Thermoelasticity of Fe$^{3+}$- and Al-bearing bridgmanite
We report \textit{ab initio} (LDA + U$_{sc}$) calculations of thermoelastic properties of ferric iron (Fe$^{3+}$)- and aluminum (Al)-bearing bridgmanite (MgSiO$_3$ perovskite), the main Earth forming phase, at relevant pressure and temperature conditions and compositions. Three coupled substitutions, namely, [Al]$_{Mg}$-[Al]$_{Si}$, [Fe$^{3+}$]$_{Mg}$-[Fe$^{3+}$]$_{Si}$, and [Fe$^{3+}$]$_{Mg}$-[Al]$_{Si}$ have been investigated. Aggregate elastic moduli and sound velocities are successfully compared with limited experimental data available. In the case of the [Fe$^{3+}$]$_{Mg}$-[Fe$^{3+}$]$_{Si}$ substitution, the high-spin (S=5/2) to low-spin (S=1/2) crossover in [Fe$^{3+}$]$_{Si}$ induces a volume collapse and elastic anomalies across the transition region. However, the associated anomalies should disappear in the presence of aluminum in the most favorable substitution, i.e., [Fe$^{3+}$]$_{Mg}$-[Al]$_{Si}$. Calculated elastic properties along a lower mantle model geotherm suggest that the elastic behavior of bridgmanite with simultaneous substitution of Fe$_{2}$O$_3$ and Al$_{2}$O$_3$ in equal proportions or with Al$_{2}$O$_3$ in excess should be similar to that of (Mg,Fe$^{2+}$)SiO$_3$ bridgmanite. Excess Fe$_{2}$O$_3$ should produce elastic anomalies though.