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.

Note that the Larmor frequency is sometimes defined as \small v_L=\frac{1}{T}, which is then \small v_L=\frac{\vert\gamma_e B\vert}{2\pi}.



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