Parametric instabilities in 3D periodically focused beams with space charge

Abstract

Parametric resonances of beam eigenmodes with a periodic focusing system under the effect of space charge-also called structural instabilities-are the collective counterparts to parametric resonances of single particles or of mechanical systems. Their common feature is that an exponential instability is driven by a temporal modulation of a system parameter. Thus, they are complementary to the more commonly considered space charge single particle resonances, where space charge pseudo-multipole terms are assumed to exist already at finite level in the initial distribution. This article elaborates on the characteristics of such parametric instabilities in 3D bunched beams-as typical in linear accelerators-for modes of second (envelope), third and fourth order, including the transverse coupled "sum envelope instabilities" recently discovered for 2D beams. Noteworthy results are the finding that parametric resonances can be in competition with single particle resonances of twice the order due to overlapping stopbands; furthermore the surprisingly good applicability of the stopband positions and widths obtained from previously published 2D linearised Vlasov stability theory to the 3D non-Kapchinskij-Vladimirskij particle-in-cell code studies presented here.

Intensity limitations due to space charge for bunch compression in synchrotrons

“…[25]) with a fourth order single particle space charge resonance has more recently been studied in straight lattices and for short bunches [20,[26][27][28].…”

“…It is pointed out in Refs. [26,28] that for a well-matched Gaussian beam transported for a short distance the fourthorder resonance can dominate over the envelope instability, since the fourth-order resonance readily occurs from the beginning due to the presence of the "pseudo-octupole" in the initial density profile. In contrast, for the envelope instability, it takes more time to "amplify" the initial noise and develop the collective instability.…”

A study of coherent and incoherent resonances in high intensity beams using a linear Paul trap

“…Additionally, the transverse distribution of the beam itself is thought to determine whether a beam is excited coherently. Hofmann, in [10], shows that KV distributions can be excited coherently by higher order modes, but that for more realistic beam models, such as waterbag or Gaussian distributions, the spread in particle tunes may lead to Landau damping of coherent modes higher than m=2. For these higher order modes the significant overlap between the incoherent tunes in the distribution and the tune of the coherent resonance can lead to energy transfer from the coherent excitation of the distribution to the incoherent excitation of particles within the beam, so that coherent oscillations never form [11].…”

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].…”

“…However, most of those theoretical studies were based on a two-dimensional model. Three-dimensional macroparticle simulations were carried out for a bunched beam under the guidance of the two-dimensional envelope instability model [16,19]. It was found in Ref.…”