Coherent Oscillations Produced by the Rapid Skirting of an Integral Resonance

Although it is well known that accelerating a cyclotron beam through an integral resonance generates a coherent oscillation of the beam, it is much less widely known that such oscillations can also be produced just by rapidly skirting the same resonance. The latter will occur whenever the perturbing field or the radial (vertical) tune changes so rapidly in the vicinity of the resonance that the accelerating beam cannot smoothly follow the resultant non-adiabatic shift of the equilibrium orbit. A straightforward analysis of these effects leads to a simple formula for the resultant growth of the oscillation amplitude. This formula is applied to an example recently found in superconducting cyclotrons when the radial tune dips rapidily, but briefly, down close to unity. Applications to other cyclotrons are also considered.