Spin states and hyperfine interactions of iron incorporated in MgSiO$_3$ post-perovskite

Using density functional theory + Hubbard $U$ (DFT+$U$) calculations, we investigate the spin states and nuclear hyperfine interactions of iron incorporated in magnesium silicate (MgSiO$_3$) post-perovskite (Ppv), a major mineral phase in the Earth's D'' layer, where the pressure ranges from about 120 to 135 GPa. In this pressure range, ferrous iron (Fe$^{2+}$) substituting for magnesium at the dodecahedral (A) site remains in the high-spin (HS) state; intermediate-spin (IS) and low-spin (LS) states are highly unfavorable. As to ferric iron (Fe$^{3+}$), which substitutes magnesium at the A site and silicon at the octahedral (B) site to form (Mg,Fe)(Si,Fe)O$_3$ Ppv, we find the combination of HS Fe$^{3+}$ at the A site and LS Fe$^{3+}$ at the B site the most favorable. Neither A-site nor B-site Fe$^{3+}$ undergoes a spin-state crossover in the D'' pressure range. The computed iron quadrupole splittings are consistent with those observed in M{\"o}ssbauer spectra. The effects of Fe$^{2+}$ and Fe$^{3+}$ on the equation of state of Ppv are found nearly identical, expanding the unit cell volume while barely affecting the bulk modulus.