Space-charge limits in linear accelerators

This report presents equations that allow an approximate evaluation of the limiting beam current for a large class of radio-frequency linear accelerators, which use quadrupole strong focusing. Included are the Alvarez, the Wideroe, and the radio-frequency quadrupole linacs. The limiting-current formulas are presented for both the longitudinal and the transverse degrees of freedom by assuming that the average space-charge force in the beam bunch arises from a uniformly distributed charge within an azimuthally symmetric three-dimensional ellipsoid. The Mathieu equation is obtained as an approximate, but general, form for the transverse equation of motion. The smooth-approximation method is used to obtain a solution and an expression for the transverse current limit. The form of the current-limit formulas for different linac constraints is discussed.