Anharmonicity in mineral physics: A physical interpretation

Abstract

Mineral physicists often refer to the quasi‐harmonic approximation. This approximation accepts the intrinsically anharmonic effects of positive thermal expansion coefficient, α, and pressure‐dependent bulk modulus, KT. However, for insulators at high temperatures, the classical specific heat, CV, and the product αKT 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, ϕ(r), with atomic spacing, r. CV and KT are reasonably explained by harmonic bonds, that is, ϕ ∝ (r − a)2, where a is the equilibrium value of r, so that ϕ″ = d2ϕ/dr2 is the only derivative considered. Thermal pressure or expansion and pressure dependence of KT depend on ϕ‴ (referred to as first‐order anharmonic effects). Temperature dependence of CV requires ϕiv and is a second‐order anharmonic effect. The temperature variation of αKT arises from ϕv, a third‐order effect. By calculating the Grüneisen parameter, γ, as the ratio of thermal pressure to thermal energy, we relate its anharmonicity to a ratio of derivatives of ϕ. For all commonly used finite strain theories, this gives a temperature variation of γ at high temperature and constant volume systematically less than that of CV. Anharmonicity of CV, which has been more comprehensively studied, may be a few percent at 2000 K, but decreases strongly with compression, so that anharmonicity of both γ and CV is negligible (less than 1% in the case of γ under deep‐Earth conditions).