Diffusion mechanism of particle beams in the presence of phase modulation in double RF systems

The particle diffusion mechanism due to phase modulation on one of the RF cavities in a double RF system is studied. We find that the Einstein relation is satisfied if there is a global chaos in the phase space. On the other hand, the existence of dominant parametric resonances forces the particles streaming along the separatrix, which may result in an anomalous diffusion process. The particle tracking simulations based on the Hamiltonian for the double RF system is employed to reveal the essential diffusion mechanism. Also for the first time, a coherent octupole mode has been observed in the bunch beam excitation. The phase space evolution of the octupole mode is displayed.