\pi)^d} \int_{BZ}
d\vec{k} \,e^{-\beta\varepsilon(\vec{k})},
\label{b1}$$ where we have replaced the sum over $\vec{k}$ by an integral over the Brillouin zone. In contrast to the continuum ideal gas, $b_1$ is not simply related to a thermal wavelength $\lambda$ as $$b_1=v_0/\lambda^{d}.
\label{b1l}$$ A formula of this type only results when the temperature is so low that $\varepsilon(\vec{k})$ may be replaced by its low-momentum behaviour $$\varepsilon(\vec{k}) \approx \frac{t}{2} \sum_{\delta} (\vec{k}
\cdot \vec{r}_\delta )^2 \equiv \frac{\hbar^2 k^2}{2m^*},
\label{lowk}$$ defining an effective mass $m^*$ for the vacancies. For $d$-dimensional hypercubic lattices we find $$m^*=\frac{\hbar^2}{2 t a^2},
\label{mstar}$$ where $a$ is the lattice constant. Using (\[lowk\]) in (\[b1\]) and extending the $\vec{k}$-integral beyond the Brillouin zone to infinity (as is allowed for large $\beta$ or small $T$) one finds (\[b1l\]) with $$\lambda^2=h^2/2\pi m^* k_BT.
\label{lambda}$$

The second term $b_2$ in (\[vircs\]) is our main concern in this paper. We find from (\[expa\]) and (\[vircs\]) $$\begin{aligned}
b_2&=&\frac{1}{N}\left( Z_2 - \frac{Z_1^2}{2}\right)\nonumber\\
&=&
\frac{1}{N}\left\{ \mathop{\rm Tr}\nolimits_2 e^{-\beta({\cal H}
-2 \Delta )} -
\frac{1}{2}\left( \mathop{\rm Tr}\nolimits_1 e^{-\beta({\cal H}_0 -
\Delta)}\right)^2\right\}.\nonumber\\
\label{b2}\end{aligned}$$ We rewrite this expression by adding and subtracting the contribution of the non-interacting two-vacancy system. So we define $$b_2^{\rm int}=\frac{1}{N} \mathop{\rm Tr}\nolimits_2\left[
e^{-\beta({\cal H}-2\Delta)}-
e^{-\beta({\cal H}_0 -2\Delta)}\right]
\label{b2int}$$ as 