The influence of density distribution on the stability of beams

We examine the effect of various density distributions in four-dimensional phase space and their projections in real and velocity space on the stability of continuous beams in alternating-gradient transport lines using particle-following computer simulations. We discuss the susceptibility of three different distributions (Kapchinskii-Vladimirskii, bicylinder, and thermal) to third- and higher-order mode instabilities. These distributions are all uniform in real space, but their velocity distributions are different; they also react differently to structure resonances. Velocity distributions of high-current beams tend to evolve to a peaked Gaussian-like form. Is there a specific velocity distribution that is stable and, therefore, the preferred injection distribution for minimizing emittance growth? Forced smoothness or uniformity in real space is necessary for setting up particle simulations of high-current beams so that spurious charge-redistribution emittance growth can be avoided. Is forced smoothness also desirable in four dimensions for continuous beams and possibly in six dimensions for bunched beams? We consider these and related questions