An extended semi-analytical approach for thermoelasticity of monoclinic crystals: application to diopside

Elastic properties of minerals at high temperature and pressure are essential for constraining the chemical composition and dynamics of the Earth's interior. In spite of great advancements in high-pressure experimental techniques, it is still a considerable challenge to measure the elasticity of minerals at the conditions of the Earth's deep mantle. Elastic coefficients at high pressure and temperature can be also obtained by the ab initio molecular dynamics simulations (AIMD) or the quasi-harmonic approximation (QHA). Although the conventional QHA method demands less computational effort than AIMD and can give the elastic tensor in a continuous temperature range, it still requires considerable computational resources and human labor because of the extensive number of calculations of vibrational density of states (VDoS) in strained configurations.

Wu and Wentzcovitch (2011) developed an alternative semi-analytical method (SAM-C_ij) that reduces the computational to less than one-tenth of the conventional QHA method. Compared with the conventional method, which requires calculations of VDoS for numerous strained configurations (e.g. at least 15 for an orthorhombic crystal) for each volume, SAM-C_ij only requires VDoS at equilibrium configuration by extracting the strain dependence of phonon frequencies from the volume dependence of phonon frequencies. This method has been successfully applied to several minerals in the mantle.

However, in its original formulation, SAM-C_ij can only be applied to systems with orthorhombic symmetry or higher. Systems with hexagonal and trigonal symmetry can also be addressed using orthorhombic supercell calculations. However, the method cannot be applied to monoclinic or triclinic systems. For example, clinopyroxene (cpx), one of major minerals in the upper mantle, is monoclinic and has 13 independent elastic coefficients. Here we extend to monoclinic systems a semi-analytical method to compute themoelastic properties of materials (Wu and Wentzcovitch, 2011) and apply it to diopside (MgCaSi₂O­₆), a mineral with monoclinic symmetry and an end-member of clinopyroxene. Using ab initio results for structural and vibrational properties, we obtain the density, the elastic tensor, single crystal elastic anisotropies, and seismic wave velocities of diopside over a wide pressure and temperature range. Our results agree well with previous available experimental results. Compared with other minerals in the upper mantle, clinopyroxene has the lowest P- and S-wave velocities in the upper mantle deeper than 240 km. Combining our results with the elastic properties of other major upper-mantle minerals in the previous studies, we estimated the density and wave velocity profiles of the pyrolite model at the depth range of 220-400 km, where pyroxene is gradually dissolved into garnet. The obtained profiles agree well with seismic models (PREM and AK135) supporting the notion of a pyrolitic upper mantle.

Wu, Z., Wentzcovitch, R.M., 2011. Quasiharmonic thermal elasticity of crystals: An analytical approach. Physical Review B 83. 184115