Stability of ions in the electron beam with a gap

A gap in the bunch train is one of methods to remove ions from the beam circulating in electron storage rings. A self-consistent theory of the ion stability in the bunch train with a gap is developed out. As is shown, stability of ions increases with increase in ion core density. Under certain conditions that have been determined, the stability islands become overlapping. Therefore, the beam clearing may not be achieved. It is proved that the maximum charge neutralization the circulating current reaches in continuous beams, any gap decreases the stable density of ions. The stable density of the ion core decreases with the number of the ion oscillations with the bunch train increasing