Linear stability analysis of a class of solutions of the filament model for a stationary field electron ring accelerator

The stability of two of the short time scale solutions obtained in the filament model for a stationary field electron ring accelerator is studied by means of a linearized perturbation on the electron and ion distribution functions. The Vlasov equations for the electrons and ions are coupled. The coupling of these equations results in a Fredholm integral equation which can be converted into an infinite system of linear equations. A stability criterion is then obtained from the zeros of the truncated determinants of the coefficients of the infinite system. One of the equilibrium solutions, for which the ion density in phase space is a monotonic non-increasing function of the unperturbed ion Hamiltonian, is entirely stable. The other solution, for which this monotonicity does not hold, may be unstable, but turns out to be stable provided that both the non-monotonicity parameter and the ratio of the ion period to the electron period are not too large. In the nonmonotonic case, the ion density in physical space also does not drop off to zero monotonically from the centre of the ring. For the unstable cases, the instability is driven by a ‘spike’ in the phase-space ion density located at the maximum value of the unperturbed ion Hamiltonian.