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{{Short description|Spectral line splitting in a magnetic field}}

{{Quantum book backlink|Atomic and spectroscopy}}

The '''Zeeman effect''' ({{IPA|nl|ˈzeːmɑn|lang}}) is the splitting of a [[Spectral line|spectral line]] into several components in the presence of a static [[Magnetic field|magnetic field]]. It arises from the interaction of the external field with the magnetic moment of the atomic electron, associated with both its [[Physics:Angular momentum|orbital angular momentum]] and [[Spin|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 [[Biography:Pieter Zeeman|Pieter Zeeman]], who discovered it in 1896 and later shared the 1902 Nobel Prize in Physics with [[Biography:Hendrik Lorentz|Hendrik Lorentz]].<ref name=PaisInward>{{Cite book |last=Pais |first=Abraham |title=Inward bound: of matter and forces in the physical world |date=2002 |publisher=Clarendon Press [u.a.] |isbn=978-0-19-851997-3 |edition=Reprint |location=Oxford}}</ref><ref name="ZeemanNobel">{{Cite web |last=Pieter |first=Zeeman |date=1902 |title=Pieter Zeeman Nobel Lecture |url=https://www.nobelprize.org/prizes/physics/1902/zeeman/lecture/ |url-status=live |archive-url=https://web.archive.org/web/20181115204904/https://www.nobelprize.org/prizes/physics/1902/zeeman/lecture/ |archive-date=2018-11-15 |access-date=2024-03-01 |website=The Nobel Prize}}</ref>

The Zeeman effect is the magnetic analogue of the [[Physics:Stark effect|Stark effect]], which is the splitting of spectral lines by an [[Physics:Electric field|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.<ref>{{Cite journal |last=Schad |first=Thomas A. |last2=Petrie |first2=Gordon J.D. |last3=Kuhn |first3=Jeffrey R. |last4=Fehlmann |first4=Andre |last5=Rimmele |first5=Thomas |last6=Tritschler |first6=Alexandra |last7=Woeger |first7=Friedrich |last8=Scholl |first8=Isabelle |last9=Williams |first9=Rebecca |last10=Harrington |first10=David |last11=Paraschiv |first11=Alin R. |last12=Szente |first12=Judit |date=2024-09-13 |title=Mapping the Sun’s coronal magnetic field using the Zeeman effect |url=https://www.science.org/doi/10.1126/sciadv.adq1604 |journal=Science Advances |language=en |volume=10 |issue=37 |doi=10.1126/sciadv.adq1604 |issn=2375-2548 |pmc=11421591 |pmid=39259791}}</ref>
<div style="float:right; border:1px solid #ccc; padding:4px; background:#ffffe0; margin:0 0 1em 1em; width:250px; text-align:center;">

'''[https://freetocopy.co.nl/Quantum/Animations%20of%20atomic%20physics/Explanation_of_how_the_magnetic_field_on_a_star_affects_the_light_emitted.mp4 ▶ Play movie]'''
[[File:Zeeman-effect.jpg|250px]]
<div style="font-size:90%; margin-top:4px;">
Explanation of how the magnetic field on a star affects the light emitted
</div>
</div>

[[File:ZeemanEffectIllus.png|thumb|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]].]]
[[File:Breit-rabi-Zeeman-en.svg|thumb|420px|Zeeman splitting of the 5s level of [[Rubidium|{{sup|87}}Rb]], including fine structure and hyperfine structure splitting. Here <math>F=J+I</math>, where <math>I</math> is the nuclear spin (for {{sup|87}}Rb, <math>I=\frac{3}{2}</math>).]]

==Discovery==
In 1896 Zeeman learned that his laboratory possessed one of [[Biography:Henry Augustus Rowland|Henry Augustus Rowland]]'s highest-resolution [[Physics:Diffraction grating|diffraction gratings]]. Inspired by [[Biography:James Clerk Maxwell|James Clerk Maxwell]]’s account of [[Biography:Michael Faraday|Michael Faraday]]’s unsuccessful attempts to influence light using magnetism, Zeeman asked whether improved spectroscopic methods might reveal such an effect.<ref name=PaisInward/>{{rp|75}}

Using a slit source and a flame containing [[Chemistry:Sodium|sodium]], he first observed a slight broadening of the sodium lines when a strong magnet was energized around the flame.<ref name=PaisInward/>{{rp|76}} When he switched to [[Chemistry:Cadmium|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.<ref name=PaisInward/>{{rp|77}} In his Nobel lecture Zeeman described the apparatus and displayed the spectrographic images directly.<ref name="ZeemanNobel" />

==Normal and anomalous Zeeman effect==
Historically, one distinguishes between the '''normal''' Zeeman effect and the '''anomalous''' Zeeman effect.<ref>{{cite journal |last1=Preston |first1=Thomas |title=Radiation phenomena in a strong magnetic field |journal=The Scientific Transactions of the Royal Dublin Society |date=1898 |volume=6 |pages=385–391 |url=https://babel.hathitrust.org/cgi/pt?id=mdp.39015035446916;view=1up;seq=481 |series=2nd series}}</ref> 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 [[Physics:Landé g-factor|Landé g-factor]]. [[Biography:Wolfgang Pauli|Wolfgang Pauli]] famously remarked on the difficulty of understanding it before the full quantum-mechanical framework was available.<ref>"Niels Bohr's Times: In Physics, Philosophy, and Polity" By Abraham Pais, page 201.</ref>

At stronger magnetic fields the splitting is no longer linear, and eventually the atom enters the [[Physics:Paschen–Back effect|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

<math display="block">
H = H_0 + V_\text{M},
</math>

where <math>H_0</math> is the unperturbed atomic Hamiltonian and

<math display="block">
V_\text{M} = -\vec{\mu}\cdot\vec{B},
</math>

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

<math display="block">
\vec{\mu} \approx -\frac{\mu_\text{B} g \vec{J}}{\hbar},
</math>

where <math>\mu_\text{B}</math> is the [[Physics:Bohr magneton|Bohr magneton]], <math>\vec{J}</math> is the total electronic angular momentum, and <math>g</math> is the Landé factor.

A more complete expression separates orbital and spin parts:

<math display="block">
\vec{\mu} = -\frac{\mu_\text{B}(g_l\vec{L}+g_s\vec{S})}{\hbar},
</math>

with <math>g_l=1</math> and <math>g_s\approx 2.0023193</math>. 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 <math>\vec{L}</math> and <math>\vec{S}</math> are not separately conserved; only the total angular momentum

<math display="block">
\vec{J}=\vec{L}+\vec{S}
</math>

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

<math display="block">
E_\text{Z}^{(1)}=\mu_\text{B}g_JB_\text{ext}m_j,
</math>

where <math>m_j</math> is the magnetic quantum number and <math>g_J</math> is the Landé factor. In LS coupling,

<math display="block">
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)}.
</math>

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

<math display="block">
g_J = 1 \pm \frac{g_S-1}{2l+1}.
</math>

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

<math display="block">
2\,^2\!P_{1/2}\to 1\,^2\!S_{1/2}
</math>

and

<math display="block">
2\,^2\!P_{3/2}\to 1\,^2\!S_{1/2}.
</math>

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

<math display="block">
\begin{align}
g_J &= 2 && \text{for } 1\,^2\!S_{1/2},\\
g_J &= \frac{2}{3} && \text{for } 2\,^2\!P_{1/2},\\
g_J &= \frac{4}{3} && \text{for } 2\,^2\!P_{3/2}.
\end{align}
</math>

[[Image:Zeeman p s doublet.svg|right|300px]]

The different values of <math>g_J</math> 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.

{| class="wikitable" style="text-align:center"
|+ Dipole-allowed Lyman-alpha transitions in the weak-field regime
! Initial state<br /><math>(n=2,l=1)</math><br /><math>|j,m_j\rangle</math>
! Final state<br /><math>(n=1,l=0)</math><br /><math>|j,m_j\rangle</math>
! Energy perturbation
|-
| <math>\left|\frac{1}{2},\pm\frac{1}{2}\right\rangle</math>
| <math>\left|\frac{1}{2},\pm\frac{1}{2}\right\rangle</math>
| <math>\mp\frac{2}{3}\mu_\text{B}B</math>
|-
| <math>\left|\frac{1}{2},\pm\frac{1}{2}\right\rangle</math>
| <math>\left|\frac{1}{2},\mp\frac{1}{2}\right\rangle</math>
| <math>\pm\frac{4}{3}\mu_\text{B}B</math>
|-
| <math>\left|\frac{3}{2},\pm\frac{3}{2}\right\rangle</math>
| <math>\left|\frac{1}{2},\pm\frac{1}{2}\right\rangle</math>
| <math>\pm\mu_\text{B}B</math>
|-
| <math>\left|\frac{3}{2},\pm\frac{1}{2}\right\rangle</math>
| <math>\left|\frac{1}{2},\pm\frac{1}{2}\right\rangle</math>
| <math>\mp\frac{1}{3}\mu_\text{B}B</math>
|-
| <math>\left|\frac{3}{2},\pm\frac{1}{2}\right\rangle</math>
| <math>\left|\frac{1}{2},\mp\frac{1}{2}\right\rangle</math>
| <math>\pm\frac{5}{3}\mu_\text{B}B</math>
|}

==Strong-field limit: Paschen–Back effect==
The [[Physics:Paschen–Back effect|Paschen–Back effect]] is the strong-field limit of Zeeman splitting.<ref>{{cite journal |last1=Paschen |first1=F. |last2=Back |first2=E. |title=Liniengruppen magnetisch vervollständigt |journal=Physica |date=1921 |volume=1 |pages=261–273 |trans-title=Line groups magnetically completed [i.e., completely resolved] |language=German}} Available at: [https://www.lorentz.leidenuniv.nl/history/proefschriften/Physica/Physica_1_1921_05391.pdf Leiden University (Netherlands)]</ref> When the magnetic perturbation becomes much larger than the spin–orbit interaction, the orbital and spin angular momenta effectively decouple. Then <math>m_l</math> and <math>m_s</math> become the appropriate quantum numbers, and the energy is approximately

<math display="block">
E_z = E_0 + B_z\mu_\text{B}(m_l+g_sm_s).
</math>

Because electric-dipole selection rules require

<math display="block">
\Delta s=0,\quad \Delta m_s=0,\quad \Delta l=\pm 1,\quad \Delta m_l=0,\pm 1,
</math>

the spectrum reduces to three principal components corresponding to <math>\Delta m_l=0,\pm 1</math>. In hydrogen, finer corrections from residual spin–orbit coupling and relativity can still be included perturbatively.<ref>{{cite book |author=Griffiths, David J. |title=Introduction to Quantum Mechanics |date=2004 |publisher=Prentice Hall |isbn=0-13-111892-7 |edition=2nd |page=280 |oclc=40251748}}</ref>

===Example: Lyman-alpha transition in hydrogen===
Ignoring fine-structure corrections, the strong-field Lyman-alpha transitions may be listed as follows:

{| class="wikitable"
|+ Dipole-allowed Lyman-alpha transitions in the strong-field regime
! Initial state<br /><math>(n=2,l=1)</math><br /><math>|m_l,m_s\rangle</math>
! Initial energy perturbation
! Final state<br /><math>(n=1,l=0)</math><br /><math>|m_l,m_s\rangle</math>
! Final energy perturbation
|-
| <math>\left|1,\frac{1}{2}\right\rangle</math>
| <math>+2\mu_\text{B}B_z</math>
| <math>\left|0,\frac{1}{2}\right\rangle</math>
| <math>+\mu_\text{B}B_z</math>
|-
| <math>\left|0,\frac{1}{2}\right\rangle</math>
| <math>+\mu_\text{B}B_z</math>
| <math>\left|0,\frac{1}{2}\right\rangle</math>
| <math>+\mu_\text{B}B_z</math>
|-
| <math>\left|1,-\frac{1}{2}\right\rangle</math>
| <math>0</math>
| <math>\left|0,-\frac{1}{2}\right\rangle</math>
| <math>-\mu_\text{B}B_z</math>
|-
| <math>\left|-1,\frac{1}{2}\right\rangle</math>
| <math>0</math>
| <math>\left|0,\frac{1}{2}\right\rangle</math>
| <math>+\mu_\text{B}B_z</math>
|-
| <math>\left|0,-\frac{1}{2}\right\rangle</math>
| <math>-\mu_\text{B}B_z</math>
| <math>\left|0,-\frac{1}{2}\right\rangle</math>
| <math>-\mu_\text{B}B_z</math>
|-
| <math>\left|-1,-\frac{1}{2}\right\rangle</math>
| <math>-2\mu_\text{B}B_z</math>
| <math>\left|0,-\frac{1}{2}\right\rangle</math>
| <math>-\mu_\text{B}B_z</math>
|}

==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,

<math display="block">
H = hA\vec{I}\cdot\vec{J} - \vec{\mu}\cdot\vec{B}
</math>

or equivalently

<math display="block">
H = hA\vec{I}\cdot\vec{J} + (\mu_\text{B}g_J\vec{J}+\mu_\text{N}g_I\vec{I})\cdot\vec{B},
</math>

where <math>A</math> is the hyperfine constant, <math>\mu_\text{N}</math> is the [[Physics:Nuclear magneton|nuclear magneton]], and <math>g_I</math> is the nuclear g-factor. For weak fields, the states are naturally described in the <math>|F,m_F\rangle</math> basis; for strong fields, the uncoupled <math>|m_I,m_J\rangle</math> basis is more appropriate.

For the important case <math>J=\frac{1}{2}</math>, the Hamiltonian can be solved analytically, giving the '''Breit–Rabi formula'''.<ref>{{cite book |last1=Woodgate |first1=Gordon Kemble |title=Elementary Atomic Structure |date=1980 |publisher=Oxford University Press |location=Oxford, England |pages=193–194 |edition=2nd}}</ref><ref>{{cite journal |last1=Breit |first1=G. |last2=Rabi |first2=I.I. |title=Measurement of nuclear spin |journal=Physical Review |date=1931 |volume=38 |issue=11 |pages=2082–2083 |doi=10.1103/PhysRev.38.2082.2 |bibcode=1931PhRv...38.2082B }}</ref> Writing

<math display="block">
x \equiv \frac{B(\mu_\text{B}g_J-\mu_\text{N}g_I)}{h\Delta W},
\qquad
\Delta W = A\left(I+\frac{1}{2}\right),
</math>

the level shifts are

<math display="block">
\Delta E_{F=I\pm 1/2}
=
-\frac{h\Delta W}{2(2I+1)}
+\mu_\text{N}g_I m_F B
\pm
\frac{h\Delta W}{2}
\sqrt{1+\frac{2m_F x}{I+1/2}+x^2}.
</math>

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

==Applications==

===Astrophysics===
[[File:Sunzeeman1919.png|thumb|right|200px|Zeeman effect on a sunspot spectral line]]
[[Biography:George Ellery Hale|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 [[Physics:Solar magnetogram|magnetograms]] of the Sun and to infer magnetic-field geometries in stars.<ref>{{Cite journal |last=Schad |first=Thomas A. |last2=Petrie |first2=Gordon J.D. |last3=Kuhn |first3=Jeffrey R. |last4=Fehlmann |first4=Andre |last5=Rimmele |first5=Thomas |last6=Tritschler |first6=Alexandra |last7=Woeger |first7=Friedrich |last8=Scholl |first8=Isabelle |last9=Williams |first9=Rebecca |last10=Harrington |first10=David |last11=Paraschiv |first11=Alin R. |last12=Szente |first12=Judit |date=2024-09-13 |title=Mapping the Sun’s coronal magnetic field using the Zeeman effect |url=https://www.science.org/doi/10.1126/sciadv.adq1604 |journal=Science Advances |language=en |volume=10 |issue=37 |doi=10.1126/sciadv.adq1604 |issn=2375-2548 |pmc=11421591 |pmid=39259791}}</ref><ref>{{Cite journal |last=Kochukhov |first=Oleg |date=December 2021 |title=Magnetic fields of M dwarfs |journal=The Astronomy and Astrophysics Review |volume=29 |issue=1 |pages=1 |doi=10.1007/s00159-020-00130-3 |arxiv=2011.01781 |issn=0935-4956}}</ref>

===Laser cooling===
The Zeeman effect is fundamental in [[Physics:Laser cooling|laser cooling]], especially in the [[Physics:Magneto-optical trap|magneto-optical trap]] and the [[Physics:Zeeman slower|Zeeman slower]].<ref>{{Cite journal |last=Bowden |first=William |last2=Gunton |first2=Will |last3=Semczuk |first3=Mariusz |last4=Dare |first4=Kahan |last5=Madison |first5=Kirk W. |date=2016-04-18 |title=An adaptable dual species effusive source and Zeeman slower design demonstrated with Rb and Li |url=https://pubs.aip.org/aip/rsi/article-abstract/87/4/043111/361124/An-adaptable-dual-species-effusive-source-and?redirectedFrom=fulltext |journal=Review of Scientific Instruments |volume=87 |issue=4 |pages=043111 |doi=10.1063/1.4945567 |issn=0034-6748 |arxiv=1509.07460 }}</ref>

===Spintronics===
Zeeman-energy-mediated coupling between spin and orbital motion is exploited in [[Spintronics|spintronics]], for example in electric dipole spin resonance in quantum dots.<ref>Y. Tokura, W. G. van der Wiel, T. Obata, and S. Tarucha, Coherent single electron spin control in a slanting Zeeman field, Phys. Rev. Lett. '''96''', 047202 (2006)</ref>

===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.<ref>{{cite AV media |people=Verdiell, Marc (CuriousMarc) |date=October 31, 2022 |title=How an Atomic Clock Really Works, Round 2: Zeeman Alignment |type=YouTube video |language=English |url=https://www.youtube.com/watch?v=xTy1kY_wtsY |access-date=March 11, 2023}}</ref>

===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.<ref>{{cite journal |last1=Thalau |first1=Peter |last2=Ritz |first2=Thorsten |last3=Burda |first3=Hynek |last4=Wegner |first4=Regina E. |last5=Wiltschko |first5=Roswitha |title=The magnetic compass mechanisms of birds and rodents are based on different physical principles |journal=Journal of the Royal Society Interface |date=18 April 2006 |volume=3 |issue=9 |pages=583–587 |pmc=1664646 |doi=10.1098/rsif.2006.0130 |pmid=16849254 }}</ref>

==Demonstrations==
[[File:Zeeman effect demo.svg|thumb|Diagram of a Zeeman effect demonstration]]
A classroom demonstration can be made by placing a sodium vapor source in a powerful [[Physics:Electromagnet|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.<ref name=demo_followup>{{Citation |title=Candle flame is repelled by magnets (and Zeeman follow-up) |url=https://youtube.com/watch/JV4Fk3VNZqs?si=U20jHpiTGt0G71pu |access-date=2024-02-27 |language=en}}</ref>

The vapor may be produced either in a sealed heated sodium tube or by introducing [[Chemistry:Sodium chloride|salt]] into a [[Organization:Bunsen burner|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.<ref name=demo_followup/><ref>{{Cite journal |last=Kox |first=A J |date=1997-05-01 |title=The discovery of the electron: II. The Zeeman effect |url=https://iopscience.iop.org/article/10.1088/0143-0807/18/3/003 |journal=European Journal of Physics |volume=18 |issue=3 |pages=139–144 |doi=10.1088/0143-0807/18/3/003 |bibcode=1997EJPh...18..139K |s2cid=53414643 |issn=0143-0807 |url-access=subscription }}</ref><ref>{{Cite journal |last1=Suzuki |first1=Masatsugu Sei |last2=Suzuki |first2=Itsuko S. |date=2011 |title=Lecture Note on Senior Laboratory Zeeman effect in Na, Cd, and Hg |url=https://www.researchgate.net/publication/269929968 |journal=ResearchGate}}</ref>

==See also==
{{#invoke:PhysicsQC|tocHeadingAndList|Physics:Quantum basics/See also}}

==References==
{{reflist|3}}

{{Author|Harold Foppele}}
{{Sourceattribution|Physics:Quantum Zeeman effect|1}}

Physics:Quantum Zeeman effect - Revision history

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