We have investigated the magnetization as a function of $T$ and $H$ of a spin system with both isotropic Heisenberg exchange and dipole-dipole interactions for $S\ge \frac{1}{2}$ and a hexagonal crystal structure. The Green's functions for such a system, recently derived by Becker, are decoupled in a first-order random-phase approximation. For both Heisenberg-dipole and simple-dipole crystals, we find a lowering of the magnetization below saturation at $T=0\mathrm{}$. The Curie-Weiss temperature $\theta$ and the ordering temperature $T_c$ are calculated and compared with experimental values on Gd${\mathrm{Cl}}_3$. In the limit of zero exchange, we obtain a condition for the type of lattice and shape of domains which make ferromagnetic ordering possible.

• Received 9 April 1969

DOI:https://doi.org/10.1103/PhysRevB.1.314