Constant coupling approximation for Heisenberg ferromagnetism

Abstract

Synopsis A general expression is derived for the partition function of a Heisenberg ferromagnetic spin system with isotropic coupling between nearest neighbouring spins, in terms of a pair density matrix of a pair of nearest neighbouring spins. With the help of this pair density matrix an effective Hamiltonian H e for a pair of spins is introduced, and it is shown that H e contains only an isotropic coupling term of the Heisenberg type, an anisotropic coupling term of the Ising type, and a term representing an effective field acting on the two spins of the pair. The familiar molecular field approximation is obtained by assuming that the two coupling terms in H e are zero. Bij minimizing the free energy, it follows that the effective field occurring in H e is then equal to the Weiss molecular field. A next approximation is obtained by assuming that the effective isotropic coupling in H e is equal to the actual coupling between the spins, and that the effective anisotropic coupling is zero, which is shown to correspond to the limiting behaviour of H e for high temperatures. This “constant coupling” approximation constitutes a straightforward generalization of the quasi-chemical method for an Ising spin system to the case of Heisenberg coupling. The thermodynamic properties of the spin system, and in particular the critical data, are calculated on the basis of this constant coupling approximation, and numerical results are given for lattices with co-ordination number 6, 8 and 12. The theory is compared to the cluster method of P. R. Weiss, and it is shown that in the present theory the difficulty of an anti-Curie point does not occur, so that the constant coupling approximation gives useful results at all temperatures.