Envelope instability and the fourth order resonance

Abstract

The well-known envelope instability or the second order even collective mode [I. Hofmann, Phys. Rev. E 57, 4 (1998)] and the fourth order resonance $4\ensuremath{\sigma}=360\ifmmode^\circ\else\textdegree\fi{}$ due to the nonlinear space charge effect in high intensity beams have been studied previously. A wide stop band around 15\ifmmode^\circ\else\textdegree\fi{} is found in a pure periodic focusing channel. In addition, it is illustrated that the fourth order resonance dominates over the envelope instability and practically replaces it in the stop band [D. Jeon et al., Phys. Rev. ST Accel. Beams 12, 054204 (2009)]. In this paper, for a continuous beam with remarkable space charge, our 2D self-consistent particle-in-cell simulation work with the code topopic shows these two kinds of effects respectively in a periodic focusing defocusing (FD) channel. For a fixed tune depression $\ensuremath{\eta}=0.8$, a stop band with a width of almost 15\ifmmode^\circ\else\textdegree\fi{} is also demonstrated. Moreover, it is confirmed that analytical results of the rms envelope instability diagram are a valid tool to interpret the width of the stop band. Emittance growth rates in stop band are also well explained. It is found that, for a nearly rms matched beam, the emittance growth in the stop band is almost proportional to the saturation time of the nonlinear instability of the envelope, which happens in a quick manner and takes only a few FD cells. In contrast, the fourth order resonance is independent of rms matching and will be accompanied by beam evolution as ``a long term effect'' once the related mechanism is excited.