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Various different classical models of electrons including their spin degree of freedom are commonly applied to describe the
coupled dynamics of relativistic electron motion and spin precession in strong electromagnetic fields. The spin dynamics is
usually governed by the Thomas-Bargmann-Michel-Telegdi equation [1, 2] in these models, while the electron’s orbital motion
follows the (modified) Lorentz force and a spin-dependent Stern-Gerlach force. Various classical models can lead to different
or even contradicting predictions how the spin degree of freedom modifies the electron’s orbital motion when the electron
moves in strong electromagnetic fields. This discrepancy is rooted in the model-specific energy dependency of the spin
induced relativistic Stern-Gerlach force acting on the electron. The Frenkel model [3, 4] and the classical Foldy-Wouthuysen
model 5 are compared exemplarily against each other and against the quantum mechanical Dirac equation in order to identify
parameter regimes where these classical models make different predictions [6, 7].
Our theoretical results allow for experimental tests of these models. In the setup of the longitudinal Stern-Gerlach effect,
the Frenkel model and classical Foldy-Wouthuysen model lead in the relativistic limit to qualitatively different spin effects
on the electron trajectory. Furthermore, it is demonstrated that in tightly focused beams in the near infrared the effect of the
Stern-Gerlach force of the Frenkel model becomes sufficiently large to be potentially detectable in an experiment. Among
the classical spin models, the Frenkel model is certainly prominent for its long history and its wide application. Our results,
however, suggest that the classical Foldy-Wouthuysen model is superior as it is qualitatively in better agreement with the
quantum mechanical Dirac equation.
In ultra strong laser setups at parameter regimes where effects of the Stern-Gerlach force become relevant also radiation
reaction effects are expected to set in. We incorporate radiation reaction classically via the Landau-Lifshitz equation and
demonstrate that although radiation reaction effects can have a significant effect on the electron trajectory, the Frenkel model
and the classical Foldy-Wouthuysen model remain distinguishable also if radiation reaction effects are taken into account.
Our calculations are also suitable to verify the Landau-Lifshitz equation for the radiation reaction of electrons and other spin
one-half particles.
1. Thomas, L. H., “I. The kinematics of an electron with an axis,” The London, Edinburgh, and Dublin Philosophical
Magazine and Journal of Science 3(13), 1–22 (1927).
2. Bargmann, V., Michel, L., and Telegdi, V. L., “Precession of the polarization of particles moving in a homogeneous
electromagnetic field,” Phys. Rev. Lett. 2(10), 435–436 (1959).
3. Frenkel, J., “Die Elektrodynamik des rotierenden Elektrons,” Z. Phys. 37(4–5), 243–262 (1926).
4. Frenkel, J., “Spinning electrons,” Nature (London) 117(2949), 653–654 (1926).
5. Silenko, A. J., “Foldy-Wouthyusen transformation and semiclassical limit for relativistic particles in strong external
fields,” Phys. Rev. A 77(1), 012116 (2008).
6. Wen, M., Bauke, H., and Keitel, C. H., “Identifying the Stern-Gerlach force of classical electron dynamics,” Sci. Rep. 6,
31624 (2016).
7. Wen, M., Keitel, C. H., and Bauke, H., “Spin one-half particles in strong electromagnetic fields: spin effects and
radiation reaction,” arXiv:1610.08951 (2016).
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