Thermoelastic properties of some cubic close-packed lattices

Abstract

A method is described whereby the strain-dependence of the first few even moments of the vibrational frequency distribution of a crystal can be used to calculate the elastic constants of moderately anharmonic crystals. From the elastic constants at T = 0 the limiting Debye temperature ΘC0 can also be obtained, to the first approximation in the anharmonicity. Detailed calculations at T = 0 are carried out for cubic close-packed lattices with a number of different central force potentials. For the 6-12 potential, contributions of the zero-point energy to the shear constants are less than those calculated by Salter and by Leibfried and Ludwig, who relied on the strain-dependence of the second moment only; and departures from the Cauchy relation are less severe. The bulk modulus is in fair agreement with earlier calculations, including the variational treatment of Bernardes. Anharmonic corrections to the quasi-harmonic value of ΘC0 are appreciable, but less than those estimated by Leibfried and Ludwig and by Wallace. For a more general potential, (r), between nearest neighbours only, general expressions are derived for the cij and ΘC0 in terms of the derivatives (n)(r). A positive (iv)(r) always tends to increase the cij above the values calculated from the static lattice, while the effect of (iii)(r) is usually to decrease them. The full expression for ΘC0 includes not only the leading terms calculated by Flinn and Maradudin, but also additional terms which are of comparable magnitude. The results are applied to the thermoelastic properties of the rare gas solids. With parameters chosen to fit the molar volume and cohesive energy, both all-neighbour and nearest-neighbour 6-12 models give bulk moduli at T = 0 in good agreement with experimental values for Xe and Ne; the predicted value for Kr is 3.21(±5%) × 1010 dyn cm-2, and for Ar 2.61(±5%) dyn cm-2. Close agreement is obtained with a recent measurement of the transverse wave velocity in polycrystalline argon. Anharmonic corrections to quasi-harmonic values for ΘC0 are found to be positive, being approximately 1.5%, 2%, 4% and 16% for Xe, Kr, Ar and Ne respectively. Roughly similar corrections will be needed for the exp-6 potential, but negative corrections are found for the Guggenheim and McGlashan argon potential, which turns out to be quite incompatible with the experimental ΘCO. At high temperatures formal expressions are derived for the adiabatic and isothermal elastic constants, but their evaluation requires a rather uncertain extrapolation. For the 6-12 potential it appears that vibrational contributions to the shear elastic constants are small at high temperatures, and that Cad44 < Cad12. The model deviates quite strongly from the experimental equation of state of xenon, especially at the highest temperatures.