Beam stability in a stellatron

Collective effects in the l=2 stellatron [Phys. Rev. Lett. 50, 507 (1983)], a high-current electron accelerator, are studied. A thin-beam model is employed, and only long-wavelength, low-frequency modes are considered. The eigenvalue problem is formulated in general and solved analytically in the smooth approximation; comparison with a numerical solution of the eigenvalue problem is excellent. It is found that the dispersion relation in the smooth approximation is identical in form to that for the modified betatron, under a simple substitution. An analytical expression for the transition energy, obtained earlier by a simple dynamical argument, is confirmed. The stellarator field is found to reduce the growth rate of the negative mass instability.