*g*-factor (physics)

A ** g-factor** (also called

**or**

*g*value**dimensionless magnetic moment**) is a dimensionless quantity that characterizes the magnetic moment and angular momentum of an atom, 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.

## Contents

## DefinitionEdit

### Dirac particleEdit

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]}

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

*is the spin angular momentum of the particle (with magnitude*

**S***ħ*/2 for Dirac particles).

### Baryon or nucleusEdit

Protons, neutrons, nuclei and other composite baryonic particles have magnetic moments arising from their spin (both the spin and magnetic moment may be zero, in which case the *g*-factor is undefined). 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

where * μ* is the magnetic moment of the nucleon or nucleus resulting from its spin,

*g*is the effective

*g*-factor,

*is its spin angular momentum,*

**I***μ*

_{N}is the nuclear magneton,

*e*is the elementary charge and

*m*

_{p}is the proton rest mass.

## CalculationEdit

### Electron *g*-factorsEdit

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*-factorEdit

The most famous of these is the *electron spin g-factor* (more often called simply the *electron g-factor*), *g*_{e}, defined by

where **μ**_{s} is the magnetic moment resulting from the spin of an electron, * S* is its spin angular momentum, and is the Bohr magneton. In atomic physics, the electron spin

*g*-factor is often defined as the

*absolute value*or

*negative*of

*g*

_{e}:

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

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 of a cyclotron:

#### Electron orbital *g*-factorEdit

Secondly, the *electron orbital g-factor*, *g*_{L}, is defined by

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

which, since *g*_{L} = 1, is *μ*_{B}*m*_{l}

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

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

#### Total angular momentum (Landé) *g*-factorEdit

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***is its total angular momentum, and*

**S***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*-factorEdit

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

where * μ* is the magnetic moment resulting from the muon’s spin,

*is the spin angular momentum, and*

**S***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 3318416(13), compared to the theoretical prediction of 2.0023318361(10). 2.002^{[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 Muon *g*−2 experiment will use it to make more precise measurements of muon *g*-factor.^{[7]}

## Measured *g*-factor valuesEdit

Particle | Symbol | g-factor |
Uncertainty |
---|---|---|---|

electron | g_{e} |
31930436182 −2.002 | 00000000052 0.000^{[8]} |

muon | g_{μ} |
3318418 −2.002 | 0000013 0.000^{[9]} |

neutron | g_{n} |
08545 −3.826 | 00090 0.000^{[10]} |

proton | g_{p} |
694702 +5.585 | 000017 0.000^{[11]} |

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

## Notes and referencesEdit

**^**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.^{[dead link]}**^**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. Bibcode:1952PhRv...85..259L. doi:10.1103/PhysRev.85.259.**^**Hagiwara, K.; Martin, A. D.; Nomura, Daisuke; Teubner, T. (2007). "Improved predictions for*g*−2 of the muon and*α*_{QED}(*M*^{2}_{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 Value: Electron g factor".*NIST*. Retrieved 5 November 2017.**^**"CODATA Value: Muon g factor".*NIST*. Retrieved 5 November 2017.**^**"CODATA Value: Neutron g factor".*NIST*. Retrieved 5 November 2017.**^**"CODATA Value: Proton g factor".*NIST*. Retrieved 5 November 2017.**^**"CODATA values of the fundamental constants".*NIST*.