Abstract
We study the single-particle dynamics in a general and parametrized alternating-gradient cell with zero chromaticity using the Lie algebra method. To our surprise, the first-order perturbation of the sextupoles largely determines the dynamics away from the major resonances. The dynamic aperture can be estimated from the topology and geometry of the phase space. In the linearly normalized phase space, it is scaled according to $\overline{A}\ensuremath{\propto}\ensuremath{\phi}\sqrt{L}$, where $\ensuremath{\phi}$ is the bending angle and $L$ the length of the cell. For the 2 degrees of freedom with equal betatron tunes, the analytical perturbation theory leads us to the invariant or quasi-invariant tori, which play an important role in determining the stable volume in the four-dimensional phase space.
Highlights
The linear motion of the particles in the modern storage rings was well understood by Courant and Snyder in the theory of the alternating-gradient synchrotron [1]
The nonlinearity in a periodic system like a storage ring generates nonlinear resonances [2,3,4], which play an important role in the single-particle dynamics
It is well known [7] that the dynamics of a charged particle in accelerators can be described by its canonical coordinates z 1⁄4 ðx; px; y; py;δ;lÞ, where δ 1⁄4 ðp − p0Þ=p0 is the relative momentum deviation and l 1⁄4 vt, v is the velocity of the particle, and t is the time of flight
Summary
The linear motion of the particles in the modern storage rings was well understood by Courant and Snyder in the theory of the alternating-gradient synchrotron [1]. The particle motion becomes nonlinear because of the sextupoles introduced for chromatic compensations. It has been successfully applied to the analysis of an isolated resonance [8,9] It fails to describe all nonlinear resonances in general due to the so-called problem of the small denominators resulting in the divergence of the perturbation series [10]. It shows that the nonresonance tori in an integrable system will be distorted but preserved, provided that the perturbation is sufficiently small.
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