Abstract
The space-charge-induced resonance band just above the betatron phase advance of 90\ifmmode^\circ\else\textdegree\fi{} 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\ifmmode^\circ\else\textdegree\fi{} 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.
Highlights
Owing to the recent progress of accelerator technologies, the performance of high-intensity hadron machines is getting better and better
If the incoherent resonance condition in Eq (2) really applies to the whole beam, the machine working point must be chosen such that the necktie does not cross any low-order single-particle resonance lines defined by Eq (1)
In WARP simulations, we check the E-values of individual particles at injection to separate the contribution from large-amplitude particles to the rms emittance. Those tail particles defined at the entrance are excluded from the emittance evaluation thereafter, which clarifies whether the emittance of the core part grows or keeps the initial level
Summary
Owing to the recent progress of accelerator technologies, the performance of high-intensity hadron machines is getting better and better. The dangerous linear parametric resonance driven by the Coulomb self-field (SF) potential can be excited near quarter integer tunes. This effect, sometimes referred to as envelope instability, has been discussed by many researchers mostly with linear transport designs in mind [12–18]. In a ring of odd symmetry, such as the hadron synchrotrons at the Japan Proton Accelerator Research Complex (J-PARC) [1,26], the severe SF-driven instability of the linear coherent mode may take place near quarter-integer tunes as opposed to the conventional incoherent picture that only predicts weak fourthorder resonances there.
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