Magnetic Moments and g-Factors

A magnetic moment measures how an object couples to a magnetic field. For an angular momentum $\mathbf J$, the magnetic moment is often proportional to $\mathbf J$:

$$\boldsymbol\mu = \gamma\mathbf J,$$

where $\gamma$ is the gyromagnetic ratio. For a charged particle with $q\ne0$, a $g$ factor rewrites this proportionality in a dimensionless way:

$$\boldsymbol\mu = g\frac{q}{2m}\mathbf J.$$

For neutral composite particles the charge-based parametrization is unsuitable: a neutron can have a magnetic moment even though its total charge is zero. Nuclear moments are usually defined instead by $\boldsymbol\mu=g_I\mu_N\mathbf I/\hbar$, with a signed $g_I$.

The sign and interpretation depend on the charge convention, the angular momentum being used, and the physical system. For electrons one usually writes $q=-e$ with $e>0$, so the magnetic moment often points opposite to the angular momentum.

The conventions below apply to the linear magnetic response. The orbital current-loop derivation belongs to Magnetic Moments from Orbital Motion. The Hamiltonian dynamics of a spin in a magnetic field are developed in Spin in Magnetic Fields and Larmor Precession. The historical role of magnetic moments is treated in Magnetic Moments, the perturbative hierarchy in Zeeman Effect as a Perturbation Example, and atomic line patterns and Breit–Rabi energies in Zeeman Effect in Atoms.

Magnetic Energy

The magnetic-dipole interaction is

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

If

$$\boldsymbol\mu=\gamma\mathbf J,$$

then

$$H_{\mathrm{mag}} = -\gamma\,\mathbf J\cdot\mathbf B.$$

For a uniform field of magnitude $B\geq0$ along $z$,

$$\mathbf B=B\hat{\mathbf z},$$

an eigenstate of $J_z$ has

$$J_z\lvert j,m\rangle = \hbar m\lvert j,m\rangle,$$

and the magnetic energy is

$$E_m = -\gamma\hbar B\,m.$$

Adjacent magnetic sublevels are separated by

$$\Delta E = \hbar|\gamma|B$$

in the simplest linear Zeeman model.

Magneton Scales

For electron-scale magnetic moments, the natural unit is the Bohr magneton:

$$\mu_B = \frac{e\hbar}{2m_e}, \qquad e>0.$$

For nuclear magnetic moments, the analogous scale is the nuclear magneton:

$$\mu_N = \frac{e\hbar}{2m_p}.$$

Because $m_p\gg m_e$, nuclear magnetic moments are typically much smaller than electron magnetic moments on the Bohr-magneton scale. This simple mass ratio is one reason electronic Zeeman effects are usually much larger than nuclear spin splittings in the same magnetic field, before system-specific enhancement or shielding effects are included.

For numerical values of common constants, see Constants.

Orbital Versus Spin g-Factors

For a charged particle’s orbital angular momentum,

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

Thus the orbital $g$ factor is

$$g_L=1$$

for ordinary orbital motion with angular momentum $\mathbf L$. For an electron, $q=-e$, so

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

Spin has its own magnetic moment. For an electron,

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

For a point spin-$1/2$ electron with minimal electromagnetic coupling at leading Dirac level,

$$g_s=2.$$

The factor of two is not obtained by treating spin as ordinary orbital motion of a small charged body. It is a spin and relativity result. In nonrelativistic quantum mechanics it is often taken as an experimentally fixed input or as inherited from the Pauli/Dirac theory.

The contrast is:

Contribution	Electron convention	Leading value
Orbital	$\boldsymbol\mu_L=-g_L\mu_B\mathbf L/\hbar$	$g_L=1$
Spin	$\boldsymbol\mu_S=-g_s\mu_B\mathbf S/\hbar$	$g_s\simeq2$

Electron Sign Convention

For an electron spin in a magnetic field,

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

If

$$\mathbf B=B\hat{\mathbf z}, \qquad B>0,$$

then

$$E_{m_s} = g_s\mu_B B\,m_s.$$

For spin $1/2$,

$$m_s=\pm\frac12,$$

so the lower-energy electron spin state has

$$m_s=-\frac12.$$

This does not mean the magnetic moment is anti-aligned with the field. It means the spin angular momentum is anti-aligned with the field while the electron magnetic moment, which is opposite to the spin, is aligned with the field.

This sign logic is one of the most common places for mistakes in spin problems.

Landé g-Factor Preview

Atomic levels often involve both orbital and spin angular momentum:

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

In the weak-field Russell–Saunders coupling regime, where an atomic level is labeled by $L,S,J,m_J$, the magnetic shift is commonly written

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

For $J>0$, the Landé factor is

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

For ordinary electron orbital and spin contributions, one usually uses

$$g_L=1, \qquad g_S\simeq2$$

at this level of approximation. This formula assumes a particular coupling regime. It is not a universal Zeeman formula, and it does not apply when the external field competes strongly with spin-orbit coupling or when different effective degrees of freedom are being used.

The field must be weak compared with separations between the coupled $J$ levels, so that mixing them can be neglected. An isolated $J=0$ level has no first-order magnetic-dipole shift; the displayed formula is undefined there and must not be used by dividing by zero. Second-order shifts can remain.

For the angular-momentum coupling background, see Spin-Orbit Coupling.

Anomalous Magnetic Moment Preview

In the positive electron-$g_s$ convention used here, the leading Dirac value is $g_s=2$, but quantum electrodynamics predicts small radiative corrections. The anomalous magnetic moment is defined by

$$a = \frac{g-2}{2}.$$

For the electron,

$$g_s = 2(1+a_e).$$

Schwinger’s first-order result is

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

where $\alpha$ is the fine-structure constant. The ellipsis hides higher-order QED, electroweak, and hadronic contributions in precision work. Here $g$ denotes the positive coefficient with the electron charge sign written separately. Tables that absorb the charge sign may quote a negative electron $g$; compare definitions before forming an anomaly. Precision predictions require more than the leading spin Hamiltonian.

The compact reference cards are Pauli Hamiltonian and Pauli Equation.

Effective g-Factors

The letter $g$ does not always mean the free-electron value. In atoms, molecules, solids, nuclei, and engineered two-level systems, an effective $g$ factor may include:

• orbital admixture,
• spin-orbit coupling,
• crystal-field effects,
• band-structure renormalization,
• hyperfine coupling,
• nuclear structure,
• many-body dressing.

An effective Hamiltonian may still have the form

$$H_{\mathrm{eff}} = -\boldsymbol\mu_{\mathrm{eff}}\cdot\mathbf B,$$

but the magnetic moment may be an emergent operator in a restricted subspace rather than the bare magnetic moment of one free particle. Identify the Hilbert space and convention before interpreting a quoted $g$ factor. For an anisotropic effective spin-$1/2$ doublet one can define

$$\mu_i=-\frac{\mu_B}{\hbar}\sum_j g_{ij}S_j, \qquad H=\frac{\mu_B}{2}(g^{\mathsf T}\mathbf B)\cdot\boldsymbol\sigma.$$

The level splitting is then $\Delta E=\mu_B|g^{\mathsf T}\mathbf B|$, so field direction matters. This is a response of the chosen doublet, not necessarily the free electron’s spin. Magnetic Moments in Matter distinguishes such moments from saturation and thermal effective moments; Ferrimagnetism explains why unequal sublattice gyromagnetic ratios separate magnetic and angular-momentum compensation.

Common Mistakes

• Replacing the electron charge $q=-e$ by $e$ without explicitly changing the sign convention.
• Assuming the electron magnetic moment points along the electron spin.
• Using $g_s\simeq2$ for orbital angular momentum; ordinary orbital motion has $g_L=1$.
• Treating the Landé formula as universal rather than as a weak-field coupling-regime result.
• Confusing the Bohr magneton and nuclear magneton.
• Forgetting that effective $g$ factors in solids, molecules, nuclei, and bands can be negative, anisotropic, or far from the free-electron value.
• Calling the anomalous magnetic moment a correction to spin itself; it is a correction to the magnetic coupling.

Exercises

1. For an electron in $\mathbf B=B\hat{\mathbf z}$ with $B>0$, which spin projection is lower in energy?

Solution

With the convention

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

the Hamiltonian is

$$H = g_s\mu_B B\frac{S_z}{\hbar}.$$

Thus

$$E_{m_s} = g_s\mu_B B\,m_s.$$

For $g_s>0$ and $B>0$, the lower energy has the smaller $m_s$:

$$m_s=-\frac12.$$

The spin is anti-aligned with $\mathbf B$, while the magnetic moment is aligned with $\mathbf B$.

2. Why does ordinary orbital angular momentum have $g_L=1$ while electron spin has leading $g_s=2$?

Solution

For orbital motion, the current-loop relation gives

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

so the coefficient multiplying $q\mathbf L/(2m)$ is $g_L=1$. Electron spin is not literal orbital circulation. Its leading magnetic moment follows from the Pauli/Dirac spin coupling and is

$$\boldsymbol\mu_S = g_s\frac{q}{2m}\mathbf S$$

with $g_s=2$ at Dirac level.

3. Use the Landé formula to find $g_J$ for $L=0$, $S=1/2$, $J=1/2$ with $g_L=1$ and $g_S=2$.

Solution

The Landé expression is

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

For $L=0$ and $J=S=1/2$, the first numerator vanishes:

$$J(J+1)+L(L+1)-S(S+1)=0.$$

The second numerator is

$$J(J+1)+S(S+1) = 2J(J+1).$$

Therefore

$$g_J=g_S=2.$$

With a more precise spin value one would get $g_J\simeq g_S$ for this ideal $L=0$ case.

4. A spin-$s$ particle has $\boldsymbol\mu=\gamma\mathbf S$ in a field $B\hat z$. What is the spacing between adjacent $m$ levels?

Solution

The energies are

$$E_m=-\gamma\hbar B\,m.$$

Adjacent $m$ values differ by one, so

$$E_{m+1}-E_m = -\gamma\hbar B.$$

The positive spacing is

$$\Delta E=\hbar|\gamma|B.$$

5. What does the anomalous magnetic moment measure conceptually?

Solution

For a minimally coupled spin-$1/2$ particle in the stated charge-sign convention, it measures the deviation of the magnetic coupling from the leading Dirac value $g=2$:

$$a=\frac{g-2}{2}.$$

For the electron, this deviation is mainly a radiative quantum-field correction to the magnetic coupling, not a change in the angular-momentum algebra of spin.

References

• C. Cohen-Tannoudji, B. Diu, and F. Laloë, Quantum Mechanics, Wiley, 1977.
• P. A. M. Dirac, “The Quantum Theory of the Electron”, Proceedings of the Royal Society A 117, 610–624 (1928).
• C. J. Foot, Atomic Physics, Oxford University Press, 2005.
• L. D. Landau and E. M. Lifshitz, Quantum Mechanics: Non-Relativistic Theory, 3rd ed., Pergamon, 1977.
• 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).