Studies of Collective Instabilities in Muon Collider Rings

Using a macroparticle model for both synchrotron motion and beta-tron motion, we provide formulae for the beam centroid shift, the tune shifts and the growth rates of the longitudinal and transverse head-tail instabilities. Both single turn and multi-turn eeects are included. The transient amplitudes based on the two-particle model are estimated. We estimate the microwave instability for the muon collider, using three sets of design parameter, and three diierent impedance models: resonator, re-sistive wall and a SLAC-like or a CEBAF-like rf accelerating structure. For momentum compaction ' 10 ?5 , the synchrotron oscillation period is about a hundred turns, the impedance of the ring is dominated by the rf cavities, and the microwave instability is well beyond threshold. For < 10 ?6 , the synchrotron oscillation period is longer than the eeective storage time, and the beam dynamics in the collider like that in a linac. For this case, the BBU instability is the most important collective eeect. The implications of using BNS damping are brieey discussed. The growth rates of the head-tail instability due to longitudinal and transverse chro-maticities, are presented. Finally we discuss brieey the beam dynamics issues related to the beam-beam interaction and a quasi-isochronous ring.