Helmholtz free energy of an anharmonic crystal to O(λ4). IV. Thermodynamic properties of Kr and Xe for the Lennard-Jones, Morse, and Rydberg potentials

We have carried out a complete calculation of the various thermodynamic properties of Kr and Xe from the (${\lambda }^4$) anharmonic perturbation theory proposed by Shukla and Cowley [Phys. Rev. B 3, 4055 (1971)], where λ is the Van Hove ordering parameter. To illustrate the effect of different potential functions on the various calculated thermodynamic properties, we have employed a nearest-neighbor central force model with the Lennard-Jones (LJ), Morse, and Rydberg interaction potentials. Along with the ${\lambda }^4$ results, we also present the results for the quasiharmonic (QH) and the ${\lambda }^2$ perturbation theory for all three potentials for each of Kr and Xe rare-gas solids. The LJ QH results are qualitative for most of the thermodynamic properties of Kr and Xe and best QH results are obtained with the Rydberg potential. The ${\lambda }^2$ perturbation theory results are poor beyond (1/3)$T_m$ ($T_m$ is the melting temperature) for the LJ potential and somewhat better for the Morse and Rydberg potentials. Best results for β (volume expansivity) and $C_p$ (specific heat at constant pressure) are obtained with the ${\lambda }^4$ theory and the Morse potential. The convergence of the perturbation expansion improves with the addition of ${\lambda }^4$ contributions and the expansion remains valid up to 40% of $T_m$ for the LJ potential. The range of expansion is much higher for the other two potentials. Whereas the curvature of most of the calculated curves for the various thermodynamic properties for the ${\lambda }^2$ theory is incorrect, the corresponding curves from the ${\lambda }^4$ theory have the correct curvature. The cancellation among the ${\lambda }^2$ and ${\lambda }^4$ contributions is most dramatic for the LJ potential and less so for the other two potentials. As T approaches $T_m$, only the Rydberg-potential results, for the ${\lambda }^2$ theory, appear to be sensible compared with the other two potentials for most of thermodynamic properties of Kr and Xe.