Dispersion relations for circular single and double dusty plasma chains

We derive dispersion relations for a system of identical particles confined in a two-dimensional annular harmonic well and which interact through a Yukawa potential, e.g., a dusty plasma ring. When the particles are in a single chain (i.e., a one-dimensional ring), we find a longitudinal acoustic mode and a transverse optical mode which show approximate agreement with the dispersion relation for a straight configuration for large radii of the ring. When the radius decreases, the dispersion relations modify: there appears an anticrossing of the modes near the crossing point resulting in a frequency gap between the lower and upper branches of the modified dispersion relations. For the double chain (i.e., a two-dimensional zigzag configuration), the dispersion relation has four branches: longitudinal acoustic and optical and transverse acoustic and optical.