# G-factor (physics)

{{#invoke:Hatnote|hatnote}} Template:Lowercase

A ** g-factor** (also called

**or**

*g*value**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

where * μ_{S}* is the total magnetic moment resulting from the spin of an electron,

*is its spin angular momentum, and*

**S***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}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.

^{[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

where * μ_{L}* is the total magnetic moment resulting from the orbital angular momentum of an electron,

*is the magnitude of its orbital angular momentum, and*

**L***μ*

_{B}is the Bohr magneton. The value of

*g*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}_{l}, the

*z*-component of the orbital angular momentum is

which, since *g _{L}* = 1, is just

*μ*

_{B}m

_{l}

#### Total angular momentum (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***is its total angular momentum, and*

**S***μ*

_{B}is the Bohr magneton. The value of

*g*is related to

_{J}*g*and

_{L}*g*by a quantum-mechanical argument; see the article Landé g-factor.

_{S}### 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,

*is the nuclear spin angular momentum,*

**I***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

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

*is the spin angular momentum, and*

**S***m*is the muon mass.

_{μ}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 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 Template:Val, compared to the theoretical prediction of Template:Val.^{[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 has been 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

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

Electron | Template:Gaps | Template:Gaps |

Muon | Template:Gaps | Template:Gaps |

Neutron | Template:Gaps | Template:Gaps |

Proton | Template:Gaps | Template:Gaps |

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

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- ↑ http://muon-g-2.fnal.gov/
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