Nondumped oscillations of electron beam in ion beam undulator with electron and ion beams with different radii

Ion beam undulator consisting of a high current relativistic electron beam propagating co-axially with a nonrelativistic ion beam in the drift tube. External radii of electron and ion beams are considered different from each other. In a steady-state situation the oscillations of electron and ion beams are characterized by lengths of standing waves M and L correspondingly. The regime with M ≪ L is suitable. Moreover, when ion beam radius less than electron beam radius, beams oscillations are dumped. When M ≫ L ion beam undulator is worked, when electron beam radius less than ion beam radius, and when ion beam radius less than electron beam radius. New effect is took place in this case. The nonlinear electric and magnetic fields are took place outside ion beam when ion beam radius less than electron beam radius. However, beautiful nondumped oscillations of electron beam are occur in the case. Now, the cause of the dumping absence is investigated