Grüneisen's Law and the Fusion Curve at High Pressure

Abstract

The correction to Grüneisen's law corresponding to the effect of the electrons at high pressure is determined, on the assumption that the lattice contribution to the pressure is small, and that the equation of state of the solid can be approximated by results of the statistical Thomas-Fermi atom model for the electron pressure. The effect of a first-order temperature perturbation, for temperatures low in the sense of the model, is included in the Thomas-Fermi theory, but exchange is neglected. For temperatures subject to the restriction noted, but high enough so that the electronic contribution to the heat capacity dominates the lattice contribution, Grüneisen's law is valid as a formal relation on the Thomas-Fermi theory, with a physical reinterpretation of the proportionality constant which enters. The fusion curve under high compression is determined from a reformulation of the Lindemann law given previously, on the similarity assumption that the ratio of the amplitude of thermal vibration to nearest neighbor distance has the same value at high pressure as at low. The fusion temperature as a function of pressure is determined to zero order from results of the zero-temperature Thomas-Fermi theory, on the assumption that Poisson's ratio is independent of temperature and pressure. It is shown that the general temperature-perturbed Thomas-Fermi theory is adequate to determine the fusion curve on this model, except for the domain of large atomic number near the Fermi-Dirac limit of high pressure, for values of Poisson's ratio which are not relatively high. Exclusive of this domain, it is proved that the predicted fusion curve is normal in the sense of Bridgman, and is not in accord with the hypotheses of Tammann or of Schames. Fusion curves predicted for the alkali elements at high pressure are in accord with experimental results of Bridgman for low pressure, in the sense that the high-pressure curves show a reversal of normal order with respect to atomic number of melting temperature in the alkali metal family, as postulated by Bridgman from extrapolation of the low-pressure curves. The theory presented yields a general method of determining the corrections, corresponding to the lattice contributions, in thermodynamic functions determined from the Thomas-Fermi model (the method is subject to a limitation on physical validity in the Fermi-Dirac limit of high pressure).