On the coarse-grained density and compressibility of a non-ideal crystal

TL;DR

This paper derives the isothermal compressibility of non-ideal crystals within density functional theory, clarifying its relation to microscopic density correlations and highlighting discrepancies with static structure factor measurements.

Contribution

It provides a detailed derivation of the crystal's compressibility from microscopic density, revealing it differs from the zero wavevector limit of density correlations.

Findings

Compressibility is not the zero wavevector limit of density correlation functions.

Static structure factor does not directly give the isothermal compressibility.

Explicit connection between microscopic density and macroscopic elastic properties.

Abstract

The isothermal compressibility of a general crystal is derived and discussed within (classical) density functional theory. Starting from the microscopic particle density, we carefully coarse grain to obtain the thermodynamic compressibility and the long wavelength limits of the correlation functions of elasticity theory. We explicitly show that the isothermal compressibility is not the wavevector to zero limit of the (total) density correlation function. The latter also cannot be obtained from the static structure factor measured in a scattering experiment.