Interplay of space-charge fourth order resonance and envelope instability

Since the finding of the space charge fourth order 4σ=360° resonance [Jeon et al., 2009 [3]], studies have been carried out to better understand the fourth order resonance and the envelope instability in high intensity linear accelerators. The questions remained unanswered under what conditions the envelope instability is excited or suppressed following the development of the fourth order resonance. Simulations suggest that the variation of σo and σ along the linac, the halo particles and the resonance islands of the fourth order resonance play an important role in exciting or suppressing the envelope instability after the fourth order resonance is manifested for initially well-matched beams. Here σ(σo) is the depressed (zero-current) phase advance per cell. The envelope instability is excited from the mismatch generated by the fourth order resonance, 1) when σo is maintained approximately constant or increases along the linac with σo>90° and 2) when the extent of the fourth order resonance shrinks, as σ increases approaching 90°. On the other hand, when σ stays almost constant (maintaining the fourth order resonance) or when σo decreases and gets close to 90° (getting out of the envelope instability), the envelope instability is suppressed and the halo particles formed by the fourth order resonance preserves a four-fold structure in the phase space.