= Classical Heisenberg mean-field partition function
For unit three-component classical <spins> and energy $-J\sum_{\langle ij\rangle}\mathbf S_i\cdot\mathbf S_j$ on a lattice of coordination $z_c$, mean-field decoupling gives $E_{\rm MF}=-z_cJS\sum_iS_{iz}+Nz_cJS^2/2$. Integrating each <spin> over <solid angle> yields
$$
Z_{\rm MF}(S)=e^{-\beta Nz_cJS^2/2}
[4\pi\sinh(\beta z_cJS)/(\beta z_cJS)]^N.
$$
The 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.
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