By Georg Heinz Hoffstaetter

This booklet examines the acceleration and garage of polarized proton beams in cyclic accelerators. uncomplicated equations of spin movement are reviewed, the invariant spin box is brought, and an adiabatic invariant of spin movement is derived. The textual content provides numerical equipment for computing the invariant spin box, and monitors the consequences in different illustrations. This publication deals a extra lucid view of spin dynamics at excessive strength than has hitherto been to be had.

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**Sample text**

The changes required in the pre-accelerator chain and in HERA-p itself are summarized in Fig. 6. This shows the polarized H− source and the polarimeters mentioned in Chap. 1 as well as the partial snake for DESY III and Siberian Snakes for PETRA. For HERA-p, the ﬂattening snakes and spin rotators, which make the spin longitudinal in the experiment, are indicated. Both the 4 or 8 Siberian Snake options for HERA-p are indicated. 5 The Invariant Torus The synchrotron radiation in proton synchrotrons that can be built today is so weak that particle motion can usually be described by a Hamiltonian.

But polarization in the core of the beam, where the oscillation amplitudes are small, will be only weakly inﬂuenced when crossing intrinsic resonances. Therefore, the average polarization of the beam is reduced. Such a reduction of polarization can be avoided by ﬁrst slowly exciting the whole beam coherently at a frequency close to the orbital tune that causes the perturbation [36], so that all particles take on relatively large oscillation amplitudes. During the acceleration, all spins then follow the adiabatic change of the polarization direction and the resonance can be crossed with little loss of polarization but with an overall ﬂip in the polarization direction.

However, in this argument it was assumed that ν0 (τ ) never takes on integer values. Because the closed-orbit spin tune changes with energy (ν0 = Gγ in a ﬂat ring), ν0 might become an integer during the acceleration process. Phenomena due to resonance between the frequencies of the two quickly changing phases φ and θ˜ could then occur. I will therefore now state an averaging theorem for systems with two quickly changing phases that allows for the crossing of resonances and apply it to spin motion on the closed orbit.