An intermediate spherical model of a ferromagnet

H. Okamoto

A. E. Patrick

A.R. Moura

Sergey E. Savel'ev, Guillermo Ramı́rez-Santiago

By mapping the hamiltonian of the spin one ferromagnet onto that of the classical spherical model we investigate the possible phase transitions and the phase diagram of the spin one ferromagnet. Similarly to what happens in the spherical model we find no phase transition in one and two dimensions. Nonetheless, for three dimensions we obtain a phase diagram in which the most important and unexpected feature is the existence of a ferromagnetic-paramagnetic transition at low temperatures.

T. H. Berlin, Mark Kac

A mathematical model, the spherical model, of a ferromagnet is described. The model is a generalization of the Ising model; and one-, two-, and three-dimensional lattices of infinite extent can be extensively discussed. A three-dimensional lattice shows ferromagnetic behavior and provides a statistical model of the Weiss phenomenological theory. The limiting free energy appears in a form which contains two of the essential features of the exactly known Ising model results in one and two dimensions. This suggests the probable form of the limiting free energy for the three-dimensional Ising model. A simplified model, the Gaussian model, is briefly discussed because this model also contains some of the significant features of the Ising model. However, the Gaussian model, unlike the spherical model, is not defined for all temperatures.

Ger Vertogen, W.A. Smit