Averaged Invariants in Storage Rings with Synchrotron Motion

TL;DR

This paper develops an analysis of coupled transverse and longitudinal particle dynamics in storage rings, deriving an averaged invariant that accounts for synchro-betatron coupling, with applications to toy and real accelerator lattices.

Contribution

It introduces a method to analyze fully coupled synchro-betatron dynamics and derives an averaged invariant applicable to generic lattices, including integrable optics designs.

Findings

The averaged invariant captures complex coupled dynamics over a synchrotron oscillation.

In a toy lattice, the analysis yields analytically tractable results.

In a real design, the invariants are shown to be periodic despite complex behavior.

Abstract

In an ideal accelerator, the single-particle dynamics can be decoupled into transverse motion -- the betatron oscillations -- and longitudinal motion -- the synchrotron oscillations. Chromatic and dispersive effects introduce a coupling between these dynamics, the so-called synchro-betatron coupling. We present an analysis of the fully coupled dynamics over a single synchrotron oscillation that leads to an averaged invariant with synchro-betatron coupling in a generic lattice. We apply this analysis to two problems: first, a toy lattice where the computations are analytically tractable, then a design for a rapid cycling synchrotron built using the integrable optics described by Danilov and Nagaitsev, showing that although there is fairly complex behavior over the course of a synchrotron oscillation, the Danilov-Nagaitsev invariants are nevertheless periodic with the synchrotron motion.