  Using evolutionary crystal structure prediction algorithm USPEX, we identify
the compositions and crystal structures of stable compounds in the Fe-S system
at pressures in the range 100-400 GPa. We find that at pressures of the Earth's
solid inner core (330-364 GPa) two compounds are stable - Fe2S and FeS. In
equilibrium with iron, only Fe2S can exist in the inner core. Using the
equation of state of Fe2S, we find that in order to reproduce the density of
the inner core by adding sulfur alone, 10.6-13.7 mol.% (6.4-8.4 wt.%) sulfur is
needed. 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 identical
mean atomic mass M in the range 52.6-53.3 results for in the inner core, which
is much higher than M = 49.3 determined for the inner core from Birch's law.
For oxygen (where the relevant stable oxide at conditions of the inner core is
Fe2O) we find the matching concentration in the range 13.2-17.2 mol.% (4.2-5.6
wt.%), which corresponds to M in the range 49.0-50.6. Combining our results and
previous works, we find that inner core density and M can be explained by only
four models (in atomic %): (a) 86%(Fe+Ni) + 14%C, (b) 84%(Fe+Ni) + 16%O, (c)
84%(Fe+Ni) + 7%S + 9%H, (d) 85%(Fe+Ni) + 6%Si + 9%H, and some of their linear
combinations (primarly, models (c) and (d)).
