ADIABATIC INVARIANTS FOR SPIN–ORBIT MOTION

It has been predicted and found experimentally that the polarization direction of particles on the closed orbit can be manipulated, without a noticeable reduction of polariza-tion, by a slow variation of magnetic fields. This feature has been used to avoid imperfection resonances where the spin precession frequency is close to a multiple of the circu-lation frequency. We report here on a proof that relates this property to an adiabatic invariant of spin motion. The proof is relatively simple since only two frequencies, the spin ro-tation frequency and the particle’s rotation frequency on the closed orbit, are involved. The invariant spin field de-scribes a periodic polarization state of a beam’s phase space distribution. This invariant spin field leads to a very useful parametrization of coupled spin and orbit dynamics. We also report on a proof showing that the invariant spin field gives rise to an adiabatic invariant of spin-orbit motion. The proof is now much more complicated since the orbital frequencies are involved. Due to this adiabatic invariance, the spin field of a polarized beam follows slow changes of the accelerator’s invariant spin field that can occur during a slow acceleration cycle. This feature is essential when high-order spin orbit resonances are crossed since it allows a reduced degree of polarization at the resonance condition to recover, to a large degree, after the resonance has been crossed.