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The study of spin dynamics in synchrotrons has evolved over the years as has the desire for achieving polarized particle beams of the highest possible beam energies. A selection of reviews of the dynamics of polarized beams may be found in [1]–[9]. In this chapter, we focus on experimental data and describe spin transport in circular accelerators and transport lines. Except where explicitly mentioned, radiative effects in electron accelerators or very high energy proton accelerators are not treated here. We begin with a review of the Thomas-BMT equation for spin motion. This will be given in terms of the SU(2) spinor representation. Spinor algebra will be introduced and applied in the description of techniques used for preserving the beam polarization during acceleration through depolarizing resonances at moderate beam energies.

Many applications of particle accelerators require beam cooling, which refers to a reduction of the beam phase space volume or an increase in the beam density via dissipative forces. In electron and positron storage rings cooling naturally occurs due to synchrotron radiation, and special synchrotron-radiation damping rings for the production of low-emittance beams are an integral part of electron-positron linear colliders. For other types of particles different cooling techniques are available. Electron cooling and stochastic cooling of hadron beams are used to accumulate beams of rare particles (such as antiprotons), to combat emittance growth (e.g., due to scattering on an internal target), or to produce beams of high quality for certain experiments. Laser cooling is employed to cool ion beams down to extremely small temperatures. Here the laser is used to induce transitions between the ion electronic states and the cooling exploits the Dopper frequency shift. Electron beams of unprecedentedly small emittance may be obtained by a different type of laser cooling, where the laser beam acts like a wiggler magnet. Finally, designs of a future muon collider rely on the principle of ionization cooling. Reference [1] gives a brief review of the principal ideas and the history of beam cooling in storage rings; a theoretical dicussion and a few practical examples can be found in [2].

From (1.12) at a fixed location we can write where φ includes the initial phase φ We then have