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.
Highlights
An understanding of space charge effects in beam dynamics of both linear and circular high intensity accelerators is crucial for the design, operation, and optimization of such accelerators
This work included the envelope instability occurring at 90° phase advance and demonstrated that similar parametric instability phenomena exist in higher order—even suggesting up to arbitrarily high order for an initial KV-distribution
This study shows that in high intensity beams a diversity of purely space charge driven parametric resonances exists —beyond the well-known envelope instability case
Summary
An understanding of space charge effects in beam dynamics of both linear and circular high intensity accelerators is crucial for the design, operation, and optimization of such accelerators. This work included the envelope instability occurring at 90° phase advance and demonstrated that similar parametric instability phenomena exist in higher order—even suggesting up to arbitrarily high order for an initial KV-distribution. In a linac environment the first test of the 90° stopband for bunched beams was undertaken in 2009 in the UNILAC high intensity heavy ion accelerator [8]. This experiment left open questions of interpretation, which were addressed in recent theoretical research clarifying the joint appearance of second order parametric (envelope instability) and fourth order single particle resonant space charge effects [9]. For simplicity we assume equal emittances in all planes and leave out emittance exchange effects, which are not driven parametrically, but by anisotropy [13]
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