  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.
