= Solution
Let $q$ be the <coordination number of a lattice>, the number of nearest neighbours of each cell away from boundaries. Define an effective <chemical potential> incorporating the cell's translational <entropy>:
$$
\widetilde\mu=\mu+k_BT\ln\frac b{\lambda^3},\qquad
\left(\frac{e^{\beta\mu}b}{\lambda^3}\right)^{\sum n_\alpha}=e^{\beta\widetilde\mu\sum n_\alpha}.
$$
Use $\langle\alpha\alpha'\rangle$ for an unordered nearest-neighbour bond, counted once. The occupation <Hamiltonian> including the chemical-potential term is
$$
\mathcal H_{\rm occ}=-4J\sum_{\langle\alpha\alpha'\rangle}n_\alpha n_{\alpha'}-\widetilde\mu\sum_\alpha n_\alpha.
$$
Under $n_\alpha=(1+\sigma_\alpha)/2$, one bond contributes
$$
-4Jn_\alpha n_{\alpha'}=-J\left(1+\sigma_\alpha+\sigma_{\alpha'}+\sigma_\alpha\sigma_{\alpha'}\right).
$$
In a regular lattice with periodic <boundary conditions>, the number of bonds is $Mq/2$, and each spin occurs in $q$ linear bond terms. Therefore
$$
\mathcal H_{\rm eff}=-J\sum_{\langle\alpha\alpha'\rangle}\sigma_\alpha\sigma_{\alpha'}
-\left(qJ+\frac{\widetilde\mu}{2}\right)\sum_\alpha\sigma_\alpha
-\frac{MqJ}{2}-\frac{M\widetilde\mu}{2}.
$$
The pair sum equals one half of the ordered sum used in the paper. Consequently the effective field is
$$
\boxed{h(T,\mu)=qJ+\frac\mu2+\frac{k_BT}{2}\ln\frac b{\lambda^3}.}
$$
Thus $\Xi=\sum_{\{\sigma\}}e^{-\beta\mathcal H_{\rm eff}}$ is the <Ising model> partition sum, including the displayed spin-independent constant if its absolute normalization is required. Dropping that constant changes $\Xi$ by a multiplicative factor but leaves spin <probabilities> and the phase-transition argument unchanged. For simple cubic cells in three dimensions, $q=6$ and the first term in $h$ is $6J$. Open boundaries replace $q$ by the local coordination near the surface; the stated uniform field is the bulk or periodic-lattice result.
Back to article page
