Shell model of assemblies of equicharged particles subject to radial confining potentials

A shell model of an assembly of N equicharged particles subject to an arbitrary radial confining potential N W(r), where W(r) is parameterized in terms of an auxiliary function Λ(t), is presented. The validity of the model requires that Λ(t) is strictly increasing and concave for any t ∈ (0, 1), Λ'(0) is infinite, and Λ(t) = -t(-1) Λ'(t)/Λ''(t) is finite at t = 0. At the bulk limit of N → ∞, the model is found to correctly reproduce the energy per particle pair and the mean crystal radius R(N), which are given by simple functionals of Λ(t) and Λ'(t), respectively. Explicit expressions for an upper bound to the cohesive energy and the large-N asymptotics of R(N) are obtained for the first time. In addition, variational formulation of the cohesive energy functional leads to a closed-form asymptotic expression for the shell occupancies. All these formulae involve the constant ξ that enters the expression -(ξ/2) n(3/2) for the leading angular-correlation correction to the minimum energy of n electrons on the surface of a sphere with a unit radius (the solution of the Thomson problem). The approximate energies, which constitute rigorous upper bounds to their exact counterparts for any value of N, include the cohesive term that is not accounted for by the mean-field (fluidlike) theory and its simple extensions but completely neglect the surface-energy correction proportional to N.