Analytic description of the phase slip effect in racetrack microtrons

Implementation of low energy injection schemes in the race-track microtron (RTM) design requires a better understanding of the longitudinal beam dynamics. Differently to the high energy case a low-energy beam will slip in phase relative to the accelerating structure phase. This phase slip is due to various features of the beam dynamics, in particular it is caused by the fact that at the first orbits the relativistic factor of the electrons beta < 1. A phase shift may also be caused by the deformation of the particle trajectories due to the complicated magnetic field profile or simply by the fringe field in the end magnets. In article [1] we proposed the concept of equilibrium or synchronous particle for the case of non-relativistic energies in RTMs. An analytic approach for the description of the synchronous phase slip was developed and explicit, though approximate, formulas which allow to define the equilibrium injection phase and to fix the parameters of the accelerator were derived. In the present article we generalize this formalism for the case of non-trivial profile of the magnetic field in the RTM end magnets. Analytic formulas are derived, their accuracy can be improved by applying certain algorithm. Two examples of application of this formalism for fixing the injection phase and RTM parameters are given.