Mechanisms for nonlinear acceleration in FFAGs with fixed RF

A signature of fixed-field acceleration is that the orbit of the beam centroid unavoidably changes with energy. The corresponding, often nonlinear, change in path length results in phase slip of the particle beam relative to a fixed-frequency accelerating waveform. Nonetheless, depending on the number and location of the fixed-points of the motion, acceleration is possible for a limited number of turns by allowing the beam to cross back and forth across the crest (cross-crest acceleration). For multiple fixed points, this asynchronous form of acceleration is facilitated by a libration path that extends from injection to extraction energy when a threshold value of accelerating voltage exceeded. Successful acceleration is accomplished when the radio frequency is made synchronous with the revolution period at the machine central orbit, or when the RF is offset in a prescribed manner that leads to staggering of phase traces on consecutive turns. The present work explores the influence of the fixed points and of RF manipulations on the longitudinal dynamics in FFAGs with a variety of path-length dependence on kinetic energy; emphasis is given to quadratic dependence as occurs in a type of accelerator currently proposed for rapid acceleration of muons.