Equilibrium vacancies with variable lattice size ($n_{\ast }\simeq N/[e^{(ϵ+pv)/(k_{B}T)}-1]$)

Consider fixed $N$ atoms and variable $n$ vacancies among $N+n$ sites, each of volume $v$, with vacancy energy $ϵ$. The configuration count is $(N+n)!/(N!n!)$. At fixed pressure and temperature, leading Stirling's approximation gives the excess Gibbs free energy $n(ϵ+pv)-k_{B}T[(N+n)\ln (N+n)-N\ln N-n\ln n]$. Its stationary point is the displayed $n_{\ast }$, and its second derivative is $k_{B}TN/[n(N+n)]>0$. Relative to the vacancy-free Gibbs free energy, the minimum is $N{k}_{B}T\ln (1-e^{-(ϵ+pv)/(k_{B}T)})$. The resulting mean-field volume is $v(N+n_{\ast })$. Normalizability requires $ϵ+pv>0$, and large vacancy fractions lie outside a realistic dilute-defect crystal model.

If the variable-site model counts every $n$ sector with weight $\left(\genfrac{}{}{0ex}{}{N+n}{n}\right)q^n$, its exact relative partition sum is $(1-q{)}^{-N-1}$. Differentiating its logarithm gives the displayed exact mean. Minimizing the leading extensive Gibbs free energy instead gives $Nq/(1-q)$, the same result to relative order $1/N$. Its exact relative equilibrium free energy is $(N+1)k_{B}T\ln (1-q)$. This correction distinguishes a thermodynamic saddle from an exact finite-system average; changing the microscopic site-creation convention can also change subextensive factors.