practical circumstances always extremely dilute, which corresponds in the spin analogy to a system in an extremely large external magnetic field. In this limit the spin analogy is of little use and in our opinion less transparent than the particle language.

Experiments show that the percentage of vacancies in a crystal is at most of the order of 1% [@Goodk], at least at temperatures of the order of 1K, where the experiments take place. As a first approximation the vacancies thus behave as an ideal gas. However, present day experiments are sufficiently accurate that effects of Bose statistics can be detected. It is one of our points that then also effects of the hard-core interaction become detectable.

In this paper we present a systematic analysis of the vacancies using $\exp (-\Delta/k_BT)$ as a small parameter, $\Delta$ being the excitation energy, or band gap, required to create a vacancy. This is equivalent to a virial expansion [@Huang] for the quantum lattice gas, and we work out the properties in detail up to the second virial coefficient. A general formula for the second virial coefficient of the hard-core Bose lattice gas is derived and evaluated for hypercubic lattices in one, two, and 