= Equilibrium vacancies with variable lattice size
{title2=$n_*\simeq N/[e^{(\epsilon+pv)/(k_BT)}-1]$}
Consider fixed $N$ atoms and variable $n$ vacancies among $N+n$ sites, each of volume $v$, with vacancy energy $\epsilon$. The configuration count is $(N+n)!/(N!n!)$. At fixed pressure and temperature, leading <Stirling's approximation> gives the excess <Gibbs free energy> $n(\epsilon+pv)-k_BT[(N+n)\ln(N+n)-N\ln N-n\ln n]$. Its stationary point is the displayed $n_*$, and its second derivative is $k_BTN/[n(N+n)]>0$. Relative to the vacancy-free Gibbs free energy, the minimum is $Nk_BT\ln(1-e^{-(\epsilon+pv)/(k_BT)})$. The resulting mean-field volume is $v(N+n_*)$. Normalizability requires $\epsilon+pv>0$, and large vacancy fractions lie outside a realistic dilute-defect crystal model.
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