Validity of the Sutherland-Lindemann law and melting temperatures in the Earth's interior

Abstract

SUMMARY Melting is a very complex physical phenomenon resulting from several combined effects. While the present development of theory does not allow us to model it in a fully satisfactory way, there is nevertheless little doubt that the Lindemann law (in fact proposed by Sutherland 20 years earlier), based on a mere mechanical instability, is incorrect in absolute terms. On the other hand, most of the theoretical approaches so far developed lead to scaling laws of Sutherland-Lindemann type, independently of the physical model used. This suggests that Sutherland-Lindemann theory may work as a semi-empirical scaling law, provided that all terms contributing to melting scale identically. While agreement with experiment in this sense appears so far unpredictable, an exhaustive test has never been performed. The present work is focused on such a test. Sutherland-Lindemann scaling law is analysed in a form tailored to be applied to the Earth's interior. The high temperatures involved would require a strongly anharmonic theory, but the relevant parameters are still unavailable. Quasi-harmonic approximation is therefore used, together with a Debye-Brillouin spectrum, in a formulation which combines practical efficiency and availability of parameters. In order to account for experimental inaccuracy we use all available data and consider only the cases with at least two independent sources. ‘Effective’ quasi-isotropic (i.e. corresponding to the equivalent isotropic continuum) elastic moduli are obtained through Voigt, Reuss, Hill and Hashin-Shtrikman averaging schemes from single-crystal laboratory data. High- and room-temperature elastic constants are used. The limited intersection of the set of substances available with both elastic constants and melting curve as a function of pressure allow the study of 12 substances. A good agreement between theory and experiment is found for all the cases considered: for metals (Al, Ag, Au, Cu, Fe), ionic crystals (KCl, NaCl, RbCl), synthetic silicates (forsterite), and minerals (bronzite, peridotite and diopside). This result supports the validity of Sutherland-Lindemann theory in Debye-Brillouin approximation as a general scaling law and justifies its use in estimating melting temperatures from elastic constant data. On the basis of the PREM model we estimate the melting curve for the whole Earth. This in turn permits the real temperature profile to be constrained as a function of composition.