Larmor precession

The above equation is the one that is used in most applications. However, a full treatment must include the effects of Thomas precession, yielding the equation (in CGS units) (The CGS units are used so that E has the same units as B):

${\displaystyle \omega _{s}={\frac {geB}{2mc}}+(1-\gamma ){\frac {eB}{mc\gamma }}={\frac {eB}{2mc}}\left(g-2+{\frac {2}{\gamma }}\right)}$

where ${\displaystyle \gamma }$ is the relativistic Lorentz factor (not to be confused with the gyromagnetic ratio above). Notably, for the electron g is very close to 2 (2.002...), so if one sets g = 2, one arrives at

${\displaystyle \omega _{s(g=2)}={\frac {eB}{mc\gamma }}}$

The spin precession of an electron in an external electromagnetic field is described by the Bargmann–Michel–Telegdi (BMT) equation [3]

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

where ${\displaystyle a^{\tau }}$, ${\displaystyle e}$, ${\displaystyle m}$, and ${\displaystyle \mu }$ are polarization four-vector, charge, mass, and magnetic moment, ${\displaystyle u^{\tau }}$ is four-velocity of electron (in a system of units in which ${\displaystyle c=1}$), ${\displaystyle a^{\tau }a_{\tau }=-u^{\tau }u_{\tau }=-1}$, ${\displaystyle u^{\tau }a_{\tau }=0}$, and ${\displaystyle F^{\tau \sigma }}$ is electromagnetic field-strength tensor. Using equations of motion,

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

one can rewrite the first term on the right side of the BMT equation as ${\displaystyle (-u^{\tau }w^{\lambda }+u^{\lambda }w^{\tau })a_{\lambda }}$, where ${\displaystyle w^{\tau }=du^{\tau }/ds}$ is four-acceleration. 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 ${\displaystyle \nabla ({\boldsymbol {\mu }}\cdot {\boldsymbol {B}})}$ can be neglected, the particle's translational motion is described by

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

The BMT equation is then written as [4]

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

The Beam-Optical version of the Thomas-BMT, from the Quantum Theory of Charged-Particle Beam Optics, applicable in accelerator optics [5] [6]

A 1935 paper published by Lev Landau and Evgeny Lifshitz predicted the existence of ferromagnetic resonance of the Larmor precession, which was independently verified in experiments by J. H. E. Griffiths (UK)[7] and E. K. Zavoiskij (USSR) in 1946.[8][9]

Larmor precession is important in nuclear magnetic resonance, magnetic resonance imaging, electron paramagnetic resonance, muon spin resonance, and neutron spin echo. It is also important for the alignment of cosmic dust grains, which is a cause of the polarization of starlight.

To calculate the spin of a particle in a magnetic field, one must in general also take into account Thomas precession if the particle is moving.

The spin angular momentum of an electron precesses counter-clockwise about the direction of the magnetic field. An electron has a negative charge, so the direction of its magnetic moment is opposite to that of its spin.

• LARMOR neutron microscope

• Georgia State University HyperPhysics page on Larmor Frequency
• Larmor Frequency Calculator