Orbits in Fixed Field Alternating Gradient Synchrotron

An analytical investigation has been made of the motion of a charged particle in an axially symmetric steady magnetic field. For a given momentum, expressions for the shape of the equilibrium orbit and betatron oscillation frequencies have been derived in terms of the field index k, the spiral parameter K, and the set of field coefficients gnm. These algebraic results have been programmed for the IBM704 computer. For a 500 MeV fixed field alternating gradient (FFAG) synchrotron designed as an injector,1 computations yield radial and vertical tunes averaged with respect to momentum of ¿r = 3.23, ¿z = 2.37 with variations of |¿¿r| ¿ 0.0001, |¿¿z| ¿ 0.0004 depending on the exact momentum. For comparison, Runge-Kutta evaluations which require a factor of six longer computing times yield average tunes of ¿r = 3.22 and ¿z = 2.49.