2 3 Ja n 19 99 III . THE PERMISSIBLE EQUILIBRIUM POLARISATION DISTRIBUTION IN A STORED PROTON BEAM

We illustrate the use of the invariant spin field for describing permissible equilibrium spin distributions in high energy spin polarised proton beams. 1 A problem and its solution Following the successful attainment of longitudinal e ± polarisation in HERA (Article II) it is natural to consider whether it would be possible to complement the polarised e ± with 820GeV polarised protons [1, 2]. As pointed out in Article I, a stored polarised proton beam can only be obtained by injecting and then accelerating a prepolarised beam provided by a suitable source. However, I comment on another concept in the Appendix. A major obstacle to reaching high energy with the polarisation intact is that the spins must negotiate groups of spin–orbit resonances every 523MeV (Article I) since the spin tune is approximately proportional to the energy. However, this problem can be ameliorated by the inclusion of Siberian Snakes [3, 4]. These are magnet systems which rotate spins by 180 degrees around an axis in the horizontal plane independently of the energy of the particle. By the installation of suitable combinations of snakes, the spin tune ν spin can be fixed at one half and then by suitable choice of orbital tunes, resonances can be avoided at all energies, assuming that the dependence of spin tune on synchrobeta amplitude is weak (Article I). Tracking simulations show that even with snakes, preservation of polarisation up to high energy is nontrivial. For example a 1 milliradian orbit deflection at 820 GeV causes a 90 degree spin rotation (Article I, Eq. (4)). Thus one should check first whether the spin distribution permitted by the requirement of equilibrium at a chosen high energy would be acceptable. There would be no point in trying to accelerate if the answer were negative. Moreover, to arrive at an answer we have the ideal tool at hand, namely the invariant spin field introduced in Article I. The measure for acceptability is the deviation ofˆn fromˆn 0 averaged across phase space. If the average deviation were, say, 60 degrees, then even with | P eq (u; s)| = 1 at each point in phase space, a polarimeter would only record about 50% polarisation. Thus the optic and ring layout must be chosen so that the deviation is minimised. The invariant spin a Updated version of a talk presented at the 15th ICFA Advanced Beam Dynamics Workshop: " Quan