Anharmonicity in mineral physics: A physical interpretation

Abstract

[1] Mineral physicists often refer to the quasi-harmonic approximation. This approximation accepts the intrinsically anharmonic effects of positive thermal expansion coefficient, alpha, and pressure-dependent bulk modulus, K-T. However, for insulators at high temperatures, the classical specific heat, C-V, and the product alphaK(T) are assumed to be independent of temperature at constant volume. Departures from this approximation are termed anharmonicity. We interpret this distinction using derivatives of the atomic potential function, phi(r), with atomic spacing, r. C-V and K-T are reasonably explained by harmonic bonds, that is, phi proportional to (r - a)(2), where a is the equilibrium value of r, so that phi" = d(2)phi/dr(2) is the only derivative considered. Thermal pressure or expansion and pressure dependence of K-T depend on phi"' (referred to as first-order anharmonic effects). Temperature dependence of C-V requires phi(iv) and is a second-order anharmonic effect. The temperature variation of alphaK(T) arises from phi(v), a third-order effect. By calculating the Gruneisen parameter, gamma, as the ratio of thermal pressure to thermal energy, we relate its anharmonicity to a ratio of derivatives of phi. For all commonly used finite strain theories, this gives a temperature variation of gamma at high temperature and constant volume systematically less than that of C-V. Anharmonicity of C-V, which has been more comprehensively studied, may be a few percent at 2000 K, but decreases strongly with compression, so that anharmonicity of both gamma and C-V is negligible (less than 1% in the case of g under deep-Earth conditions).