bohr magneton

Prerequisites: Spin. Full path (19 pages)

Intuitive
Concrete
Abstract

Intuitive

The Bohr magneton is the natural unit expressing the magnetic moment of the electron. Electrons have a magnetic moment because they have a property called spin (a sort of intrinsic angular momentum) and because they sometimes go in circles.

Concrete

The Bohr magneton is defined as

$\mu_B = \frac{e\hbar}{2m_e} \approx 9.274\times 10^{-24}\ \mathrm{J\,T^{-1}} \approx 5.788\times 10^{-5}\ \mathrm{eV\,T^{-1}} \qquad \text{(SI)},$

with $e > 0$ the elementary charge and $m_e$ the electron mass. In Gaussian units it reads $\mu_B = e\hbar/(2m_ec)$ .

Where it comes from. A charge $q$ of mass $m$ moving on a circle of radius $r$ with speed $v$ is a current loop with current $I = qv/(2\pi r)$ and area $\pi r^2$ . Its magnetic moment is

$\mu = I\cdot \pi r^2 = \frac{q v r}{2} = \frac{q}{2m}\,L,$

where $L = mvr$ is the angular momentum. The ratio $q/2m$ between magnetic moment and angular momentum is the classical gyromagnetic ratio. In quantum mechanics orbital angular momentum comes in units of $\hbar$ , so for an electron the natural size of the magnetic moment is $e\hbar/2m_e = \mu_B$ . For orbital motion, taking into account the negative charge,

$\vec\mu_L = -\frac{e}{2m_e}\vec L = -\mu_B\,\frac{\vec L}{\hbar} .$

Spin magnetic moment. The electron spin $\vec S$ also carries a magnetic moment,

$\vec\mu_S = -g_s\,\mu_B\,\frac{\vec S}{\hbar}, \qquad g_s \approx 2.0023 .$

The value $g_s = 2$ follows from the Dirac equation. The small deviation is the anomalous magnetic moment, $a_e = (g_s - 2)/2 \approx \alpha/2\pi \approx 0.00116$ at leading order in quantum electrodynamics. Since $S_z = \pm\hbar/2$ , the spin magnetic moment along a field axis is close to $\mp\mu_B$ , which is why the electron's magnetic moment is often quoted as about one Bohr magneton.

Zeeman effect. In a magnetic field $\vec B = B\hat z$ the energy of a magnetic moment is $E = -\vec\mu\cdot\vec B$ . Ignoring spin, an atomic level with magnetic quantum number $m_l$ shifts by

$\Delta E = \mu_B\, B\, m_l ,$

so a field of $1\ \mathrm{T}$ splits neighbouring levels by about $5.8\times 10^{-5}\ \mathrm{eV}$ .

For nuclei the corresponding unit is the nuclear magneton $\mu_N = e\hbar/(2m_p)$ , smaller than $\mu_B$ by the mass ratio $m_e/m_p \approx 1/1836$ .

Abstract

From the Dirac equation. The factor $g_s = 2$ appears in the nonrelativistic limit of the Dirac equation. For a particle of charge $q$ in an electromagnetic field, with kinetic momentum $\vec\pi = \vec p - q\vec A$ and $\vec p = -i\hbar\nabla$ , the leading term is the Pauli equation Hamiltonian $(\vec\sigma\cdot\vec\pi)^2/2m + q\phi$ . Using $(\vec\sigma\cdot\vec a)(\vec\sigma\cdot\vec b) = \vec a\cdot\vec b + i\vec\sigma\cdot(\vec a\times\vec b)$ and $\vec\pi\times\vec\pi = iq\hbar\vec B$ ,

$\frac{(\vec\sigma\cdot\vec\pi)^2}{2m} = \frac{\vec\pi^{\,2}}{2m} - \frac{q\hbar}{2m}\,\vec\sigma\cdot\vec B .$

For the electron, $q = -e$ , the last term is $+\mu_B\,\vec\sigma\cdot\vec B = -\vec\mu_S\cdot\vec B$ with $\vec\mu_S = -2\mu_B\vec S/\hbar$ and $\vec S = \hbar\vec\sigma/2$ , which is $g_s = 2$ . The same expansion of $\vec\pi^{\,2}$ to first order in a uniform field gives the orbital term $\mu_B\vec L\cdot\vec B/\hbar$ with $g_L = 1$ .

Form factors. In quantum field theory the coupling of a spin- $\frac12$ particle to the electromagnetic field is described by the vertex function. For on-shell particles, current conservation and Lorentz invariance restrict it to two form factors. With incoming momentum $p$ , outgoing momentum $p'$ and $q = p' - p$ (units $\hbar = c = 1$ ), the vertex is proportional to $\bar u(p')\Gamma^\mu u(p)$ with

$\Gamma^\mu = \gamma^\mu F_1(q^2) + \frac{i\sigma^{\mu\nu}q_\nu}{2m}F_2(q^2), \qquad \sigma^{\mu\nu} = \frac{i}{2}[\gamma^\mu, \gamma^\nu] .$

$F_1(0) = 1$ fixes the charge, and the gyromagnetic factor is $g = 2\big(F_1(0) + F_2(0)\big) = 2 + 2F_2(0)$ . At one loop $F_2(0) = \alpha/2\pi$ . The Bohr magneton is thus the scale set by the charge, mass and $\hbar$ of the particle, and $g$ measures the coupling in these units. A term $\bar\psi\sigma^{\mu\nu}\psi F_{\mu\nu}$ (Pauli term) in an effective Lagrangian shifts $g$ away from $2$ ; it has mass dimension $5$ , so it is not renormalizable and is not present in the QED Lagrangian, which is why $g - 2$ is a calculable loop effect there.

bohr magneton

Intuitive

Concrete

Abstract

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