Thermoelasticity of Al$^{3+}$- and Fe$^{3+}$-bearing bridgemanite

We present quasi-harmonic LDA+U calculations of thermoelastic properties of Fe$^{3+}$- and Al$^{3+}$-bearing bridgemanite (MgSiO$_3$), the main Earth forming phase, at relevant P,T conditions and compositions. Three charge-coupled substitutions, namely, Al$^{3+}$-Al$^{3+}$, Fe$^{3+}$-Fe$^{3+}$, and Fe$^{3+}$-Al$^{3+}$ have been investigated. Aggregate elastic moduli and sound velocities are successfully compared with limited experimental measurements available. The effect of the pressure induced high-spin to low-spin state change in Fe$^{3+}$ in the B-site has been investigated in great detail since it has potentially dramatic effects on seismic velocities in the Earth's lower mantle.