g-factor (physics)

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 that 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 its angular momentum quantum number and a unit of magnetic moment, usually the Bohr magneton or nuclear magneton.

Definition

Dirac particle

The spin magnetic moment of a charged, spin-1/2 particle that does not possess any internal structure (a Dirac particle) is given by^[1]

{\boldsymbol \mu }=g{e \over 2m}{\boldsymbol S},

where μ is the spin magnetic moment of the particle, g is the g-factor of the particle, e is the elementary charge, m is the mass of the particle, and S is the spin angular momentum of the particle (with magnitude ħ/2 for Dirac particles).

Baryon or nucleus

Protons, neutrons, nuclei and other composite baryonic particles have magnetic moments arising from their spin (both of which may be zero). Conventionally, the associated g-factors are defined using the nuclear magneton, and thus implicitly using the proton's mass rather than the particle's mass as for a Dirac particle. The formula used under this convention is

{\boldsymbol {\mu }}=g{\mu _{{\text{N}}} \over \hbar }{\boldsymbol {I}}=g{e \over 2m_{{\text{p}}}}{\boldsymbol {I}},

where μ is the magnetic moment of the nucleon or nucleus resulting from its spin, g is the effective g-factor, I is its spin angular momentum, μ_N is the nuclear magneton, e is the elementary charge and m_p is the proton rest mass.

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 }}_{{{\text{s}}}}=g_{{\text{e}}}{\mu _{{\text{B}}} \over \hbar }{\boldsymbol {S}}

where μ_s is the 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_{{\text{s}}}=|g_{{\text{e}}}|=-g_{{\text{e}}}.

The z-component of the magnetic moment then becomes

\mu _{{\text{z}}}=-g_{{\text{s}}}\mu _{{\text{B}}}m_{{\text{s}}}

The value g_s is roughly equal to 2.002319, and is known to extraordinary precision.^[2]^[3] The reason it is not precisely two is explained by quantum electrodynamics calculation of the anomalous magnetic dipole moment.^[4] The spin g-factor is related to spin frequency for a free electron in a magnetic field from cyclotron:

\nu _{s}={\frac {g}{2}}\nu _{c}

Electron orbital g-factor

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

{\boldsymbol {\mu }}_{{\text{L}}}=-g_{{\text{L}}}{\mu _{{\mathrm {B}}} \over \hbar }{\boldsymbol {L}},

where μ_L is the magnetic moment resulting from the orbital angular momentum of an electron, L is its orbital angular momentum, and μ_B is the Bohr magneton. For an infinite mass nucleus, 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 _{{\text{z}}}=g_{{\text{L}}}\mu _{{\text{B}}}m_{{\text{l}}}

which, since g_L = 1, is μ_Bm_l

For a finite mass nucleus, there is an effective g value^[5]

$g_{L}=1-{\frac {1}{M}}$

where M is the ratio of the nuclear mass to the electron mass.

Total angular momentum (Landé) g-factor

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

{\boldsymbol {\mu }}=-g_{{\text{J}}}{\mu _{{\text{B}}} \over \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.

Muon g-factor

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

{\boldsymbol \mu }=g{e \over 2m_{\mu }}{\boldsymbol {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.

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 is exactly explained by the Standard Model. The muon g-factor can, in theory, be affected by physics beyond the Standard Model, so it has been measured very precisely, in particular at the Brookhaven National Laboratory. In the E821 collaboration final report in November 2006, the experimental measured value is 7000200233184160000♠2.0023318416(13), compared to the theoretical prediction of 7000200233183610000♠2.0023318361(10).^[6] This is a difference of 3.4 standard deviations, suggesting that beyond-the-Standard-Model physics may be having an effect. The Brookhaven muon storage ring has been transported to Fermilab where the g−2 experiment will use it to make more precise measurements of muon g-factor.^[7]

Measured g-factor values

Currently accepted NIST g-factor values ^[8]
Particle	Symbol	g-factor	Uncertainty
electron	g_e	2999799768069563818♠−2.00231930436182	6987520000000000000♠0.00000000000052
muon	g_μ	2999799766815820000♠−2.0023318418	6991130000000000000♠0.0000000013
neutron	g_n	2999617391455000000♠−3.82608545	6993900000000000000♠0.00000090
proton	g_p	7000558569470199999♠+5.585694702	6992170000000000000♠0.000000017

The electron g-factor is one of the most precisely measured values in physics, with a relative standard uncertainty of 6987260000000000000♠2.6×10⁻¹³.

Notes and references

↑ Povh, Bogdan; Rith, Klaus; Scholz, Christoph; Zetsche, Frank (2013-04-17). Particles and Nuclei. ISBN 978-3-662-05023-1.
↑ Gabrielse, Gerald; Hanneke, David (October 2006). "Precision pins down the electron's magnetism". CERN Courier. 46 (8): 35–37.
↑ 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.
↑ Lamb, Willis E. (1952-01-15). "Fine Structure of the Hydrogen Atom. III". Physical Review. 85 (2): 259–276. doi:10.1103/PhysRev.85.259.
↑ Hagiwara, K.; Martin, A. D.; Nomura, Daisuke; Teubner, T. (2006-11-08). "Improved predictions for g−2 of the muon and α_QED(M2
Z)". Physics Letters B. 649 (2–3): 173–179. arXiv:hep-ph/0611102. Bibcode:2007PhLB..649..173H. doi:10.1016/j.physletb.2007.04.012.
↑ "Muon g-2". Muon-g-2.fnal.gov. Retrieved 2015-05-08.
↑ "CODATA values of the fundamental constants". NIST.