On understanding spin-flip synchrotron radiation and the transverse polarization of electrons in storage rings

A mainly didactic discussion is given of the mechanism for the gradual build up of transverse polarization of electrons and positrons in storage rings. The history and basic results are reviewed briefly. Then an intuitive explanation of the polarization in terms of spontaneous emission via a nonrelativistic magnetic dipole transition in a moving inertial frame is presented and criticized. This simple treatment contains a large part of the essential physics, but not all. It is surprisingly successful for electrons and positrons ($g=2\mathrm{}$), becomes exact for large $g$ factors of either sign, but fails badly for particles with $g$ factors of the range $0\lesssim g\lesssim 1.5$. The failure occurs because here the spin-magnetic-moment system cannot be treated even approximately in isolation from the orbital motion. A correct semiclassical description of radiation by a spin system is then given, in direct analogy with semiclassical radiation theory for charged particles ignoring spin. The classical equation of motion for a spin in relativistic motion, derived originally by Thomas, is used to obtain an effective Hamiltonian of interaction of a spin with electromagnetic fields. Emission and absorption of radiation is then described by replacing the classical electromagnetic fields with the appropriately normalized photon fields. The resulting formulas are applicable to charged particles of arbitrary $g$ factor. Expressions are given for the differential spectra in angle and in frequency for numbers of photons and for radiated power, as well as the previously known results for the total transition rates. These results seem of physical interest only for $g=2\mathrm{}$ but serve useful pedagogic purposes, refuting some of the expectations of the naive explanation. The various differential spectra are treated in detail for $g=2\mathrm{}$ and compared with the corresponding spectra for ordinary synchrotron radiation.