Effects of overlapping parametric resonances on the particle diffusion process

Fermilab, Box 500, Batavia, Illinois 60510~Received 19 April 1999!The evolution of the beam distribution in a double-rf system with a phase modulation on either the primaryor secondary rf cavity was measured. We find that the particle diffusion process obeys the Einstein relation ifthe phase space becomes globally chaotic. When dominant parametric resonances still exist in the phase space,particles stream along the separatrices of the dominant resonance, and the beam width exhibits characteristicoscillatory structure. The particle-tracking simulations for the double-rf system are employed to reveal theessential diffusion mechanism. Coherent octupolar motion has been observed in the bunch beam excitation.The evolution of the longitudinal phase space in the octupole mode is displayed. @S1063-651X~99!00111-7#PACS number~s!: 29.27.Bd, 41.75.2i, 05.45.2aI. INTRODUCTION