Calculation of Lattice-Parameter Changes with Hydrostatic Pressure from Third-Order Elastic Constants

Abstract

Part I describes the calculation of the first and second pressure derivatives of the lattice parameters from the second- and third-order elastic constants (or from the second-order constants and their pressure derivatives). These derivatives determine only the linear and quadratic terms in the expansions of the lattice parameters as power series in the pressure. The quadratic expressions obtained by truncating the power series are not suitable for extrapolation beyond a few tens of kilobars. Part II gives extrapolation formulas that turn out to be far better than the truncated power-series expansions. For quartz, magnesium, and cadmium, the resulting extrapolation from ultrasonic data taken at low pressures is found to be in reasonable agreement with x-ray determinations of the lattice parameters out to well over 100 kbar. Appendix A includes derivations of all the formulas needed to make the extrapolation from ultrasonic and thermal data. Complications arise not only from the distinction between adiabatic and isothermal coefficients, but also from the distinction between “effective” and “thermodynamic” coefficients.