A matrix approach to bunch shape oscillation damping

APPROACH TO BUNCH SHAPEIt is well knovm that if the bunch area in the Jongi tudinal phase space is not matched with the bucket trajectories, the filamentation process increases the phase space area occupied by the beam, and then•reduces density.Longitudinal mismatch can be easily detected on the :z signal of a pick-up electrode the envelope of bunch signals is modulated at twice the synchrotron frequency.enAs H was shown by H. G. Herewardl " one can use information from this detected signal to modulate the stable phase and consequently the longitudinal focusing of the RF system.As a result one can achieve damping of these bunch shape oscillations provided they are coherent for all the bunches.In Hereward 1 s theory, mismatch of bunches is characterized by the second moment of the particle distribution in longitudinal phase space.In this paper we shall use a less general, but perhaps simpler description of the beam in terms of ellipses.