Interacting topological defects on frozen topographies

We propose and analyze an effective free energy describing the physics of d isclination defects in particle arrays constrained to move on an arbitrary two-dimensional surface. At finite temperature the physics of interacting d isclinations is mapped to a Laplacian sine-Gordon Hamiltonian suitable for numerical simulations. We discuss general features of the ground state and thereafter specialize to the spherical case. The ground state is analyzed a s 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 much greater than a, where R is the radius of the sphere an d a is the particle spacing. For large core energies there are 12 disclinat ions 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-->infinity, is also addressed. Fina lly we discuss the application of our results to the classic Thomson proble m of finding the ground state of electrons distributed on a two sphere.