Dipolar and Quadrupolar Ordering in Magnetic Crystals

Abstract

Consider an array of interacting electric dipoles, for instance polar molecules such as HCl. Bilinear interactions between these dipoles may induce a dipolar ordering below a transition temperature T _d: each dipole i then has a mean orientation along some direction +z _i. Neglecting the possibility of non-collinearity and sub-lattice structure, suppose that the ordering is ferroelectric. The order parameter of the transition is the polarization <cosθ>, θ being the angle between a dipole and the polarization axis +z. Similarly, consider an array of interacting electric quadrupoles, for instance centro-symmetric linear molecules such as H ₂, having no dipolar moment. Interactions between these quadrupoles may induce some kind of ordering below a transition temperature T _q. In particular, if all quadrupoles are aligned by positive interactions, the ordering is called ferroquadrupolar. The order parameter is then the orientation <cos²θ - 1/3>, θ being the angle between the quadrupoles and the orientation axis z.