Lawrence Berkeley National Laboratory

In this note, we develop in closed form the dynamical $SO(3)$ spin map determined by the Thomas-BMT equation for a charged particle with general gyromagnetic ratio in a uniform longitudinal magnetic and electric field. This field configuration provides an idealized model of uniform particle acceleration with transverse (solenoidal) confinement. The spin 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. The dynamical evolution of these phase space variables is also described. This allows exact transport of both phase space and spin variables over long distances in such a field region, without the need for numerical integration.