Perturbation of Radial Oscillations in Superconducting Cyclotrons Due to Asymmetric Dee Voltages or Phases

Like many other cyclotrons, the new superconducting cyclotrons frequently operate with ?r close to unity, and must actually accelerate the beam through the ?r = 1 resonance in the central region and again just prior to extraction. The spiral shaped dees in these machines are designed to operate with equal voltages and prescribed relative phases, and deviations from these values significantly perturb the radial oscillations. An analysis of these effects, including those produced by the spiral electric gaps, is outlined here and the results compared with those from a magnetic field bump. Possible applications to the control of orbit centering are discussed and illustrated with data from detailed orbit computations.