\[rat2\] and \[rat3\].

Conclusion
==========

The pressure and density of a gas of quantum particles on a lattice can be expressed in fugacity expansions, $$\begin{aligned}
p&=&\frac{1}{\beta v_0} \sum_{\ell=1}^{\infty} b_{\ell}
e^{-\ell\beta\Delta},
\label{pee}\\
n&=&\frac{1}{v_0} \sum_{\ell=1}^{\infty} \ell b_{\ell}
e^{-\ell\beta\Delta}.
\label{en}\end{aligned}$$ For bosons with a hard-core interaction, the first two coefficients, $b_1$ and $b_2$, can be calculated. It turns out that the effect of the hard-core interaction on $b_2$ depends strongly on temperature and on the transfer integral $t$. For small $\beta t$ it is fermionic in character, as far as $b_2$ is concerned. For large $\beta t$ the effect of the hard-core disappears, and only the Bose character remains.

The system of vacancies that exists in solid $^4$He should behave to a good approximation like this simple model: the vacancies move through the crystal lattice by a tunneling process, they are bosons, and they have a hard-core repulsion. The gas of vacancies has been experimentally studied by probing the attenuation and the velocity shift of sound in a $^4$He crystal [@Goodk]. Very pure hcp Helium was used, so that the effects of both the phonons and the delocalized, bosonic vacancies could be observed. Treating the vacancies as a gas of free particles, it 