Emittance growth rates for displaced beams

Emittance growth rates have been previously analyzed for nonuniform beams in linear channels and for initially uniform mismatched beams in nonlinear channels. These studies were for centered beams. Additional emittance growth can arise in cases where the beam is initially displaced. The purpose of this study is to obtain growth rates for displaced beams. This work differs from studies involving random displacement of electrodes. Our analysis assumes instead that the focusing system is perfectly aligned but that the beam is initially displaced with respect to the equilibrium axis. If the focusing force is slightly nonlinear, we find a gradual transfer of the potential energy of beam displacement into kinetic energy associated with emittance growth. We present explicit results for the emittance growth distance as a function of the nonlinearity of the channel. These results will have practical importance for designers of accelerators and transport systems when setting realistic tolerances for initial beam alignment. These tolerances will depend on the nonlinearity and the length of the system