Transverse oscillation of particles in the vicinity of resonances for a cyclotron

Abstract

Transverse oscillation is an important issue in beam dynamics of cyclotrons and can be described by the Mathieu equation. We review the standard form of the Mathieu equation, d 2 u dθ 2 þ ðδ þ ε · cos 2θÞu ¼ 0, and propose a modification of the method of multiple scales (i.e., a perturbation method) so that the asymptotic analytical solutions of the Mathieu equation can be computed in the stable and unstable regions for both δ ≥ 0 and δ < 0. This method was applied to the nonlinear transverse oscillation equations for a cyclotron. Analytical solutions for transverse oscillation in the stable and unstable regions (i.e., vicinity of the resonances) were obtained, and the accuracy of these analytical solutions was confirmed by their close agreement with the direct numerical integration. Useful results such as the analytical solution of the transverse oscillation frequency, increasing rate of the amplitude in unstable regions, and the resonance width were also derived; the stable condition and driving terms of the resonances can be obtained from the analytical solutions.