Correlation functions in classical solids

Abstract

By invoking thermodynamic potentials as generating functions for hierarchies of correlation functions, we develop a description of solids written in the same statistical language used to describe inhomogeneous fluids. Important constraints then follow from consideration of the symmetries of the crystalline solid. Considerable insight into the two-particle density is obtained by appealing to the harmonic model of the solid, which motivates the idea of parametrizing the correlation functions using parameters unique to each lattice site. By paralleling the derivation of the Ornstein-Zernike equation we are led to an equivalent relation for the solid between the parameters of the direct correlation function and the parameters of the two-particle density. By similarly paralleling the derivation of Percus identity, we develop an equation for the parametrization of correlation functions of a solid analogous to the hypernetted-chain equation of inhomogeneous fluids. The harmonic model of the solid thus emerges from the appropriate limit of the hypernetted-chain equation for an extremely inhomogeneous fluid.