The trapped ion–electron instability in an electron storage ring with a gap in the bunch train

The trapped ion–electron instability in an electron storage ring is studied for a broad distribution of natural ion oscillation frequencies, termed ion bounce frequencies. A gap in the train of electron bunches may be used to create bands of unstable ion bounce frequencies, so that an ion species is ejected from regions of the ring where its horizontal or vertical bounce frequency is unstable. Expressions are obtained for the reduction in the incoherent tune shifts and trapped ion–electron instability growth rates resulting from the gap.