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

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 \<SUP>-</SUP>gg<SUB>||</SUB>(T) and \<SUP>-</SUP>gg ⊥(T) have been calculated. In magnesium the theoretical curve for is in good agreement with experiment but the theoretical curve for \<SUP>-</SUP>gg<SUB>||</SUB>(T) is about 10 per cent higher than the experimental values over the entire temperature range. In zinc the \<SUP>-</SUP>gg<SUP>||</SUP>(T) vs. T curve exhibits a steep maximum at low temperatures. While the shapes of the theoretically calculated lattice Gruneisen functions \<SUP>-</SUP>gg<SUB>||</SUB>(T) and 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 \<SUP>-</SUP>gg<SUB>||</SUB>(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

Anharmonic properties of the hexagonal metals, magnesium, zinc and beryllium - I. Lattice dynamics and third order elastic constants

The lattice dynamics and the anharmonic properties of the hexagonal metals magnesium, zinc and beryllium have been worked out utilising the approach of Keating. In Magnesium, the dispersion curves are fitted using ten second order parameters and the six second order elastic constants are evaluated. The ten third order elastic constants are calculated using three third order parameters. The experimental measurements on the pressure derivatives of the second order elastic constants in magnesium are in good agreement with the calculated values. The dispersion relations and the second order elastic constants in zinc are fitted using twelve second order parameters. The third order elastic constants of zinc evaluated with the use of three anharmonic parameters are in good agreement with experiment. The dispersion curves and the second order elastic constants of beryllium are obtained using twelve second order parameters. The three third order parameters are obtained from the measured pressure derivatives of three of the second order elastic constants and the third order elastic constants of this metal are computed. The anisotropic thermal expansion of these metals on the present model will be discussed in Section 2 of this paper