Rigid Disks and Spheres at High Densities. III

Abstract

A formal derivation of an asymptotic expansion for the Helmholtz free energy FN of a system of N ν-dimensional rigid spheres of diameter σ is given, which has the form, FN/NkBT ∼ν ln (λ/σ)-νln(τ-1)+C+D(τ-1)+2(r-1)2+⋯, r→1 where kB is Boltzmann's constant, λ is the mean thermal de Broglie wavelength, and τ= V/V0 is the reduced volume of a system of volume V, close-packed volume V0, at temperature T. Formal expressions for the constants C, D, and E are derived, and the cell-cluster technique is applied to the calculation of C and D for the hexagonal-close-packed and face-centered-cubic lattices. The results are C HCP= 1.7786846⋯, CFCC=1.7795003⋯, D HCP=0.54187⋯, DFCC=0.557994⋯ A value for D for the two-dimensional triangular lattice is found to be DT.=0. 098993⋯. Agreement with computer experiments is generally good.