  There has been a long debate on the stable phase of iron under the Earth's
inner core conditions. Because of the solid-liquid coexistence at the inner
core boundary, the thermodynamic stability of solid phases directly relates to
their melting temperatures, which remains considerable uncertainty. In the
present study, we utilized a semi-empirical potential fitted to
high-temperature ab initio data to perform a thermodynamic integration from
classical systems described by this potential to ab initio systems. This method
provides a smooth path for thermodynamic integration and significantly reduces
the uncertainty caused by the finite-size effect. Our results suggest the hcp
phase is the stable phase of pure iron under the inner core conditions, while
the free energy difference between the hcp and bcc phases is tiny, on the order
of 10s meV/atom near the melting temperature.
