Electron storage rings operating at high energies have proven to be invaluable source of synchrotron radiation.
Two and even three simultaneous beams of particles have been observed at different light source facilities worldwide. So called alpha-buckets were studied at Metrological Light Source (MLS) in Berlin (Germany), SOLEIL facility in France, DIAMOND light source in UK, NSLS ring in Brookhaven (USA). It is widely recognized that alpha buckets are general phenomena which is not restricted to a certain storage ring. However, earlier measurements showed essential limitations on parameters of alpha-buckets, strong dependence on high order (sextupole and octupoles) magnetic field imperfections and associated fast decay of electron current as well as reduced life time. Also a rising relevance of high order non-linear longitudinal beam dynamics is associated with new generation of diffraction limited light sources (DLSR) approaches, which all suffer nonlinear momentum compaction factor. A large variety of future generation electron synchrotrons require a comprehensive investigation of the physical processes involved into the operation of such rings.
In this paper, we present review of high order non-linear longitudinal beam dynamics based on the longitudinal equations of motion and Hamiltonian expanded to a high order of the momentum compaction factor.
Roots of the third order equation with a free term are derived in a form suitable for analytical estimations. The momentum independent term of orbit lengthening due to particle transverse excursions is estimated and taken into account. The results from simulations were benchmarked against existing experiments at the Karlsruhe Research Accelerator (KARA), SOLEIL and MLS rings. Parameters of three simultaneous beams and alpha buckets at MLS and SOLEIL have been reproduced with high accuracy. General conditions for stable operation of alpha buckets are presented. Based on analytical formulas and computer simulations, studies of longitudinal motion at KARA have been performed with an objective to estimate feasibility of filling and storing of beam in α-buckets. A Computer model was used to describe the behavior and dynamics of simultaneous beams in the KARA storage ring.