Equilibrium and linear stability analysis of a class of solutions of a thin ring model for a stationary field electron ring accelerator

A class of equilibrium solutions is developed for a thin-ring model of a stationary field electron ring accelerator during the slow-moving electron ring stage. The problem is viewed as a generalized version of the Bernstein-Greene-Kruskal problem, leading to the solution of electron and ion charge densities. Linear stability analysis shows that this class of equilibrium solutions is linearly stable under J = 0 perturbations of all values of the ratio of the speed of the most energetic ions to the speed of the most energetic electrons. The values of the oscillation frequencies and the expressions for their corresponding eigenfunctions are found to exhibit convergence