Fast spin tracking using a Magnus expansion

Spin motion in particle accelerators obeys the Thomas-Bargmann-Michel-Telegdi (T-BMT) equation. Due to the structure of the T-BMT equation, the spin-transfer quaternion of a magnet is generally a nonlinear function of the entrance coordinates even if the phase-space motion is linear. This nonlinear function can be written as a Dyson expansion, for example as employed in the program SPRINT, which normalized the first-order expansion of the spin-transfer quaternion. Alternatively, this nonlinear function can be written as a Magnus expansion. This paper points out that in cases where the phase-space coordinates change little, as is generally the case for accelerator elements, the Magnus expansion is a much more appropriate method to describe the nonlinear spin motion because this expansion terminates after the first term when the phase-space coordinates are constant. We will demonstrate, with several examples, that an approximation based on the Magnus expansion leads to very good agreement with time-consuming numerical integration, and to significantly better agreement than obtained with historical codes like SPRINT.