Coulomb crystals of few particles: Closed-form expressions for equilibrium energies and geometries, and vibrational force constants
Abstract 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.
A method of solving the eikonal equation, in either flat or curved space–times, with arbitrary Cauchy data, is extended to the case of data given on a characteristic surface. We find a beautiful relationship between the Cauchy and characteristic data for the same solution, namely they are related by a Legendre transformation. From the resulting solutions, we study and describe the wave-front singularities that are associated with their level surfaces (the characteristic surfaces or “big wave fronts”).