Betatron Oscillations in the Synchrotron

Abstract

Betatron oscillations in the University of Michigan synchrotron have been investigated by exciting these oscillations with a transverse electric field. The results indicate that the betatron oscillation is composed of a group of component frequencies which are separated by the frequency of synchrotron phase oscillation. The suggested explanation of the observed splitting is the modulation of the betatron oscillation frequency by the synchroton oscillations.

Averaged invariants in storage rings with synchrotron motion

“…Using the Yoshida identity transforms this to a fourth order operation, reducing the numerically induced error in 𝐻 ⊥ conservation. As we can see in figure (1), 𝐻 ⊥ is conserved on the order of .2%. This provides a baseline for our assessment, and any variation in 𝐻 ⊥ varying above this level should indicate a physical outcome rather than a numerical artifact of our integration scheme.…”

“…For the following simulations, we use parameters designed to emulate the tunes of the iRCS lattice described in section 5 below. In particular, we use the numerical values in table (1). For these parameters, the lattices have the same linear betatron tune, synchrotron tune, and linear chromaticity.…”

“…The slow change in the momentum offset of a particle's trajectory suggests that there will also be a slow change in the chromaticity -the synchrotron motion will modulate the betatron oscillations with a frequency of the betatron phase. Synchro-betatron coupling therefore can lead to sidebands in the betatron motion located at 𝜈 ⊥ ±𝑚𝜈 𝑠 for betatron tune 𝜈 ⊥ and synchrotron tune 𝜈 𝑠 [1,2]. Because of the coupling in the system, the synchrotron motion modifies the usual transverse action-angle variables [3].…”

“…Synchro-betatron coupling can lead to complex coupled dynamics. The impact of synchrobetatron coupling has been well-studied for linear alternating gradient focusing lattices [1][2][3], but their influence on more novel lattice designs, such as nonlinear integrable optics [4][5][6][7], has yet to be studied in detail.…”

Electron Model Fixed Field Alternating Gradient Accelerator

“…2). Although more than the 8 A summary of the design parameters is given in Table I. TABLE I in the original design to arrive at a magnet geometry.…”