  Iron represents the principal constituent of the Earth's core, but its
high-pressure melting diagram remains ambiguous. Here we present a simple
analytical approach to predict the melting properties of iron under deep-Earth
conditions. In our model, anharmonic free energies of the solid phase are
directly determined by the moment expansion technique in quantum statistical
mechanics. This basis associated with the Lindemann criterion for a vibrational
instability can deduce the melting temperature. Moreover, we correlate the
thermal expansion process with the shear response to explain a discontinuity of
atomic volume, enthalpy, and entropy upon melting. Our numerical calculations
are quantitatively consistent with recent experiments and simulations. The
obtained results would improve understanding of the Earth's structure,
dynamics, and evolution.
