The electrostatic instability of rippled, magnetically focused, non-neutral, cylindrical beams of charged particles

The stability of rippled non-neutral beams of charged particles produced under conditions in which centrifugal, space charge and magnetic forces are not balanced, is investigated. The beams are considered to be cold and assumed to partly fill a conducting pipe in the presence of an axial uniform magnetic field. The calculations hold for small ripple amplitudes and long-wavelength electrostatic perturbations (or, alternatively, thin beams). The rippled equilibrium beams considered in this work are found to support unstable, azimuthally symmetric, surface electrostatic modes. The growth rates are proportional to the relative ripple amplitude and to the effective beam plasma frequency.