The Application of the Bethe-Peierls Method to Ferromagnetism

The method introduced by Bethe and Peierls for treating the problem of order-disorder in alloys is applied to the problem of ferromagnetism. The method is first applied to the Ising model of the spin in which case the treatment is much the same as it is for the alloy problem, as has already been pointed out by Peierls. The correct treatment of the spin is used for spin values of ½ and 1 per atom. The critical temperatures of different types of lattices are investigated. The method gives results in agreement with the rigorous results of the Bloch spin-wave theory in that only three-dimensional lattices are found to be ferromagnetic. The values of the critical temperatures of these lattices are found to lie between the values predicted by the two Heisenberg approximations. The discontinuity of the specific heat at the critical temperature is computed for a body-centered lattice and for the two values of the spin. The magnitude of the discontinuity is larger than that predicted by Heisenberg's first approximation. The magnitude for the spin 1 is 3.4 $3.4k$ per atom and compares favorably with the experimental value for iron. The susceptibility is computed as a function of the temperature above the critical point. The variation of the susceptibility with temperature does not obey the Curie-Weiss Law but displays some curvature. This curvature explains qualitatively the difference between the "paramagnetic" and ferromagnetic critical temperatures and also helps remove some of the discrepancy between the number of Bohr magnetons per atom as measured at high and low temperatures.