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

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 $\frac{E_0}{N}\left\{\right\}\left\{\sim \right\}\{\rho \to {\rho }_0\}\frac{{\hslash }^2}{2m}A{({\rho }^{-\frac{1}{3}}-{{\rho }_0}^{-\frac{1}{3}})}^{-2\mathrm{}},$ 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 $A_{\mathrm{sc}}={\pi }^2$. For more realistic close-packed systems we estimate $A_{\mathrm{cp}}\approx 10 \mathrm{to} 15$. 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 $A\approx 15.7$, the hydrogen data $A\approx 15.9$. The fitted hard-core diameters, 1.73 Å and 1.90 Å, respectively, are smaller than expected from accepted potentials; this is discussed. Thermodynamics of the simple cubic system give $c_v\propto T$ 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.