g-factor (physics)

A Template:Mvar-factor (also called Template:Mvar value) is a dimensionless quantity that characterizes the magnetic moment and angular momentum of a whole atom, a particle, or a nucleus. It is the ratio of the magnetic moment (or, equivalently, the gyromagnetic ratio) of a particle to that expected of a classical particle of the same charge and angular momentum. In nuclear physics, the nuclear magneton replaces the classically expected magnetic moment (or gyromagnetic ratio) in the definition. The two definitions coincide for the proton.

Because the Template:Mvar-factor can be measured very precisely, and also calculated very precisely from theoretical models, small discrepancies in particles' measured and predicted Template:Mvar-factors are used as tests for theories in particle physics, in particular the Standard Model.

Definition

Dirac particle

The spin magnetic moment of a charged, Template:Nobr particle that does not possess any internal structure (a Dirac particle) is given by $$𝝁=g\frac{e}{2m}𝐒,$$ where Template:Mvar is the spin magnetic moment of the particle, Template:Mvar is the Template:Mvar-factor of the particle, Template:Mvar is the elementary charge, Template:Mvar is the mass of the particle, and Template:Math is the spin angular momentum of the particle (with magnitude Template:Sfrac for Dirac particles).

Baryon or nucleus

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 Template:Mvar-factor is undefined). Conventionally, the associated Template:Mvar-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 $$𝝁=g\frac{{\mu }_{𝖭}}{\hslash }𝐈=g\frac{e}{2m_{𝗉}}𝐈,$$ where Template:Mvar is the magnetic moment of the nucleon or nucleus resulting from its spin, Template:Mvar is the effective Template:Mvar-factor, Template:Math is its spin angular momentum, Template:Mvar_N is the nuclear magneton, Template:Mvar is the elementary charge, and Template:Mvar_p is the proton rest mass.

Calculation

Electron Template:Mvar-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 Template:Mvar-factors.

Electron spin Template:Mvar-factor

The electron spin Template:Math-factor, Template:Mvar_e, defined by $${𝝁}_{𝗌}=g_{𝖾}\frac{{\mu }_{𝖡}}{\hslash }𝐒,$$ where Template:Mvar_s is the magnetic moment resulting from the spin of an electron, Template:Math is its spin angular momentum, and Template:MvarTemplate:Sub = Template:Sfrac is the Bohr magneton.

The value Template:Mvar_e is almost equal to -2. The reason it is not precisely -2 is explained by quantum electrodynamics calculation of the anomalous magnetic dipole moment. The measured value, Template:Physconst is known to extraordinary precision – one part in Template:10^.

Positive electron g-factors

In atomic physics, the electron spin Template:Mvar-factor is often defined with a positive sign and the negative sign is included in the relationship between the Template:Mvar-factor and the Template:Mvar component of the magnetic moment: $${\mu }_{𝗓}=-g_{𝗌}{\mu }_{𝖡}{m}_{𝗌},$$ where $\hslash m_{𝗌}$ are the eigenvalues of the Template:Mvar operator, meaning that Template:Mvar_s can take on values ±Template:Small. This convention has issues in complex cases and has not been adopted by the CODATA standard.

Electron orbital Template:Mvar-factor

Secondly, the electron orbital Template:Math-factor Template:Mvar_{Template:Mvar} is defined by $${𝝁}_L=-g_L\frac{{\mu }_{𝖡}}{\hslash }𝐋,$$ where Template:Mvar_{Template:Mvar} is the magnetic moment resulting from the orbital angular momentum of an electron, Template:Math is its orbital angular momentum, and Template:Mvar_B is the Bohr magneton. For an infinite-mass nucleus, the value of Template:Mvar_{Template:Mvar} 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 Template:Mvar_{Template:Math}, the Template:Mvar component of the orbital magnetic moment is $${\mu }_z=-g_L{\mu }_{\text{B}}{m}_{\ell };$$ since Template:Mvar_{Template:Mvar} = 1, the result is Template:Nobr

For a finite-mass nucleus, there is an effective Template:Mvar value $$g_L=1-\frac{1}{M},$$ where Template:Mvar is the ratio of the nuclear mass to the electron mass.

Total angular momentum (Landé) Template:Mvar-factor

Thirdly, the Landé g-factor Template:Mvar_{Template:Mvar} is defined by $$|{𝝁}_J|=g_J\frac{{\mu }_{\text{B}}}{\hslash }|𝐉|,$$ where Template:Mvar_{Template:Mvar} is the total magnetic moment resulting from both spin and orbital angular momentum of an electron, Template:Nobr is its total angular momentum, and Template:Mvar_B is the Bohr magneton. The value of Template:Mvar_{Template:Mvar} is related to Template:Mvar_{Template:Mvar} and Template:Mvar_s by a quantum-mechanical argument; see the article [[Landé g-factor|Landé Template:Mvar-factor]]. Template:Mvar_{Template:Mvar} and Template:Math vectors are not collinear, so only their magnitudes can be compared.

Muon Template:Mvar-factor

If supersymmetry is realized in nature, there will be corrections to Template:Mvar−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 Template:Mvar–2.

The muon, like the electron, has a Template:Mvar-factor associated with its spin, given by the equation $$𝝁=g\frac{e}{2m_{\mu }}𝐒,$$ where Template:Mvar is the magnetic moment resulting from the muon's spin, Template:Math is the spin angular momentum, and Template:Mvar_μ is the muon mass.

That the muon Template:Mvar-factor is not quite the same as the electron Template:Mvar-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 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 Template:Mvar-factors for electrons and muons is exactly explained by the Standard Model. The muon Template:Mvar-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. This is a difference of Template:Nobr suggesting that beyond-the-Standard-Model physics may be a contributory factor. The Brookhaven muon storage ring was transported to Fermilab where the [[Muon g−2|Muon Template:Mvar–2 experiment]] used it to make more precise measurements of muon Template:Mvar-factor. On April 7, 2021, the [[Muon g−2|Fermilab Muon Template:Mvar−2 collaboration]] presented and published a new measurement of the muon magnetic anomaly. When the Brookhaven and Fermilab measurements are combined, the new world average differs from the theory prediction by Template:Nobr

Sign conventions

The g-factors typically represent a linear relationship between a magnetic moment and an angular momentum. In particle physics and nuclear magnetic resonance, a positive g-factor means the magnetic moment and angular momentum point are parallel; a negative g-factor means anti-parallel. However, in electron spin resonance anti-parallel orientation is assigned a positive value. The sign convention is important in cases where multiple moments are present in a single problem, for example in high resolution Zeeman effect studies of molecules. The negative-means-antiparallel convention has been adopted by the CODATA standard organization.

Recommended Template:Mvar-factor values

CODATA recommended Template:Mvar-factor values

Particle	Symbol	Template:Mvar-factor	Relative standard uncertainty
electron	Template:Mvare	Template:Physconst	Template:Physconst
muon	Template:Mvarμ	Template:Physconst	Template:Physconst
proton	Template:Mvarp	+Template:Physconst	Template:Physconst
neutron	Template:Mvarn	Template:Physconst	Template:Physconst

The electron Template:Mvar-factor is one of the most precisely measured values in physics.

See also

• Anomalous magnetic dipole moment
• Electron magnetic moment
• [[Landé g-factor|Landé Template:Mvar-factor]]

External links

• Template:Commons category-inline