g-factor

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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.

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[edit] Calculation

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[edit] Special cases

[edit] 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:

[edit] Electron spin g-factor

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

\boldsymbol{\mu}_S=-g_S \mu_\mathrm{B} (\boldsymbol{S}/\hbar)

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

\boldsymbol{\mu}_z=-g_S \mu_\mathrm{B} m_s

The value gS is roughly equal to two, and is known to extraordinary accuracy.[1][2] The reason it is not precisely two is explained by quantum electrodynamics.[3]

[edit] Electron orbital g-factor

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

\boldsymbol{\mu}_L=g_L \mu_\mathrm{B} (\boldsymbol{L}/\hbar)

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 μB is the Bohr magneton. The value of gL 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 ml, the z-component of the orbital angular momentum is

\boldsymbol{\mu}_z=g_L \mu_\mathrm{B} m_l

which, since gL = 1, is just μBml

[edit] Landé g-factor

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

\boldsymbol{\mu}=g_J \mu_\mathrm{B} (\boldsymbol{J}/\hbar)

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 μB 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.

[edit] 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}=g \mu_\mathrm{p} (\boldsymbol{I}/\hbar)

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

[edit] Muon g-factor

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.
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

\mathbf{\mu}=g (e\hbar/(2m_\mu)) (\mathbf{S}/\hbar)

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 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[4]. This is a difference of 3.4 standard deviations, suggesting beyond-the-Standard-Model physics may be having an effect.

[edit] Measured g-factor values

Elementary Particle g-factor Uncertainty
Electron ge 2.002 319 304 3622 0.000 000 000 0015
Neutron gn -3.826 085 46 0.000 000 90
Proton gp 5.585 694 701 0.000 000 056
Muon gμ 2.002 331 8396 0.000 000 0012
Currently accepted NIST g-factor values[1]

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.

[edit] Notes and references

  1. ^ See CERN courier article
  2. ^ B Odom, D Hanneke, B D'Urso and G Gabrielse (2006). "New measurement of the electron magnetic moment using a one-electron quantum cyclotron". Physical Review Letters 97 (3): 030801. doi:10.1103/PhysRevLett.97.030801. 
  3. ^ S J Brodsky, V A Franke, J R Hiller, G McCartor, S A Paston and E V Prokhvatilov (2004). "A nonperturbative calculation of the electron's magnetic moment". Nuclear Physics B 703 (1-2): 333-362. doi:10.1016/j.nuclphysb.2004.10.027. 
  4. ^ Hagiwara, K.; Martin, A. D. and Nomura, Daisuke and Teubner, T. (2006). "Improved predictions for g-2 of the muon and alpha(QED)(M(Z)**2)" (subscription required). 

[edit] See also