Characterization of overlapping betatron resonances above the phase advance of 90° per cell

The space-charge-induced resonance band just above the betatron phase advance of 90 degrees per cell is of great practical importance. The instability is closely related to the linear collective mode and can thus give rise to severe emittance growth at high beam density. In a circular machine, this type of second-order resonance occurs not only at half-integer tunes but also near quarter-integer tunes, depending on the lattice superperiodicity. Self-consistent numerical simulations are carried out to elucidate the resonance feature above the 90 degrees cell tune. The present results suggest the existence of three different resonance mechanisms working there; namely, the fourth-order incoherent resonance in the beam tail, the second-order and fourth-order coherent resonances in the beam core. It is reconfirmed that no serious emittance growth occurs even if particles deep inside the core satisfy the incoherent resonance condition. The recently proposed stop-band diagram, free from the concept of incoherent tune spread, appears to be consistent with the numerical observations.