A critical examination of the Holstein-Primakoff spin-wave technique has yielded explicit criteria for the stability of ferromagnetic arrays. These criteria have been applied to point dipoles on the three primitive cubic lattices for the shape most favorable to ferromagnetism, that of a long, thin sample. The results are: (1) dipolar ferromagnetism cannot occur in the simple cubic lattice; (2) the ferromagnetic state can be the lowest, or at least a metastable, state on the face-centered cubic and body-centered cubic lattices. The zero-point energies of ferromagnetic states are calculated for the $〈100〉$, $〈011〉$, and $〈111〉$, directions, yielding two anisotropy constants, $K_1$ and $K_2$, for each lattice. The Holstein-Primakoff Hamiltonian is shown not to be diagonal for spin waves in the pathological region, i.e., of wavelengths comparable to the dimensions of the specimens. The properties of this region are shown to be unimportant in energy considerations but may effect stability. They are of great importance in effects such as ferromagnetic resonance which involve long-wavelength spin waves.

• Received 11 March 1955

DOI:https://doi.org/10.1103/PhysRev.99.1135