Spectral properties of classical two-dimensional clusters

We present a study of the spectral properties such as the energy spectrum, the eigenmodes, and the density of states of a classical finite system of two-dimensional charged particles which are confined by a quadratic potential. Using the method of Newton optimization we obtain the ground state and the metastable states. For a given configuration the eigenvectors and eigenfrequencies for the normal modes are obtained using the Householder diagonalization technique for the dynamical matrix whose elements are the second derivative of the potential energy. For small clusters the lowest excitation corresponds to an intershell rotation. The energy barrier for such rotations is calculated. For large clusters the lowest excitation consists of a vortex/antivortex pair. The Lindeman melting criterion is used to calculate the order-disorder transition temperature for intershell rotation and intershell diffusion. The value of the transition temperature at which intershell rotation becomes possible depends very much on the configuration of the cluster, i.e., the distribution of the particles between the different shells. Magic numbers are associated with clusters which are most stable against intershell rotation. The specific heat of the cluster is also calculated using the Monte Carlo technique, which we compare with an analytical calculation where effects due to anharmonicity are incorporated.