G-factor (physics)

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 electron spin g-factor, g_e, defined by $${\displaystyle {\mathit{\mu }}_{\mathsf{s}}=g_{\mathsf{e}}\frac{{\mu }_{\mathsf{B}}}{\hslash }\mathbf{S},}$$ where μ_s is the magnetic moment resulting from the spin of an electron, S is its spin angular momentum, and μ_B = ⁠e ħ/2 m_e⁠ is the Bohr magneton.

The value g_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, −2.00231930436092(36)‍ is known to extraordinary precision – one part in 10¹³.

Positive electron g-factors

In atomic physics, the electron spin g-factor is often defined with a positive sign and the negative sign is included in the relationship between the g-factor and the z component of the magnetic moment: $${\displaystyle {\mu }_{\mathsf{z}}=-g_{\mathsf{s}}{\mu }_{\mathsf{B}}{m}_{\mathsf{s}},}$$ where ${\displaystyle \hslash m_{\mathsf{s}}}$ are the eigenvalues of the S_z operator, meaning that m_s can take on values ±⁠1/2⁠. This convention has issues in complex cases and has not been adopted by the CODATA standard.

Electron orbital g-factor

Secondly, the electron orbital g-factor g_L is defined by $${\displaystyle {\mathit{\mu }}_L=-g_L\frac{{\mu }_{\mathsf{B}}}{\hslash }\mathbf{L},}$$ 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 magnetic moment is $${\displaystyle {\mu }_z=-g_L{\mu }_{\text{B}}{m}_{\ell };}$$ since g_L = 1, the result is −μ_B m_ℓ .

For a finite-mass nucleus, there is an effective g value $${\displaystyle g_L=1-\frac{1}{M},}$$ 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 $${\displaystyle |{\mathit{\mu }}_J|=g_J\frac{{\mu }_{\text{B}}}{\hslash }|\mathbf{J}|,}$$ where μ_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, 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. μ_J and J vectors are not collinear, so only their magnitudes can be compared.

Muon g-factor

The muon, like the electron, has a g-factor associated with its spin, given by the equation $${\displaystyle \mathit{\mu }=g\frac{e}{2m_{\mu }}\mathbf{S},}$$ 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 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 −2.0023318416(13), compared to the theoretical prediction of −2.00233183620(86). This is a difference of 3.4 standard deviations, 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 experiment used it to make more precise measurements of muon g-factor. On April 7, 2021, the Fermilab Muon g−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 4.2 standard deviations.

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 g-factor values

The electron g-factor is one of the most precisely measured values in physics.