  Our first-principles calculations show that both the compressional and shear
waves of hcp-Fe become elastically isotropic under the high temperatures of
Earth inner core conditions, with the variation in sound velocities along
different angles from the c axis within 1%. We computed the thermoelasticity at
high pressures and temperatures from quasiharmonic linear response
linear-muffin-tin-orbital calculations in the generalized-gradient
approximation. The calculated anisotropic shape and magnitude in hcp-Fe at
ambient temperature agree well with previous first-principles predictions, and
the anisotropic effects show strong temperature dependences. This implies that
other mechanisms, rather than the preferential alignment of the hcp-Fe crystal
along the Earth rotation axis, account for the seismic P-wave travel time
anomalies. Either the inner core is not hcp iron, and/or the seismologically
observed anisotropy is caused by inhomogeneity, i.e. multiple phases.
