Coulomb crystals of few particles: Closed-form expressions for equilibrium energies and geometries, and vibrational force constants

Analytical expressions for equilibrium energies and geometries, and the corresponding vibrational force constants are derived for clusters of between three and eight particles interacting through a pairwise potential of the form ϵ ( r ) = ( 1 / 3 ) ( r 2 + 2 r - 1 ) . In the case of five-, seven-, and eight-particle clusters, these expressions are functions of single parameters that are the larger real roots of respective twelfth-degree polynomials.