Short description: Spectral line splitting in a magnetic field

The Zeeman effect (nl) is the splitting of a spectral line into several components in the presence of a static magnetic field. It arises from the interaction of the external field with the magnetic moment of the atomic electron, associated with both its orbital angular momentum and spin. Because different magnetic sublevels shift by different amounts, a single spectral line can separate into several components. The effect is named after the Dutch physicist Pieter Zeeman, who discovered it in 1896 and later shared the 1902 Nobel Prize in Physics with Hendrik Lorentz.^[1]^[2]

The Zeeman effect is the magnetic analogue of the Stark effect, which is the splitting of spectral lines by an electric field. It is important both conceptually and practically: the spacing between Zeeman components depends on magnetic-field strength, so the effect is widely used to measure magnetic fields in atoms, plasmas, sunspots, and stars.^[3]

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Explanation of how the magnetic field on a star affects the light emitted

The spectral lines of a mercury vapor lamp at wavelength 546.1 nm, showing the anomalous Zeeman effect. (A) Without magnetic field. (B) With magnetic field, spectral lines split as transverse Zeeman effect. (C) With magnetic field, split as longitudinal Zeeman effect. The spectral lines were obtained using a Fabry–Pérot interferometer.

Zeeman splitting of the 5s level of ⁸⁷Rb, including fine structure and hyperfine structure splitting. Here $F = J + I$ , where $I$ is the nuclear spin (for ⁸⁷Rb, $I = \frac{3}{2}$ ).

Discovery

In 1896 Zeeman learned that his laboratory possessed one of Henry Augustus Rowland's highest-resolution diffraction gratings. Inspired by James Clerk Maxwell’s account of Michael Faraday’s unsuccessful attempts to influence light using magnetism, Zeeman asked whether improved spectroscopic methods might reveal such an effect.^[1]^: 75

Using a slit source and a flame containing sodium, he first observed a slight broadening of the sodium lines when a strong magnet was energized around the flame.^[1]^: 76 When he switched to cadmium, the line splitting became unmistakable. Lorentz showed that the effect could be interpreted in terms of his electron theory, and the discovery rapidly became one of the major empirical foundations of early electron physics.^[1]^: 77 In his Nobel lecture Zeeman described the apparatus and displayed the spectrographic images directly.^[2]

Normal and anomalous Zeeman effect

Historically, one distinguishes between the normal Zeeman effect and the anomalous Zeeman effect.^[4] The normal effect occurs when the total electron spin is zero and produces a simple triplet pattern. The anomalous effect occurs when the total spin is nonzero; it was called “anomalous” because the electron spin had not yet been discovered when the phenomenon was first studied.

Later quantum theory explained the anomalous effect through spin and the Landé g-factor. Wolfgang Pauli famously remarked on the difficulty of understanding it before the full quantum-mechanical framework was available.^[5]

At stronger magnetic fields the splitting is no longer linear, and eventually the atom enters the Paschen–Back effect regime, where spin–orbit coupling is effectively broken by the external field. In modern literature, the distinction between “normal” and “anomalous” is used less often, and authors typically speak simply of the Zeeman effect.

Theoretical presentation

The Hamiltonian of an atom in an external magnetic field can be written as

$H = H_{0} + V_{M},$

where $H_{0}$ is the unperturbed atomic Hamiltonian and

$V_{M} = - \vec{μ} \cdot \vec{B},$

is the perturbation due to the magnetic field. The atomic magnetic moment is dominated by the electronic contribution, so to a good approximation

$\vec{μ} \approx - \frac{μ_{B} g \vec{J}}{ℏ},$

where $μ_{B}$ is the Bohr magneton, $\vec{J}$ is the total electronic angular momentum, and $g$ is the Landé factor.

A more complete expression separates orbital and spin parts:

$\vec{μ} = - \frac{μ_{B} (g_{l} \vec{L} + g_{s} \vec{S})}{ℏ},$

with $g_{l} = 1$ and $g_{s} \approx 2.0023193$ . In weak fields the magnetic perturbation is smaller than the fine-structure interaction, so it can be treated perturbatively. This is the usual Zeeman regime. In stronger fields, the perturbation exceeds LS coupling and the Paschen–Back regime is reached.

Weak-field Zeeman effect

If spin–orbit coupling dominates over the external magnetic interaction, then $\vec{L}$ and $\vec{S}$ are not separately conserved; only the total angular momentum

$\vec{J} = \vec{L} + \vec{S}$

remains a good quantum number. The resulting first-order energy shift is

$E_{Z}^{(1)} = μ_{B} g_{J} B_{ext} m_{j},$

where $m_{j}$ is the magnetic quantum number and $g_{J}$ is the Landé factor. In LS coupling,

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

For a single electron above closed shells, with $s = \frac{1}{2}$ and $j = l \pm s$ , this simplifies to

$g_{J} = 1 \pm \frac{g_{S} - 1}{2 l + 1} .$

Example: Lyman-alpha transition in hydrogen

The Lyman-alpha transition in hydrogen involves the fine-structure transitions

$2^{2} P_{1 / 2} \to 1^{2} S_{1 / 2}$

and

$2^{2} P_{3 / 2} \to 1^{2} S_{1 / 2} .$

In a weak external magnetic field, the $1^{2} S_{1 / 2}$ and $2^{2} P_{1 / 2}$ levels split into two substates each, while the $2^{2} P_{3 / 2}$ level splits into four. Their Landé factors are

$\begin{aligned} g_{J} & = 2 & for 1^{2} S_{1 / 2}, \\ g_{J} & = \frac{2}{3} & for 2^{2} P_{1 / 2}, \\ g_{J} & = \frac{4}{3} & for 2^{2} P_{3 / 2} . \end{aligned}$

The different values of $g_{J}$ mean that the Zeeman splitting is not the same for all orbitals. The fine-structure splitting exists even without a magnetic field, but the magnetic field produces an additional splitting of each fine-structure level.

Dipole-allowed Lyman-alpha transitions in the weak-field regime
Initial state $(n = 2, l = 1)$ $\| j, m_{j} ⟩$	Final state $(n = 1, l = 0)$ $\| j, m_{j} ⟩$	Energy perturbation
$\| \frac{1}{2}, \pm \frac{1}{2} ⟩$	$\| \frac{1}{2}, \pm \frac{1}{2} ⟩$	$\mp \frac{2}{3} μ_{B} B$
$\| \frac{1}{2}, \pm \frac{1}{2} ⟩$	$\| \frac{1}{2}, \mp \frac{1}{2} ⟩$	$\pm \frac{4}{3} μ_{B} B$
$\| \frac{3}{2}, \pm \frac{3}{2} ⟩$	$\| \frac{1}{2}, \pm \frac{1}{2} ⟩$	$\pm μ_{B} B$
$\| \frac{3}{2}, \pm \frac{1}{2} ⟩$	$\| \frac{1}{2}, \pm \frac{1}{2} ⟩$	$\mp \frac{1}{3} μ_{B} B$
$\| \frac{3}{2}, \pm \frac{1}{2} ⟩$	$\| \frac{1}{2}, \mp \frac{1}{2} ⟩$	$\pm \frac{5}{3} μ_{B} B$

Strong-field limit: Paschen–Back effect

The Paschen–Back effect is the strong-field limit of Zeeman splitting.^[6] When the magnetic perturbation becomes much larger than the spin–orbit interaction, the orbital and spin angular momenta effectively decouple. Then $m_{l}$ and $m_{s}$ become the appropriate quantum numbers, and the energy is approximately

$E_{z} = E_{0} + B_{z} μ_{B} (m_{l} + g_{s} m_{s}) .$

Because electric-dipole selection rules require

$Δ s = 0, Δ m_{s} = 0, Δ l = \pm 1, Δ m_{l} = 0, \pm 1,$

the spectrum reduces to three principal components corresponding to $Δ m_{l} = 0, \pm 1$ . In hydrogen, finer corrections from residual spin–orbit coupling and relativity can still be included perturbatively.^[7]

Example: Lyman-alpha transition in hydrogen

Ignoring fine-structure corrections, the strong-field Lyman-alpha transitions may be listed as follows:

Dipole-allowed Lyman-alpha transitions in the strong-field regime
Initial state $(n = 2, l = 1)$ $\| m_{l}, m_{s} ⟩$	Initial energy perturbation	Final state $(n = 1, l = 0)$ $\| m_{l}, m_{s} ⟩$	Final energy perturbation
$\| 1, \frac{1}{2} ⟩$	$+ 2 μ_{B} B_{z}$	$\| 0, \frac{1}{2} ⟩$	$+ μ_{B} B_{z}$
$\| 0, \frac{1}{2} ⟩$	$+ μ_{B} B_{z}$	$\| 0, \frac{1}{2} ⟩$	$+ μ_{B} B_{z}$
$\| 1, - \frac{1}{2} ⟩$	$0$	$\| 0, - \frac{1}{2} ⟩$	$- μ_{B} B_{z}$
$\| - 1, \frac{1}{2} ⟩$	$0$	$\| 0, \frac{1}{2} ⟩$	$+ μ_{B} B_{z}$
$\| 0, - \frac{1}{2} ⟩$	$- μ_{B} B_{z}$	$\| 0, - \frac{1}{2} ⟩$	$- μ_{B} B_{z}$
$\| - 1, - \frac{1}{2} ⟩$	$- 2 μ_{B} B_{z}$	$\| 0, - \frac{1}{2} ⟩$	$- μ_{B} B_{z}$

Intermediate field and the Breit–Rabi formula

In atoms with hyperfine structure, both the hyperfine interaction and the Zeeman interaction must be considered. In the magnetic-dipole approximation,

$H = h A \vec{I} \cdot \vec{J} - \vec{μ} \cdot \vec{B}$

or equivalently

$H = h A \vec{I} \cdot \vec{J} + (μ_{B} g_{J} \vec{J} + μ_{N} g_{I} \vec{I}) \cdot \vec{B},$

where $A$ is the hyperfine constant, $μ_{N}$ is the nuclear magneton, and $g_{I}$ is the nuclear g-factor. For weak fields, the states are naturally described in the $| F, m_{F} ⟩$ basis; for strong fields, the uncoupled $| m_{I}, m_{J} ⟩$ basis is more appropriate.

For the important case $J = \frac{1}{2}$ , the Hamiltonian can be solved analytically, giving the Breit–Rabi formula.^[8]^[9] Writing

$x \equiv \frac{B (μ_{B} g_{J} - μ_{N} g_{I})}{h Δ W}, Δ W = A (I + \frac{1}{2}),$

the level shifts are

$Δ E_{F = I \pm 1 / 2} = - \frac{h Δ W}{2 (2 I + 1)} + μ_{N} g_{I} m_{F} B \pm \frac{h Δ W}{2} \sqrt{1 + \frac{2 m_{F} x}{I + 1 / 2} + x^{2}} .$

This formula is especially useful in alkali atoms and in atomic-clock physics.

Applications

Astrophysics

Zeeman effect on a sunspot spectral line

George Ellery Hale was the first to identify the Zeeman effect in solar spectra, thereby demonstrating strong magnetic fields in sunspots. Today the effect is used to produce magnetograms of the Sun and to infer magnetic-field geometries in stars.^[10]^[11]

Laser cooling

The Zeeman effect is fundamental in laser cooling, especially in the magneto-optical trap and the Zeeman slower.^[12]

Spintronics

Zeeman-energy-mediated coupling between spin and orbital motion is exploited in spintronics, for example in electric dipole spin resonance in quantum dots.^[13]

Metrology

Hyperfine transition-based atomic clocks can shift when exposed to magnetic fields. Such shifts are monitored and corrected through Zeeman measurements during clock calibration and degaussing procedures.^[14]

Biology

One proposed explanation of avian magnetoreception invokes magnetic effects on retinal proteins and has sometimes been discussed in relation to Zeeman-type energy shifts.^[15]

Demonstrations

A classroom demonstration can be made by placing a sodium vapor source in a powerful electromagnet and viewing a sodium lamp through the magnet opening. With the magnet off, sodium vapor absorbs the lamp light strongly; with the magnet on, the absorption line splits, changing the transmission and allowing more light to pass.^[16]

The vapor may be produced either in a sealed heated sodium tube or by introducing salt into a Bunsen burner flame. Care is required, because the magnetic field can also influence the flame itself; this complication was already relevant in Zeeman’s original work.^[16]^[17]^[18]

Table of contents (138 articles)

Index

Core theory

Foundations
Conceptual and interpretations
Mathematical structure and systems
Atomic and spectroscopy
Wavefunctions and modes
Quantum dynamics and evolution
Measurement and information
Quantum information and computing

Applications and extensions

Quantum optics and experiments
Open quantum systems
Quantum field theory
Statistical mechanics and kinetic theory
Condensed matter and solid-state physics
Plasma and fusion physics
Timeline
Advanced and frontier topics

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Physics:Quantum Zeeman effect

Discovery

Normal and anomalous Zeeman effect

Theoretical presentation

Weak-field Zeeman effect

Example: Lyman-alpha transition in hydrogen

Strong-field limit: Paschen–Back effect

Example: Lyman-alpha transition in hydrogen

Intermediate field and the Breit–Rabi formula

Applications

Astrophysics

Laser cooling

Spintronics

Metrology

Biology

Demonstrations

See also

Table of contents (138 articles)

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