Exceptional orbits: a new class of spin trajectories in circular accelerators

We claim there is a new class of spin trajectories in circular accelerators, which are not accounted for in current theories of spin dynamics. We call them exceptional orbits. On such orbits the spin motion drives a secular growth of the orbital amplitude(s). Our results do not by themselves prove the spatial separation of spin states of a relativistic charged particle beam, but our findings are consistent with those of other workers. Our findings do invalidate the unconditional use of the semiclassical approximation to treat the orbital motion, to diagonalize the spin–orbit Hamiltonian. We analyze a simple model in detail to demonstrate our ideas. We review the canonical transformation theory of spin dynamics in storage rings, and extend it to account for exceptional orbits. We also examine how exceptional orbits affect map-based algorithms. We show that the algorithms can sometimes break down, in ways not hitherto suspected. We show that on an exceptional orbit the secular rate of spin phase advance depends explicitly on the orbital arc-length, in contradiction to claims that the spin tune depends only on the orbital amplitudes.