Temperature Dependence of Interatomic Forces

Abstract

In computer modelling of defects in crystalline materials it is common practice to use a pair potential or N-body potential to describe the energy of the system as a function of the atom positions. The equilibrium atomic positions at 0 K are determined by minimization of the total potential energy with respect to each atomic coordinate in the model. Having obtained an equilibrium structure at 0 K it is natural to ask how the structure changes when the temperature is increased and what are the thermodynamic properties of the defect. A popular approach is to use molecular dynamics or Monte Carlo simulations to explore the classical phase space of the system and derive thermal averages of the atom positions and thermodynamic functions. Atoms are assumed to interact via the same potential as they were at 0K and temperature enters the simulations either through the classical kinetic energies in molecular dynamics or the acceptance criterion of trial configurations in Monte Carlo. The burden of this paper is to describe an alternative approach to determining equilibrium structure and thermodynamic properties at a finite temperature. In this method the (Helmholtz) free energy of the system for a given (thermally averaged) atomic configuration and interatomic potential is expressed in the harmonic approximation. The total free energy comprises this harmonic term and the potential energy arising from the interatomic potential. The total free energy is minimized at a given temperature by varying the volume of the system and the positions of atoms within it. During the relaxation the frequencies of vibrational normal modes change and thus the free energy arising from them also changes. This procedure is often called the quasiharmonic approximation to distinguish it from the harmonic approximation in which the same lattice vibrational free energy is calculated but for fixed atom positions that remain the same as those at 0K.