Two particle model for studying the effects of space-charge force on strong head-tail instabilities

In this paper, we present a new two particle model for studying the strong head-tail instabilities in the presence of the space-charge force.It is a simple expansion of the well-known two particle model for strong head-tail instability and is still analytically solvable.No chromaticity effect is included.It leads to a formula for the growth rate as a function of the two dimensionless parameters: the space-charge tune shift parameter (normalized by the synchrotron tune) and the wakefield strength, ϒ.The three-dimensional contour plot of the growth rate as a function of those two dimensionless parameters reveals stopband structures.Many simulation results generally indicate that a strong head-tail instability can be damped by a weak spacecharge force, but the beam becomes unstable again when the space-charge force is further increased.The new two particle model indicates a similar behavior.In weak space-charge regions, additional tune shifts by the space-charge force dissolve the mode coupling.As the space-charge force is increased, they conversely restore the mode coupling, but then a further increase of the space-charge force decouples the modes again.This mode coupling/decoupling behavior creates the stopband structures.