Current limitations due to instabilities in modified and conventional betatrons

Current thresholds for longitudinal and transverse instabilities are calculated for beams of specified dimensions in conventional(Bθ= 0) and modified (Bθ ≠ 0) betatrons, using simple models for the longitudinal and transverse impedances. Self field effects of the beam are included and lead to a competitive effect between the stabilization mechanism and instability growth. This competition can result in a multi-valuedness in the limiting current vs. beam energy spread plot, even for conventional betatrons. Accessibility of the various limiting current levels appears to depend upon the rate of beam injection. The stabilizing effects of betatron oscillations are discussed and written as the sum of three, contribution to an effective energy spread. We find that the presence of a strong toroidal field can significantly improve the current carrying capacity of the accelerator.