Dynamics of three simultaneously stored beams in a storage ring

In this study, the observation of three simultaneously rotating beams in the SOLEIL storage ring is reported. This event occurred in November 2007 while operating in a low momentum compaction factor mode. The dynamics of the three beams is simulated using the longitudinal equations of motion and the longitudinal phase-space Hamiltonian which are extended to include higher-order terms of momentum compaction factor up to the third order. The effect of the transverse oscillation amplitude of the particles is also included in this work. It is shown that this term, which is experimentally very difficult to compensate for very low momentum compaction factor optics, can strongly affect the longitudinal beam dynamics. Finally, an extended formula of the variation of synchrotron frequency with respect to the relative variation of the rf frequency is derived and is used to deduce the higher-order terms of the momentum compaction factor from experimental data.