Spread-frequency model of the fast beam-ion instability in an electron storage ring

When a gap in the electron bunch train prevents the trapping of ions, a transverse electron-ion instability may result from the ions created and lost during a single passage of the bunch train. A spread-frequency model is used to study this instability when the ions have a broad distribution of natural oscillation frequencies about the center of the electron beam. A growing disturbance saturates from Balakin-Novokhatsky-Smirnov damping at approximately the same time, and with the same total growth, as in the case without an ion frequency spread. At the tail of the bunch train, an unstable disturbance is amplified by a factor ∼exp(ω_{i}τ_{b}) before saturation occurs, where ω_{i} is a typical ion oscillation frequency and τ_{b} is the duration of the bunch train. Initially, the instability displays exponential growth in time, unlike the case where the ion-frequency spread is neglected. For a broad distribution of ion frequencies, instability may be prevented by a betatron damping rate that exceeds the incoherent betatron frequency shift induced by ions at the tail of the bunch train