Strength of higher-order spin-orbit resonances

Abstract

When polarized particles are accelerated in a synchrotron, the spin precession can be periodically driven by Fourier components of the electromagnetic fields through which the particles travel. This leads to resonant perturbations when the spin-precession frequency is close to a linear combination of the orbital frequencies. When such resonance conditions are crossed, partial depolarization or spin flip can occur. The amount of polarization that survives after resonance crossing is a function of the resonance strength and the crossing speed. This function is commonly called the Froissart-Stora formula. It is very useful for predicting the amount of polarization after an acceleration cycle of a synchrotron or for computing the required speed of the acceleration cycle to maintain a required amount of polarization. However, the resonance strength could in general only be computed for first-order resonances and for synchrotron sidebands. When Siberian Snakes adjust the spin tune to be 1 2 , as is required for high energy accelerators, first-order resonances do not appear and higherorder resonances become dominant. Here we will introduce the strength of a higher-order spin-orbit resonance, and also present an efficient method of computing it. Several tracking examples will show that the so computed resonance strength can indeed be used in the Froissart-Stora formula. HERA-p is used for these examples which demonstrate that our results are very relevant for existing accelerators.

Quasiperiodic spin-orbit motion and spin tunes in storage rings

“…s can vary with the beam energy and with J. This is confirmed by simulations in SPRINT [6,8,9,[11][12][13]23,25], hence the name ''amplitude dependent spin tune.'' To study the dependence of J on J (recall Remark 5 of Sec.…”

“…Nevertheless, it seems that it usually exists for storage rings of interest. This is supported by a large amount of numerical work in which, for the cases studied, it was possible to construct at least a very good approximation to an ISF [6,8,9,[11][12][13]23,25]. Note that in these simulations hard edge and some thin lens representations of fields were used so that the A's were not smooth; see Remark 16 in Sec.…”

“…Studies of the effect on polarization of real resonance crossing can be found in [11][12][13]23]. The consequences for the polarization of crossing first order resonances are usually quantified using the Froissart-Stora formula [54].…”

“…For this and the other results in this paper it has been convenient to prescribe that the A's are smooth. But the demonstration of ISFs and spin tunes in [6,8,9,[11][12][13]23,25] indicates that the smoothness condition can often be relaxed. For convenience, in the remainder of this section we will use the term ISF in this spirit.…”

“…Furthermore, the fact that the integral vanishes at one position does not mean that it vanishes at other positions. We prefer to reserve the term ''depolarization'' for a definitely irreversible loss of polarization such as occurs in the presence of noise or for an effectively irreversible loss of the kind that can occur during resonance crossing [11]. Although we have introduced n via the notion of equilibrium, the integral R dJ eq Jj R dnj also contains useful information when the spin distribution is not in equilibrium:…”