Adiabatic invariance of spin-orbit motion in accelerators

It has been predicted and found experimentally that the polarization direction of particles on the closed orbit of a circular accelerator can be manipulated, without a noticeable reduction of polarization, by means of 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 circulation frequency. As a first step we show that this property is related to an adiabatic invariant of spin motion. The proof is relatively simple since it involves only two frequencies, the spin-rotation frequency and the particle’s rotation frequency on the closed orbit. The invariant spin field (ISF) describes a periodic polarization state of a beam’s phase-space distribution. This ISF leads to a very useful parametrization of coupled spin and orbit dynamics. We prove that this ISF gives rise to an adiabatic invariant of spin-orbit motion. This proof is much more complicated since the orbital frequencies are involved. Because of this adiabatic invariance, a beam’s spin field follows slow changes of the accelerator’s ISF that can occur during a slow acceleration cycle. This feature is essential when high-order spin-orbit resonances are crossed, since it allows polarization that has been reduced at the resonance condition to be recovered, to a large degree, after the resonances have been crossed.