Study the spontaneous magnetization of the Heisenberg model in the mean field approximation and examine the dependence of \(\bar{L}_{0}\) on \(T\) (i) in the neighborhood of the critical temperature where \(\left(1-T / T_{C}\right) \ll 1\) and (ii) at sufficiently low temperatures where \(T / T_{c} \ll 1\). Compare these results with the corresponding ones, namely, (12.5.14) and (12.5.15) for the Ising model.

[Note: In this connection, it may be pointed out that, at very low temperatures, the experimental data do not agree with the theoretical formula derived here. We find instead a much better agreement with the formula \(\bar{L}_{0}=\left\{1-A(k T / J)^{3 / 2}\right\}\), where \(A\) is a numerical constant (equal to 0.1174 in the case of a simple cubic lattice). This formula is known as Bloch's \(T^{3 / 2}\)-law and is derivable from the spin-wave theory of ferromagnetism; see Wannier (1966), Section 15.5.]