ELECTROMAGNETIC BEAM STABILITY IN A BETATRON WITH HELICAL QUADRUPOLE MAGNETIC-FIELD

A Vlasov treatment is developed to investigate the beam stability in a betatron with stellarator windings. The general formulation includes effects of finite beam temperature and radius. An analytic dispersion relation is derived. In the limit of cold beam and small beam radius, the general dispersion relation reduces to simpler form, which exhibits instability due to frequency matching between the transverse electric (TE) beam pipe cavity modes and the Doppler shifted electron oscillations in a combined longitudinal and helical quadrupole magnetic field. In particular, an internal beam mode with azimuthal mode number v = 0 is found to be unstable for typical parameters of the Spiral Line Induction Accelerator design [Bull. Am. Phys. Soc. 34, 2160 (1989)] © 1990 American Institute of Physics.