PFC / JA - 93 - 29 Properties of the Kapchinskij - Vladimirskij Equilibrium and Envelope Equation for an Intense Charged - Particle Beam in a Periodic Focusing Field

The properties of the Kapchinskij-Vladimirskij (K-V) equilibrium and envelope equation are examined for an intense charged-particle beam propagating through an applied periodic solenoidal focusing magnetic field including the effects of the self-electric and selfmagnetic fields associated with the beam space-charge and current. It is found that the beam emittance is proportional to the maximum canonical angular momentum achieved by the particles within the K-V distribution. The Poincar4 mapping technique is used to determine systematically the axial dependence of the radius of the matched (equilibrium) beam and to explore nonlinear resonances in the nonequilibrium beam envelope oscillations. Certain correlations are found between the nonlinear resonances and wellknown instabilities for the K-V equilibrium. It is shown, for the first time, that the nonequilibrium beam envelope oscillations exhibit chaotic behavior for periodic focusing magnetic fields and sufficiently high beam densities, and that there exists a uniquely matched beam in the parameter regime of practical interest, i.e., ao < 90*, where ao is the phase advance over one axial period of the focusing field in the absence of spacecharge effects. The nonlinear resonances and chaotic behavior in the nonequilibrium beam envelope oscillations may play an important role in mismatched or multiple beam transport, including emittance growth and beam halo formation and evolution. PACS numbers: 07.77.+p, 29.27.Eg, 41.75.-i, 52.25.Wz