Spin decoherence rate in a homogenous all-electric ring II: Synchrotron oscillations

In a previous paper, I derived analytical expressions for the spin decoherence rate for motion in a homogenous weak focusing all-electric storage ring. In this paper, I extend the formalism to include an rf cavity and synchrotron oscillations. I perform a canonical transformation to separate the radial motion into betatron and dispersion terms, up to third order terms. (The canonical transformation is actually for rings of arbitrary structure without transverse coupling.) I apply the results to derive the spin decoherence rate due to synchrotron oscillations in a homogenous weak focusing all-electric storage ring. Tracking results display good agreement between numerical simulations and theory.