High-Temperature Susceptibility of an Exchange-Interaction Model of Ferromagnetism

The Schrödinger exchange interaction for arbitrary spin is used to construct a Hamiltonian for a nearest-neighbor model of ferromagnetism. The general high-temperature series for the low-field susceptibility is developed, and the first three terms of the series, for arbitrary crystal lattices, are calculated. The Curie temperature obtained from setting ${\chi }^{-1\mathrm{}}=0\mathrm{}$ and taking the first two terms (molecular-field theory) or three terms ("Heisenberg Gaussian" approximation), is given by $\frac{\mathrm{}(2S+1)\mathrm{}{k}_B{T}_C}{J}=z or \frac{4}{[1-{(1-\frac{8}{z})}^{\frac{1}{2}}]},$ respectively. The functional dependence on $z$ (number of nearest neighbors) of the latter result, for arbitrary $S$, is exactly the same as originally obtained by Heisenberg for $S=\frac{1}{2}$. In general, the exchange model appears to disorder at a lower temperature than the Heisenberg model, as could be anticipated from the higher degeneracy of the ground state of a pair of spins interacting with the exchange Hamiltonian compared to the Heisenberg Hamiltonian.