Magnetic Moments in External Fields and the Zeeman Effect

Put a sodium lamp above the poles of a strong electromagnet. The single yellow line of its emission spectrum fractures into several closely spaced lines. This splitting, discovered by Pieter Zeeman in 1896, is one of the cleanest windows quantum mechanics ever opened onto the inner architecture of the atom. It shows directly and quantitatively that angular momentum is quantized. It also shows that electrons carry an intrinsic magnetic moment associated with spin. Finally, it shows that the response of an atom to a magnetic field depends on the exact way orbital and spin angular momenta combine. The physics is deceptively simple to state: a magnetic moment $\boldsymbol{\mu}$ sitting in a field $\mathbf{B}$ has energy $-\boldsymbol{\mu}\cdot\mathbf{B}$ . So degenerate atomic sublevels that differ only in the orientation of their angular momentum acquire different energies once the field is switched on. What makes the problem rich — and what separates a first course from real mastery — is that the moment being oriented is not simply proportional to the total angular momentum $\mathbf{J}$ . Orbital motion and electron spin contribute to the magnetic moment with different proportionality constants (g-factors of 1 and 2 respectively). So when spin-orbit coupling has already welded $\mathbf{L}$ and $\mathbf{S}$ into a single vector $\mathbf{J}$ , the net moment is not collinear with $\mathbf{J}$ in general. The resolution is to project the true moment onto $\mathbf{J}$ , since that is the only vector that survives time-averaging under fast precession. This produces the Landé $g$ -factor, and with it the full richness of the "anomalous" Zeeman effect. This article develops the Zeeman Hamiltonian from the classical energy of a magnetic dipole. It builds the Landé $g$ -factor by vector projection and confirms it by degenerate perturbation theory. It also works out both limits: the historically first-understood "normal" triplet splitting (spin-zero systems, or the high-field Paschen–Back limit) and the "anomalous" multiplet splitting that actually dominates real atomic spectra. Selection rules, relative intensities, and the crossover between weak-field (Zeeman) and strong-field (Paschen–Back) regimes are treated with full derivations. Non-trivial worked numbers are given too, not just formulas to memorize. Beyond its historical importance, the Zeeman effect is the direct conceptual ancestor of every magnetic-resonance measurement used today. The energy-level splitting it describes is exactly what is exploited today. So is the way the g-factor encodes the microscopic mixture of orbital and spin character in a state. This exploitation happens, at drastically different field and frequency scales, in nuclear magnetic resonance and electron spin resonance. Understanding Zeeman splitting properly is therefore a prerequisite not only for atomic spectroscopy but for magnetometry. It is also essential for astrophysical field measurements (Zeeman splitting of solar and stellar spectral lines), and the entire technology of MRI and EPR spectrometers.

Key takeaways

A magnetic moment $\boldsymbol\mu$ in a field $\mathbf{B}$ has interaction energy $H' = -\boldsymbol\mu\cdot\mathbf{B}$ , lifting the $(2J+1)$ -fold degeneracy of a level into equally spaced $m_J$ sublevels.
Orbital and spin moments carry different gyromagnetic ratios ( $g_L=1$ , $g_S\approx 2$ ), so the total moment is not parallel to $\mathbf{J}$ when $\mathbf{L}$ and $\mathbf{S}$ are coupled.
Projecting the moment onto $\mathbf J$ (valid because $\mathbf L$ and $\mathbf S$ precess rapidly about $\mathbf J$ ) yields the Landé $g$ -factor, $g_J = 1+\dfrac{J(J+1)+S(S+1)-L(L+1)}{2J(J+1)}$ .
The "normal" Zeeman effect (three equally spaced lines) appears only when $S=0$ ; almost all real atoms show the "anomalous" pattern with $g_J\neq1$ and unequal splittings.
At fields strong enough that the Zeeman energy exceeds the spin-orbit splitting, $\mathbf L$ and $\mathbf S$ decouple and the pattern reverts to a simple triplet — the Paschen–Back effect.
Key idea: the Zeeman splitting pattern is a direct spectroscopic readout of the Landé $g$ -factor, which itself encodes how orbital and spin angular momentum are mixed inside a coupled atomic state.

Where this lives today

The Zeeman effect is far from a historical curiosity confined to sodium lamps. In astrophysics, Zeeman splitting of spectral lines (especially the iron and other metal lines with large, well-tabulated $g_J$ values) is the primary tool for measuring magnetic fields on the Sun, in sunspots, in magnetic white dwarfs, and in star-forming molecular clouds via the Zeeman splitting of the 21 cm hydrogen hyperfine line and OH/CN maser lines — exactly the nuclear/hyperfine Zeeman formalism worked out above, scaled to astronomically weak fields but astronomically long integration times. In atomic, molecular, and optical physics, Zeeman splitting is the working principle behind the Zeeman slower, a standard piece of apparatus in every laser-cooling and Bose–Einstein-condensation laboratory: a spatially varying magnetic field is used to Zeeman-shift an atomic transition into resonance with a fixed-frequency laser as atoms decelerate, compensating the changing Doppler shift. Optical pumping and magneto-optical traps rely on the same $g_J$ -weighted sublevel structure to select circular polarizations that drive specific $\Delta m_J$ transitions. In chemistry and biochemistry, the direct descendants of this physics are electron spin resonance (ESR/EPR), used to study free radicals, transition-metal centers in metalloproteins, and spin-labeled probes in membrane biophysics, and — at a different energy and coupling scale — nuclear magnetic resonance (NMR), the workhorse of structure determination for small molecules, proteins, and (as MRI) clinical diagnostic imaging. Both techniques measure exactly the Zeeman-split sublevel spacings derived here (with electronic vs. nuclear $g$ -factors and Bohr vs. nuclear magnetons respectively), and the quantum mechanical basis of NMR and ESR is the direct generalization of this Hamiltonian to spin systems coupled further by chemical shielding and hyperfine/exchange interactions. In materials science and metrology, the Zeeman effect underlies optically pumped magnetometers (used in geophysical surveying and biomagnetism, including magnetoencephalography), and precision atomic-clock and quantum-computing platforms use carefully chosen "clock states" — combinations of $m_F$ sublevels engineered so that their first-order Zeeman shift cancels — to suppress magnetic-field sensitivity, a direct practical exploitation of the $g_F$ formalism developed in the hyperfine worked example.

Ask the tutor: How does the Landé g-factor formula change when hyperfine coupling (nuclear spin) is included alongside spin-orbit coupling, and how does that connect to the Breit–Rabi formula used in atomic clocks?

The physical picture

Classically, a current loop of magnetic moment $\boldsymbol\mu$ placed in a uniform field $\mathbf B$ experiences a torque $\boldsymbol\mu\times\mathbf B$ that tries to align it with the field, and it has potential energy

E = -\boldsymbol\mu\cdot\mathbf B.

Quantum mechanically an atomic electron's orbital motion generates a magnetic moment $\boldsymbol\mu_L = -\dfrac{e}{2m_e}\mathbf L = -\mu_B\mathbf L/\hbar$ (in SI units, with $\mu_B = e\hbar/2m_e$ the Bohr magneton), exactly as derived from the classical current-loop analogy in the treatment of Orbital Magnetic Moments and Space Quantization. Spin contributes an anomalous moment $\boldsymbol\mu_S = -g_S\mu_B\mathbf S/\hbar$ with $g_S \approx 2.0023$ (we use $g_S=2$ throughout, the Dirac-equation value, since higher-order QED corrections are a fraction of a percent). The total electronic magnetic moment operator is therefore

\boldsymbol\mu = -\frac{\mu_B}{\hbar}\left(\mathbf L + 2\mathbf S\right) = -\frac{\mu_B}{\hbar}\left(\mathbf J + \mathbf S\right).

Because the coefficient multiplying $\mathbf S$ is twice that multiplying $\mathbf L$ , this vector is not parallel to $\mathbf J = \mathbf L+\mathbf S$ unless $S=0$ or $L=0$ . Yet in an atom with spin-orbit coupling already established, $\mathbf J$ is the only good angular momentum (its magnitude and $z$ -projection are conserved to good approximation), and $\mathbf L,\mathbf S$ individually precess rapidly around $\mathbf J$ . Any component of $\boldsymbol\mu$ perpendicular to $\mathbf J$ averages to zero over this fast precession, leaving only the projection along $\mathbf J$ as physically meaningful in a weak external field. This geometric fact — illustrated below — is the entire content of the Landé $g$ -factor.

Figure

Vector model: L and S precessing about J, with μ not parallel to J

Once the atom is placed in an external field $\mathbf B = B\hat z$ , this projected moment interacts with the field to give the Zeeman Hamiltonian

H_{\text{Zeeman}} = g_J\,\mu_B\, B\, \frac{J_z}{\hbar},

whose eigenvalues on the $|J,m_J\rangle$ states of a fine-structure level are

E(m_J) = g_J\,\mu_B\,B\,m_J,\qquad m_J = -J,-J+1,\dots,+J.

This is the master result: a $(2J+1)$ -fold degenerate level splits into $2J+1$ equally spaced sublevels, with spacing $g_J\mu_B B$ set entirely by the Landé factor. The figure below shows how a single spectral line (a transition between two fine-structure terms) becomes a multiplet once each level is Zeeman-split and the electric-dipole selection rule $\Delta m_J = 0,\pm1$ is applied.

Figure

Zeeman splitting of a P→S transition with unequal g-factors

Notice immediately that because $g_J$ differs between the two combining levels (4/3 for $^2P_{3/2}$ , 2 for $^2S_{1/2}$ ), the six allowed transitions are not equally spaced — this asymmetric splitting pattern, ubiquitous in real atoms with nonzero spin, is exactly what nineteenth-century spectroscopists labeled "anomalous," even though it is now understood as the generic case and the equally-spaced triplet is the special one.

The core ideas

The Zeeman Hamiltonian. For an atomic state described by quantum numbers $L,S,J,m_J$ , the perturbation due to a weak external field along $z$ is

H' = \frac{\mu_B}{\hbar}\left(\mathbf L + 2\mathbf S\right)\cdot\mathbf B = \frac{\mu_B B}{\hbar}\left(L_z + 2S_z\right).

where: $\mu_B = e\hbar/2m_e = 9.274\times10^{-24}\ \mathrm{J/T}$ is the Bohr magneton, $B$ is the applied field magnitude, and $L_z,S_z$ are the $z$ -components of orbital and spin angular momentum operators. Reading: this is simply $-\boldsymbol\mu\cdot\mathbf B$ written in terms of the operators that generate the moment, with the factor of 2 on spin already built in.

The Landé $g$ -factor. Within a fixed $J$ -multiplet (good quantum numbers $L,S,J$ ), the effective Hamiltonian reduces to

H'_{\text{eff}} = g_J\,\mu_B B\, \frac{J_z}{\hbar},\qquad g_J = 1 + \frac{J(J+1) + S(S+1) - L(L+1)}{2J(J+1)}.

where: $L,S,J$ are the orbital, spin, and total angular momentum quantum numbers of the term (Russell–Saunders coupling assumed). Reading: $g_J=1$ recovers the pure-orbital case ( $S=0$ ), $g_J=2$ recovers the pure-spin case ( $L=0$ ), and intermediate values interpolate smoothly — the "1" out front is the orbital contribution, the fraction is the extra spin contribution weighted by how much spin character survives the vector coupling.

Zeeman sublevel energies and splitting. The energy of a $|J,m_J\rangle$ sublevel, and the spacing between adjacent sublevels, are

E(m_J) = g_J\,\mu_B\,B\,m_J,\qquad \Delta E_{\text{adjacent}} = g_J\,\mu_B\,B.

where: $m_J$ runs in integer steps from $-J$ to $+J$ . Reading: the entire multiplet fans out linearly in $B$ (weak-field regime) with a single characteristic slope $g_J\mu_B B$ ; measuring that slope spectroscopically is a direct measurement of $g_J$ , hence of the internal $L,S,J$ composition of the term.

Selection rules and transition frequencies. For an electric-dipole transition between an upper term $(L',S',J',g_J')$ and lower term $(L,S,J,g_J)$ , allowed lines obey $\Delta m_J = 0,\pm1$ and have frequency shift from the field-free line

h\,\Delta\nu = \left(g_J'\,m_J' - g_J\,m_J\right)\mu_B B.

where: $\Delta\nu$ is measured relative to the zero-field transition frequency $\nu_0$ . Reading: because $g_J'\neq g_J$ in general, the shifts for different $(m_J,m_J')$ pairs with the same $\Delta m_J$ are not identical, producing the characteristic unequal spacing of the anomalous pattern; only when $g_J'=g_J$ (true, e.g., whenever $S=0$ on both terms) does the pattern collapse to the equally-spaced normal triplet.

Derivation AThe Landé g-factor by vector projection

Goal. Show that within a state of fixed $J$ , the magnetic moment operator $\boldsymbol\mu = -(\mu_B/\hbar)(\mathbf L+2\mathbf S)$ behaves, for the purpose of computing energies, as if it were $-g_J(\mu_B/\hbar)\mathbf J$ , and derive the formula for $g_J$ .

Step 1 — Why only the $\mathbf J$ -projection survives. In the absence of the external field, spin-orbit coupling $\xi(r)\,\mathbf L\cdot\mathbf S$ is present and conserves $J,m_J,L,S$ but not $m_L,m_S$ individually. In this coupled basis, $\mathbf L$ and $\mathbf S$ are not separately diagonal; instead they precess about the fixed direction of $\mathbf J$ (a standard result of the vector model, since $[\mathbf L,\,\mathbf L\cdot\mathbf S]\neq 0$ but $\mathbf J$ commutes with the spin-orbit term). A weak external field only samples the time-averaged moment, and the component of any vector perpendicular to $\mathbf J$ averages to zero under this precession. Only the component of $\mathbf L+2\mathbf S$ along $\mathbf J$ contributes to first-order energy shifts.

Step 2 — Projection formula. The projection of a vector operator $\mathbf V$ onto $\mathbf J$ , within a subspace of fixed $J$ , is captured by the projection theorem:

\langle J m_J|\mathbf V|J m_J\rangle = \frac{\langle \mathbf V\cdot\mathbf J\rangle}{J(J+1)\hbar^2}\,\langle J m_J|\mathbf J|J m_J\rangle.

Apply this with $\mathbf V = \mathbf L + 2\mathbf S = \mathbf J + \mathbf S$ :

\langle \mathbf L+2\mathbf S\rangle_{\text{eff}} = \frac{\langle(\mathbf J+\mathbf S)\cdot\mathbf J\rangle}{J(J+1)\hbar^2}\,\mathbf J = \left(1 + \frac{\langle\mathbf S\cdot\mathbf J\rangle}{J(J+1)\hbar^2}\right)\mathbf J.

Step 3 — Evaluate $\langle\mathbf S\cdot\mathbf J\rangle$ . Since $\mathbf J = \mathbf L+\mathbf S$ ,

\mathbf L^2 = (\mathbf J-\mathbf S)^2 = \mathbf J^2 - 2\mathbf J\cdot\mathbf S + \mathbf S^2 \quad\Rightarrow\quad \mathbf J\cdot\mathbf S = \frac{1}{2}\left(\mathbf J^2+\mathbf S^2-\mathbf L^2\right).

Taking expectation values in the $|L,S,J,m_J\rangle$ state,

\langle \mathbf S\cdot\mathbf J\rangle = \frac{\hbar^2}{2}\Big[J(J+1)+S(S+1)-L(L+1)\Big].

Step 4 — Assemble $g_J$ . Substituting into Step 2,

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

This is the Landé formula. It is symmetric in the roles played by $L$ and $S$ except for the explicit sign, reflecting that spin ( $g_S=2$ ) and orbital ( $g_L=1$ ) moments enter the original vector asymmetrically.

Step 5 — Sanity limits. If $S=0$ (all moment is orbital), $J=L$ and $g_J = 1 + \dfrac{0+L(L+1)-L(L+1)}{2L(L+1)} = 1$ , recovering the normal (orbital-only) result. If $L=0$ (all moment is spin, e.g. an S-term), $J=S$ and $g_J = 1+\dfrac{S(S+1)+S(S+1)-0}{2S(S+1)} = 2$ , recovering the pure-spin g-factor. Both limits check out exactly, confirming the derivation.

Go deeper · Derivation B · 2 minDegenerate perturbation theory confirms the diagonal result

Derivation BDegenerate perturbation theory confirms the diagonal result

Goal. Verify, without appeal to the semiclassical vector-precession argument, that $H'=(\mu_B B/\hbar)(L_z+2S_z)$ is diagonal in the coupled basis $|L,S,J,m_J\rangle$ to first order, with eigenvalue $g_J\mu_B B m_J$ — i.e. that ordinary first-order degenerate perturbation theory reproduces the Landé result.

Step 1 — Set up the problem. The zero-field Hamiltonian $H_0 + \zeta\,\mathbf L\cdot\mathbf S$ (fine structure already included) is diagonal in the coupled basis $|LSJm_J\rangle$ with energy $E_J$ depending only on $J$ (for fixed $L,S$ ). This basis is exactly the "good" basis required for degenerate perturbation theory when the perturbation $H'\propto B$ is turned on, provided $H'$ does not connect states of different $J$ at first order — which must be checked, since $H'$ involves $L_z+2S_z = J_z + S_z$ , and $S_z$ alone is not diagonal in $J$ .

Step 2 — Off-diagonal matrix elements in $J$ . The operator $S_z$ can connect $|L,S,J,m_J\rangle$ to $|L,S,J\pm1,m_J\rangle$ (same $m_J$ , since $S_z$ commutes with $J_z$ ). These off-diagonal terms exist, but they only produce second-order corrections in $B$ (mixing between fine-structure levels separated by the spin-orbit energy $\Delta E_{SO}$ ), scaling as $(\mu_B B)^2/\Delta E_{SO}$ . As long as $\mu_B B \ll \Delta E_{SO}$ (the weak-field/Zeeman regime, typically true for laboratory fields up to several tesla against fine-structure splittings of $10^{-3}$ – $10^{-2}$ eV), these are negligible and first-order degenerate perturbation theory within a single $J$ -multiplet is valid.

Step 3 — Diagonal matrix element. Within the $2J+1$ -dimensional degenerate subspace of fixed $J$ , first-order perturbation theory instructs us to diagonalize $H'$ restricted to that subspace. Because $H'$ commutes with $J_z$ (as $[L_z+2S_z,J_z]=0$ , since both $L_z$ and $S_z$ individually commute with $J_z=L_z+S_z$ ), $H'$ is automatically diagonal in the $m_J$ basis — no further diagonalization is needed, and the eigenvalues are simply the diagonal matrix elements

E^{(1)}(m_J) = \frac{\mu_B B}{\hbar}\langle J,m_J|L_z+2S_z|J,m_J\rangle.

Step 4 — Evaluate using the projection result. From the previous derivation, $\langle L_z + 2S_z\rangle = g_J \langle J_z\rangle/\hbar\cdot\hbar = g_J\hbar m_J$ (since $\langle J_z\rangle = \hbar m_J$ exactly, a good quantum number). Substituting,

E^{(1)}(m_J) = \frac{\mu_B B}{\hbar}\cdot g_J\hbar m_J = g_J\mu_B B\, m_J.

This matches the vector-model result exactly, confirming that the projection-theorem shortcut is not an approximation beyond the weak-field assumption itself — it is the exact first-order degenerate-perturbation-theory answer.

Step 5 — Regime of validity flagged explicitly. The derivation shows the only approximation made is neglecting $J$ -mixing, valid precisely when the Zeeman energy is small compared to fine-structure splitting. When this fails (fields comparable to or larger than the spin-orbit interaction), $J$ ceases to be a good quantum number and one must return to diagonalizing $H'$ in the uncoupled $|m_L,m_S\rangle$ basis — this is the Paschen–Back regime, treated in the worked examples below.

Worked examples

Example 1 — Foundational: the sodium D₂ line, ${}^2P_{3/2}\to{}^2S_{1/2}$ . Sodium's principal series arises from $3p\to3s$ transitions. Compute the Landé factors and the Zeeman splitting pattern. For the upper term $^2P_{3/2}$ : $L=1$ , $S=1/2$ , $J=3/2$ .

g_J = 1+\frac{\frac{3}{2}\cdot\frac{5}{2}+\frac{1}{2}\cdot\frac{3}{2}-1\cdot2}{2\cdot\frac{3}{2}\cdot\frac{5}{2}} = 1+\frac{\frac{15}{4}+\frac{3}{4}-2}{\frac{15}{2}} = 1+\frac{\frac{18}{4}-2}{\frac{15}{2}} = 1+\frac{\frac{10}{4}}{\frac{15}{2}} = 1+\frac{10}{4}\cdot\frac{2}{15} = 1+\frac{20}{60} = \frac{4}{3}.

For the lower term $^2S_{1/2}$ : $L=0$ , $S=1/2$ , $J=1/2$ , giving the pure-spin value $g_J=2$ (matches the limiting case checked in the derivation). Take a modest laboratory field $B = 0.60\ \mathrm{T}$ (chosen non-round to keep the arithmetic honest). With $\mu_B = 9.274\times10^{-24}\ \mathrm{J/T}$ ,

\mu_B B = 9.274\times10^{-24}\times0.60 = 5.564\times10^{-24}\ \mathrm{J} = 3.474\times10^{-5}\ \mathrm{eV}.

For the upper level ( $g_J=4/3$ ), the four sublevels $m_J = -3/2,-1/2,+1/2,+3/2$ have energies

E(m_J) = \frac{4}{3}(3.474\times10^{-5}\ \mathrm{eV})\,m_J = (4.632\times10^{-5}\ \mathrm{eV})\,m_J,

i.e. $-6.948\times10^{-5},\ -2.316\times10^{-5},\ +2.316\times10^{-5},\ +6.948\times10^{-5}$ eV for $m_J=-3/2,-1/2,+1/2,+3/2$ . For the lower level ( $g_J=2$ ), the two sublevels have

E(m_J) = 2(3.474\times10^{-5}\ \mathrm{eV})\,m_J = (6.948\times10^{-5}\ \mathrm{eV})\,m_J,

giving $-3.474\times10^{-5}$ eV ( $m_J=-1/2$ ) and $+3.474\times10^{-5}$ eV ( $m_J=+1/2$ ). Applying $\Delta m_J = 0,\pm1$ , the six allowed transitions have energy shifts relative to the field-free D₂ line ( $h\Delta\nu = g_J'm_J'\mu_BB - g_Jm_J\mu_BB$ , all in units of $10^{-5}$ eV):

$m_J'$ (P₃/₂)	$m_J$ (S₁/₂)	$\Delta m_J$	Shift ( $10^{-5}$ eV)
$+3/2$	$+1/2$	$+1$	$6.948-3.474=+3.474$
$+1/2$	$+1/2$	$0$	$2.316-3.474=-1.158$
$+1/2$	$-1/2$	$+1$	$2.316+3.474=+5.790$
$-1/2$	$+1/2$	$-1$	$-2.316-3.474=-5.790$
$-1/2$	$-1/2$	$0$	$-2.316+3.474=+1.158$
$-3/2$	$-1/2$	$-1$	$-6.948+3.474=-3.474$

The six shifts, $\pm3.474,\ \pm5.790,\ \pm1.158$ (in $10^{-5}$ eV), are manifestly not equally spaced — this is the anomalous Zeeman pattern in numbers, directly traceable to $g_J = 4/3 \neq g_J = 2$ .

Limiting case check. If we (incorrectly, but instructively) set $S=0$ for both levels so that $g_J=1$ throughout, every shift in the table above collapses to $\pm\mu_BB\,\Delta m_J = 0,\pm3.474\times10^{-5}$ eV — exactly the three equally spaced lines of the normal Zeeman triplet. This confirms the formula degenerates correctly to the historically "normal" case when spin is absent.

Example 2 — Applied: EPR-relevant free radical / high-field decoupling (Paschen–Back onset). Consider a hypothetical one-electron ion in a $^2P$ term (say $L=1,S=1/2$ ) with a fine-structure splitting between $^2P_{3/2}$ and $^2P_{1/2}$ of $\Delta E_{SO} = 2.0\times10^{-3}$ eV — a realistic order of magnitude for a light atom. At what field does the Zeeman energy become comparable to the fine structure, forcing a crossover from the anomalous-Zeeman (weak-field, $J$ good) regime to the Paschen–Back (strong-field, $L,S$ decoupled) regime? Set the criterion $\mu_B B \sim \Delta E_{SO}$ :

B_c = \frac{\Delta E_{SO}}{\mu_B} = \frac{2.0\times10^{-3}\times1.602\times10^{-19}\ \mathrm{J}}{9.274\times10^{-24}\ \mathrm{J/T}} = \frac{3.204\times10^{-22}}{9.274\times10^{-24}}\ \mathrm{T} = 34.6\ \mathrm{T}.

Below roughly 35 T, the atom is in the anomalous-Zeeman regime treated by the Landé formula; at and above it, $\mathbf L$ and $\mathbf S$ decouple from each other and separately precess about $\mathbf B$ instead of about each other. In this Paschen–Back limit, $m_L$ and $m_S$ (not $m_J$ ) become the good quantum numbers, and the energy becomes

E(m_L,m_S) = \mu_B B\,(m_L + 2m_S) + \zeta\, m_L m_S,

where the last term is the (now perturbative) residual spin-orbit contribution ( $\zeta$ the spin-orbit coupling constant). To leading order in $B$ , the spectrum reduts to the normal triplet pattern again — this is why the Paschen–Back limit is often summarized as "the anomalous effect reverts to normal at high field." Note this is a genuinely different physical regime from the $S=0$ limiting check in Example 1: there $g_J=1$ identically at all fields because there is no spin to decouple; here the recovery of a normal-looking triplet is a high-field decoupling effect that only becomes exact asymptotically as $B\to\infty$ (or more precisely $\mu_BB \gg \Delta E_{SO}$ ). Real EPR (electron spin resonance) experiments on transition-metal or radical species deliberately work at fields (typically 0.33 T at X-band) where the electronic Zeeman splitting of unpaired-spin sublevels, $\Delta E = g\mu_B B$ with an effective $g$ -factor close to but shifted from the free-electron value $g_e = 2.0023$ by spin-orbit admixture, is probed directly by microwave absorption — the same Zeeman Hamiltonian derived above, simply reinterpreted for a single unpaired spin rather than a full atomic multiplet.

Example 3 — Exam/competition level: nuclear Zeeman effect and the hyperfine-coupled g-factor. A hydrogen atom in its ground electronic state ( $L=0,S=1/2$ , so $J=1/2$ ) also has a nuclear spin $I=1/2$ (the proton). The total angular momentum is $F=I+J$ , taking values $F=0,1$ . In a weak field (weaker than the hyperfine splitting, by analogy with the weak-field criterion established above), the appropriate "super-Landé" factor for the $F=1$ hyperfine level is

g_F = g_J\,\frac{F(F+1)+J(J+1)-I(I+1)}{2F(F+1)} + g_I\,\frac{F(F+1)+I(I+1)-J(J+1)}{2F(F+1)},

an exact structural analogue of the Landé formula with $(J,S,L)\to(F,J,I)$ for the electronic part and $(F,I,J)$ for the nuclear part. Here $g_J = 2$ (pure spin, $L=0$ , computed exactly as in the $L=0$ limiting case above) and the proton's nuclear g-factor is $g_I \approx 5.586$ (with a minus sign convention issue absorbed here into treating $g_I$ as the proton's positive $g$ -factor referenced to the nuclear magneton $\mu_N = \mu_B/1836$ , so that the second term is suppressed by roughly $1/1836$ relative to the first — this is why nuclear Zeeman shifts are three orders of magnitude smaller than electronic ones, foreshadowing why NMR requires far higher fields or far more sensitive detection than EPR to see comparable splittings). For $F=1$ : $J=1/2,I=1/2$ ,

g_F^{(e)} = 2\times\frac{1\cdot2+\frac12\cdot\frac32-\frac12\cdot\frac32}{2\cdot1\cdot2} = 2\times\frac{2}{4} = 1.

The electronic contribution to $g_F$ is exactly 1 (an elegant special case, since $I=J=1/2$ makes the last two terms in the numerator cancel). The nuclear correction term,

g_F^{(n)} = 5.586\times\frac{2+\frac{3}{4}-\frac{3}{4}}{4}\times\frac{\mu_N}{\mu_B} = 5.586\times\frac{2}{4}\times\frac{1}{1836.15} = \frac{2.793}{1836.15} = 1.521\times10^{-3},

so $g_F \approx 1.00152$ — the nuclear correction shifts the effective $g$ -factor by about 0.15%, a tiny but real and measurable effect (it underlies the precision measurements that determine $g_I$ for various nuclei). Take $B=0.10$ T (a realistic weak-field NMR-adjacent scale, well below the $\sim$ 0.05 T hyperfine-decoupling field for hydrogen's 1420 MHz hyperfine splitting):

\Delta E_{\text{adjacent}} = g_F\,\mu_B B \approx 1.00152\times9.274\times10^{-24}\times0.10 = 9.288\times10^{-25}\ \mathrm{J} = 5.798\times10^{-6}\ \mathrm{eV}.

Limiting case check. As $\mu_N/\mu_B\to0$ (i.e., treating the nucleus as infinitely heavy so its magnetic moment vanishes), $g_F\to g_F^{(e)}=1$ exactly — the nuclear Zeeman contribution disappears identically, exactly as physically required, and the formula reduces cleanly to a pure-electronic Landé-type result.

Limiting cases, scope & common traps

Regime / limit	Condition	Result
Normal Zeeman effect	$S=0$ (e.g. singlet states, or artificially ignoring spin)	$g_J=1$ for all levels in a transition; three equally spaced lines ( $\pi,\sigma^\pm$ )
Anomalous Zeeman effect	$S\neq0$ , weak field ( $\mu_BB\ll\Delta E_{SO}$ )	$g_J$ from Landé formula, generally different for upper/lower terms; unequal splittings
Paschen–Back effect	Strong field ( $\mu_BB\gg\Delta E_{SO}$ )	$L,S$ decouple; $m_L,m_S$ good quantum numbers; pattern reverts to a normal-looking triplet, spin-orbit becomes a small perturbative correction
Intermediate field	$\mu_BB\sim\Delta E_{SO}$	Neither basis diagonalizes $H$ ; must diagonalize the full $(2J+1)(\dots)$ -dimensional matrix numerically (Breit–Rabi–type treatment)
Nuclear Zeeman / hyperfine field	Weak field vs. hyperfine splitting	Analogous $g_F$ formula; nuclear term suppressed by $\mu_N/\mu_B\sim1/1836$
$J=0$ level	Any $L=S$ (e.g. ${}^1S_0, {}^3P_0$ )	No Zeeman splitting at all to first order — a single, non-degenerate $m_J=0$ sublevel

Common trapCommon misconception: treating the magnetic moment as always antiparallel and collinear with

\mathbf J

It is tempting to say "the atom has moment $\mu = g_J\mu_B\sqrt{J(J+1)}$ pointing along $-\mathbf J$ " as though this were an exact operator statement. It is only the effective, time-averaged moment relevant once $\mathbf L$ and $\mathbf S$ have been forced by spin-orbit coupling to precess about $\mathbf J$ . The instantaneous, microscopic moment $-(\mu_B/\hbar)(\mathbf L+2\mathbf S)$ is never actually parallel to $\mathbf J$ ; the projection is a physical simplification valid specifically in the weak-field, good- $J$ regime, and it breaks down once fields are strong enough to decouple $\mathbf L$ from $\mathbf S$ (the Paschen–Back limit).

The chemist behind it

Origins · 1 minPieter Zeeman and how this was discovered

Pieter Zeeman

1865–1943 · University of Amsterdam / Leiden, late 19th–early 20th century

Pieter Zeeman was a Dutch experimental physicist working under Heike Kamerlingh Onnes and, crucially, alongside Hendrik Lorentz, whose classical electron theory Zeeman set out to test. In 1896, defying an instruction from his supervisor to abandon the (initially unsuccessful) search, Zeeman placed a sodium flame between the poles of a powerful Ruhmkorff electromagnet and observed the broadening, then splitting, of the D lines. Lorentz immediately supplied the classical theoretical framework — an oscillating electron in a magnetic field should have its motion split into three frequencies — which matched Zeeman's initial (normal) triplet observations and earned the two men the 1902 Nobel Prize in Physics jointly, one of the earliest Nobel recognitions of quantum-adjacent atomic physics. It would take another quarter-century, and the full development of electron spin by Uhlenbeck and Goudsmit together with the vector-coupling and Landé g-factor formalism, to explain why most atoms actually show more than three lines — the "anomalous" effect that had puzzled spectroscopists precisely because Lorentz's classical theory could not account for half-integer angular momentum quantum numbers.

History

- 1896 — Pieter Zeeman observes magnetic broadening/splitting of the sodium D lines in Leiden.

- 1897 — Lorentz's classical oscillating-electron theory explains the normal triplet splitting and predicts the sign/magnitude of the effect (used shortly after to help establish the electron's charge-to-mass ratio).

- 1898–1912 — Preston, Runge, and others catalogue "anomalous" multiplet patterns in alkali and other spectra that Lorentz's theory cannot explain.

- 1925 — Uhlenbeck and Goudsmit propose electron spin; Landé derives the empirical

g

-factor formula from the vector model, resolving the anomalous patterns.

- 1927 onward — Full quantum-mechanical (Pauli/Dirac spin) treatment confirms

g_S\approx2

and places the Landé formula on rigorous footing via degenerate perturbation theory.

What exactly is the difference between the normal and anomalous Zeeman effect?

The normal Zeeman effect is the historically first-observed pattern: a spectral line splits into exactly three equally spaced components (one unshifted $\pi$ line, two symmetric $\sigma^\pm$ lines) when a magnetic field is applied. It occurs whenever both combining atomic terms have zero total spin, $S=0$ , so that $g_J=1$ identically and the moment is purely orbital. The anomalous Zeeman effect is what happens for virtually every real atom with unpaired spin: because $g_J$ depends on $L,S,J$ through the Landé formula and differs between the upper and lower terms of a transition, the multiplet splits into more than three lines with unequal spacings. "Anomalous" is a historical label, not a statement about rarity — it is in fact the generic, expected behavior once electron spin is included, and the "normal" case is the special one.

Where does the factor of 2 (g_S≈2) in the spin magnetic moment actually come from?

Nonrelativistic quantum mechanics does not predict it; it emerges from the Dirac equation, the relativistic wave equation for spin-1/2 particles, which in its non-relativistic (Pauli) limit produces a magnetic coupling term with gyromagnetic ratio exactly 2 for a structureless spin-1/2 particle. Quantum electrodynamics refines this further, giving the famous anomalous magnetic moment correction $g_S = 2(1+\alpha/2\pi+\cdots)\approx2.0023$ , one of the most precisely verified predictions in all of physics. For essentially all chemical and atomic-spectroscopy purposes, $g_S=2$ is an excellent approximation, and it is the value used throughout the Landé formula and Zeeman-splitting calculations in this article.

How do I know whether a given laboratory field is in the "weak-field" Zeeman regime or the Paschen–Back regime?

Compare the Zeeman energy scale $\mu_B B$ to the fine-structure (spin-orbit) splitting $\Delta E_{SO}$ between adjacent $J$ levels of the same $L,S$ term. If $\mu_B B \ll \Delta E_{SO}$ , $J$ remains a good quantum number, $\mathbf L$ and $\mathbf S$ stay strongly coupled, and the Landé $g_J$ formalism applies (anomalous Zeeman regime). If $\mu_B B \gtrsim \Delta E_{SO}$ , the external field dominates over the internal spin-orbit coupling, $\mathbf L$ and $\mathbf S$ decouple from each other and separately align with $\mathbf B$ , and you are in the Paschen–Back regime. For light atoms with fine structure of order $10^{-3}$ eV, the crossover field is tens of tesla; for heavier atoms with larger spin-orbit splitting, correspondingly larger fields are needed to reach Paschen–Back behavior.

Does the Zeeman effect apply to nuclear spins too, and is that relevant to NMR?

Yes — the same $-\boldsymbol\mu\cdot\mathbf B$ Hamiltonian applies to nuclear magnetic moments, simply replacing the Bohr magneton with the much smaller nuclear magneton $\mu_N = \mu_B/1836$ and the electronic $g$ -factor with a nucleus-specific nuclear $g$ -factor (5.586 for the proton). This is precisely the physical basis of nuclear magnetic resonance: nuclear spin sublevels split in an external field with spacing $g_N\mu_N B$ , and the resonant absorption of radiofrequency photons connecting adjacent sublevels is the NMR signal. Because $\mu_N$ is about 1836 times smaller than $\mu_B$ , achieving comparably large splittings in NMR requires far stronger fields (several tesla) than are needed for electronic (EPR) Zeeman splittings at similar detection sensitivity.

Why is the g-factor sometimes written as a fraction like 4/3 or 6/5 rather than an integer?

Because the Landé formula is a weighted combination of the pure-orbital value ( $g_L=1$ ) and the pure-spin value ( $g_S=2$ ), with weights fixed by how much of the total angular momentum $J$ is "made of" $L$ versus $S$ in that particular coupled state. Since $L(L+1)$ , $S(S+1)$ , and $J(J+1)$ are generally not simple multiples of each other, the resulting combination is a rational but non-integer number for most terms with both nonzero $L$ and nonzero $S$ . Only in the pure limits ( $S=0$ giving $g_J=1$ , or $L=0$ giving $g_J=2$ ) does the formula collapse to an integer.

What role does the electric-dipole selection rule Δm_J = 0, ±1 play in the observed splitting pattern?

It restricts which pairs of Zeeman sublevels (one in the upper term, one in the lower term) can be connected by an allowed radiative transition, and it also fixes the polarization of each resulting line: $\Delta m_J=0$ transitions are linearly polarized along the field ( $\pi$ lines, unobservable when viewed along the field axis), while $\Delta m_J=\pm1$ transitions are circularly polarized ( $\sigma^\pm$ lines) and are what one observes when viewing along the field direction (transverse vs. longitudinal Zeeman observation geometries actually reveal different subsets of lines, a classic diagnostic used historically to confirm the vector nature of the moment). Without this selection rule, far more lines than the observed three (normal) or six-plus (anomalous, per multiplet pair) would nominally connect the sublevels.

Can the Zeeman effect occur for a J=0 level?

No first-order splitting occurs. A $J=0$ level has only a single state, $m_J=0$ , so there is no degeneracy to lift and the Landé formula is formally singular (division by $J(J+1)=0$ ) but physically moot since the multiplicity is 1. Such levels (e.g. ${}^1S_0$ ground states of many-electron atoms with closed subshells, or ${}^3P_0$ ) are magnetically inert to first order; any residual field dependence appears only via second-order (diamagnetic or Van Vleck paramagnetic) corrections that couple $J=0$ to nearby $J\neq0$ levels through the field, an effect distinct from the Zeeman splitting discussed here.