Transverse deflections in a cavity due to the short-range longitudinal wake

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Consider an ultra-relativistic electron bunch passing through a (cylindrically symmetric) multi-cell linac cavity that is filled with fundamental mode rf. It is well known that this bunch--on entering the cavity--experiences a focusing kick, and--on exiting the cavity--a defocusing kick, even though the mode is cylindrically symmetric. The effects of these kicks in linacs tend to be significant only in low energy regions. Tracking computer programs such as MAD [1] and LIAR [2] include a simple model of these kicks, one based on calculations of W.H. Panofsky [3]. According to this model the effect is represented by two thin lenses positioned at the ends of the cavity, with the strength of the lenses dependent on the accelerating gradient in the cavity. However, a beam will itself excite wakefields that modify its energy gain in a cavity, a modification that depends also on longitudinal position within the bunch. The program LIAR extends Panofsky's rf kick model to include this modification to the effective gradient experienced by different parts of the beam. In this report we investigate how the wakefields affect the rf cavity kicks. In particular, we are interested in the case when the wakefields are a significant perturbation to the problem,more » such as when, for example, the beam traverses an empty cavity (one with no rf). In this report we have shown that one can calculate the transverse kicks when one knows the time-dependent variation of the longitudinal wake forces on axis. The variation in gradient due to wakefields, however, will in general differ from that due to normal rf acceleration. In particular, transients at the ends of structures, and--for constant gradient structures--an increase in gradient amplitude from beginning to end of the cavity, will mean that the model of focusing/defocusing edges, used for rf acceleration, will be inaccurate. Finally, we conclude that the method LIAR uses to treat the effect of rf focusing in the presence of wakefields on beam orbit is approximately correct.« less