where $$\label{f_S}
f_{S}(h_i) = \frac{1}{S} \frac{\sum\limits_{l=-S}^{S} l \rme^{\beta h_i l}}{\sum\limits_{n = -S}^{S} \rme^{\beta h_i n}} = B_S(\beta h_i S),$$ $B_S(x)$ is the Brillouin function, $\beta = (k_\mathrm{B} T)^{-1}$, $T$ is temperature, $h_{i} = | {\mathbf{h}}_{i} |$, and $h_i^{(3)}$ is the $(3)$-component of the Weiss mean field for lattice site $i$, $$\label{h_i}
\textbf{h}_i = \left( \Omega, 0, \frac{J_{1}}{S} \sum_{i'} S^{(3)}_{i'} + \frac{J_\mathrm{an}}{S} S^{(3)}_i + \mu E_{i} \right).$$ The summation $\sum_{i'}$ is a sum over the nearest neighbours of site $i$, and therefore depends on whether pseudospin $i$ is in a surface or bulk layer.

We linearize equation  to obtain the condition that ensures ferroelectricity. In the uniform case, $$\label{h_uniform}
{\mathbf{h}} = \left( \Omega, 0, \frac{J_{0}}{S} S^{(3)} + \mu E \right),$$ where $$\label{J0}
J_0 = ZJ_1 + J_\mathrm{an},$$ for coordination number $Z$. At zero-temperature, $f_{S}(h_{i}) \rightarrow 1$, and from equation  the model therefore predicts a paraelectric-ferroelectric phase transition when $$S^{(3)} = \frac{J_0S^{(3)}}{\Omega}.$$ From this one sees that, for any $S$, a ferroelectric transition occurs at nonzero temperature only when $J_0 > \Omega$. In the case of a paraelectric like STO, $J_0 < \Omega$.

![Inverse dielectric susceptibility versus temperature for SrTiO$_{3}$, modelled using three-, four- and five-component pseudospins. The fitting parameters were found 