Anharmonic properties of the hexagonal metals, magnesium, zinc and beryllium—II. Thermal expansion

Abstract

The generalized Gruneisen parameters γ″ = − ∂ log ω ∂ϵ″ and γ′ = − ∂ log ω ∂ϵ′ have been calculated for various normal mode frequencies in magnesium, zinc and beryllium using the model outlined in part I. The temperature dependence of the effective Gruneisen functions \\ ̄ gg ∥(T) and \\ ̄ gg ⊥(T) have been calculated. In magnesium the theoretical curve for \\ ̄ gg ⊥(T) is in good agreement with experiment but the theoretical curve for \\ ̄ gg ∥(T) is about 10 per cent higher than the experimental values over the entire temperature range. In zinc the \\ ̄ gg ∥(T) vs. T curve exhibits a steep maximum at low temperatures. While the shapes of the theoretically calculated lattice Gruneisen functions \\ ̄ gg ∥(T) and \\ ̄ gg ⊥(T) curves are similar to the experimentally observed variation of the total Gruneisen functions, the theoretical values are much higher than the experimental values. The reason for the discrepancy is the extreme sensitivity of the GPs of the low frequency modes corresponding to wave vectors lying on the vertical edge of the Brillouin zone. Perhaps a direct measurement of the pressure dependence of these frequencies in zinc by inelastic scattering of neutrons will provide more reliable values for the anharmonic parameters. The low temperature limit of \\ ̄ gg ∥(T) which depends on the TOE constants of the material does not agree with the value obtained by Barron and Munn in their analysis of the thermal expansion data. Possibly their choice of the electronic Gruneisen parameters is not correct. In beryllium there is no reliable experimental data to make a comparison with the calculated temperature dependence of the lattice Gruneisen functions. The variation of the generalised GPs of the elastic modes with the direction of propagation is illustrated by polar diagrams in the three metals.