Stability of the phase motion in race-track microtrons

Abstract

We model the phase oscillations of electrons in race-track microtrons by means of an area preserving map with a fixed point at the origin, which represents the synchronous trajectory of a reference particle in the beam. We study the nonlinear stability of the origin in terms of the synchronous phase —the phase of the synchronous particle at the injection. We estimate the size and shape of the stability domain around the origin, whose main connected component is enclosed by an invariant curve. We describe the evolution of the stability domain as the synchronous phase varies. We also clarify the role of the stable and unstable invariant curves of some hyperbolic (fixed or periodic) points.Peer ReviewedPostprint (author's final draft

Racetrack Microtron—Pushing the Limits

“…The most dangerous nonlinear resonances causing beam losses are the third-order and forth-order resonances situated for ν = 1 at the synchronous phase values ϕ s = 25.5 • and ϕ s = 17.7 • , respectively. If the number of orbits is not small, 10 or more, and if ϕ s is close to the resonance, then the stable phase oscillation region narrows sharply [25][26][27]. Previous studies have also revealed quite non-trivial properties of the stable phase oscillation region in RTMs.…”

“…Additionally, the stability region includes stable elliptic islands that are separated from the longitudinal acceptance which, in terms of the theory of dynamical systems, is a connected to a local stability domain that includes the synchronous trajectory (see Ref. [27] for details).…”