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.

Tensor agebra
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 aτ is the polarization four vector, and uτ is four velocity of electron. The electromagnetic field tensor Fτσ 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


 * $$m\frac{du^{\tau}}{ds} = e F^{\tau \sigma}u_{\sigma},$$

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
 * $${\;\,dS^\alpha\over d\tau}={e\over m}\bigg[{g\over2}F^{\alpha\beta}S_\beta+\left({g\over2}-1\right)u^\alpha\left(S_\lambda F^{\lambda\mu}U_\mu\right)\bigg]\;,$$

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

Geometric algebra
The BMT equation in geometric algebra is


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