Finite-size corrections to the free energy of crystalline solids

TL;DR

This paper investigates finite-size effects on the free energy of crystalline solids, providing corrections and estimates for defect-free crystals and analyzing the impact of vacancies on free energy.

Contribution

It introduces explicit finite-size corrections to the free energy of crystals and estimates the free energy of defect-free and vacancy-containing crystals.

Findings

Finite-size corrections depend linearly on 1/N after leading corrections.

Estimated free energy of a defect-free hard-sphere crystal near coexistence.

Vacancies significantly reduce the free energy compared to defect-free crystals.

Abstract

We analyse the finite-size corrections to the free energy of crystals with a fixed center of mass. When we explicitly correct for the leading ($\ln N/N$) corrections, the remaining free energy is found to depend linearly on 1/N. Extrapolating to the thermodynamic limit ($N\to\infty$), we estimate the free energy of a defect free crystal of particles interacting through an $r^{-12}$-potential. We also estimate the free energy of perfect hard-sphere crystal near coexistence: at $\rho \sigma^{3}$=1.0409, the excess free energy of a defect-free hard-sphere crystal is $5.91889(4)kT$ per particle. This, however, is not the free energy of an equilibrium hard-sphere crystal. The presence of an finite concentration of vacancies results in a reduction of the free energy that is some two orders of magnitude larger than the present error estimate.