Anisotropic mean-square displacements in two-dimensional colloidal crystals of tilted dipoles

Superparamagnetic colloidal particles confined to a flat horizontal air-water interface in an external magnetic field, which is tilted relative to the interface, form anisotropic two-dimensional crystals resulting from their mutual dipole-dipole interactions. Using real-space experiments and harmonic lattice theory we explore the mean-square displacements of the particles in the directions parallel and perpendicular to the in-plane compo- nent of the external magnetic field as a function of the tilt angle. We find that the anisotropy of the mean-square displacement behaves nonmonotonically as a function of the tilt angle and does not correlate with the structural anisotropy of the crystal. It is common wisdom that a one-component classical many-body system consisting of particles at constant density that interact, e.g., via a pairwise-additive repulsive inverse- power potential, freezes into a periodic crystal lattice at zero temperature f1g. At finite temperatures and prior to melting, the crystal is still stable but the particles perform small- amplitude excursions from their equilibrium positions. The averaged mean-square displacement around the equilibrium lattice sites, which is a quantitative measure of these particle excursions, plays a key role in describing the bulk melting process of the crystal: the traditional Lindemann rule f2,3g states that a solid melts if the root mean-square displacement exceeds about 10% of the lattice constant. This phenomeno- logical rule is a good estimate for melting in three spatial dimensions. In two dimensions, however, mean-square dis- placements are diverging f4g, but fluctuations of the relative distance of nearest neighbors can be nevertheless used to establish a bulk Lindemann melting rule f5g. In this paper, we investigate the anisotropy of the mean-