Skobeltsyn Institute of Nuclear Physics
Lomonosov Moscow State University

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. We generalize the concept of equilibrium or synchronous
particle for the case of non-relativistic energies. An analytical approach for the
description of the synchronous phase slip is developed and explicit, though
approximate, formulas which allow defining the equilibrium injection phase and fixing
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 analytical
approach is checked by direct numerical computations and was shown to be quite
satisfactory. Explicit examples of injection schemes and fixing of RTM global
parameters are presented.