Larmor precession of an electron

Larmor precession is the change in orientation of the axis of the magnetic moment $\small \boldsymbol{\mathit{\mu}}$ of a particle with respect to the axis of an external magnetic field. Consider an electron at rest with intrinsic spin of $\small \boldsymbol{\mathit{s}}$ in a $\small z$ -directional uniform magnetic field $\small \boldsymbol{\mathit{B}}$ . The magnetic field interacts with the electron’s magnetic moment and generates a torque $\small \boldsymbol{\mathit{\tau}}=\boldsymbol{\mathit{\mu}}\times\boldsymbol{\mathit{B}}$ (see diagram below).

If $\small \boldsymbol{\mathit{\mu}}$ is the spin magnetic moment of an electron, eq61 becomes $\small \boldsymbol{\mathit{\mu}}=\gamma_e\boldsymbol{\mathit{s}}$ , where $\small \gamma_e$ is the gyromagnetic ratio of the electron. Taking the derivative on both sides of this equation with respect to $\small t$ ,

$\small \frac{d\boldsymbol{\mathit{\mu}}}{dt}=\gamma_e\frac{d\boldsymbol{\mathit{s}}}{dt}$

From eq64 and eq71, we have $\small \frac{d\boldsymbol{\mathit{L}}}{dt}=\boldsymbol{\mathit{\mu}}\times \boldsymbol{\mathit{B}}$ , whose spin analogue is $\small \frac{d\boldsymbol{\mathit{s}}}{dt}=\boldsymbol{\mathit{\mu}}\times \boldsymbol{\mathit{B}}$ . So,

$\small \frac{d\boldsymbol{\mathit{\mu}}}{dt}=\gamma_e\, \boldsymbol{\mathit{\mu}}\times \boldsymbol{\mathit{B}}$

With reference to the above diagram, the change in arc length with respect to $\small t$ is $\small \frac{d\mu}{dt}=\mu sin\theta\frac{d\phi}{dt}$ . Hence,

$\small \gamma_e\mu Bsin\theta=\mu sin\theta\frac{d\phi}{dt}$

$\small \int_{0}^{2\pi}d\phi=\gamma_eB\int_{0}^{T}dt$

$\small \omega_L=\vert\gamma_e B\vert=\frac{eB}{m_e}\; \; \; \; \; \; \; \; 149$

where $\small T$ is the period, $\small m_e$ is the mass of an electron and $\small \omega_L=\frac{2\pi}{T}$ is the Larmor frequency, which is defined as a positive value.