Double stop-band structure near half-integer tunes in high-intensity rings

Abstract

This paper addresses a detailed experimental study of collective instability bands generated near every half-integer tune per lattice period by coherent dipole and quadrupole resonances. Both instabilities appear side by side or overlap each other but are mostly separable because the dipole resonance often creates a narrower stop band accompanied by more severe particle losses. The separation of these low-order resonance bands becomes greater as the beam intensity increases. In principle, the double stop-band structure can be formed even without machine imperfections when the beam's initial phase-space profile is deviated from the ideal stationary distribution. The tabletop ion-trap system called "S-POD" is employed to experimentally demonstrate the parameter dependence of the double stop-band structure. Numerical simulations are also performed for comparison with experimental observations.

Empirical condition of betatron resonances with space charge

“…24 in Appendix B and Ref. [31]), no second peak has been found in a linear stop band, except for the one generated by the coherent dipole (m ¼ 1) mode [39]. The breathing mode does not appear to be excited seriously in a quadrupole channel.…”

“…9(b) and 9(c) for comparison. The sawtooth-like configuration of the emittance-growth curves in the waterbag and parabolic cases are well known and has been observed repeatedly in S-POD experiment [31,[35][36][37][38][39][40][41]. It is formed because the beam stays in a resonance band for a longer period at a larger κ 2 , thus experiencing more emittance growth.…”

“…It also has nothing to do with the double stop-band structure reported in Ref. [39], which is formed near every half-integer tune by the overlapping of two coherent resonances associated with the dipole (m ¼ 1) and quadrupole (m ¼ 2) oscillation modes. We have no such overlapping with the strong dipole mode here.…”

“…We have emphasized the importance of the parametric resonance condition (7) since its discovery [27,31,[35][36][37][38][39][40][41], presenting not only systematic numerical simulation data but also various experimental observations from the novel tabletop apparatus "S-POD" (Simulator of Particle Orbit Dynamics) at Hiroshima University [42][43][44]. S-POD reproduces, within a very compact ion trap [45], the collective behavior of an intense beam propagating through a wide range of AG lattice configurations; see Appendix B for some information about S-POD.…”

“…The horizontal and vertical quadrupole resonances (m ¼ 2) at every half-integer tune per lattice period are accompanied by a strong dipole (m ¼ 1) resonance [39]. Since the dipole stop band is very narrow and ideally has no space-charge-induced shift ðC 1 ¼ 0Þ, it is always located close to half-integer lines, ν 0xð0yÞ ¼ N sp =2; N sp ; 3N sp =2; ….…”