Suppose that each atom of a crystal lattice can be in one of several internal states (which may be denoted by the symbol \(\sigma\) ) and the interaction energy between an atom in state \(\sigma^{\prime}\) and its nearest neighbor in state \(\sigma^{\prime \prime}\) is denoted by \(u\left(\sigma^{\prime}, \sigma^{\prime \prime}\right)\left\{=u\left(\sigma^{\prime \prime}, \sigma^{\prime}\right)\right\}\). Let \(f(\sigma)\) be the probability of an atom being in a particular state \(\sigma\), independently of the states in which its nearest neighbors are. The interaction energy and the entropy of the lattice may then be written as

\[E=\frac{1}{2} q N \sum_{\sigma^{\prime}, \sigma^{\prime \prime}} u\left(\sigma^{\prime}, \sigma^{\prime \prime}\right) f\left(\sigma^{\prime}\right) f\left(\sigma^{\prime \prime}\right)\]

and \[S / N k=-\sum_{\sigma} f(\sigma) \ln f(\sigma)\]
respectively. Minimizing the free energy \((E-T S)\), show that the equilibrium value of the function \(f(\sigma)\) is determined by the equation

\[f(\sigma)=C \exp \left\{-(q / k T) \Sigma_{\sigma^{\prime}} u\left(\sigma, \sigma^{\prime}\right) f\left(\sigma^{\prime}\right)\right\}\]
where \(C\) is the constant of normalization. Further show that, for the special case \(u\left(\sigma^{\prime}, \sigma^{\prime \prime}\right)=\) \(-J \sigma^{\prime} \sigma^{\prime \prime}\), where \(\sigma\) can be either +1 or -1 , this equation reduces to the Weiss equation (12.5.11), with \(f(\sigma)=\frac{1}{2}\left(1+\bar{L}_{0} \sigma\right)\).