Quantum Hard-Sphere Gas in the Limit of High Densities with Application to Solidified Light Gases

Abstract

The properties of a quantum hard-sphere gas in the limit of high densities are investigated, with particular emphasis on the ground-state energy per particle. This has the asymptotic form $\frac{{E}_{0}}{N}{}{\ensuremath{\sim}}{\ensuremath{\rho}\ensuremath{\rightarrow}{\ensuremath{\rho}}_{0}}\frac{{\ensuremath{\hbar}}^{2}}{2m}A{({\ensuremath{\rho}}^{\ensuremath{-}\frac{1}{3}}\ensuremath{-}{{\ensuremath{\rho}}_{0}}^{\ensuremath{-}\frac{1}{3}})}^{\ensuremath{-}2},$ as deduced from the Heisenberg principle, and is independent of particle statistics. A model of hard spheres arranged in a simple cubic lattice is solved by reduction to the known one-dimensional case, and gives ${A}_{\mathrm{sc}}={\ensuremath{\pi}}^{2}$. For more realistic close-packed systems we estimate ${A}_{\mathrm{cp}}\ensuremath{\approx}10 \mathrm{to} 15$. This form connects smoothly to the well-known low-density gas-parameter expansions. Phonon properties in the Debye approximation are derived. The model is applied to the zero-point kinetic energies of hexagonal-centered-cubic (hcp) $^{3}\mathrm{He}$, $^{4}\mathrm{He}$, ${\mathrm{H}}_{2}$, and ${\mathrm{D}}_{2}$, as determined from pressure data. The helium data give $A\ensuremath{\approx}15.7$, the hydrogen data $A\ensuremath{\approx}15.9$. The fitted hard-core diameters, 1.73 \AA{} and 1.90 \AA{}, respectively, are smaller than expected from accepted potentials; this is discussed. Thermodynamics of the simple cubic system give ${c}_{v}\ensuremath{\propto}T$ for both bosons and fermions, which may explain the anomalous (non-Debye) heat capacitics of solid $^{3}\mathrm{He}$ and $^{4}\mathrm{He}$ at low temperatures.