Systematic Approximations to the Partition Function for Crystalline Solids

Abstract

The two-dimensional harmonic system based upon a triangular lattice with nearest-neighbor interactions only is used as a test case to compare several systematic approximation schemes for obtaining the partition function of a crystalline solid. Attention is focused on the high-temperature limit where the free energy function FN has the form FN/NkBT = −2lnT*+D. T* is a reduced temperature. Comparisons are made on the results obtained for the additive constant D. The various schemes examined are (1) a modified cell-cluster expansion, (2) a sequence of m×∞ strips, i.e., extended tunnel models, (3) extrapolation of the properties of m×m systems with fixed boundary conditions, and (4) extrapolation of the properties of m×m systems with periodic boundary conditions. The results demonstrate that a cell-cluster expansion through sixth order (i.e., including all six-particle figures) gives a better estimate of the free energy than a 10×∞ system, a 12×12 system under fixed boundary conditions, or an 11×11 system under periodic boundary conditions. The exact value of D to four significant figures was obtained by extrapolating the results of the m×m systems with periodic boundary conditions, and the result is 0.8256.