Classical crystal formation of dipoles in two dimensions

We consider a two-dimensional layer of dipolar particles in the regime of strong dipole moments. Here we can describe the system using classical methods and determine the crystal structure that minimizes the total energy. The dipoles are assumed to be aligned by an external field and we consider different orientations of the dipolar moments with respect to the two-dimensional plane of motion. We observe that when the orientation angle changes away from perpendicular and towards the plane, the crystal structure will change from a hexagonal form to one that has the dipoles sitting in equidistant rows, i.e. a striped configuration. In addition to calculating the crystal unit cell, we also consider the phonon spectrum and the speed of sound. As the orientation changes away from perpendicular the phonon spectrum develops local minima that are a result of the deformation to the crystal structure.