Estimate of the longitudinal self electric field of an ion beam

The self electric field on the axis of an ion beam of radius a in a conducting cylinder of radius b is given. The formula assumes that the radius of the conducting cylinder is much smaller than the length of the ion beam (b much less than L) and that E/sub z/ is calculated away from the edge of the beam (vertical bar z vertical bar less than vertical bar L/2 vertical bar). For the HIDE parameters, i.e., loosely speaking a 50 TW, 100 JK, multi-GeV heavy ion beam, the assumption b much less than L is no longer valid and hence this equation cannot be used. Since for an unneutralized heavy ion beam it is necessary to apply ramp voltages to compensate for the longitudinal self fields it is desirable to know exactly what these fields are. Here, exact expressions for E/sub z/ on the axis of the ion beam are obtained and are compared under different circumstances with the approximate results given by the equation