Stability of Orbits in Combined Betatron and Azimuthal Magnetic Fields

In a cylindrically symmetric magnetic field the requirement for a stable betatron orbit is that there be a minimum in the vector potential component Aθ, i.e., the induction electric field strength. The problem of stability when an azimuthal or Bθ field is added to the betatron field is considered. It is shown that when Bθ is present there must be an extremum in Aθ for stable motion. This criteria includes the pure betatron field as a special case so that the Schmidt beam current formula still applies. Schmidt's result is extended to include resistive as well as runaway currents, and its application to the various plasma betatron experiments is discussed.