Summation of free-energy diagrams of an anharmonic crystal and equation of state for a Lennard-Jones solid

Abstract

We present a method of calculating the Helmholtz free energy of an anharmonic crystal. The exact expression for $F,$ obtained by summing an infinite series of contributions, from all the loops and bubbles (quartic and cubic contributions to the self-energy of the Green's function), is evaluated numerically and the equation of state results for a Lennard-Jones solid are compared with the ${\ensuremath{\lambda}}^{2}$ perturbation theory (PT) which contains only the lowest order cubic and quartic contributions. It is shown that the infinite sum results are considerably improved over the ${\ensuremath{\lambda}}^{2}$ PT results for higher temperatures. Next we have presented a powerful ansatz approach of evaluating the same sum. The numerical results from this method are shown to be identical to the exact sum except at near ${T}_{m}$ where they are very slightly different. The ansatz method is then extended to the higher order ${\ensuremath{\lambda}}^{4}$ diagrams and here too the numerical results are found to be improved over the results from the ${\ensuremath{\lambda}}^{4}$ PT. The ansatz procedure is then extended to the propagator renormalization and the numerical results obtained seem to have the best agreement with the results of classical Monte Carlo simulations.