Inequality linking vacancy-formation energy and rigidity in a liquid at melting point from a density-independent pair potential theory

Johnson has recently used an explicit form of the Daw-Baskes embedded-atom model to display a clearcut correlation between vacancy-formation energy in a close-packed crystal and the shear modulus. Here, a complementary treatment for the hot crystal, based on statistical thermodynamics, is shown to lead to an inequality linking these same quantities at the melting temperature, with an appropriate definition of rigidity in the liquid phase. The inequality rests on three main assumptions: (i) use of a density-independent pair potential theory, (ii) neglect of density dependence of the liquid pair correlation function, and (iii) absence of appreciable ionic relaxation around the vacancy.