Space-Charge Effects on the Quadrupole Focusing System in Low-Energy Proton Linear Accelerators

A general computer program has been used to calculate the stability diagram for (+)(-)(+)(-) quadrupole arrangements when the space charge is uniformly distributed in an elliptic cylinder or an ellipsoid. Linearized equations for longitudinal and transverse envelopes of the bunch have been solved simultaneously in order to find the velocity dependence of the phase damping.