Simulations of the longitudinal instability in the SLC damping rings

A longitudinal, single bunch instability has been observed in the SLC damping rings. Beyond a threshold current of 3×10¹⁰ the energy spread of the beam increases and a “saw-tooth instability” appears. The latter term is meant to describe a rather complicated phenomenon, depending on both current and rf voltage. In one form it describes a cycle that includes a quick increase in bunch length, over a time on the order of a synchrotron period, and then a much slower return to the original length, over a time on the order of a radiation damping time. Although the total relative change in length is only about 10% the resulting unpredictability of the beam properties in the rest of the SLC accelerator makes it difficult, if not impossible, to operate the SLC above the threshold current. With the goal of trying to understand this instability the simulations that are the subject of this paper were begun. Bunch lengthening calculations have been performed before for the SLC damping rings, to obtain the average bunch shape as function of current. The wakefields of all the important vacuum chamber components were first obtained numerically. The dominant elements were found to be many small discontinuities-bellows, masks, transitions, etc.-elements that are inductive to the beam. Once the total wakefield had been obtained, and the threshold current was known (from measurements), the average bunch shapes were found by means of a potential well calculation. The bunch shapes obtained in this way were found to agree very well with measurement results. In this paper we investigate the single bunch behavior of the SLC damping rings using time domain tracking and also a Vlasov equation approach. Since the earlier bunch length calculations the damping ring vacuum chamber has been modified, by sleeving the bellows. Our results will, therefore, include the effects of this modification