Rigid Spheres Near Close Packing: Tunnel Model and a Lower Bound on the Partition Function

Abstract

In the high‐density limit, the Helmholtz free energy function for an assembly of N rigid spheres at temperature T has the asymptotic form lim τ→1 F N / Nk B T ≃ 3 ln (σ / λ) − 3 ln (τ − 1) + C + ··· , where k B is Boltzmann's constant, σ is the sphere diameter, λ the mean thermal de Broglie wavelength, and τ = V / V 0 where V 0 is the close‐packed volume of the hard‐sphere crystal whose bulk volume is V . The value of the constant C for the tunnel model in a face‐centered cubic lattice is found to be C = 1.60145 ··· , as compared with the single‐particle free‐volume cell value C = 1.5629 ··· . In addition an upper bound for C in a face‐centered lattice is calculated, with the result C 3 < 3.5045··· , a previous result being C 3 < 3.6423··· .