  The solid inner core of the Earth is predominantly composed of iron alloyed
with several percent Ni and some lighter elements, Si, S, O, H, and C being the
prime candidates. There have been a growing number of papers investigating C
and H as possible light elements in the core, but the results are
contradictory. Here, using ab initio simulations, we study the Fe-C and Fe-H
systems at inner core pressures (330-364 GPa). Using the evolutionary structure
prediction algorithm USPEX, we have determined the lowest-enthalpy structures
of possible carbides (FeC, Fe2C, Fe3C, Fe4C, FeC2, FeC3, FeC4 and Fe7C3) and
hydrides (Fe4H, Fe3H, Fe2H, FeH, FeH2, FeH3, FeH4) and have found that Fe2C
(Pnma) is the most stable iron carbide at pressures of the inner core, while
FeH, FeH3 and FeH4 are stable iron hydrides at these conditions. For Fe3C, the
cementite structure (Pnma) and the Cmcm structure recently found by random
sampling are less stable than the I-4 and C2/m structures found here. We found
that FeH3 and FeH4 adopt chemically interesting thermodynamically stable
structures, in both compounds containing trivalent iron. The density of the
inner core can be matched with a reasonable concentration of carbon, 11-15
mol.percent (2.6-3.7 wt.percent) at relevant pressures and temperatures. This
concentration matches that in CI carbonaceous chondrites and corresponds to the
average atomic mass in the range 49.3-51.0, in close agreement with inferences
from the Birch's law for the inner core. Similarly made estimates for the
maximum hydrogen content are unrealistically high, 17-22 mol.percent (0.4-0.5
wt.percent), which corresponds to the average atomic mass in the range
43.8-46.5. We conclude that carbon is a better candidate light alloying element
than hydrogen.
