Lawrence Berkeley National Laboratory

In this note, we develop in closed form the time-evolution $SO(3)$ spin map determined by the Thomas-BMT equation for a charged particle with general gyromagnetic ratio in a uniform magnetic field. This result is then applied to determine the corresponding $s$-evolution spin map in the curvilinear Frenet-Serret coordinate frame associated with the transport of a charged-particle beam through an ideal sector dipole (bending magnet). The map is expressed in a factorized Lie-algebraic (axis-angle) form, making it straightforward to determine the corresponding $SU(2)$ or quaternion representation. To facilitate numerical implementation, the Lie generators are expressed as functions of the six phase space variables commonly used for charged particle transport in $s$-based tracking codes. This capability allows exact transport of both phase space and spin variables over an entire ideal sector dipole, without the need for numerical integration. The results can also help to validate alternative spin tracking approaches.