Coherent resonance stop bands in alternating gradient beam transport

Abstract

An extensive experimental study is performed to confirm fundamental resonance bands of an intense hadron beam propagating through an alternating gradient linear transport channel. The present work focuses on the most common lattice geometry called "FODO" or "doublet" that consists of two quadrupoles of opposite polarities. The tabletop ion-trap system "S-POD" (Simulator of Particle Orbit Dynamics) developed at Hiroshima University is employed to clarify the parameter-dependence of coherent beam instability. S-POD can provide a non-neutral plasma physically equivalent to a chargedparticle beam in a periodic focusing potential. In contrast with conventional experimental approaches relying on large-scale machines, it is straightforward in S-POD to control the doublet geometry characterized by the quadrupole filling factor and drift-space ratio. We verify that the resonance feature does not essentially change depending on these geometric factors. A few clear stop bands of low-order resonances always appear in the same pattern as previously found with the sinusoidal focusing model. All stop bands become widened and shift to the higher-tune side as the beam density is increased. In the spacecharge-dominated regime, the most dangerous stop band is located at the bare betatron phase advance slightly above 90 degrees. Experimental data from S-POD suggest that this severe resonance is driven mainly by the linear self-field potential rather than by nonlinear external imperfections and, therefore, unavoidable at high beam density. The instability of the third-order coherent mode generates relatively weak but noticeable stop bands near the phase advances of 60 and 120 degrees. The latter sextupole stop band is considerably enhanced by lattice imperfections. In a strongly asymmetric focusing channel, extra attention may have to be paid to some coupling resonance lines induced by the Coulomb potential. Our interpretations of experimental data are supported by theoretical predictions and systematic multiparticle simulations.

Empirical condition of betatron resonances with space charge

“…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.…”

“…(6) had been given explicitly. Since it is hopeless to solve the 2D Vlasov-Poisson equations mathematically for arbitrary AG lattices, a plausible conjecture was made recently by the Hiroshima group [31]. The proposed 2D resonance condition has a remarkably simple form, in spite of the complex collective process behind:…”

“…(8) was first reached in Ref. [31] through careful theoretical considerations and systematic S-POD experiments [48,49]. Hofmann added the parametric factor 1=2 to the right-hand side of Eq.…”

“…II, we introduce the 2D coherent resonance condition recently conjectured in Ref. [31]. We then proceed to systematic particle-in-cell (PIC) simulations.…”

Three-dimensional envelope instability in periodic focusing channels

“…It has been studied theoretically [1][2][3][4][5][6][7][8] and experimentally [9][10][11] since the 1980s. In recent years, there was growing interest in further understanding this instability and other structural resonances [12][13][14][15][16][17][18][19][20][21][22]. Some of those studies were summarized in a recently published monograph [23].…”