A METHOD TO OBTAIN THE FREQUENCY OF THE LONGITUDINAL DIPOLE OSCILLATION FOR MODELING AND CONTROL IN SYNCHROTRONS WITH SINGLE OR DOUBLE HARMONIC RF SYSTEMS

In a heavy-ion synchrotron the bunched particle beams can perform longitudinal oscillations of several modes. These oscillations are dampedby Landau damping, but can become unstable if driven accordingly. Furthermore, Landau damping is accompanied by filamentation which increases the longitudinal beam emittance and thus reduces the beam quality. To stabilize the beam and to keep the emittance low, control measures are taken. However, any controller design requires knowledge about the oscillation frequency of the mode which is to be damped. For a single harmonic RF voltage and a very small bunch, the longitudinal equations of motion of the particle can be linearized, and all the particles oscillate with approximately the same synchrotron frequency which equals the frequency of the dipole oscillation. For a larger bunch, or for a double harmonic RF system introducing nonlinearities around the reference point, this is no longer valid. In this work, we present a method to obtain the frequency of the dipole oscillation for larger bunches with a single or double harmonic RF system and show how this can be used for a state space controller design.