  We study the ground state properties of classical Coulomb charges interacting
with a 1/r potential moving on a plane but confined either by a circular hard
wall boundary or by a harmonic potential. The charge density in the continuum
limit is determined analytically and is non-uniform. Because of the non-uniform
density there are both disclinations and dislocations present and their
distribution across the system is calculated and shown to be in agreement with
numerical studies of the ground state (or at least low-energy states) of N
charges, where values of N up to 5000 have been studied. A consequence of these
defects is that although the charges locally form into a triangular lattice
structure, the lattice lines acquire a marked curvature. A study is made of
conformal crystals to illuminate the origin of this curvature. The scaling of
various terms which contribute to the overall energy of the system of charges
viz, the continuum electrostatic energy, correlation energy, surface energy
(and so on) as a function of the number of particles N is determined. "Magic
number" clusters are those at special values of N whose energies take them
below the energy estimated from the scaling forms and are identified with
charge arrangements of high symmetry.
