Thomson rings in a disk

We discuss the basic principles of a self-organization of a finite number of charged particles interacting via the 1/r Coulomb potential in a disk geometry. Our approach 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 a high accuracy the equilibrium configurations of a few hundreds charged particles. For $n\gtrsim 200$ we predict the formation of a hexagonal core and valence circular rings for the centered configurations.