G-factor

:"For the acceleration-related quantity in mechanics, see "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 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

pecial cases

Electron "g"-factors

There are three magnetic moments associated with an electron: One from its spin angular momentum, one from its 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", "g_S" (more often called simply the "electron g-factor", "g"_e), defined by

:

where "μ_S" is the total magnetic moment resulting from the spin of an electron, "S" is the magnitude of its spin angular momentum, and "μ"_B is the Bohr magneton. The "z"-component of the magnetic moment then becomes

:

The value "g_S" is roughly equal to two, and is known to extraordinary accuracy. [ [http://cerncourier.com/main/article/46/8/20 See CERN courier article] ] [cite journal
author = B Odom, D Hanneke, B D'Urso and G Gabrielse
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] The reason it is not "precisely" two is explained by quantum electrodynamics. [cite journal
author = S J Brodsky, V A Franke, J R Hiller, G McCartor, S A Paston and E V Prokhvatilov
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]

Electron orbital "g"-factor

Secondly, the "electron orbital g-factor", "g_L", is defined by

:

where "μ_L" 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 "g_L" is exactly equal to one, by a quantum-mechanical argument analogous to the derivation of the classical magnetogyric ratio. For an electron in an orbital with a magnetic quantum number m_l, the "z"-component of the orbital angular momentum is

:

which, since "g_L" = 1, is just "μ"_Bm_l

Landé "g"-factor

Thirdly, the "Landé g-factor", "g_J", is defined by

:

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 "g_J" is related to "g_L" and "g_S" 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

:

where "μ" is the magnetic moment resulting from the nuclear spin, "I" is the nuclear spin angular momentum, and "μ"_p is the nuclear magneton.

Muon "g"-factor

The muon, like the electron has a "g"-factor from its spin, given by the equation

:$mathbf\{mu\}=g\; (ehbar/(2m\_mu))\; (mathbf\{S\}/hbar)$

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

The muon "g"-factor can 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 2.0023318416 with an uncertainy of 0.0000000013, compared to the theoretical prediction of 2.0023318361 with an uncertainty of 0.0000000010 [cite journal
last = Hagiwara
first = K.
coauthors = Martin, A. D. and Nomura, Daisuke and Teubner, T.
title = Improved predictions for g-2 of the muon and alpha(QED)(M(Z)**2)
date = 2006
url = http://arxiv.org/abs/hep-ph/0611102
format = subscription required] . This is a difference of 3.4 standard deviations, suggesting beyond-the-Standard-Model physics may be having an effect.

Measured "g"-factor values

It should be noted that the electron g-factor is one of the most precisely measured values in physics, with its uncertainty beginning at the twelfth decimal place.

Notes and references

ee also

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