Thomson rings in a disk

We discuss the basic principles of self-organization of a finite number of charged particles interacting via the 1/r Coulomb potential in disk geometry. The analysis is based on the cyclic symmetry and periodicity of the Coulomb interaction between particles located on several rings. As a result, a system of equations is derived, which allows us readily to determine with high accuracy the equilibrium configurations of a few hundred charged particles. For n greater than or similar to 200, we predict the formation of a hexagonal core and valence circular rings for the centered configurations.