Abstract
We derive a four-component spinor wave function for an electron in a helical undulator which, in the relativistic limit, successfully reproduces all of the results of the classical calculation for the radiation angular distribution, polarization, and energy spectrum. This wave function also allows the nonclassical calculation of the spin flip. For electron energies below the several hundred GeV range, the spin-flip probability is negligible, but for higher energies and high undulator strengths it cannot be neglected if beam polarization is to be preserved, even though nonflip radiation still greatly dominates the radiation intensity. The anomalous magnetic moment ${a}_{e}$ is seen to play a dominant role in the helical undulator spin-flip process. The probability of spin flip is shown to have a ${\ensuremath{\gamma}}^{5}$ dependence on electron energy. For high energy electrons, the direction of spin flip is independent of the handedness of the undulator. As a result, at sufficiently high energy, a polarized electron or positron beam rapidly depolarizes by spontaneous radiation in the undulator. Because of the high correlation between the direction of spin flip and the handedness of the spin-flip radiation, we conjecture that it may be possible to polarize electrons by using the intense circularly polarized photons in the helical undulator to stimulate spin flip.
Highlights
Relativistic electron beams passing through helical undulators are an increasingly common source of circularly polarized x rays [1]
In a 1987 Physical Review Letters (PRL) [6,7], we showed that a simpler treatment could obtain the same results
We introduce a form of the four-component spinor wave function which is well suited to the electron relativistic kinematic limit, ) 1
Summary
Relativistic electron beams passing through helical undulators are an increasingly common source of circularly polarized x rays [1]. We derive the fourcomponent spinor wave function for the Dirac equation including the phenomenological anomalous moment for an electron/positron in an ideal undulator field in the relativistic limit. In contrast with the case of circular storage rings, here we ignore stochastic radiative effects, fluctuations, and damping, since the electrons make only a single pass in the undulator With these assumptions, it can be shown that the spontaneous radiation in which the spin is not flipped leads to the. The Dirac equation is solved in the high energy limit, with a phenomenological anomalous moment term added [Eq (1) below] As it is written, the electromagnetic fields in Eq (1) include only the ideal classical magnetic field of the undulator. There are two physical effects of the anomalous moment: (1) when the electron magnetic moment interacts with the classical magnetic field, possibly affecting the orbital motion (this turns out to be negligible), and (2) when the electron magnetic moment interacts with the radiation vector potential to modify photon absorption or emission
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