Zeeman Effect

Why the Zeeman effect matters

Pieter Zeeman noticed in 1896 that the yellow sodium D-line, when its source was placed near a strong electromagnet, broadened in a way that could be resolved into discrete components. Lorentz immediately gave a classical explanation in terms of orbiting electrons; their joint Nobel followed in 1902. What looked like a curiosity of spectroscopy turned out to be the first quantitative window into the magnetic structure of atoms — and a century later it remains the workhorse for measuring magnetic fields anywhere photons can reach us.

• Astrophysical magnetometry. Practically every magnetic-field measurement in the universe — sunspots, stellar dynamos, accretion disks, white dwarfs, neutron-star atmospheres, the interstellar medium via 21 cm OH/H₂O masers — rests on Zeeman line splitting or its polarization signatures. Telescopes don't carry magnetometers; spectrographs do.
• NMR and MRI. The same physics, scaled to nuclei: nuclear Zeeman splitting of spin-½ protons in a 1.5 T MRI scanner is ΔE ≈ 60 MHz × h. Tissue contrast is a difference in proton relaxation times in that splitting; chemical shift NMR — the cornerstone of molecular structure determination — is a tiny perturbation on the nuclear Zeeman ladder.
• Atomic clocks. Cesium and rubidium clocks use hyperfine transitions that are exquisitely Zeeman-sensitive; clocks operate at "magic" field values where the second-order Zeeman shift cancels the first-order, yielding fractional stabilities below 10⁻¹⁵.
• Laser cooling and trapping. Magneto-optical traps (MOTs) work because Zeeman shifts make atoms preferentially absorb counter-propagating laser light — a position-dependent restoring force built directly out of ΔE = μ_B g_J m_J B in a quadrupole field.
• Quantum sensing. NV centers in diamond, atomic vapor magnetometers (SERF magnetometers reach 1 fT/√Hz), and SQUID alternatives all read out via Zeeman-shifted transitions. Brain magnetometry (MEG) and unexploded-ordnance detection both rely on this.

From classical Lorentz triplets to the Landé formula

Lorentz's classical model treated the radiating electron as a damped oscillator. In a uniform B-field, an electron's circular motion picks up a Larmor precession at frequency ω_L = eB/2m_e. The original line ν₀ splits into three: ν₀ (unshifted, π component, polarized parallel to B) and ν₀ ± ω_L/2π (shifted, σ⁺ and σ⁻, circularly polarized when viewed along B). This normal triplet is what Zeeman first observed in cadmium.

The anomalous patterns stayed unexplained for nearly thirty years. Sommerfeld and Landé built phenomenological formulas — the Landé g-factor was originally a fitted number — but its origin was opaque. Goudsmit and Uhlenbeck's 1925 hypothesis of electron spin (s = ½, g_s ≈ 2) closed the gap. The full splitting in the weak-field (LS-coupling) regime is ΔE = μ_B g_J m_J B, where g_J encodes the projection of the total magnetic moment onto J. Dirac's 1928 relativistic equation later predicted g_s = 2 exactly; QED corrections measured to twelve decimal places put g_s = 2.00231930436118…, the most precisely tested prediction in physics.

Common misconceptions

• "Zeeman always splits a line into three." Only the normal Zeeman effect, and only for transitions between singlet states (S = 0). Most atoms have unpaired spin and exhibit anomalous splittings — sodium's D₁ line splits into 4 components, D₂ into 6.
• "It's a tiny effect that needs huge magnets." At B = 1 T, the splitting is roughly 14 GHz — easily resolved by a standard grating spectrometer. NMR routinely operates at sub-tesla fields and resolves chemical shifts of parts per million.
• "You need strong B to see it." The Hanle effect detects fields of microgauss in stellar atmospheres via depolarization of scattered resonance lines. Atomic vapor magnetometers reach attotesla sensitivity in the lab, leveraging Zeeman precession in the Earth-field regime.
• "Lorentz's classical theory was wrong." It correctly predicts the normal Zeeman effect — that's why the 1902 Nobel was joint. It only fails when spin enters, which is unsurprising for an 1896 theory built before spin was conceived.
• "The anomalous Zeeman effect is rare." It's the rule, not the exception. The terminology survives for historical reasons.
• "Paschen-Back is just a stronger Zeeman." It's a different physical regime: L and S decouple from each other and each couples directly to B. The splitting pattern, polarization rules, and selection rules all change.

Selection rules and polarization

Zeeman transitions obey Δm_J ∈ {−1, 0, +1}. The Δm_J = 0 component (π) is linearly polarized parallel to B and absent when viewed along B. The Δm_J = ±1 components (σ⁺ and σ⁻) are circularly polarized when viewed along B, and linearly polarized perpendicular to B when viewed transversely. Solar spectroscopists exploit these rules constantly — circularly-polarized Stokes V profiles in absorption lines map the line-of-sight magnetic field, while linearly-polarized Stokes Q and U profiles map the transverse component.

Numbers worth keeping in your head

• μ_B = 9.274 × 10⁻²⁴ J/T, equivalently 1.4 MHz/G or 14 GHz/T.
• Sodium D-line splitting at 1 T: ~14 GHz (compare to D₁–D₂ fine-structure splitting of 516 GHz).
• Sunspot umbra: ~3000 G; quiet sun granulation: ~1–10 G; Earth's field: ~0.5 G.
• Neutron-star surface fields: 10⁸–10¹⁵ G (magnetars). Atomic structure breaks down above ~10⁹ G — atoms become cigar-shaped along B.
• Cesium clock magic field: ~3 G; second-order Zeeman shift coefficient ~427.45 Hz/T².