Analysis of the longitudinal collective behavior in a 50GeV×50GeV muon collider ring

Simulations of the longitudinal instability in the 50GeV×50GeV muon collider ring have been performed. Operation of the ring close to the slippage factor η_{1}≃10^{-6}, such that synchrotron motion is frozen, minimizes the need for rf to maintain the bunch length. However, there is still an energy spread due to the bunch wake. For design parameters of the ring, this induced energy is too large and must be controlled. This paper demonstrates that the bunch wake may be compensated for by two rf cavities with low rf voltages. These studies were made at the nominal design point, and sensitivities to errors were explored. It is seen that the small energy spread of the beam (δE/E=3×10^{-5}) in the 50GeV×50GeV muon collider ring can be maintained during the 1000 turn lifetime of the muons. Controlled beam dynamics requires proper choice of rf parameters (rf voltage, rf frequency, and phase offset) for two cavities; these parameters depend on the ring design through the impedance, beam pipe radius, and momentum compaction. The simulation also shows that the computation of wake field using bins of variable width (each with a constant number of macroparticles in each bin) gives an accurate wake and also yields reduced computing time compared to an evaluation of the wake as the direct sum over the wakes of all preceding macroparticles