Magnetic Moments

Magnetic moments made angular momentum visible. Spectral splittings, Stern–Gerlach beam separation, magnetic resonance, and precision $g$-factor measurements all rely on the coupling between magnetic moments and magnetic fields.

Historically, magnetic moments were also a diagnostic tool. They exposed the limits of purely orbital models, motivated electron spin, and later became one of the precision triumphs of QED.

Orbital Magnetic Moments

A charged particle moving in an orbit acts like a current loop and produces a magnetic moment. In quantum mechanics, orbital angular momentum $\mathbf L$ gives an orbital magnetic moment. For a nonrelativistic particle with charge $q$ and mass $m$, with orbital motion measured about a fixed origin,

$$\boldsymbol\mu_L = \frac{q}{2m}\mathbf L.$$

For an electron, write $q=-e$ with $e>0$ and define the Bohr magneton

$$\mu_B = \frac{e\hbar}{2m_e}.$$

Then the electron’s orbital magnetic moment is

$$\boldsymbol\mu_L = -\mu_B \frac{\mathbf L}{\hbar}.$$

The minus sign is physical. The electron’s magnetic moment points opposite to its orbital angular momentum because the electron charge is negative.

Spin Magnetic Moment

Electron spin also carries a magnetic moment. With the same $e>0$ convention,

$$\boldsymbol\mu_S = -g_s\mu_B \frac{\mathbf S}{\hbar}.$$

The dimensionless number $g_s$ is the spin $g$ factor. At the leading Dirac level for a point electron,

$$g_s=2.$$

That factor of two is historically crucial. A naive orbital analogy would not have produced the observed electron spin magnetic moment. The spin magnetic moment was part of why spin could explain anomalous Zeeman patterns and fine-structure data.

The magnetic interaction energy is

$$H_{\mathrm{mag}} = -\boldsymbol\mu\cdot\mathbf B.$$

For spin alone, this becomes

$$H_{\mathrm{spin}} = g_s\mu_B \frac{\mathbf S\cdot\mathbf B}{\hbar} = \frac{g_s\mu_B}{2} \boldsymbol\sigma\cdot\mathbf B$$

for a spin-$1/2$ electron. The sign follows from using $q=-e$ for the electron.

g-Factors

A $g$ factor tells how strongly an angular momentum contributes to a magnetic moment. The simplest orbital electron contribution has effective $g_L=1$, while the leading spin contribution has $g_s=2$.

Atomic spectra often involve total angular momentum

$$\mathbf J=\mathbf L+\mathbf S.$$

With $g_L=1$, $g_s=2$, negligible hyperfine structure, and Russell–Saunders coupling, a level with $J>0$ has Landé factor

$$g_J = 1+ \frac{ J(J+1)+S(S+1)-L(L+1) }{ 2J(J+1) }.$$

The apparent $0/0$ at $J=0$ is not a prediction: an isolated nondegenerate $J=0$ level has no first-order electronic Zeeman shift, although second-order shifts can occur. This formula is not universal. It assumes a particular angular-momentum coupling scheme and weak-field perturbative regime. Its historical importance is that it connected observed magnetic splittings to the mixture of orbital and spin angular momentum in atomic states.

The common weak-field Zeeman shift is written schematically as

$$\Delta E = g_J\mu_B B m_J.$$

Different systems, particles, effective bands, nuclei, and coupling regimes have different $g$ factors. Treating $g$ as always equal to $2$ is a common error.

Experimental Signatures

Magnetic moments show up in several characteristic ways:

• an inhomogeneous field produces Stern–Gerlach deflection through a force proportional to a magnetic-moment projection;
• a uniform weak field shifts or splits energy levels through Zeeman coupling;
• oscillating fields drive magnetic resonance transitions between spin states;
• precision spectroscopy and trapped-particle measurements determine $g$ factors with extraordinary sensitivity;
• deviations from simple $g=2$ behavior reveal interactions, radiative corrections, and effective-medium physics.

The Stern–Gerlach experiment made discrete magnetic-moment projection visible in a beam. The anomalous Zeeman effect made clear that orbital magnetism alone was not enough. Magnetic resonance later turned spin magnetic moments into a central experimental tool.

QED Precision Preview

Dirac’s relativistic electron theory predicts $g_s=2$ for a point electron at leading order. QED predicts small radiative corrections. The anomalous magnetic moment is commonly written

$$a_e = \frac{g_e-2}{2}.$$

Here $g_e$ is the positive magnitude, with the electron charge sign already in the magnetic moment. Schwinger’s first-order QED result is

$$a_e = \frac{\alpha}{2\pi} +\cdots,$$

where $\alpha$ is the fine-structure constant and the ellipsis denotes higher-order corrections. Modern measurements and calculations of the electron magnetic moment are among the sharpest tests of QED.

This precision story is a preview. The present chapter uses it only to show that magnetic moments remained central after the old spectroscopic puzzles were solved.

Common Misconceptions

• A magnetic moment is not the same thing as angular momentum; it is proportional to angular momentum only after specifying the particle, charge, mass, and $g$ factor.
• The electron magnetic moment points opposite to electron angular momentum because the electron is negatively charged.
• Spin magnetic moment is not explained by literal spinning charge.
• The orbital and spin $g$ factors are not the same.
• Zeeman splittings require a specified coupling regime; there is no single universal Zeeman formula.
• The anomalous magnetic moment is not an experimental mistake; it is a precision quantum-field effect.
• Effective $g$ factors in atoms, solids, or nuclei need not equal the free-electron value.

Exercises

1. For an electron, is the orbital magnetic moment parallel or antiparallel to $\mathbf L$?

Solution

It is antiparallel. With $e>0$,

$$\boldsymbol\mu_L = -\mu_B \frac{\mathbf L}{\hbar}.$$

The minus sign comes from the electron’s negative charge.

2. In a field $\mathbf B=B\hat{\mathbf z}$, use $\boldsymbol\mu_S=-g_s\mu_B\mathbf S/\hbar$ to find the spin magnetic energy for $S_z=+\hbar/2$.

Solution

The spin moment component is

$$\mu_{S,z} = -g_s\mu_B \frac{1}{2}.$$

The magnetic energy is

$$E = -\mu_{S,z}B = \frac{g_s\mu_B B}{2}.$$

3. Use the Landé formula to find $g_J$ for an $L=0$, $S=1/2$, $J=1/2$ state.

Solution

Substitute $J(J+1)=3/4$, $S(S+1)=3/4$, and $L(L+1)=0$:

$$g_J = 1+ \frac{3/4+3/4}{2(3/4)} = 1+1 = 2.$$

4. Ignoring the anomaly, find the electron spin transition frequency in a $0.10\,\mathrm T$ field. Use $\mu_B/h\simeq14.0\,\mathrm{GHz/T}$.

Solution

The two spin energies are $\pm g_s\mu_B B/2$, so the gap is $g_s\mu_B B$, not half of that. Thus $\nu=g_s\mu_B B/h\simeq2.80\,\mathrm{GHz}$ for $g_s=2$. The cyclic frequency is $\nu$; the angular frequency is $2\pi\nu$.

References

• P. A. M. Dirac, “The Quantum Theory of the Electron,” Proceedings of the Royal Society A 117, 610-624, 1928, DOI: 10.1098/rspa.1928.0023.
• C. J. Foot, Atomic Physics, Oxford University Press, 2005.
• W. Gerlach and O. Stern, “Der experimentelle Nachweis der Richtungsquantelung im Magnetfeld,” Zeitschrift für Physik 9, 349-352, 1922, DOI: 10.1007/BF01326983.
• W. Pauli, “Zur Quantenmechanik des magnetischen Elektrons,” Zeitschrift für Physik 43, 601-623, 1927.
• J. J. Sakurai and J. Napolitano, Modern Quantum Mechanics, 3rd ed., Cambridge University Press, 2020.
• J. Schwinger, “On Quantum-Electrodynamics and the Magnetic Moment of the Electron,” Physical Review 73, 416-417, 1948, DOI: 10.1103/PhysRev.73.416.
• G. E. Uhlenbeck and S. Goudsmit, “Spinning Electrons and the Structure of Spectra,” Nature 117, 264-265, 1926, DOI: 10.1038/117264a0.
• P. Zeeman, “On the Influence of Magnetism on the Nature of the Light Emitted by a Substance,” Philosophical Magazine 43, 226-239, 1897.