Calculation of the Temperature Dependence of the Second-Order Elastic Constants of fcc Ar, Kr, and Xe Using a Two-Body Short-Range Interatomic Potential

The temperature dependence of the second-order elastic constants of fcc Ar, Kr, and Xe have been studied using phenomenological ($m-6$) Lennard-Jones potentials acting between nearest neighbors and fitted to the zero-temperature zero-pressure lattice constant and the sublimation energy. The theory of the second-order elastic constants is briefly reviewed, and the second-order strain dependence of the quasiharmonic free-energy density is derived using perturbation theory. The resulting expressions for the temperature dependence of the elastic constants involve both third- and fourth-order force constants, and require a single scan of the Brillouin zone, which is carried out to give an accuracy of about 1%. We find that the contribution of the third-order force constants is comparable to that of the fourth-order ones and in some cases predominates. Adiabatic and isothermal results are presented both for constant volume (0°K $M$ volume) and for the experimentally observed equilibrium $M$ volume. We did not solve the equation of state. Wave velocities derived from our elastic constants are compared with recent experiments on crystals of Ar and Kr. Our results show that there are large anharmonic contributions to the temperature dependence of the elastic constants. Consequently, our Ne results are quantitatively unrealiable (except at $T=0\mathrm{}$) and we omit them. For $T>\mathrm{}\sim \frac{{\Theta }_{\infty }}{2}$ there are indications of the breakdown of perturbation theory. The way our results may be affected by including higher-order anharmonic terms is illustrated by considering the temperature dependence of the zero-pressure bulk modulus of Ar.