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Using the evolutionary crystal structure predictionalgorithm USPEX, we identify the compositions and crystalstructures of thermodynamically stable compounds in the Fe ±S system at pressures in the range of 100 ± 400 GPa. We findthat at pressures in the Earth's solid inner core (330 ± 364 GPa)two compounds are stable – Fe2S and FeS. In equilibrium withiron, only Fe2S can exist in the inner core. Using the equation ofstate of Fe2S, we find that, in order to reproduce the density ofthe inner core by adding sulfur alone, 10.6 ± 13.7 mol.% (6.4 ±8.4 wt.%) sulfur is needed. An analogous calculation for silicon(where the only stable compound at inner core pressures is FeSi)reproduces the density of the inner core with 9.0 ± 11.8 mol.%(4.8 ± 6.3 wt.%) silicon. In both cases, a virtually identicalmean atomic massMMin the range of 52.6 ± 53.3 results forthe inner core, which is much higher thanMMà49:3 inferred forthe inner core from Birch's law. In the case of oxygen (allowingfor the equilibrium coexistence of suboxide Fe2O with ironunder core conditions), the inner core density can be ex-plained by the oxygen content of 13.2 ± 17.2 mol.% (4.2 ±5.6 wt.%), which corresponds toMMbetween 49.0 and 50.6 

With the motivation of searching for new superconductors in the Mg–B system, we performed ab initio evolutionary searches for all the stable compounds in this binary system in the pressure range of 0–200 GPa. We found previously unknown, yet thermodynamically stable, compositions MgB 3 and Mg 3 B 10 . Experimentally known MgB 2 is stable in the entire pressure range 0–200 GPa, while MgB 7 and MgB 12 are stable at pressures below 90 GPa and 35 GPa, respectively. We predict a reentrant behavior for MgB 4 , which becomes unstable against decomposition into MgB 2 and MgB 7 at 4 GPa and then becomes stable above 61 GPa. We find ubiquity of phases with boron sandwich structures analogous to the AlB 2 -type structure. However, with the exception of MgB 2 , all other magnesium borides have low electron–phonon coupling constants λ of 0.32–0.39 and are predicted to have T c below 3 K.