User:Maschen/Bargmann–Michel–Telegdi equation

In physics, Bargmann–Michel–Telegdi (BMT) equation describes the spin precession of an electron in an external electromagnetic field. It is named after Valentine Bargmann, Louis Michel, and Valentine Telegdi.

Background

A particle with spin angular momentum s has a corresponding magnetic moment μ, which will interact with an external magnetic field B. The classical equation of motion for the spin of a particle subject follows from "torque equals rate of change of angular momentum" (rotation analogue of Newton's second law)

${\frac {d\mathbf {s} }{dt}}={\boldsymbol {\mu }}\times \mathbf {B}$

and the spin and magnetic moment are proportional to each other

$\mathbf {s} \propto {\boldsymbol {\mu }}$

This applies to the rest frame of the particle. A covariant generalization to any frame is given by the BMT equation.

Equations

Tensor agebra

Further information: Tensor and Four vector

For an electron of electric charge e, mass m, magnetic moment μ, the equation is

${\frac {da^{\tau }}{ds}}={\frac {e}{m}}u^{\tau }u_{\sigma }F^{\sigma \lambda }a_{\lambda }+2\mu (F^{\tau \lambda }-u^{\tau }u_{\sigma }F^{\sigma \lambda })a_{\lambda },$

where a^τ is the polarization four vector, and u^τ is four velocity of electron. The electromagnetic field tensor F^τσ is externally applied. We have the relations

$a^{\tau }a_{\tau }=-u^{\tau }u_{\tau }=-1$

$u^{\tau }a_{\tau }=0$

Using the Lorentz force, the general equations of motion for any charged particle

$m{\frac {du^{\tau }}{ds}}=eF^{\tau \sigma }u_{\sigma },$

one can rewrite the first term on the right side of the BMT equation as $(-u^{\tau }w^{\lambda }+u^{\lambda }w^{\tau })a_{\lambda }$, where

$w^{\tau }=du^{\tau }/ds$

is the four acceleration of the electron. This term describes Fermi–Walker transport and leads to Thomas precession. The second term is associated with Larmor precession.

When electromagnetic fields are uniform in space or when gradient forces like $\nabla ({\boldsymbol {\mu }}\cdot {\boldsymbol {B}})$ can be neglected, the particle's translational motion is described by

${du^{\alpha } \over d\tau }={e \over m}F^{\alpha \beta }u_{\beta }\;.$

The BMT equation is then written as

${\;\,dS^{\alpha } \over d\tau }={e \over m}{\bigg [}{g \over 2}F^{\alpha \beta }S_{\beta }+\left({g \over 2}-1\right)u^{\alpha }\left(S_{\lambda }F^{\lambda \mu }U_{\mu }\right){\bigg ]}\;,$

The Beam-Optical version of the Thomas-BMT, from the Quantum Theory of Charged-Particle Beam Optics, applicable in accelerator optics

Geometric algebra

Further information: Geometric algebra

The BMT equation in geometric algebra is

${\frac {ds}{dt}}={\frac {e}{m}}F\cdot s+(g-2){\frac {e}{2m}}(F\cdot s)\wedge vv$

References

• Jackson, J. D. (1975) [1962]. Classical Electrodynamics (2nd ed.). John Wiley & Sons. pp. 556–558. ISBN 0-471-43132-X.
• V.B. Berestetskii, E.M. Lifshitz, L.P. Pitaevskii (1982). Quantum Electrodynamics. Vol. Vol. 4 (2nd ed.). Butterworth-Heinemann. p. 154. ISBN 978-0-7506-3371-0. {{cite book}}: |volume= has extra text (help)CS1 maint: multiple names: authors list (link)
• Misner, Charles W.; Thorne, Kip S.; Wheeler, John Archibald (1973). Gravitation. San Francisco: W. H. Freeman. ISBN 978-0-7167-0344-0.