  We propose and analyze an effective free energy describing the physics of
disclination defects in particle arrays constrained to move on an arbitrary
two-dimensional surface. At finite temperature the physics of interacting
disclinations is mapped to a Laplacian Sine-Gordon Hamiltonian suitable for
numerical simulations. We then specialize to the case of a spherical crystal at
zero temperature. The ground state is analyzed as a function of the ratio of
the defect core energy to the Young's modulus. We argue that the core energy
contribution becomes less and less important in the limit R >> a, where R is
the radius of the sphere and a is the particle spacing. For large core energies
there are twelve disclinations forming an icosahedron. For intermediate core
energies unusual finite-length grain boundaries are preferred. The complicated
regime of small core energies, appropriate to the limit R/a goes to infinity,
is also addressed. Finally we discuss the application of our results to the
classic Thomson problem of finding the ground state of electrons distributed on
a two-sphere.
