Accurate and efficient particle tracking through Siberian Snakes is crucial to building comprehensive accelerator simulation model. At the Alternating Gradient Synchrotron (AGS) and Relativistic Heavy Ion Collider (RHIC), Siberian Snakes are traditionally modeled in MAD-X by Taylor map matrices generated at specific current and energy configurations. This method falls short during ramping due to the nonphysical jumps between matrices. Another common method is to use grid field maps for the Snakes, but field map files are usually very large and thus cumbersome to use. In this work, we apply a new method called the Generalized Gradient (GG) map formalism to model complex fields in Siberian Snakes. GG formalism provides an analytic function in x and y for which automatic differentiation, i.e. Differential Algebra or Truncated Power Series Algebra can find accurate high order maps. We present simulation results of the Siberian Snakes in both the AGS and RHIC using the Bmad toolkit for accelerator simulation, demonstrating that GG formalism provides accurate particle tracking results.

Current and historic tracking studies of the RHIC accelerator lattice find difficulty in explaining the transmission efficiency of spin polarization from the AGS extraction to RHIC storage energies. In this paper, we discuss mechanisms that result in resonant depolarizing behavior, beyond the usual intrinsic and imperfection resonance structures. In particular, the focus of this paper will be on higher-order resonances that become apparent in the presence of snakes. The set of conditions that identify higher-order spin-orbit resonances are 𝜈 = 𝑗0 + 𝑗 ⃗ ⋅𝑄⃗for integers (𝑗0, 𝑗) ∈ ℤ^4, where 𝜈 is the spin tune and 𝑄⃗ contains the orbit tunes. Note that we do not use the closed-orbit spin tune 𝜈0 but rather the amplitude-dependent spin tune 𝜈(𝐽𝑥, 𝐽𝑦, 𝐽𝑧) that depends on the phase-space amplitudes. While Sibrian snakes keep 𝜈0 at 1/2, the amplitude-dependent spin tune can deviate from 1/2 and can cross resonances during acceleration.

The Generalized Gradient (GG) formalism of Venturini and Dragt for describing static magnetic or electric fields has been implemented in the Bmad toolkit for accelerator simulations. In conjunction with this, a new method for calculating GG derivatives from a field table has been developed which avoids some of the problems of the Venturini and Dragt method. Generalized gradients are also implemented in the PTC toolkit developed by Etienne Forest which is interfaced to Bmad. This allows for construction of spin/orbital Taylor maps useful for nonlinear analysis and rapid tracking.