Pressure and density of vacancies in solid ⁴He

Crystals of $^4$He contain vacancies that move around by a quantum mechanical hopping process. The density and pressure of these vacancies can be experimentally studied. The accuracy of the experiments is high enough to detect the effect of the Bose statistics of the vacancies. In this paper we examine the effect of the hard-core repulsion between the vacancies, which should also have a measurable effect on their behaviour. We set up a virial expansion for a lattice gas of hard-core particles, and calculate the second virial coefficient. It turns out that the vacancies behave as ideal Bose particles at low temperatures, but that the hard-core interaction makes them behave more and more like fermions as the temperature increases.