Thermodynamic properties of MgSiO$_3$ majorite and phase transitions near 660-km depth in MgSiO$_3$ and Mg$_2$SiO$_4$: a first principles study*

Thermodynamic properties of MgSiO$_3$ tetragonal majorite have been calculated at high $P$-$T$ within the quasiharmonic approximation based on DFT using both LDA and GGA. The LDA results compare exceptionally well with measured thermodynamic properties. A classical Monte Carlo simulation based on a cluster expansion method demonstrates that disorder between Mg and Si in the octahedral sites in majorite does not occur below 3600 K within 30 GPa. The calculated phase boundaries between majorite (mj), perovskite (pv), and ilmenite (il) MgSiO$_3$ agree much better with experiments by using GGA than by LDA. The Clapeyron slopes (CS) predicted by GGA and LDA are close to each other: $0.9$ -- $1.7$ MPa/K for mj-pv, 6.9 -- 7.9 MPa/K for mj-il, and $-7$ -- $-3$ MPa/K for il-pv transition. The triple point predicted by GGA is at $21.8 \pm 1$ GPa and $1840 \pm 200$ K which is $\sim$400 K lower than most experiments. Our calculations also reveal that wadsleyite decomposes to an assemblage of majorite plus periclase above 2280 K with a large negative CS ($-22$ -- $-12$ MPa/K) and that ringwoodite decomposes to ilmenite plus periclase below 1400 K (1.2 MPa/K). The geophysical implications to mantle convection and the composition of the Earth's transition zone will also be discussed.