The Zeeman Effect

An external magnetic field breaks the rotational symmetry that makes the $(2j+1)$ orientations of an atomic level degenerate. The field selects an axis, and the energy of a state comes to depend on how its angular momentum projects onto that axis. The resulting splitting of spectral lines, discovered by Zeeman in 1896, was the first laboratory access to the internal magnetic structure of the atom, and the pattern of the split lines encodes the coupling of spin and orbital motion more directly than any zero-field measurement. This lesson treats the weak-field regime, where the Zeeman energy is small compared with the fine-structure splitting; the strong-field and intermediate cases are the subject of the next lesson.

The atomic magnetic moment

A charged particle in a bound orbit carries a magnetic moment proportional to its angular momentum. For the orbital motion of an electron of charge $-e$ and mass $m_e$, the classical gyromagnetic relation gives

with orbital $g$-factor $g_L=1$ and the Bohr magneton ${\mu }_B$ setting the scale of atomic magnetism. Its CODATA value is ${\mu }_B=9.2740\times 10^{-24}{\text{J T}}^{-1}=5.7884\times 10^{-5}{\text{eV T}}^{-1}$,¹ so a one-tesla field shifts levels by tens of microelectronvolts, or equivalently ${\mu }_B/h=13.996{\text{GHz T}}^{-1}$.

Spin contributes a moment that is anomalously large by a factor $g_s$:

The value $g_s\approx 2$ is the prediction of the Dirac equation; the small excess is the QED anomalous moment. Combining the two contributions, and taking $g_s=2$ for now,

The factor of two on the spin term is the entire source of the anomalous Zeeman effect: were $g_s=1$, the moment would be proportional to $\vec{L}+\vec{S}=\vec{J}$ and every level would split with the same, classical, spacing.

The Zeeman Hamiltonian from minimal coupling

The interaction is not postulated but follows from replacing the canonical momentum with the gauge-covariant one, $\vec{p}\to \vec{p}+e\vec{A}$, in the kinetic energy. For a uniform field $\vec{B}=B\hat{z}$ the symmetric-gauge vector potential is $\vec{A}=\frac{1}{2}\vec{B}\times \vec{r}$, and

The cross term reproduces the orbital moment coupling $-{\vec{\mu }}_L\cdot \vec{B}$. Adding the spin moment coupling $-{\vec{\mu }}_S\cdot \vec{B}$ by hand (it emerges automatically only from the Dirac equation) gives the paramagnetic Zeeman Hamiltonian

The final rearrangement, using $\vec{J}=\vec{L}+\vec{S}$, isolates the piece that is diagonal in $J_z$ (trivial) from the piece proportional to $S_z$ (the source of all the structure).

Comparing the two field-dependent terms, the paramagnetic term is linear in $B$ and the diamagnetic term quadratic; their ratio is $H_{\text{dia}}/H_Z\sim {\mu }_{B}B⟨r^2⟩/(2{\hslash }^2/m_e)$, negligible until $B$ reaches $10^5\text{T}$ for ground states.

Weak field: the anomalous Zeeman effect

When $H_Z$ is small compared with the fine-structure splitting, the good zero-order states are the fine-structure eigenstates $|n\ell sj{m}_j⟩$, in which $\vec{L}$ and $\vec{S}$ are locked into a fixed $\vec{J}$ that precesses slowly about the field. First-order perturbation theory gives the shift as the expectation value of $H_Z$ in these states,

The first term is elementary; the second requires $⟨S_z⟩$ inside a state of definite $j$, where $\vec{S}$ is not separately conserved. The projection theorem (a corollary of the Wigner-Eckart theorem for vector operators) supplies it: within a manifold of fixed $j$, every vector operator $\vec{V}$ has the same matrix elements as its projection onto $\vec{J}$,

The physical content is the vector model: $\vec{S}$ precesses rapidly about $\vec{J}$, so only its component along $\vec{J}$ survives the time average, and that surviving component points along $\vec{J}$ with weight $(\vec{S}\cdot \vec{J})/J^2$.

To evaluate $\vec{S}\cdot \vec{J}$, write $\vec{L}=\vec{J}-\vec{S}$ and square: $L^2=J^2+S^2-2\vec{S}\cdot \vec{J}$, so

Then $⟨S_z⟩=\frac{⟨\vec{S}\cdot \vec{J}⟩}{j(j+1){\hslash }^2}{m}_j\hslash$, and the shift takes the form $E_Z^{(1)}=g_J{\mu }_{B}B{m}_j$ with the Landé g-factor

for $g_s=2$; keeping the exact $g_s$ replaces the $1$ by $\frac{1}{2}(1+g_s)$ and scales the second term correspondingly.²

The equal spacing is the signature of the weak-field regime: the field enters only through $J_z$, whose eigenvalues are equally spaced, and the coefficient $g_J$ is a pure number fixed by the angular-momentum quantum numbers.

Worked Landé factors for the sodium doublet

The sodium D lines connect the ground term $3s{}^2{S}_{1/2}$ to the two fine-structure components $3p{}^2{P}_{1/2}$ and $3p{}^2{P}_{3/2}$. Their Landé factors follow directly:

Because the three factors differ, the split sublevels of the upper and lower terms are not uniformly spaced relative to one another, and the transition resolves into more than three lines. This is the anomalous Zeeman pattern, which puzzled spectroscopists for three decades until spin supplied the extra factor of two.

The normal Zeeman triplet

When both the upper and lower levels have $g_J=1$, which happens for pure singlet states ($s=0$, so $j=\ell$ and the Landé formula collapses to unity), the sublevel spacing is ${\mu }_{B}B$ in every level. The dipole selection rule $\Delta m_j=0,\pm 1$ then produces transition frequencies

so the line splits into exactly three components at ${\nu }_0$ and ${\nu }_0\pm {\mu }_{B}B/h$, regardless of the individual $m_j$ values. The shift in wavenumbers is the Lorentz unit ${\mu }_{B}B/hc=0.4669{\text{cm}}^{-1}{\text{T}}^{-1}$. This is the normal Zeeman effect, and it reproduces exactly the classical Lorentz calculation of an electron oscillator in a field, in which the field splits the oscillation into one unshifted linear mode and two counter-rotating circular modes.

Polarization and the transverse/longitudinal patterns

The three families of $\Delta m_j$ carry distinct polarizations, fixed by the angular momentum the emitted photon must remove.

• $\pi$ components ($\Delta m_j=0$). The dipole operator component is $z$, the emission is linearly polarized parallel to $\vec{B}$, and it vanishes when the atom is viewed along the field (a dipole radiates nothing along its own axis).
• ${\sigma }^{\pm }$ components ($\Delta m_j=\pm 1$). The dipole operator components are $x\pm iy$; the photon carries $\pm \hslash$ of angular momentum along $\vec{B}$. Viewed along the field (longitudinal) these appear circularly polarized in opposite senses; viewed across the field (transverse) they appear linearly polarized perpendicular to $\vec{B}$.

The selection rules and polarizations are not independent additions: both follow from the same dipole matrix element $⟨f|\vec{r}|i⟩$ decomposed into spherical components, the $q=0$ component driving $\pi$ and the $q=\pm 1$ components driving ${\sigma }^{\pm }$. A measurement of the polarization of each Zeeman component therefore fixes the sign of $\Delta m_j$ and, through the line positions, the Landé factors of both levels.

Magnitude and regime of validity

The whole treatment rests on $H_Z\ll H_{\text{fs}}$. Fine-structure splittings run from $\sim 10^{-4}\text{eV}$ in hydrogen $n=2$ to $\sim 2\times 10^{-3}\text{eV}$ for the sodium D doublet, while ${\mu }_{B}B$ reaches $10^{-4}\text{eV}$ only near $B\approx 2\text{T}$. Weak-field theory therefore holds for the sodium D lines up to a few tesla but fails much earlier for the small hydrogen fine structure, where the Paschen-Back regime is reached in ordinary laboratory fields. The crossover, and the diagonalization that bridges the two limits, is developed next.