Larmor precession

In physics, Larmor precession (named after Joseph Larmor) is the precession of the magnetic moments of electrons, atomic nuclei, and atoms about an external magnetic field. The magnetic field exerts a torque on the magnetic moment,

$\vec{\Gamma} = \vec{\mu}\times\vec{B}= \gamma\vec{J}\times\vec{B}$

where $\vec{\Gamma}$ is the torque, $\vec{\mu}$ is the magnetic dipole moment, $\vec{J}$ is the angular momentum vector, $\vec{B}$ is the external magnetic field, $\times$ symbolizes the cross product, and $\ \gamma$ is the gyromagnetic ratio which gives the proportionality constant between the magnetic moment and the angular momentum.

Larmor frequency

The angular momentum vector $\vec{J}$ precesses about the external field axis with an angular frequency known as the Larmor frequency,

ω = − γB

where ω is the angular frequency, $\gamma=\frac{-e g}{2m}$ is the gyromagnetic ratio, and B is the magnitude of the magnetic field and g is the g-factor (normally 1, except for in quantum physics).

Simplified, this becomes:

$\omega = \frac{egB}{2m}$

where ω is the Larmor frequency, m is mass, e is charge, and B is applied field.

Each isotope has a unique Larmor frequency for NMR spectroscopy, which is tabulated here.

Bargmann-Michel-Telegdi equation

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

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

where a^τ, e, m, and μ are polarization four-vector, charge, mass, and magnetic moment, u^τ is four-velocity of electron, a^τa_τ = − u^τu_τ = − 1, u^τa_τ = 0, and F^τσ is electromagnetic field-strength tensor. Using equations of motion,

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

one can rewrite the first term in the right side of the BMT equation as ( − u^τw^λ + u^λw^τ)a_λ, where w^τ = du^τ / ds is four-acceleration. This term describes Fermi-Walker transport and leads to Thomas precession. The second term is associated with Larmor precession.

Applications

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) and E. K. Zavoiskij (USSR) in 1946.

Larmor precession is important in nuclear magnetic resonance, electron paramagnetic resonance and muon spin resonance.

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

See also

• Rabi cycle

External links

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