Magnetic Dipole Energy in Hexagonal Crystals

The magnetic fields at dipole positions in two hexagonal arrays of parallel dipoles are computed by two methods. One of these methods is also used to find the field in hexagonal close packing at a point on the hexagonal axis near a vacated dipole position. The hexagonal close-packed arrangement with axial ratio $c=2\mathrm{}\times \frac{6^{\frac{1}{2}}}{3}$ (as for spheres) is found to be magnetically stable by a small margin when magnetization is along the hexagonal axis. The general expressions for the energy per dipole in homogeneously strained hexagonal arrays are solved and applied to two arrays, one simple hexagonal ($H$), the other close-packed hexagonal ($\mathrm{HC}$), both with $c=2\mathrm{}\times \frac{6^{\frac{1}{2}}}{3}$. The results are applicable in cases where $c$ differs slightly from this value.