The elastic constants of condensed matter: A direct-correlation function approach

We describe a simple, systematic and physically transparent method for calculating the elastic constants of condensed matter. This approach is equally useful when applied to such diverse materials as alkali halides and nematic and smectic A liquid crystals, as we report in this paper. Our analysis involves regarding the periodic density of the ordered phase to be representable as a small perturbation to the uniform density distribution of the corresponding fluid phase. We implement this idea by making use of recent work on the density wave theory of freezing and the statistical mechanics of nonuniform systems. The theory makes the role of the structure of the medium explicit while leaving the role of the intermolecular potential implicit. We find, for example, that the elastic constants of an alkali halide crystal can be expressed in terms of the curvatures of the Fourier transforms of the charge–charge and number–number direct correlation functions of the corresponding fluid evaluated at the reciprocal lattice vectors of the Bravais lattice. We also report molecular representations for the Frank elastic constants of a nematic liquid crystal and a smectic A liquid crystal.