Bose-Einstein Gas with Repulsive Interactions: Hard Spheres at High Density

A method developed by the authors for treating Bose-Einstein particles at low temperature is specialized to the case of hard sphere interactions. It is shown that the integral equations determining the energy can in this case be replaced by an integral involving the unknown excitation spectrum. The energy is first determined as a power series in ${\left(\rho a^3\right)}^{\frac{1}{2}}$, where $\rho$ is the density and $a$ the hard-sphere radius. This solution is shown to diverge badly at high density. A result is then obtained which is valid over the full range of density, reducing to the exact expressions at low density and to a good approximation at high density. The excitation spectrum of the system is shown to have a peculiar nonmonotonic relation between energy and momentum very similar to that deduced by Landau from the low-temperature properties of ${\mathrm{He}}^4$.