Transverse Beam Dynamics in the Modified Betatron

摘要

The linearized equations governing the motion of the center of a beam about its equilibrium position in a modified betatron, as well as equations governing the motion of an individual particle about the beam center, are presented and solved. Self field effects, including toroidal hoop stresses and wall image forces, are included in the analysis. All fields, both self and applied, are assumed to be azimuthally symmetric but are allowed to have arbitrary time dependences. The solutions to the equations of motion are analyzed for stability and conditions for stability are obtained. Further study of the solutions illustrates two phenomena of experimental interest: (1) the unavoidable traversal of a finite 'instability gap' in parameter space during acceleration; and (2) the adiabatic increase in the amplitude of the betatron oscillations during removal of the toroidal magnetic field, prior to beam ejection. By careful design, the effects of these phenomena can be reduced to insignificant levels in an actual accelerator. (Author)