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{{Short description|The spin of an electron}}
{{hatnote|"Electron spin"  redirects here. See also [[Electron spin resonance]] and [[Spin (physics)]].}}

In [[atomic physics]], the '''electron magnetic moment''', or more specifically the '''electron magnetic dipole moment''', is the [[magnetic moment]] of an [[electron]] caused by its intrinsic properties of [[spin (physics)|spin]] and [[electric charge]]. The value of the electron magnetic moment is approximately {{val|-9.284764|e=-24|ul=J/T}}.  The electron magnetic moment has been measured to an accuracy of 7.6 parts in 10<sup>13</sup>.<ref>{{cite journal
 |authors=B. Odom, D. Hanneke, B. D’Urso, G. Gabrielse
 |title=New Measurement of the Electron Magnetic Moment Using a One-Electron Quantum Cyclotron
 |journal=Phys. Rev. Lett.
 |volume=97
 |issue=3
 |pages=030801
 |year=2006
 |doi=10.1103/physrevlett.97.030801
 |pmid=16907490
 |bibcode=2006PhRvL..97c0801O
}}</ref>

==Magnetic moment of an electron==

The electron is a [[charged particle]] with charge −1''e'', where ''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 [[electric charge|electrically charged]] body creates a [[magnetic dipole]] with [[magnet#Modelling magnets|magnetic poles]] of equal magnitude but opposite [[wikt:magnetic polarity|polarity]]. This analogy does hold, since an electron indeed 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]] depending on its orientation with respect to the field.

If the electron is visualized as a classical [[charged particle]] literally rotating about an axis with [[angular momentum]] '''L''', its magnetic dipole moment '''μ''' is given by

:<math>\boldsymbol{\mu} = \frac{-e}{2m_\text{e}} \mathbf{L},</math>

where ''m''<sub>e</sub> is the [[electron]] [[rest mass]]. Note that the [[angular momentum]] '''L''' in this equation may be the spin angular momentum, the orbital angular momentum, or the total angular momentum. It turns out that the classical result is off by a proportional factor for the [[spin magnetic moment]]. As a result, the classical result is corrected by multiplying it with a [[dimensionless quantity|dimensionless]] correction factor ''g'', known as the [[g-factor (physics)|g-factor]]:

:<math>\boldsymbol{\mu} = g \frac{-e}{2m_\text{e}} \mathbf{L}.</math>

It is usual to express the magnetic moment in terms of the [[reduced Planck constant]] ''ħ'' and the [[Bohr magneton]] ''μ''<sub>B</sub>:

:<math>\boldsymbol{\mu} = -g \mu_\text{B} \frac{\mathbf{L}}{\hbar}.</math>

Since the [[Bohr magneton|magnetic moment is quantized]] in units of ''μ''<sub>B</sub>, correspondingly the [[Angular momentum quantum number|angular momentum is quantized]] in units of ''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>
 \langle p_f | j^\mu | p_i \rangle = \bar u(p_f) \left\{ F_1(q^2) \gamma^\mu +\frac{i \sigma^{\mu\nu}}{2m_{\rm e}}q_\nu F_2(q^2)+i\epsilon^{\mu\nu\rho\sigma}\sigma_{\rho\sigma}q_\nu F_3(q^2)+\frac 1{2m_{\rm 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_{\rm 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> \mu = (F_1(0)+F_2(0))/2m_{\rm e}</math>  is its static magnetic dipole moment, and <math> -F_3(0)/2m_{\rm e}</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>A. Mahajan and A. Rangwala.  [https://books.google.com/books?id=_tXrjggX7WwC&pg=PA419&lpg=PA419&dq=%22intrinsic+dipole+moment%22+and+electron+%22Bohr+magneton%22 Electricity and Magnetism], p. 419 (1989).</ref> It is

:<math>\boldsymbol{\mu}_\text{s} = -g_\text{s} \mu_\text{B} \frac{\mathbf{S}}{\hbar}.</math>

Here '''S''' is the electron spin angular momentum. The spin [[g-factor (physics)|g-factor]] is approximately two: ''g''<sub>s</sub> ≈ 2. The magnetic moment of an electron is approximately twice what it should be in classical mechanics. The factor of two implies that the electron appears to be twice as effective in producing a magnetic moment as the corresponding classical charged body.

The spin magnetic dipole moment is approximately one ''μ''<sub>B</sub> because {{nowrap|''g'' ≈ 2}} and the electron is a spin-1/2 particle ({{nowrap|1=''S'' = ''ħ''/2}}):

:<math>\mu_\text{S} \approx 2\frac{e \hbar}{2m_\text{e}c}\frac{\frac{\hbar}{2}}{\hbar} = \mu_\text{B}.</math> {{dubious|date=September 2015}}

The ''z'' component of the electron magnetic moment is

:<math>(\boldsymbol{\mu}_\text{s})_z = -g_\text{s} \mu_\text{B} m_\text{s},</math>

where ''m''<sub>s</sub> is the [[spin quantum number]]. Note that '''μ''' is a ''negative'' constant multiplied by the [[spin (physics)|spin]], so the magnetic moment is ''[[Antiparallel (mathematics)|antiparallel]]'' to the spin angular momentum.

The spin [[g-factor (physics)|g-factor]] {{nowrap|1=''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)|g-factor]] has been experimentally determined to have the value

:{{val|2.00231930436182|(52)}}.<ref>http://physics.nist.gov/cgi-bin/cuu/Value?eqae|search_for=electron+magnetic+moment</ref>

Note that it is only two thousandths larger than 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]]. In fact, one famous triumph of the [[quantum electrodynamics]] theory is the accurate prediction of the electron g-factor. The most accurate value for the electron magnetic moment is

:{{val|-9.284764620|(57)|e=-24|u=J/T}}.<ref>http://physics.nist.gov/cgi-bin/cuu/Value?muem</ref>

===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 '''L'''. Then the orbital magnetic dipole moment is

:<math>\boldsymbol{\mu}_L = -g_\text{L}\mu_\text{B} \frac{\mathbf{L}}{\hbar}.</math>

Here ''g<sub>L</sub>'' is the electron orbital [[g-factor (physics)|g-factor]] and ''μ''<sub>B</sub> is the [[Bohr magneton]]. The value of ''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 '''J''' by a similar equation:

:<math>\boldsymbol{\mu}_J = g_J \mu_\text{B} \frac{\mathbf{J}}{\hbar}.</math>

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

==Example: hydrogen atom==
For a [[hydrogen]] atom, an [[electron]] occupying the [[atomic orbital]] Ψ<sub>''n'',''ℓ'',''m''</sub>, the [[magnetic dipole moment]] is given by

:<math>\mu_L = -g_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 '''L''' is the orbital [[angular momentum]], ''n'', ''ℓ'' and ''m'' are the [[principal quantum number|principal]], [[azimuthal quantum number|azimuthal]] and [[magnetic quantum number|magnetic]] [[quantum numbers]] respectively.
The ''z'' component of the orbital magnetic dipole moment for an electron with a [[magnetic quantum number]] ''m<sub>ℓ</sub>'' is given by

:<math>(\boldsymbol{\mu}_L)_z = -\mu_\text{B} m_\ell.</math>

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

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

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 ''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 parts, corresponding to atoms with ''L<sub>z</sub>'' = −1, 0, and +1. The conclusion is that silver atoms have net intrinsic angular momentum of {{frac|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>H = \frac{1}{2m} \left [ \boldsymbol{\sigma}\cdot \left ( \mathbf{p} - \frac{e}{c}\mathbf{A} \right ) \right ]^2 + e\phi.</math>

Here '''A''' is the [[magnetic potential]] and ''ϕ'' the [[electric potential]], both representing the [[electromagnetic field]], and '''σ''' = (''σ<sub>x</sub>'', ''σ<sub>y</sub>'', ''σ<sub>z</sub>'') 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>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]] ''ħ'' = ''c'' = 1)

:<math>\left [ -i\gamma^\mu\left ( \partial_\mu + ieA_\mu \right ) + m \right ] \psi = 0\,</math>

where <math>\scriptstyle \gamma^\mu</math> are the [[gamma matrices]] (known as [[Dirac matrices]]) and ''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 ''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>\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>\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>\begin{align}
  E - e\phi &\approx mc^2 \\
          p &\approx m v
\end{align}</math>

and so the second equation may be written

:<math>\psi_- \approx \frac{1}{2mc} \boldsymbol{\sigma} \cdot \left( \mathbf{p} - \frac{e}{c}\mathbf{A} \right) \psi_+</math>

which is of order ''v/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> \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 ''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>Source: Journal of Mathematical Physics, 52, 082303 (2011) ({{cite web |url=http://jmp.aip.org/resource/1/jmapaq/v52/i8/p082303_s1 |title=Archived copy |accessdate=2012-04-26 |url-status=dead |archiveurl=https://archive.is/20120718231440/http://jmp.aip.org/resource/1/jmapaq/v52/i8/p082303_s1 |archivedate=2012-07-18 }} or http://akhmeteli.org/wp-content/uploads/2011/08/JMAPAQ528082303_1.pdf )</ref>

==Measurement==
The existence of the [[anomalous magnetic moment]] of the electron has been detected experimentally by [[Nuclear magnetic resonance|magnetic resonance]] method. 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>[http://journals.aps.org/pr/abstract/10.1103/PhysRev.74.250 [[Polykarp Kusch]], H. M. Foley]</ref><ref>[http://journals.aps.org/pr/abstract/10.1103/PhysRev.73.412 intrinsic moment of the electron]</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)|g-factor]].

<math> \nu_s = \frac{g}{2} \nu_c</math>

<math>  \frac{g}{2} = \frac{\bar{\nu_c + \nu_a}}{\nu_c}</math>

==See also==
* [[Electron precipitation]]
* [[Bohr magneton]]
* [[Nuclear magnetic moment]]
* [[Neutron magnetic moment]]
* [[Proton magnetic moment]]
* [[Anomalous magnetic dipole moment]]
* [[Electron electric dipole moment]]
* [[Fine structure]]
* [[Hyperfine structure]]

==References==
<references />

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

{{DEFAULTSORT:Electron Magnetic Dipole Moment}}
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