Self-consistent study of space-charge-driven coupling resonances

In this paper we present a comprehensive analysis of the self-consistent, collective behavior associated with the space-charge-driven (Montague) coupling resonance near 2Q_x-2Q_y=0, including the effect of linear coupling. Based on analytical work and particle-in-cell simulation in the 2D coasting beam limit, we derive scaling laws for stop-band widths and growth rates, which may be applied to circular machines as well as to linear accelerators. For slow crossing of the stop bands, we find a strong directional dependence. In the case of crossing from below—assuming that the rising tune pertains to the direction, in which the initial emittance is the larger one—the emittance exchange is a smooth and fully reversible process. For crossing from above, we encounter a discontinuous behavior, which disappears largely, if an external linear coupling is applied.