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 4σ=360° due to the nonlinear space charge effect in high intensity beams have been studied previously. A wide stop band around 15° 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 η=0.8, a stop band with a width of almost 15° 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.