Thermodynamics, statistical thermodynamics, and computer simulation of crystals with vacancies and interstitials

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Abstract
Molecular-dynamics and Monte Carlo simulations of fluids and of crystals are typically performed using periodic boundary conditions and small numbers of particles, i.e., small compared with Avogadro’s number. For fluids, if the dimensions of the simulated system are large enough compared with the correlation length of the material, the simulation can provide intensive thermodynamic properties that are very close to those for the thermodynamic limit of large system size. For crystals, however, the periodic boundary conditions and the crystal structure impose a constraint that makes it impossible, in practice, for the system to develop an equilibrium concentration of vacancies. The effect of this constraint must be taken into account when inferring thermodynamic properties from the results of computer simulations of crystals. We develop the thermodynamic and statistical thermodynamic theory of a crystal subject to such a constraint. The results include a set of stability conditions that must be satisfied and a criterion for equilibrium. We also describe a method for calculating the equilibrium concentration of vacancies in a crystal from computer simulations.