 Classical Heisenberg mean-field partition function

For unit three-component classical spins and energy $-J{\sum }_{⟨ij⟩}{\mathbf{S}}_i\cdot {\mathbf{S}}_j$ on a lattice of coordination $z_c$, mean-field decoupling gives $E_{\mathrm{MF}}=-z_{c}JS{\sum }_i{S}_{iz}+N{z}_{c}J{S}^2/2$. Integrating each spin over solid angle yieldsThe constant term corrects double counting. A normalized angular measure removes $(4\pi {)}^N$ but does not change the self-consistency equation. The lattice dimension and spin-space dimension are different data.