Pressure derivatives of shear and bulk moduli from the thermal Grüneisen parameter and volume-pressure data |
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| Authors: | AM Hofmeister HK Mao |
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| Institution: | 1 Department of Earth and Planetary Sciences, Washington University, St. Louis, MO 63130, USA
2 Geophysical Laboratory, 5251 Broad Branch Rd. NW, Washington D.C. 20015, USA |
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| 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 (γi) of the longitudinal acoustic (LA) mode equals the thermodynamic parameter (γth) for monatomic solids. The success of the Debye model indicates that γLA = γth is generally true. Available elasticity data for crystalline solids contain 30 reliable measurements, covering 10 structures, of the pressure derivatives of the bulk (KS) and the shear (G) moduli. For these phases, the measured values of γth and γLA agree well. Other solids in the database have disparate γLA values, suggesting large experimental uncertainties within which γLA = γth. This relationship allows inference of the pressure (P) derivative of the shear modulus (∂G/∂P = G′) from widely available measurements of γth, the isothermal bulk modulus (KT), ∂KT/∂P, and G. We predict G′ as 1.55 for stishovite, 1.6 to 2.15 for MgSiO3 ilmenite, 1.0 for γ-Mg1.2Fe0.8SiO4, and 0 for FeS (troilite). Similarly, G′ measured for MgSiO3 perovskite suggests that KS′ = 4, corroborating volume-pressure data. For many materials, pairs of G′ and KS′ = ∂KS/∂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 KS′ and G′. The best values for pressure derivatives can be ascertained because the trend defined by measured pairs of G′ with KS′ intersects the relationship of G′ to K′ defined by γLA = γth at a steep angle. Our results for isostructural series show linear correlations of KS′ with KS and of G′ with G. Values of KS′ 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. |
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