of the interaction have no other structure than the conservation of the total incoming and outgoing momentum (up to a reciprocal lattice vector), a feature which is of great advantage in solving the two-particle problem.

The virial expansion
====================

The grand partition function of the vacancy system is given by $$\Xi = \mathop{\rm Tr}\nolimits e^{-\beta {\cal H}},
\label{xi}$$ with ${\cal H}$ given by (\[ham\]), $\beta=1/k_BT$, and $\mathop{\rm Tr}\nolimits$ stands for the trace over all symmetrized states. As ${\cal H}$ conserves the number of vacancies, $\Xi$ can be expanded as $$\Xi = \sum_{n=0}^{\infty} Z_n e^{-n\beta \Delta},
\label{expa}$$ where $Z_n$ is the canonical partition sum for $n$ vacancies excluding the contribution from the gap $\Delta$. Of course, $Z_0=1$, and $$Z_1=\mathop{\rm Tr}\nolimits_1 e^{-\beta({\cal H}_0-\Delta)} =
\sum_{\vec{k}} e^{-\beta
\varepsilon (\vec{k})},
\label{z1}$$ because for one vacancy no hard-core effects enter. $\mathop{\rm Tr}\nolimits_n$ is the trace over $n$-vacancy states.

For $\ln \Xi$ we may deduce from (\[expa\]) $$\beta p N v_0 =\ln \Xi =N \sum_{\ell=1}^{\infty} b_{\ell} e^{-\ell
\beta
\Delta},
\label{vircs}$$ where $v_0$ is the volume of the unit cell, $p$ is the pressure of the vacancies, and the $b_{\ell}$ are the fugacity expansion coefficients. The first of these, $b_1$, reads $$b_1(\beta) = \frac{1}{N} Z_1 = \frac{v_0}{(2 