Focusing limit of a cyclotron: Axial betatron instability against beam dynamics approach

In an isochronous relativistic cyclotron, axial defocusing of a beam caused by the radial growth of the isochronous magnetic field is compensated by the azimuthal field gradient introduced by sectors. The focusing capabilities of sectors set the maximum ion energy obtainable from the machine. Usually, the focusing limit of a machine is determined by using the criterion for axial beam instability evolving from the equations of betatron oscillations. The obtained value of the focusing limit is approximate because the equations of betatron oscillations it originates from are approximate as well. The accurate value of the focusing limit is obtained by simulating accelerated beam dynamics in the extraction region. It is shown that the focusing limit of a cyclotron resulting from the two methods could differ for more than 9%. The suggested third method for focusing limit computation relies on the beam dynamics simulation along the critical equilibrium orbit rather than the acceleration orbit and thus it is less time consuming although equally accurate.