cubic crystalline materials is derived on the basis of a quasiharmonic-anisotropic-continuum model. It is assumed that the strain dependence of the frequencies of oscillation in a given material is independent of wavelength. The results are expressed in terms of the properties of the static lattice on the assumption that higher-than-first-order effects in the (vibrational) strain for the actual lattice can be neglected. It is found for the present model that the temperature-dependent part of the adiabatic second-order elastic constants is not proportional to the vibrational energy, contrary to previous results based on simplifying assumptions. Mathematical expressions in terms of the second-, third-, and fourth-order elastic constants are given for three temperature regions. These are for high temperatures, where the second-order elastic constants change linearly with temperature; for low temperatures, where they change as the fourth power of the temperature; and for the difference between the actual values at 0 K and the values extrapolated back to 0 K from the high-temperature linear region, this difference being due to the effect of zero-point oscillations. The theory should be most accurate for the low-temperature (fourth-power) region, mainly because the assumption regarding wavelength independence is not required, as only long-wavelength modes are effective in this region. In succeeding articles applications to various materials for purposes of comparison with experiments will be made.