Parametric resonance of trapped ions in electron storage rings

Stability of trapped ions in electron storage rings is discussed in detail in view of the parametric resonance induced by modulated beam currents and also equidistant bunched beam currents. The equation of motion of the ions is analysed with Hamiltonian formalism in considering the nonlinear electric field produced by the electron beam. It is shown that the resonant condition is determined by the oscillation frequency of the trapped ions at the Fourier component of the beam current, and that the nonlinear field suppresses the maximum amplitude reached by the trapped ions at the resonance. The trapped ions expand around the electron beam very quickly at and near the resonant condition, but they are not lost because of the nonlinearity of the field. The expansion depends much on the strength of the Fourier component. The longitudinal motion of the trapped ions and the variation of electron beam size along the circumference are also taken into account. Current dependence of the stability is discussed, and the critical mass is deduced by the resonance condition. Numerical discussions are made for several storage rings.