Molecular Field in the Spherical Model

The Heisenberg model of ferromagnetism is replaced by a classical model in which the interaction between a pair of neighboring atoms is $-2\mathrm{JS}(S+1\mathrm{})\mathrm{}{\varepsilon }_i·{\varepsilon }_j$, where $S$ is the spin of any atom, $J$ is the exchange integral, and the ${\varepsilon }_i$ are classical unit vectors. The spherical model is then used to evaluate the molecular field acting on any atom $i$. This effective field is found to have the generalized Weiss form, $H+W\left(T\right)M+W^{\prime }\mathrm{}\left(T\right)\mathrm{}{M}^{\prime }{y_i}^{\pi }$, where $H$ is the magnetic field, $M$ the magnetization, and $M^{\prime }$ the antiferromagnetic order (in units of magnetization). The coefficient ${y_i}^{\pi }$ changes sign from one sublattice to another. The "Weiss" coefficients $W\left(T\right)\mathrm{}$ and $W^{\prime }\mathrm{}\left(T\right)\mathrm{}$ are slowly temperature-dependent and obey $\frac{\mathrm{dW}}{\mathrm{dT}}>0\mathrm{}$; $\frac{d{W}^{\prime }}{\mathrm{dT}}>0\mathrm{}$.