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{{lowercase}}
:''For the acceleration-related quantity in mechanics, see [[g-force|''g''-force]].''

A '''''g''-factor''' (also called '''''g'' value''' or '''dimensionless magnetic moment''') is a dimensionless quantity which characterizes the [[magnetic moment]] and [[gyromagnetic ratio]] of a particle or [[atomic nucleus|nucleus]]. It is essentially a proportionality constant that relates the observed magnetic moment μ of a particle to the appropriate [[angular momentum]] [[quantum number]] and the appropriate fundamental quantum unit of magnetism, usually the [[Bohr magneton]] or [[nuclear magneton]].

==Calculation==

=== Electron ''g''-factors ===
There are three magnetic moments associated with an electron: One from its [[Spin (physics)|spin angular momentum]], one from its [[azimuthal quantum number|orbital angular momentum]], and one from its total angular momentum (the quantum-mechanical sum of those two components). Corresponding to these three moments are three different ''g''-factors:

==== Electron spin ''g''-factor ====
The most famous of these is the ''electron spin g-factor'' (more often called simply the ''electron g-factor''), ''ge'', defined by

:<math> \boldsymbol{\mu}_S = \frac{g_e\mu_\mathrm{B}}{\hbar}\boldsymbol{S}</math>

where '''''μS''''' is the total magnetic moment resulting from the spin of an electron, '''''S''''' is its [[spin (physics)|spin]] angular momentum, and ''μ''B is the [[Bohr magneton]]. In atomic physics, the electron spin ''g''-factor is often defined as the ''absolute value'' or ''negative'' of ''ge'':

:<math>g_S = |g_e| = -g_e.</math>

The ''z''-component of the magnetic moment then becomes

:<math> \mu_z=-g_S \mu_\mathrm{B} m_s</math>

The value ''gS'' is roughly equal to 2.002319, and is known to extraordinary precision.<ref>{{cite journal|url=http://cerncourier.com/main/article/46/8/20 |title=Precision pins down the electron's magnetism|journal=CERN Courier|volume=46|issue=8|month=October|pages=35–37|first1=Gerald|last1= Gabrielse |first2= David|last2= Hanneke|year=2006}}</ref><ref>
{{cite journal
| title = New measurement of the electron magnetic moment using a one-electron quantum cyclotron
| year = 2006
| journal = [[Physical Review Letters]]
| volume = 97
| issue = 3
| pages = 030801
| doi = 10.1103/PhysRevLett.97.030801
| pmid=16907490
| bibcode=2006PhRvL..97c0801O
| last1 = Odom
| first1 = B.
| last2 = Hanneke
| first2 = D.
| last3 = d’Urso
| first3 = B.
| last4 = Gabrielse
| first4 = G.
}}</ref> The reason it is not ''precisely'' two is explained by [[quantum electrodynamics]] calculation of the [[anomalous magnetic dipole moment]].<ref>
{{cite journal
| title = A nonperturbative calculation of the electron's magnetic moment
| year = 2004
| journal = [[Nuclear Physics B]]
| volume = 703
| issue = 1–2
| pages = 333–362
| doi = 10.1016/j.nuclphysb.2004.10.027
|arxiv = hep-ph/0406325 |bibcode = 2004NuPhB.703..333B
| last1 = Brodsky
| first1 = S
| last2 = Franke
| first2 = V
| last3 = Hiller
| first3 = J
| last4 = McCartor
| first4 = G
| last5 = Paston
| first5 = S
| last6 = Prokhvatilov
| first6 = E }}</ref>

==== Electron orbital ''g''-factor ====
Secondly, the ''electron orbital g-factor'', ''gL'', is defined by

:<math> \boldsymbol{\mu}_L = -\frac{g_L \mu_\mathrm{B}}{\hbar}\boldsymbol{L}</math>

where '''''μL''''' is the total magnetic moment resulting from the orbital angular momentum of an electron, '''''L''''' is the magnitude of its orbital angular momentum, and ''μ''B is the [[Bohr magneton]]. The value of ''gL'' is exactly equal to one, by a quantum-mechanical argument analogous to the derivation of the [[magnetogyric ratio|classical magnetogyric ratio]]. For an electron in an orbital with a [[magnetic quantum number]] ml, the ''z''-component of the orbital angular momentum is

:<math>\mu_z=g_L \mu_\mathrm{B} m_l</math>

which, since ''gL'' = 1, is just ''μ''Bml

====Total angular momentum (Landé) ''g''-factor ====
Thirdly, the ''[[Landé g-factor]]'', ''gJ'', is defined by
:<math> \boldsymbol{\mu} = -\frac{g_J \mu_\mathrm{B} }{\hbar}\boldsymbol{J}</math>

where '''''μ''''' is the total magnetic moment resulting from both spin and orbital angular momentum of an electron, '''''J''''' = '''''L'''''+'''''S''''' is its total angular momentum, and ''μ''B is the [[Bohr magneton]]. The value of ''gJ'' is related to ''gL'' and ''gS'' by a quantum-mechanical argument; see the article [[Landé g-factor]].

=== Nucleon and nucleus ''g''-factors ===
Protons, neutrons, and many nuclei have spin and magnetic moments, and therefore associated ''g''-factors. The formula conventionally used is
:<math> \boldsymbol{\mu} = \frac{g \mu_\mathrm{N}}{\hbar}\boldsymbol{I}</math>

where '''''μ''''' is the magnetic moment resulting from the nuclear spin, '''''I''''' is the nuclear spin angular momentum, ''μ''N is the [[nuclear magneton]], and ''g'' is the effective ''g''-factor.

=== Muon ''g''-factor ===
[[Image:The muon g-2.svg|frame|If [[supersymmetry]] is realized in nature, there will be corrections to g-2 of the muon due to loop diagrams involving the new particles. Amongst the leading corrections are those depicted here: a [[neutralino]] and a [[smuon]] loop, and a [[chargino]] and a muon [[sneutrino]] loop. This represents an example of "beyond the Standard-Model" physics that might contribute to g-2.]]
The muon, like the electron has a ''g''-factor from its spin, given by the equation
:<math>\mathbf{\mu} = \frac{ge}{2m_\mu}\mathbf{S}</math>

where '''''μ''''' is the magnetic moment resulting from the muon’s spin, '''''S''''' is the spin angular momentum, and ''mμ'' is the muon mass.

The fact that the muon g-factor is not quite the same as the electron g-factor is mostly explained by quantum electrodynamics and its calculation of the [[anomalous magnetic dipole moment]]. Almost all of the small difference between the two values (99.96% of it) is due to a well-understood lack of a heavy-particle diagrams contributing to the probability for emission of a photon representing the magnetic dipole field, which are present for muons, but not electrons, in QED theory. These are entirely a result of the mass difference between the particles.

However, not all of the difference between the g-factors for electrons and muons are exactly explained by the quantum electrodynamics [[Standard Model]]. The muon ''g''-factor can, at least in theory, be affected by physics [[beyond the Standard Model]], so it has been measured very precisely, in particular at the [[Brookhaven National Laboratory]]. As of November 2006, the experimentally measured value is {{val|2.0023318416|(13)}}, compared to the theoretical prediction of {{val|2.0023318361|(10)}}.<ref>
{{cite journal
| last1 = Hagiwara
| first1 = K. |last2= Martin,|first2= A. D. |last3= Nomura|first3= Daisuke |last4= Teubner|first4= T.
| title = Improved predictions for g-2 of the muon and alpha(QED)(M(Z)**2)
| year = 2006
| doi = 10.1016/j.physletb.2007.04.012
| journal = Physics Letters B
| volume = 649
| issue = 2–3
| pages = 173–179
|arxiv = hep-ph/0611102 | bibcode=2007PhLB..649..173H
}}</ref> This is a difference of 3.4 [[standard deviation]]s, suggesting beyond-the-Standard-Model physics may be having an effect. The Brookhaven muon storage ring is being transported to Fermilab where the g-2 experiment will use it to make more precise measurements of muon g-factor.<ref>http://muon-g-2.fnal.gov/</ref>

== Measured ''g''-factor values ==
{| class="wikitable" style="text-align:center"
|-
! Particle !! ''g''-factor !! Uncertainty
|-
! Electron <math>g_\mathrm{e}</math>
| {{gaps|−2.002|319|304|361|53}} || {{gaps|0.000|000|000|000|53}}
|-
! Neutron <math>g_\mathrm{n}</math>
| {{gaps|−3.826|085|45}} || {{gaps|0.000|000|90}}
|-
! Proton <math>g_\mathrm{p}</math>
| {{gaps|5.585|694|713}} || {{gaps|0.000|000|046}}
|-
! Muon <math>g_{\mu}</math>
| {{gaps|−2.002|331|8414}} || {{gaps|0.000|000|0012}}
|-
|+Currently accepted [[NIST]] ''g''-factor values <ref>{{cite web|work=NIST|title=CODATA values of the fundamental constants|url=http://physics.nist.gov/cgi-bin/cuu/Category?view=html&All+values.x=80&All+values.y=11}}</ref>
|}

The electron ''g''-factor is one of the most precisely measured values in physics, with a relative standard uncertainty of 2.6 x 10−13.

==Notes and references==
{{reflist}}

==See also==
*[[anomalous magnetic dipole moment]]
*[[Electron magnetic dipole moment]]
*[http://physics.nist.gov/cuu/Constants/codata.pdf CODATA recommendations 2006]

[[Category:Atomic physics]]
[[Category:Nuclear physics]]
[[Category:Particle physics]]
[[Category:Physical constants]]

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