Relaxation oscillations of the synchrotron motion caused by narrow-band impedances

Although the linearized theory of small amplitude synchrotron oscillations and the instability thresholds derived from it have long been understood, there is no satisfactory description of the large amplitude highly nonlinear synchrotron motion of a bunched beam. With an appropriate tuning of the RF cavity impedance, large amplitude, low frequency, self-sustained relaxation oscillations of this synchrotron motion are generated. This paper presents detailed experimental data on such behavior, tracking code results that reproduce the important characteristics, and a simple analytical model that explains the key features of the relaxation oscillation: growth of the instability, saturation of the oscillation, breakup of the bunch, and subsequent damping of the system back to the beginning of the next cycle of the relaxation oscillation.