Larmor precession

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In physics, Larmor precession, named after Joseph Larmor refers to the precession of the magnetic moments of electrons, atomic nuclei, and atoms around the direction of 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, $\vec{J}$ is the angular momentum, $\vec{B}$ is the external magnetic field, $\times$ is the cross product, and $\ \gamma$ is the gyromagnetic ratio which gives the proportionality constant between the magnetic moment and the angular momentum.

If we consider the case of a static magnetic field,

$\vec{B}=B_0\hat{z}$

we find that the angular momentum vector $\vec{J}$ precesses about the z axis with an angular frequency known as the Larmor Frequency,

$\ \omega_{0} = \gamma B_0$

producing a gyroscopic motion, much like the spinning of a top.

It should be noted that in the above discussion we used the total electronic angular momentum vector $\vec{J}$ , but the equations above hold equally well for spin angular momentum of the electron $\vec{S}$ ,the orbital angular momentum of the electron $\vec{L}$ , the spin angular momentum of the nucleus $\vec{I}$ , or the total angular momentum of the atom $\vec{F}$ .

The gyromagnetic ratio is in general different for each type of angular momentum that may be considered, but using the following formula we may find the factor of interest.

$\gamma = \frac{g\mu_{B}}{\hbar}$ ,

where $\ g$ is the appropriate Landé g-factor, $\ \mu_{B}$ is the Bohr magneton, and $\hbar$ is the reduced Planck's constant. For an electron, the gyromagnetic ratio is approximately 2.8 MHz / Gauss.

A famous 1935 paper published by Lev Landau and Evgeny Lifshitz predicted the existence of ferromagnetic resonance of the Larmor precession, which was verified experimentally and independently by J. H. E. Griffiths (Great Britan) and E.K Zavoiskij (U.S.S.R.) in 1946.

The concept of Larmor precession is used in nuclear magnetic resonance.