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 Λ-dimensional rigid spheres of diameter Σ is given, which has the form, FN / NkBT ∼ lim τ → 1 ν ln(λ / σ) − ν ln(τ − 1) + C + D(τ − 1) + E(τ − 1)2 + ···, where kB is Boltzmann's constant, λ is the mean thermal de Broglie wavelength, and θ = 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 CHCP = 1.7786846···, CFCC = 1.7795003···, DHCP = 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.