Parametric instabilities in 3D periodically focused beams with space charge

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.

Figure 1

Evolution of KV-envelopes versus cell number for $k_{0,x,y}=100°$, $k_{x,y}=82°$.

Figure 2

Complete stopband with relative growth of rms envelopes for fixed $k_{0x,y}=100°$ as function of $k_{x,y}$.

Figure 3

Stopbands of second order (top, envelope instability), third order (center) and fourth order (bottom) parametric resonances as function of a dimensionless intensity parameter. Each curve relates intensity to $k_{x,y}$ for a fixed value of $k_{0,x,y}$ (in degrees)(source: Ref. [1]).

Figure 4

rms emittances versus cell number for $k_{0,x,y}=120°$, $k_{x,y}=80°$ (inserts showing phase space plots at indicated cells) for KV beam (top graph) and for waterbag distribution beam (bottom graph).

Figure 5

rms phase advances versus cell number for waterbag example of Fig. 4.

Figure 6

rms emittances with phase space inserts for $k_{0,x,y}=120°$, $k_{x,y}=73°$ for KV beam (top graph) and waterbag distribution beam (bottom graph).

Figure 7

Density in $y$ with contour plots for $k_{0,x,y}=120°$, $k_{x,y}=73°$ for waterbag distribution beam.

Figure 8

Top graph: KV-envelopes versus cell number for $k_{0,x}=60°$, $k_{0,y}=140°$; bottom graph: simulation results for rms emittances versus cell number for waterbag distribution, with phase space inserts at cell 120.

Figure 9

rms emittances versus cell number for $k_{0,x}=60°$, $k_{0,y}=145°$ for waterbag distribution, with $x-y$ real space inserts at consecutive cells 200 and 201.

Figure 10

Top graph: rms emittances versus cell number for $k_{0,x,y}=90°$, $k_{x,y}=41.5°$ for waterbag distribution. Bottom graph: phase space plots at cells 5 and 45.

Figure 11

rms emittances versus cell number for $k_{0,x,y}=70°$, $k_{x,y}=35°$ and waterbag distribution, with $x-x^{\prime }$ insert.

Figure 12

Schematic stability diagram showing the location of transverse space charge driven 2nd to 4th order parametric resonances; and relevant 4th and 6th order single particle resonances (hatched, uncoloured bars).