Pressure derivatives of shear and bulk moduli from the thermal Grüneisen parameter and volume-pressure data

Abstract

Grüneisen’s parameters are central to studies of Earth’s interior because these link elastic data to thermodynamic properties through the equation of state and can be measured using either microscopic or macroscopic techniques. The original derivation requires that the mode Grüneisen parameter (γ i) of the longitudinal acoustic (LA) mode equals the thermodynamic parameter (γ th) for monatomic solids. The success of the Debye model indicates that γ LA = γ th is generally true. Available elasticity data for crystalline solids contain 30 reliable measurements, covering 10 structures, of the pressure derivatives of the bulk ( K S) and the shear ( G) moduli. For these phases, the measured values of γ th and γ LA agree well. Other solids in the database have disparate γ LA values, suggesting large experimental uncertainties within which γ LA = γ th. This relationship allows inference of the pressure ( P) derivative of the shear modulus (∂ G/∂ P = G′) from widely available measurements of γ th, the isothermal bulk modulus ( K T ), ∂ K T /∂ P, and G. We predict G′ as 1.55 for stishovite, 1.6 to 2.15 for MgSiO 3 ilmenite, 1.0 for γ-Mg 1.2Fe 0.8SiO 4, and 0 for FeS (troilite). Similarly, G′ measured for MgSiO 3 perovskite suggests that K S ′ = 4, corroborating volume-pressure data. For many materials, pairs of G′ and K S ′ = ∂ K S /∂ P from independent elasticity studies of a given phase define a nearly linear trend, suggesting systematic errors. Non-hydrostatic conditions and/or pressure calibration likely cause the observed variance in K S ′ and G′. The best values for pressure derivatives can be ascertained because the trend defined by measured pairs of G′ with K S ′ intersects the relationship of G′ to K′ defined by γ LA = γ th at a steep angle. Our results for isostructural series show linear correlations of K S ′ with K S and of G′ with G. Values of K S ′ are nearly 4 for high-pressure phases, which is consistent with the harmonic oscillator model, whereas G′ has a wide range of −1 to 3. Hence, inference of a detailed mineralogy inside the Earth is best constrained by comparing seismic determinations of shear moduli to laboratory measurements.