Why charges go to the surface: a generalized Thomson problem

– We study a variant of the generalized Thomson problem in which n particles are confined to a neutral sphere and interacting by a 1/r potential. It is found that for γ ≤ 1 the electrostatic repulsion expels all the charges to the surface of the sphere. However for γ > 1 and n > nc(γ) occupation of the bulk becomes energetically favorable. It is curious to note that the Coulomb law lies exactly on the interface between these two regimes. In a recent paper [1] Bowick et al. studied a system of particles confined to the surface of a sphere and interacting by a repulsive 1/r potential with 0 < γ < 2. They called this “the generalized Thomson problem”. It is interesting, however, to recall that the original Thomson problem was posed as a model of a classical atom [2]. Thus, n electrons were supposed to be confined in the interior of a sphere with a uniform neutralizing background, the so called “plum pudding” model of an atom. The Thomson problem, which is still unsolved, is then to find the ground state of electrons inside the sphere. In the absence of a neutralizing background the electrostatic repulsion between the particles “dynamically” drives the charges to the surface. This significantly simplifies the calculations by reducing the search of the ground state from the three dimensions down to two [3]. But what if instead of the Coulomb potential electrons interacted by a 1/r potential? Would they still go to the surface or prefer to stay in the bulk? This question was not addressed in the paper of Bowick et al. who have a priory confined their particles to reside on the surface. It is clear that for a small number of charges, mutual repulsion will force them to the surface. What happens, however, as the concentration of particles increases? To answer this question we compare the electrostatic energy of the configuration in which all n particles are on the surface of a sphere with a configuration in which n− 1 particles are at the surface and one particle is located at the center of a sphere. The electrostatic energy of n particles of charge q interacting through a generalized Coulomb potential q/ǫr , with dielectric constant ǫ, confined to the surface of a sphere with radius a can be obtained by considering the electrostatic energy of the two dimensional one component plasma (OCP) F n , i.e. charges on (∗) E-mail: [email protected] (∗∗) E-mail: [email protected]