1.3.1 Gyromagnetic precession

It is known from quantum mechanics that there is a proportionality relationship between the magnetic spin momentum $\mu$ and angular momentum $\mathbf{L}$ of electrons. This relationship can be expressed as

$$\mbox{\boldmath$\mu$} =-\gamma\mathbf{L}\quad,$$ (1.72)

where $\gamma=2.21\times10^5$ m A$^{-1}$ s$^{-1}$ is the absolute value of the gyromagnetic ratio

$$\gamma=\frac{g \vert e\vert}{2m_e c} \quad;$$ (1.73)

$g\simeq 2$ is the Landé splitting factor, $e=-1.6\times 10^{-19}$ C is the electron charge, $m_e=9.1\times 10^{-31}$ kg is the electron mass and $c=3\times10^8$ m/s is the speed of light. By applying the momentum theorem one can relate the rate of change of the angular momentum to the torque exerted on the particle by the magnetic field $\mathbf{H}$:

$$\frac{d\mathbf{L}}{dt}= \mbox{\boldmath$\mu$} \times\mathbf{H}\quad.$$ (1.74)

By using Eq. (1.72), one ends up with a model which describes the precession of the spin magnetic moment around the field:

$$\frac{d\mbox{\boldmath$\mu$}}{dt}=-\gamma\mbox{\boldmath$\mu$}\times\mathbf{H}\quad.$$ (1.75)

The frequency of precession is the Larmor frequency

$$f_L=\frac{\gamma H}{2\pi} \quad.$$ (1.76)

Eq. (1.75) can be written for each spin magnetic moment within the elementary volume $dV_\textbf{r}$:

$$\frac{d\mbox{\boldmath$\mu$}_j}{dt}=-\gamma\mbox{\boldmath$\mu$}_j\times\mathbf{H}\quad,$$ (1.77)

where now the magnetic field $\mathbf{H}$ is intended to be spatially uniform. Now, by taking the volume average of both sides of the latter equation, one has:

$$\frac{1}{dV_\textbf{r}} \frac{d\sum_j\mbox{\boldmath$\mu$}_j}{dt... ...amma \frac{\sum_j\mbox{\boldmath$\mu$}_j}{dV_\textbf{r}}\times\mathbf{H}\quad,$$ (1.78)

and, therefore, recalling the definition (1.1) of magnetization vector field $\textbf{M}$, we end up with the following continuum gyromagnetic precession model:

$$\frac{\partial \textbf{M}}{\partial t}=-\gamma\textbf{M}\times\mathbf{H}\quad.$$ (1.79)