g-factor (physics)

The spin magnetic moment of a charged, spin-1/2 particle that does not possess any internal structure (a Dirac particle) is given by{{cite book |url=https://books.google.com/books?id=HC_qCAAAQBAJ&pg=PA74 |title=Particles and Nuclei |isbn=978-3-662-05023-1 |last1=Povh |first1=Bogdan |last2=Rith |first2=Klaus |last3=Scholz |first3=Christoph |last4=Zetsche |first4=Frank |date=2013-04-17|publisher=Springer }}

$$\backslash boldsymbol\; \backslash mu\; =\; g\; \{e\; \backslash over\; 2m\}\; \backslash mathbf\; S\; ,$$

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 S is the spin angular momentum of the particle (with magnitude ħ/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 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

$$\backslash boldsymbol\{\backslash mu\}\; =\; g\; \{\backslash mu\_\backslash text\{N\}\; \backslash over\; \backslash hbar\}\; \{\backslash mathbf\{I\}\}\; =\; g\; \{e\; \backslash over\; 2\; m\_\backslash text\{p\}\}\; \backslash mathbf\{I\}\; ,$$

where μ is the magnetic moment of the nucleon or nucleus resulting from its spin, g is the effective g-factor, I is its spin angular momentum, μ_N is the nuclear magneton, e is the elementary charge, and m_p is the proton rest mass.

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:

The most known of these is the electron spin g-factor (more often called simply the electron g-factor) g_e, defined by

\boldsymbol{\mu}_\text{s} = g_\text{e} \frac{\mu_\text{B}}{\hbar} \mathbf{S},

where μ_s is the magnetic moment resulting from the spin of an electron, S is its spin angular momentum, and μ{{sub|B}} = eħ/2m{{sub|e}} is the Bohr magneton. In atomic physics, the electron spin g-factor is often defined as the absolute value of g_e:

g_\text{s} = |g_\text{e}| = -g_\text{e}.

The z component of the magnetic moment then becomes

\mu_\text{z} = -g_\text{s} \mu_\text{B} m_\text{s},

where $\backslash hbar\; m\_\backslash text\{s\}$ are the eigenvalues of the S_z operator, meaning that m_s can take on values $\backslash pm\; 1/2$.{{Cite book |last=Griffiths |first=David J. |title=Introduction to Quantum Mechanics |last2=Schroeter |first2=Darrell F. |date=2018 |publisher=Cambridge University Press |isbn=978-1-107-18963-8 |edition=3rd |location=Cambridge}}

The value g_s is roughly equal to 2.002319 and is known to extraordinary precision{{snd}} one part in 10¹³.{{Cite journal |last1=Fan |first1=X. |last2=Myers |first2=T. G. |last3=Sukra |first3=B. A. D. |last4=Gabrielse |first4=G. |date=2023-02-13 |title=Measurement of the Electron Magnetic Moment |url=https://link.aps.org/doi/10.1103/PhysRevLett.130.071801 |journal=Physical Review Letters |volume=130 |issue=7 |pages=071801 |doi=10.1103/PhysRevLett.130.071801 |pmid=36867820 |arxiv=2209.13084 |bibcode=2023PhRvL.130g1801F }} The reason it is not precisely two is explained by quantum electrodynamics calculation of the anomalous magnetic dipole moment.

{{cite journal

| title = A nonperturbative calculation of the electron's magnetic moment

| year = 2004

| journal = Nuclear Physics B

| volume = 703

| issue = 1–2

| pages = 333–362

| last1 = Brodsky

| first1 = S

| last2 = Franke

| first2 = V

| last3 = Hiller

| first3 = J

| last4 = McCartor

| first4 = G

| last5 = Paston

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| last6 = Prokhvatilov

| first6 = E

| doi = 10.1016/j.nuclphysb.2004.10.027

| arxiv = hep-ph/0406325

| bibcode = 2004NuPhB.703..333B

| s2cid = 118978489

}}

Secondly, the electron orbital g-factor g_L is defined by

\boldsymbol{\mu}_L = -g_L \frac{\mu_\text{B}}{\hbar} \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

\mu_z = -g_L \mu_\text{B} m_l,

which, since g_L = 1, is −μ_Bm_l.

For a finite-mass nucleus, there is an effective g value{{Cite journal | last=Lamb|first=Willis E. | date=1952-01-15 | title=Fine Structure of the Hydrogen Atom. III | journal=Physical Review | volume=85 | issue=2 | pages=259–276 | doi=10.1103/PhysRev.85.259 | pmid=17775407 | bibcode=1952PhRv...85..259L}}

g_L = 1 - \frac{1}{M},

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

Thirdly, the Landé g-factor g_J is defined by

|\boldsymbol{\mu}_J| = g_J \frac{\mu_\text{B}}{\hbar} |\mathbf{J}|,

where μ_J is the total magnetic moment resulting from both spin and orbital angular momentum of an electron, {{nowrap|1=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.

File:The muon g-2.svg is realized in nature, there will be corrections to g−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 g–2.]]

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

$$\backslash boldsymbol\; \backslash mu\; =\; g\; \{e\; \backslash over\; 2m\_\backslash mu\}\; \backslash mathbf\{S\}\; ,$$

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 {{val|2.0023318416|(13)}}, compared to the theoretical prediction of {{val|2.00233183620|(86)}}.

{{cite journal

| last1 = Hagiwara

| first1 = K. |last2= Martin|first2= A. D. |last3= Nomura|first3= Daisuke |last4= Teubner|first4= T.

| title = Improved predictions for g−2 of the muon and α_QED(M{{sup sub|2|Z}})

| year = 2007

| doi = 10.1016/j.physletb.2007.04.012

| journal = Physics Letters B

| volume = 649

| issue = 2–3

| pages = 173–179

| arxiv = hep-ph/0611102 | bibcode=2007PhLB..649..173H

| s2cid = 118565052

}} 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.

{{cite journal

| author = B. Abi

| display-authors = etal

| collaboration = Muon g−2 collaboration

| title = Measurement of the Positive Muon Anomalous Magnetic Moment to 0.46 ppm

| date = 7 April 2021

| doi = 10.1103/PhysRevLett.126.141801 |issn=0031-9007

| journal = Physical Review Letters

| volume = 126

| issue = 14

| pages = 141801

| pmid = 33891447

| arxiv = 2104.03281 | bibcode = 2021PhRvL.126n1801A

| s2cid = 233169085

}} When the Brookhaven and Fermilab measurements are combined, the new world average differs from the theory prediction by 4.2 standard deviations.

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

• Anomalous magnetic dipole moment
• Electron magnetic moment
• Landé g-factor

{{Reflist}}

• [http://physics.nist.gov/cuu/Constants/codata.pdf CODATA recommendations 2006]

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• {{Cite journal |last1=Gwinner |first1=Gerald |last2=Silwal |first2=Roshani |date=June 2022 |title=Tiny isotopic difference tests standard model of particle physics |url=https://www.nature.com/articles/d41586-022-01569-3 |journal=Nature |language=en |volume=606 |issue=7914 |pages=467–468 |doi=10.1038/d41586-022-01569-3|pmid=35705815 |s2cid=249710367 |url-access=subscription }}

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