Longitudinal beam dynamics with phase slip in race-track microtrons

Implementation of low-energy injection schemes in race-track microtron (RTM) designs requires a better understanding of the longitudinal beam dynamics. Unlike the high-energy case a low-energy beam slips in phase with respect to the accelerating field phase so that the standard notion of synchronous particle is not applicable. In the article, we generalize the concept of synchronous particle for the case of non-relativistic energies. An analytic approach for the description of the synchronous phase slip is developed and explicit, though approximate, formulas which allow to determine the equilibrium injection phase and to fix the parameters of the accelerator are derived. The approximation can be improved in a systematic way by calculating higher-order corrections. The precision of the analytic approach is checked by direct numerical computations and is shown to be quite satisfactory. Explicit examples of injection schemes and fixing of RTM global parameters are presented. We also address the issue of stability of synchrotron oscillations around the generalized synchronous trajectory and introduce the notion of critical energy.