Stability of beams hollow in longitudinal phase space

Intense beams, ~100 μA, can be accumulated in storage rings by means of charge exchange injection together with a sweep of the closed orbit and/or incoming beam parameters. Such `painting´ schemes can result in 6-D charge distributions peaked at the synchronous orbit and stable under space charge forces. Some of these schemes, however, may produce, at some point during the procedure, charge distributions that are hollow in longitudinal phase space. Under these conditions tracking simulations incorporating longitudinal space charge effects suggest that a dipole instability may develop. The development time depends on the total charge and its distribution. It is found that hollow beams may be practicable provided that the total charge is less than a critical value. Initial growth rates are given