Comparison of Monte Carlo and anharmonic-lattice-dynamics results for the thermodynamic properties and atomic mean-square displacement of Xe using the Morse potential

Monte Carlo (MC) and anharmonic-lattice-dynamics [the ${\lambda }^2$ and ${\lambda }^4$ perturbation-theory (PT)] calculations of the thermodynamic properties of Xe are presented for the temperature range 60–160 K using a nearest-neighbor central-force (NNCF) model of the fcc crystal with atoms interacting via a Morse potential. In particular, we calculate the equilibrium lattice parameter at zero pressure and the corresponding specific heats at constant volume and at constant pressure, volume expansivity, adiabatic and isothermal bulk moduli, and Gru¨neisen parameter. We also calculate the atomic mean-square displacement (MSD) from the MC method and the lowest-order (${\lambda }^2$) PT for the same NNCF model and the Morse potential. For the thermodynamic properties, the MC results are found to agree more closely with the ${\lambda }^2$ PT than the ${\lambda }^4$ PT results. Similarly the MSD results from the MC method agree quite well with those from the ${\lambda }^2$ theory. This may be due to the fact that the exact solution of the Schro¨dinger equation for the vibrational states of the Morse potential for a one-dimensional or an isotropic three-dimensional model agrees exactly with the ${\lambda }^2$ PT. We show that this is indeed true by evaluating the ${\lambda }^2$ and ${\lambda }^4$ contributions to vibrational energy for the above model of the Morse potential and showing that all the ${\lambda }^4$ contributions add up to zero and that the total ${\lambda }^2$ contribution is in agreement with the solution of the Schro¨dinger equation.