Quantum Hard Disks and Hard Spheres at High Densities

Abstract

A high-density asymptotic expression for the ground-state energy per particle for a system of d-dimensional hard spheres at density ρ is given by E0 ≈ lim ρ→ρ0; (ℏ2K0/2m) (ρ−1/d−ρ0−1/d)−2, where 2πℏ = h is Planck's constant, m is the particle mass, ρ0 is the highest attainable density, and K0 is a constant that depends on the type of hard-sphere packing. Rigorous upper and lower bonds on K0 of 27.0 and 1.63 are established for hard spheres in fcc or hcp lattices, and an estimate for the lower bound on K0, believed to be accurate to within 5%, is 5.94. A cell-cluster expansion is used to approximate FN, the Helmholtz free energy, for a hard-sphere system at high density and low temperature, and the low-temperature limit for disks in a triangular lattice is N−1FN→ lim T→0E0 ≈ lim a→σ h2(5.3667 + 1.881)/2m(a − σ)2, a being the nearest-neighbor lattice spacing and σ the hard-disk diameter. The second term in the constant factor is the cell-cluster correction to the free volume approximation due to pair correlations. As an aid in formulating the general problem for hard spheres at high density it is proved that in the asymptotic limit π → ρ0, the eigenvalue problem for the region of accessible configurations becomes that of determining the frequencies of a dN-dimensional resonance cavity whose boundary consists of intersecting (dN−1)-dimensional hyperplanes.