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

Because theg-factor can be measured very precisely, and also calculated very precisely from theoretical models, small discrepancies in particles' measured and predictedg-factors are used as tests for theories inparticle physics, in particular theStandard Model.^[1]

The spin magnetic moment of a charged,spin⁠1/2⁠ particle that does not possess any internal structure (a Dirac particle) is given by^[2]$${\displaystyle \mathit{\mu }=g\frac{e}{2m}\mathbf{S},}$$whereμ is the spin magnetic moment of the particle,g is theg-factor of the particle,e is theelementary charge,m is the mass of the particle, andS is the spin angular momentum of the particle (with magnitude⁠ħ/2⁠ for Dirac particles).

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 theg-factor is undefined). Conventionally, the associatedg-factors are defined using thenuclear 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$${\displaystyle \mathit{\mu }=g\frac{{\mu }_{\mathsf{N}}}{\hslash }\mathbf{I}=g\frac{e}{2m_{\mathsf{p}}}\mathbf{I},}$$whereμ is the magnetic moment of the nucleon or nucleus resulting from its spin,g is the effectiveg-factor,I is its spin angular momentum,μ_N is the nuclear magneton,e is the elementary charge, andm_p is the proton rest mass.

There are three magnetic moments associated with anelectron: One from itsspin angular momentum, one from itsorbital angular momentum, and one from its total angular momentum (the quantum-mechanical sum of those two components). Corresponding to these three moments are three differentg-factors:

The most known of these is theelectron sping-factor (more often called simply theelectrong-factor)g_e, defined by$${\displaystyle {\mathit{\mu }}_{\mathsf{s}}=g_{\mathsf{e}}\frac{{\mu }_{\mathsf{B}}}{\hslash }\mathbf{S},}$$whereμ_s is the magnetic moment resulting from the spin of an electron,S is itsspin angular momentum, andμ_B =⁠e ħ/2m_e⁠ is theBohr magneton. In atomic physics, the electron sping-factor is often defined as theabsolute value ofg_e :$${\displaystyle g_{\mathsf{s}}=|g_{\mathsf{e}}|=-g_{\mathsf{e}}.}$$

Thez component of the magnetic moment then becomes$${\displaystyle {\mu }_{\mathsf{z}}=-g_{\mathsf{s}}{\mu }_{\mathsf{B}}{m}_{\mathsf{s}},}$$where${\displaystyle \hslash m_{\mathsf{s}}}$ are theeigenvalues of theS_zoperator, meaning thatm_s can take on values ±⁠1/2⁠.^[3]

The valueg_s is roughly equal to 2.002319 and is known to extraordinary precision – one part in 10¹³.^[4] The reason it is notprecisely two is explained byquantum electrodynamics calculation of theanomalous magnetic dipole moment.^[5]

Secondly, theelectron orbitalg-factorg_L is defined by$${\displaystyle {\mathit{\mu }}_L=-g_L\frac{{\mu }_{\mathsf{B}}}{\hslash }\mathbf{L},}$$whereμ_L is the magnetic moment resulting from the orbital angular momentum of an electron,L is its orbital angular momentum, andμ_B is theBohr magneton. For an infinite-mass nucleus, the value ofg_L is exactly equal to one, by a quantum-mechanical argument analogous to the derivation of theclassical magnetogyric ratio. For an electron in an orbital with amagnetic quantum numberm_ℓ, thez component of the orbital magnetic moment is$${\displaystyle {\mu }_z=-g_L{\mu }_{\text{B}}{m}_{\ell };}$$sinceg_L = 1, the resultis −μ_Bm_ℓ .

For a finite-mass nucleus, there is an effectiveg value^[6]$${\displaystyle g_L=1-\frac{1}{M},}$$whereM is the ratio of the nuclear mass to the electron mass.

Thirdly, theLandé g-factorg_J is defined by$${\displaystyle |{\mathit{\mu }}_J|=g_J\frac{{\mu }_{\text{B}}}{\hslash }|\mathbf{J}|,}$$whereμ_J is the total magnetic moment resulting from both spin and orbital angular momentum of an electron,J =L +S , is its total angular momentum, andμ_B is theBohr magneton. The value ofg_J is related tog_L andg_s by a quantum-mechanical argument; see the articleLandég-factor.μ_J andJ vectors are not collinear, so only their magnitudes can be compared.

Themuon, like the electron, has ag-factor associated with its spin, given by the equation$${\displaystyle \mathit{\mu }=g\frac{e}{2m_{\mu }}\mathbf{S},}$$whereμ is the magnetic moment resulting from the muon's spin,S is the spin angular momentum, andm_μ is the muon mass.

That the muong-factor is not quite the same as the electrong-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, inQED theory. These are entirely a result of the mass difference between the particles.

However, not all of the difference between theg-factors for electrons and muons is exactly explained by theStandard Model. The muong-factor can, in theory, be affected byphysics beyond the Standard Model, so it has been measured very precisely, in particular at theBrookhaven National Laboratory. In the E821 collaboration final report in November 2006, the experimental measured value is2.0023318416(13), compared to the theoretical prediction of2.00233183620(86).^[7] This is a difference of3.4standard deviations, suggesting that beyond-the-Standard-Model physics may be a contributory factor. The Brookhaven muon storage ring was transported toFermilab where theMuong–2 experiment used it to make more precise measurements of muong-factor. On April 7, 2021, theFermilab Muong−2 collaboration presented and published a new measurement of the muon magnetic anomaly.^[8] When the Brookhaven and Fermilab measurements are combined, the new world average differs from the theory prediction by4.2 standard deviations.

The electrong-factor is one of the most precisely measured values in physics.^[4]

• Anomalous magnetic dipole moment
• Electron magnetic moment
• Landég-factor

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• Particle physics
• Physical constants

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