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{{Short description|Spin of an electron}}

In [[atomic physics]], the '''electron magnetic moment''', or more specifically the '''electron magnetic dipole moment''', is the [[magnetic moment]] of an [[electron]] resulting from its intrinsic properties of [[Spin (physics)|spin]] and [[electric charge]].  The value of the electron magnetic moment (symbol ''μ''<sub>e</sub>) is {{physconst|mue|after=.}} In units of the [[Bohr magneton]] (''μ''<sub>B</sub>), it is {{val|-1.00115965218059|(13)|u=''μ''<sub>B</sub>}},<ref name=":0">{{Cite journal |last=Fan |first=X. |last2=Myers |first2=T. G. |last3=Sukra |first3=B. A. D. |last4=Gabrielse |first4=G. |date=2023-02-13 |title=Measurement of the Electron Magnetic Moment |url=https://link.aps.org/doi/10.1103/PhysRevLett.130.071801 |journal=Physical Review Letters |volume=130 |issue=7 |pages=071801 |doi=10.1103/PhysRevLett.130.071801|arxiv=2209.13084 }}</ref> a value that was measured with a relative accuracy of {{val|1.3|e=-13}}.

== Magnetic moment of an electron ==
The electron is a [[charged particle]] with charge −{{mvar|e}}, where {{mvar|e}} is the [[elementary charge|unit of elementary charge]]. Its [[angular momentum]] comes from two types of rotation: [[Spin (physics)|spin]] and [[orbital motion]]. From [[classical electrodynamics]], a rotating distribution of [[electric charge]] produces a [[magnetic dipole]], so that it behaves like a tiny [[bar magnet]]. One consequence is that an external [[magnetic field]] exerts a [[Magnetic moment#Torque on a moment|torque]] on the electron [[magnetic moment]] that depends on the orientation of this dipole with respect to the field.

If the electron is visualized as a classical rigid body in which the mass and charge have identical distribution and motion that is rotating about an axis with [[angular momentum]] {{math|'''L'''}}, its magnetic dipole moment {{mvar|'''μ'''}} is given by:
<math display="block">\boldsymbol{\mu} = \frac{-e}{2m_\text{e}}\,\mathbf{L}\,,</math>
where {{mvar|m}}<sub>e</sub> is the [[electron]] [[rest mass]]. The [[angular momentum]] '''L''' in this equation may be the spin angular momentum, the orbital angular momentum, or the total angular momentum. The ratio between the true [[spin magnetic moment]] and that predicted by this model is a [[dimensionless quantity|dimensionless]] factor {{math|''g''<sub>e</sub>}}, known as the electron [[g-factor (physics)|{{math|''g''}}-factor]]:
<math display="block">\boldsymbol{\mu} = g_\text{e}\,\frac{(-e)}{~2m_\text{e}~}\,\mathbf{L}\,.</math>

It is usual to express the magnetic moment in terms of the [[reduced Planck constant]] {{mvar|ħ}} and the [[Bohr magneton]] {{mvar|μ}}<sub>B</sub>:
<math display="block">\boldsymbol{\mu} = -g_\text{e}\,\mu_\text{B}\,\frac{\mathbf{L}}{\hbar}\,.</math>

Since the [[Bohr magneton|magnetic moment is quantized]] in units of {{mvar|μ}}<sub>B</sub>, correspondingly the [[Angular momentum quantum number|angular momentum is quantized]] in units of {{mvar|ħ}}.

=== Formal definition ===
Classical notions such as the center of charge and mass are, however,  hard to make precise for a quantum elementary particle. In practice the  definition used by experimentalists comes from the [[Form factor (quantum field theory)|form factors]] <math>F_i(q^2)</math> 
appearing in the matrix element 
<math display="block">
 \langle p_f | j^\mu | p_i \rangle = \bar u(p_f) \left[ F_1(q^2) \gamma^\mu +\frac{~i \sigma^{\mu\nu}~}{~2\,m_\text{e}~} q_\nu F_2(q^2) + i\epsilon^{\mu\nu\rho\sigma} \sigma_{\rho\sigma} q_\nu F_3(q^2) + \frac{1}{~2\,m_\text{e}~}\left(q^\mu-\frac{q^2}{2m} \gamma^\mu \right)\gamma_5 F_4(q^2) \right] u(p_i)
</math>
of the electromagnetic current operator between two on-shell states. Here <math> u(p_i)</math> and <math> \bar u(p_f)</math> are 4-spinor  solution of the [[Dirac equation]] normalized so that <math> \bar u u = 2m_\text{e}</math>, and <math>q^\mu=p^\mu_f-p^\mu_i</math> is the momentum transfer from the current to the electron. The <math> q^2 = 0</math> form factor <math> F_1(0) = -e</math> is the electron's charge, <math display="inline"> \mu = \frac{1}{2m_\text{e}}(F_1(0)+F_2(0))</math> is its static magnetic dipole moment, and <math display="inline"> -\frac{1}{2m_\text{e}} F_3(0)</math> provides the formal definion of the [[electron electric dipole moment|electron's electric dipole moment]]. The remaining form factor <math>F_4(q^2)</math> would, if non zero, be the [[anapole moment]].

=== Spin magnetic dipole moment ===
The [[spin magnetic moment]] is intrinsic for an electron.<ref>{{cite book |first1=A. |last1=Mahajan |first2=A. |last2=Rangwala |url=https://books.google.com/books?id=_tXrjggX7WwC&q=%22intrinsic+dipole+moment%22+and+electron+%22Bohr+magneton%22&pg=PA419 |title=Electricity and Magnetism |page=419 |year=1989 | isbn=9780074602256 }}</ref> It is
<math display="block">\boldsymbol{\mu}_\text{s} = -g_\text{s}\,\mu_\text{B}\,\frac{~\mathbf{S}~}{\hbar}\,.</math>

Here {{math|'''S'''}} is the electron spin angular momentum. The spin [[g-factor (physics)|{{mvar|g}}-factor]] is approximately two: <math>g_\text{s} \approx 2</math>. The factor of two indicates that the electron appears to be twice as effective in producing a magnetic moment as a charged body for which the mass and charge distributions are identical.

The spin magnetic dipole moment is approximately one {{mvar|μ}}<sub>B</sub> because <math>g_\text{s} \approx 2</math> and the electron is a spin-{{sfrac|1|2}} particle ({{nowrap|{{mvar|S}} {{=}} {{sfrac|{{mvar|ħ}}|2}}}}):

The {{mvar|z}} component of the electron magnetic moment is
<math display="block">(\boldsymbol{\mu}_\text{s})_z = -g_\text{s}\,\mu_\text{B}\,m_\text{s}\,,</math>
where {{mvar|m}}<sub>s</sub> is the [[spin quantum number]]. Note that {{math|'''μ'''}} is a ''negative'' constant multiplied by the [[Spin (physics)|spin]], so the magnetic moment is ''[[Antiparallel vectors|antiparallel]]'' to the spin angular momentum.

The spin [[g-factor (physics)|''g''-factor]] {{nowrap|{{mvar|g}}<sub>s</sub> {{=}} 2}} comes from the [[Dirac equation]], a fundamental equation connecting the electron's spin with its electromagnetic properties. Reduction of the Dirac equation for an electron in a magnetic field to its non-relativistic limit yields the Schrödinger equation with a correction term, which takes account of the interaction of the electron's intrinsic magnetic moment with the magnetic field giving the correct energy.

For the electron spin, the most accurate value for the spin [[g-factor (physics)|{{mvar|g}}-factor]] has been experimentally determined to have the value
{{block indent | em = 1.5 | text = {{physconst|ge|after=.}}}}

Note that this differs only marginally from the value from the Dirac equation. The small correction is known as the [[anomalous magnetic dipole moment]] of the electron; it arises from the electron's interaction with virtual photons in [[quantum electrodynamics]]. A triumph of the [[quantum electrodynamics]] theory is the accurate prediction of the electron ''g''-factor. The CODATA value for the electron magnetic moment is
{{block indent | em = 1.5 | text = {{physconst|mue|after=.}}}}

=== Orbital magnetic dipole moment ===
The revolution of an electron around an axis through another object, such as the nucleus, gives rise to the orbital magnetic dipole moment. Suppose that the angular momentum for the orbital motion is {{math|'''L'''}}. Then the orbital magnetic dipole moment is
<math display="block">\boldsymbol{\mu}_L = -g_\text{L}\,\mu_\text{B}\,\frac{~\mathbf{L}~}{\hbar}\,.</math>

Here {{mvar|g}}<sub>L</sub> is the electron orbital [[g-factor (physics)|{{mvar|g}}-factor]] and {{mvar|μ}}<sub>B</sub> is the [[Bohr magneton]]. The value of {{mvar|g}}<sub>L</sub> is exactly equal to one, by a quantum-mechanical argument analogous to the derivation of the [[gyromagnetic ratio|classical gyromagnetic ratio]].

=== Total magnetic dipole moment ===
The total [[magnetic dipole moment]] resulting from both spin and orbital angular momenta of an electron is related to the total angular momentum {{math|'''J'''}} by a similar equation:
<math display="block">\boldsymbol{\mu}_\text{J} = -g_\text{J}\,\mu_\text{B}\,\frac{~\mathbf{J}~}{\hbar}\,.</math>

The [[g-factor (physics)|{{mvar|g}}-factor]] {{mvar|g}}<sub>J</sub> is known as the [[Landé g-factor|Landé ''g''-factor]], which can be related to {{mvar|g}}<sub>L</sub> and {{mvar|g}}<sub>S</sub> by quantum mechanics. See [[Landé g-factor|Landé ''g''-factor]] for details.

== Example: hydrogen atom ==
For a [[hydrogen]] atom, an [[electron]] occupying the [[atomic orbital]] {{math|Ψ}}<sub>{{mvar|n,ℓ,m}}&nbsp;</sub>, the [[magnetic dipole moment]] is given by
<math display="block">\mu_\text{L} = -g_\text{L} \frac{\mu_\text{B}}{\hbar}\langle\Psi_{n,\ell,m}|L|\Psi_{n,\ell,m}\rangle = -\mu_\text{B}\sqrt{\ell(\ell + 1)}.</math>

Here {{mvar|L}} is the orbital [[angular momentum]], {{mvar|n}}, {{mvar|ℓ}}, and {{mvar|m}} are the [[principal quantum number|principal]], [[azimuthal quantum number|azimuthal]], and [[magnetic quantum number|magnetic]] [[quantum numbers]] respectively.
The {{mvar|z}} component of the orbital magnetic dipole moment for an electron with a [[magnetic quantum number]] {{mvar|m}}<sub>ℓ</sub> is given by
<math display="block">(\boldsymbol{\mu}_\text{L})_z = -\mu_\text{B} m_\ell.</math>

== History ==
The electron magnetic moment is intrinsically connected to electron spin and was first hypothesized during the early models of the atom in the early twentieth century. The first to introduce the idea of electron spin was [[Arthur Compton]] in his 1921 paper on investigations of ferromagnetic substances with X-rays.<ref name=Compton1921>{{cite journal |last=Compton |first=Arthur H. |authorlink=Arthur Compton |date=August 1921 |title=The Magnetic Electron |journal=Journal of the Franklin Institute |volume=192 |number=2 | doi=10.1016/S0016-0032(21)90917-7 |url=https://zenodo.org/record/1533754 }}</ref>{{rp|pp=145–155}}<ref>Charles P. Enz, Heisenberg's applications of quantum mechanics (1926-33) or the settling of the new land*), Department de Physique Théorique Université de Genève, 1211 Genève 4, Switzerland (10. I. 1983)</ref> In Compton's article, he wrote: "Perhaps the most natural, and certainly the most generally accepted view of the nature of the elementary magnet, is that the revolution of electrons in orbits within the atom give to the atom as a whole the properties of a tiny permanent magnet."<ref name=Compton1921/>{{rp|p=146}} 

That same year [[Otto Stern]] proposed an experiment carried out later called the [[Stern–Gerlach experiment]] in which silver atoms in a magnetic field were deflected in opposite directions of distribution. This pre-1925 period marked the [[old quantum theory]] built upon the [[Bohr model|Bohr-Sommerfeld model]] of the atom with its classical elliptical electron orbits. During the period between 1916 and 1925, much progress was being made concerning the arrangement of electrons in the [[periodic table]]. In order to explain the [[Zeeman effect]] in the Bohr atom, Sommerfeld proposed that electrons would be based on three 'quantum numbers', n, k, and m, that described the size of the orbit, the shape of the orbit, and the direction in which the orbit was pointing.<ref>Manjit Kumar, Quantum: Einstein, Bohr and the Great Debate About the Nature of Reality, 2008.</ref> [[Irving Langmuir]] had explained in his 1919 paper regarding electrons in their shells, "Rydberg has pointed out that these numbers are obtained from the series <math>N = 2(1 + 2^2 + 2^2 + 3^2 + 3^2 + 4^2)</math>. The factor two suggests a fundamental two-fold symmetry for all stable atoms."<ref>{{Cite journal |last=Langmuir |first=Irving |date=1919 |title=The arrangement of electrons in atoms and molecules |url=https://linkinghub.elsevier.com/retrieve/pii/S0016003219910970 |journal=Journal of the Franklin Institute |language=en |volume=187 |issue=3 |pages=359–362 |doi=10.1016/S0016-0032(19)91097-0}}</ref> This <math>2n^2</math> configuration was adopted by [[Edmund Stoner]], in October 1924 in his paper 'The Distribution of Electrons Among Atomic Levels' published in the Philosophical Magazine. [[Wolfgang Pauli]] hypothesized that this required a fourth quantum number with a two-valuedness.<ref>Wolfgang Pauli. Exclusion principle and quantum mechanics. Online available via ⟨http://nobelprize.org⟩{{Dead link|date=October 2022 |bot=InternetArchiveBot |fix-attempted=yes }}. Nobel Lecture delivered on December 13th 1946 for the 1945 Nobel Prize in Physics.</ref>

== Electron spin in the Pauli and Dirac theories ==

{{main article|Pauli equation|Dirac equation}}

Starting from here the charge of the electron is  {{math|''e'' < 0}}. The necessity of introducing half-integral [[Spin (physics)|spin]] goes back experimentally to the results of the [[Stern–Gerlach experiment]]. A beam of atoms is run through a strong non-uniform magnetic field, which then splits into {{mvar|N}} parts depending on the intrinsic angular momentum of the atoms. It was found that for [[silver]] atoms, the beam was split in two—the ground state therefore could not be integral, because even if the intrinsic angular momentum of the atoms were as small as possible, 1, the beam would be split into 3&nbsp;parts, corresponding to atoms with {{mvar|L}}{{sub|z}} = −1, 0, and +1. The conclusion is that silver atoms have net intrinsic angular momentum of {{sfrac|1|2}}. [[Wolfgang Pauli|Pauli]] set up a theory which explained this splitting by introducing a two-component wave function and a corresponding correction term in the [[Hamiltonian (quantum mechanics)|Hamiltonian]], representing a [[Semi-classical theory|semi-classical]] coupling of this [[wave function]] to an applied magnetic field, as so:
<math display="block">H = \frac{1}{2m} \left [ \boldsymbol{\sigma}\cdot \left ( \mathbf{p} - \frac{e}{c}\mathbf{A} \right ) \right ]^2 + e\phi.</math>

Here {{math|'''A'''}} is the [[magnetic vector potential]] and {{mvar|ϕ}} the [[electric potential]], both representing the [[electromagnetic field]], and {{mvar|'''σ'''}} = ({{mvar|σ}}{{sub|x}}, {{mvar|σ}}{{sub|y}}, {{mvar|σ}}{{sub|z}}) are the [[Pauli matrices]]. On squaring out the first term, a residual interaction with the magnetic field is found, along with the usual classical Hamiltonian of a charged particle interacting with an applied field:
<math display="block">H = \frac{1}{2m}\left ( \mathbf{p} - \frac{e}{c}\mathbf{A} \right )^2 + e\phi - \frac{e\hbar}{2mc} \boldsymbol{\sigma} \cdot \mathbf{B}.</math>

This Hamiltonian is now a 2 × 2 matrix, so the Schrödinger equation based on it must use a two-component wave function. Pauli had introduced the 2 × 2 sigma matrices as pure ''phenomenology'' — Dirac now had a ''theoretical argument'' that implied that [[Spin (physics)|spin]] was somehow the consequence of incorporating [[special relativity|relativity]] into [[quantum mechanics]]. On introducing the external electromagnetic [[4-potential]] into the Dirac equation in a similar way, known as [[minimal coupling]], it takes the form (in [[natural units]] {{mvar|ħ}} = {{mvar|c}} = 1)
<math display="block">\left [ -i\gamma^\mu\left ( \partial_\mu + i e A_\mu \right ) + m \right ] \psi = 0</math>
where <math>\gamma^\mu</math> are the [[gamma matrices]] (known as [[Dirac matrices]]) and {{mvar|i}} is the [[imaginary unit]]. A second application of the [[Dirac equation#Properties|Dirac operator]] will now reproduce the Pauli term exactly as before, because the spatial Dirac matrices multiplied by {{mvar|i}}, have the same squaring and commutation properties as the Pauli matrices. What is more, the value of the [[gyromagnetic ratio]] of the electron, standing in front of Pauli's new term, is explained from first principles. This was a major achievement of the Dirac equation and gave physicists great faith in its overall correctness. The Pauli theory may be seen as the low energy limit of the Dirac theory in the following manner. First the equation is written in the form of coupled equations for 2-spinors with the units restored:
<math display="block">\begin{pmatrix}
(mc^2 - E + e \phi) & c\sigma\cdot \left (\mathbf{p} - \frac{e}{c}\mathbf{A} \right ) \\ -c\boldsymbol{\sigma}\cdot \left ( \mathbf{p} - \frac{e}{c}\mathbf{A} \right ) & \left ( mc^2 + E - e \phi \right )
\end{pmatrix}
\begin{pmatrix} \psi_+ \\ \psi_- \end{pmatrix}
=
\begin{pmatrix} 0 \\ 0 \end{pmatrix}. </math>
so
<math display="block">\begin{align}
   (E - e\phi) \psi_+ - c\boldsymbol{\sigma} \cdot \left( \mathbf{p} - \frac{e}{c}\mathbf{A} \right) \psi_- &= mc^2 \psi_+ \\
  -(E - e\phi) \psi_- + c\boldsymbol{\sigma} \cdot \left( \mathbf{p} - \frac{e}{c}\mathbf{A} \right) \psi_+ &= mc^2 \psi_-
\end{align}</math>

Assuming the field is weak and the motion of the electron non-relativistic, we have the total energy of the electron approximately equal to its [[rest energy]], and the momentum reducing to the classical value,
<math display="block">\begin{align}
  E - e\phi &\approx mc^2 \\
          p &\approx m v
\end{align}</math>
and so the second equation may be written
<math display="block">\psi_- \approx \frac{1}{2mc} \boldsymbol{\sigma} \cdot \left( \mathbf{p} - \frac{e}{c}\mathbf{A} \right) \psi_+</math>
which is of order {{sfrac|{{mvar|v}}|{{mvar|c}}}} - thus at typical energies and velocities, the bottom components of the [[Dirac spinor]] in the standard representation are much suppressed in comparison to the top components. Substituting this expression into the first equation gives after some rearrangement
<math display="block"> \left(E - mc^2\right) \psi_+ = \frac{1}{2m} \left[ \boldsymbol{\sigma}\cdot \left( \mathbf{p} - \frac{e}{c} \mathbf{A} \right) \right]^2 \psi_+ + e\phi \psi_+</math>

The operator on the left represents the particle energy reduced by its rest energy, which is just the classical energy, so we recover Pauli's theory if we identify his 2-spinor with the top components of the Dirac spinor in the non-relativistic approximation. A further approximation gives the [[Schrödinger equation]] as the limit of the Pauli theory. Thus the Schrödinger equation may be seen as the far non-relativistic approximation of the Dirac equation when one may neglect spin and work only at low energies and velocities. This also was a great triumph for the new equation, as it traced the mysterious {{mvar|i}} that appears in it, and the necessity of a complex wave function, back to the geometry of space-time through the Dirac algebra. It also highlights why the Schrödinger equation, although superficially in the form of a diffusion equation, actually represents the propagation of waves.

It should be strongly emphasized that this separation of the Dirac spinor into large and small components depends explicitly on a low-energy approximation. The entire Dirac spinor represents an ''irreducible'' whole, and the components we have just neglected to arrive at the Pauli theory will bring in new phenomena in the relativistic regime – [[antimatter]] and the idea of creation and annihilation of particles.

In a general case (if a certain linear function of electromagnetic field does not vanish identically), three out of four components of the spinor function in the Dirac equation can be algebraically eliminated, yielding an equivalent fourth-order partial differential equation for just one component. Furthermore, this remaining component can be made real by a gauge transform.<ref>{{cite journal |first=Andrey |last=Akhmeteli |year=2011 |title=One real function instead of the Dirac spinor function |journal=Journal of Mathematical Physics |volume=52 |issue=8 |page=082303 |doi=10.1063/1.3624336 |arxiv=1008.4828 |bibcode=2011JMP....52h2303A |s2cid=119331138 |url=http://jmp.aip.org/resource/1/jmapaq/v52/i8/p082303_s1 |access-date=2012-04-26 |url-status=dead |archive-url=https://archive.today/20120718231440/http://jmp.aip.org/resource/1/jmapaq/v52/i8/p082303_s1 |archive-date=2012-07-18 |df=dmy-all}} <!-- or {{cite web |url=http://akhmeteli.org/wp-content/uploads/2011/08/JMAPAQ528082303_1.pdf |title=[no title cited] |url-status=dead |date=August 2011}} --></ref>

== Measurement ==
The existence of the [[anomalous magnetic moment]] of the electron has been detected experimentally by [[Nuclear magnetic resonance|magnetic resonance]] method.<ref name=":0" /> This allows the determination of [[hyperfine splitting]] of electron shell energy levels in atoms of [[Isotopes of hydrogen#Hydrogen-1 .28protium.29|protium]] and [[deuterium]] using the measured resonance frequency for several transitions.<ref>{{cite journal |url=http://journals.aps.org/pr/abstract/10.1103/PhysRev.73.412 |first2=Polykarp |last2=Kusch |author2-link=Polykarp Kusch |first1=H.M. |last1=Foley |date=15 February 1948 |title=Intrinsic moment of the electron |journal=Physical Review |volume=73 |issue=4 |page=412 |doi=10.1103/PhysRev.73.412 |access-date=2 April 2015 |archive-date=8 March 2021 |archive-url=https://web.archive.org/web/20210308160648/https://journals.aps.org/pr/abstract/10.1103/PhysRev.73.412 |url-status=live }}</ref><ref>{{cite journal |url=http://journals.aps.org/pr/abstract/10.1103/PhysRev.74.250 |first1=Polykarp |last1=Kusch |author1-link=Polykarp Kusch |first2=H.M. |last2=Foley |date=1 August 1948 |title=The magnetic moment of the electron |journal=Physical Review |volume=74 |issue=3 |pages=207–11 |doi=10.1103/PhysRev.74.250 |pmid=17820251 |bibcode=1948PhRv...74..250K |access-date=2 April 2015 |archive-date=22 April 2021 |archive-url=https://web.archive.org/web/20210422161042/https://journals.aps.org/pr/abstract/10.1103/PhysRev.74.250 |url-status=live }}</ref>

The [[magnetic moment]] of the electron has been measured using a one-electron quantum [[cyclotron]] and [[quantum nondemolition]] spectroscopy. The spin frequency of the electron is determined by the [[G-factor (physics)|{{mvar|g}}-factor]].
<math display="block"> \nu_\text{s} = \frac{g}{2} \nu_c</math>
<math display="block"> \frac{g}{2} = \frac{\bar{\nu}_c + \bar{\nu}_a}{\bar{\nu}_c}</math>

== See also ==
{{div col|colwidth=25em}}
* [[Spin (physics)]]
* [[Electron precipitation]]
* [[Bohr magneton]]
* [[Nuclear magnetic moment]]
* [[Nucleon magnetic moment]]
* [[Anomalous magnetic dipole moment]]
* [[Electron electric dipole moment]]
* [[Fine structure]]
* [[Hyperfine structure]]
{{div col end}}

== References ==
{{reflist|25em}}

== Bibliography ==
* {{cite book | author=Sergei Vonsovsky |title=Magnetism of Elementary Particles | url=https://archive.org/details/MagnetismOfElementaryParticles | publisher=Mir Publishers | year=1975 |author-link=Sergei Vonsovsky }}
* {{cite book | author=Sin-Itiro Tomonaga |title=The Story of Spin | publisher=University of Chicago Press | year=1997 |author-link=Sin-Itiro Tomonaga }}

{{DEFAULTSORT:Electron Magnetic Dipole Moment}}
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