High‐temperature elastic constant data on minerals relevant to geophysics

Abstract

The high‐temperature measurements of elastic constants and related temperature derivatives of nine minerals of interest to geophysical and geochemical theories of the Earth's interior are reviewed and discussed. A number of correlations between these parameters, which have application to geophysical problems, are also presented. Of especial interest is α, the volume coefficient of thermal expansion, and a section is devoted to this physical property. Here we show how α can be estimated at very high temperatures and how it varies with density. An estimate of α for Mg‐perovskite at deep‐mantle conditions is made. The formula for the Grüneisen ratio γ as a function of V and T is presented, including plots of the numerical values of γ over a wide T and V range. An example calculation of γ for MgO is made. The high‐T‐high‐P values of γ calculated here agree well with results from the ab initio method of calculation for MgO. The use of the thermoelastic parameters is reviewed, showing application to the understanding of thermal pressure, thermal expansivity, enthalpy, and entropy. We review an extrapolation formula to determine Ks, the adiabatic bulk modulus, at very high T. We show that the thermal pressure is quite linear with T up to high temperatures (∼1800 K), and, as a consequence, the anharmonic contribution to the Helmholtz free energy is sufficiently small, so that it can and should be ignored in thermodynamic calculations for mantle conditions.

Structural and elastic anisotropy of crystals at high pressures and temperatures from quantum mechanical methods: The case of Mg2SiO4 forsterite

“…Some considerations on the relative performance of DFT functionals can be made: (i) the description of the thermal expansion of forsterite below 800 K is approximately correct for all functionals; (ii) as already noticed in previous studies on different classes of materials, 28 provides the lowest thermal expansion; (iii) the generalized gradient PBE and PBEsol functionals and the global hybrid PBE0 functional do provide a reliable description of the thermal expansion up to about 1500 K while above that temperature, explicit anharmonic effect is expected to play a non-negligible role in the low-pressure thermal expansion of forsterite; and (iv) the hybrid B3LYP functional provides the largest expansion and significantly deviates from the expected linearity above about 800 K. In the lower panel of Figure 5, we report the isothermal bulk modulus, K T , of forsterite, as computed according to Eq. (7) with the same five functionals and as measured by Anderson et al 49 (full circles) and Suzuki et al 77 (empty circles). Four functionals (LDA, PBE, PBEsol and PBE0) provide a very similar description of the temperature dependence of K T , which almost perfectly matches the experimental slope up to about 1800 K while the B3LYP hybrid functional strongly deviates above about 800 K. As regards the absolute value of the isothermal bulk modulus, the PBEsol and PBE0 functionals are found to provide a satisfactory agreement with the experimental data while PBE and LDA underestimate and overestimate them, respectively.…”

“…(7)) and can be compared with measurements from X-ray diffraction experiments, for instance. In the figure, full symbols are experimental data of K T as measured by Anderson et al 49 (circles) and Suzuki et al 77 (triangles). All computed values, as obtained at PBE level, are upshifted by 6.2 GPa in the figure, in order to match their experimental counterparts as we want to highlight the correctness of their temperature (and pressure) dependence rather than of their absolute values, which are rather DFT functional-dependent, as previously shown in Figure 5.…”

“…However, their descriptions of the thermal expansion and temperature dependence of the bulk modulus are rather close to each other up to the melting point and to the experimental behavior up to about 1500 K, the only exception being the B3LYP hybrid functional that starts deviating above about 800 K. In the upper panel of Figure 5, we report the volumetric thermal expansion coefficient, α V (T), of forsterite up to 2200 K, its melting temperature being T M = 2163 K. Six experimental determinations are reported (see caption of Figure 5 for details), which show discrepancies as large as 25% among them. Three experimental datasets that agree particularly well to each other are marked with full (instead of empty) symbols and are here taken as a reference: those from Anderson et al 49 (full circles), Gillet et al 80 (full rhombi), and Fei and Saxena 81 (full triangles). Some considerations on the relative performance of DFT functionals can be made: (i) the description of the thermal expansion of forsterite below 800 K is approximately correct for all functionals; (ii) as already noticed in previous studies on different classes of materials, 28 provides the lowest thermal expansion; (iii) the generalized gradient PBE and PBEsol functionals and the global hybrid PBE0 functional do provide a reliable description of the thermal expansion up to about 1500 K while above that temperature, explicit anharmonic effect is expected to play a non-negligible role in the low-pressure thermal expansion of forsterite; and (iv) the hybrid B3LYP functional provides the largest expansion and significantly deviates from the expected linearity above about 800 K. In the lower panel of Figure 5, we report the isothermal bulk modulus, K T , of forsterite, as computed according to Eq.…”

Experimental deformation of olivine single crystals at mantle pressures and temperatures

“…Pressure was deduced from calibrant unit-cell volumes using the corresponding equations of state (for alumina and spinel: Anderson et al(1992); see also Chang and Barsch (1973) for spinel; for San Carlos olivine: Zha et al (1998)), while stress σ = σB B 1 B -B B -σB B 3 B was deduced from differences in the d spacing characterizing lattice planes in different orientations with respect to the principal stress σB B 1 B B (see Li et al (2004) for details). Stress uncertainty using these techniques is fairly large, since it depends on the accuracy on the measured d spacing -i.e., on the position of x-ray diffraction peak maxima and shifting toward higher energy on compression (smaller d spacing) -which in turn depends on several experimental factors (see Raterron et al (2007) for details).…”

Some Elastic Properties and Phase Transitions in Some Alkali Halides Using Interatomic Potential Model

“…where P and Q are the derivative operators as follows, The equations for the elastic constants are then defined by Anderson [14], where ∆ is the volume-per cell (ion pair):…”