G-factor (physics)

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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

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 (more often called simply the electron g-factor), g_e, defined by

${\boldsymbol {\mu }}_{S}={\frac {g_{e}\mu _{{\mathrm {B}}}}{\hbar }}{\boldsymbol {S}}$

where μ_S is the total magnetic moment resulting from the spin of an electron, S is its 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 g_e:

$g_{S}=|g_{e}|=-g_{e}.$

The z-component of the magnetic moment then becomes

$\mu _{z}=-g_{S}\mu _{{\mathrm {B}}}m_{s}$

The value g_S is roughly equal to 2.002319, and is known to extraordinary precision.^[1]^[2] The reason it is not precisely two is explained by quantum electrodynamics calculation of the anomalous magnetic dipole moment.^[3]

Electron orbital g-factor

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

${\boldsymbol {\mu }}_{L}=-{\frac {g_{L}\mu _{{\mathrm {B}}}}{\hbar }}{\boldsymbol {L}}$

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

$\mu _{z}=g_{L}\mu _{{\mathrm {B}}}m_{l}$

which, since g_L = 1, is just μ_Bm_l

Total angular momentum (Landé) g-factor

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

${\boldsymbol {\mu }}=-{\frac {g_{J}\mu _{{\mathrm {B}}}}{\hbar }}{\boldsymbol {J}}$

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

${\boldsymbol {\mu }}={\frac {g\mu _{{\mathrm {N}}}}{\hbar }}{\boldsymbol {I}}$

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

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

${\mathbf {\mu }}={\frac {ge}{2m_{\mu }}}{\mathbf {S}}$

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 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 2.0023318416(13), compared to the theoretical prediction of 2.0023318361(10).^[4] This is a difference of 3.4 standard deviations, 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.^[5]

Measured g-factor values

Currently accepted NIST g-factor values ^[6]
Particle	g-factor	Uncertainty
Electron $g_{{\mathrm {e}}}$	−2.00231930436153	0.00000000000053
Neutron $g_{{\mathrm {n}}}$	−3.82608545	0.00000090
Proton $g_{{\mathrm {p}}}$	5.585694713	0.000000046
Muon $g_{{\mu }}$	−2.0023318414	0.0000000012

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

↑ Gabrielse, Gerald; Hanneke, David (October 2006). "Precision pins down the electron's magnetism". CERN Courier 46 (8): 35–37. Cite uses deprecated parameters (help)
↑ Odom, B.; Hanneke, D.; d’Urso, B.; Gabrielse, G. (2006). "New measurement of the electron magnetic moment using a one-electron quantum cyclotron". Physical Review Letters 97 (3): 030801. Bibcode:2006PhRvL..97c0801O. doi:10.1103/PhysRevLett.97.030801. PMID 16907490.
↑ Brodsky, S; Franke, V; Hiller, J; McCartor, G; Paston, S; Prokhvatilov, E (2004). "A nonperturbative calculation of the electron's magnetic moment". Nuclear Physics B 703 (1–2): 333–362. arXiv:hep-ph/0406325. Bibcode:2004NuPhB.703..333B. doi:10.1016/j.nuclphysb.2004.10.027.
↑ Hagiwara, K.; Martin,, A. D.; Nomura, Daisuke; Teubner, T. (2006). "Improved predictions for g-2 of the muon and alpha(QED)(M(Z)**2)". Physics Letters B 649 (2–3): 173–179. arXiv:hep-ph/0611102. Bibcode:2007PhLB..649..173H. doi:10.1016/j.physletb.2007.04.012.
↑ http://muon-g-2.fnal.gov/
↑ "CODATA values of the fundamental constants". NIST.