G-factor (physics)

A -factor 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 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 in the definition. The two definitions coincide for the proton.
Because the -factor can be measured very precisely, and also calculated very precisely from theoretical models, small discrepancies in particles' measured and predicted -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, particle that does not possess any internal structure is given by
where ' is the spin magnetic moment of the particle, is the -factor of the particle, is the elementary charge, is the mass of the particle, and ' is the spin angular momentum of the particle.

Baryon or nucleus

Protons, neutrons, nuclei, and other composite baryonic particles have magnetic moments arising from their spin. Conventionally, the associated -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
where ' is the magnetic moment of the nucleon or nucleus resulting from its spin, is the effective -factor, ' is its spin angular momentum, _N is the nuclear magneton, is the elementary charge, and _p is the proton rest mass.

Calculation

Electron -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. Corresponding to these three moments are three different -factors:

Electron spin -factor

The most known of these is the electron spin -factor _e, defined by
where '_s is the magnetic moment resulting from the spin of an electron, ' is its spin angular momentum, and is the Bohr magneton. In atomic physics, the electron spin -factor is often defined as the absolute value of _e :
The component of the magnetic moment then becomes
where are the eigenvalues of the operator, meaning that _s can take on values ±.
The value _s is roughly equal to and is known to extraordinary precision – one part in. The reason it is not precisely two is explained by quantum electrodynamics calculation of the anomalous [magnetic dipole moment].

Electron orbital -factor

Secondly, the electron orbital -factor is defined by
where ' is the magnetic moment resulting from the orbital angular momentum of an electron, ' is its orbital angular momentum, and _B is the Bohr magneton. For an infinite-mass nucleus, the value of 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 , the component of the orbital magnetic moment is
since 1, the result
is
For a finite-mass nucleus, there is an effective value
where is the ratio of the nuclear mass to the electron mass.

Total angular momentum (Landé) -factor

Thirdly, the Landé g-factor is defined by
where ' is the total magnetic moment resulting from both spin and orbital angular momentum of an electron, is its total angular momentum, and _B is the Bohr magneton. The value of is related to and _s by a quantum-mechanical argument; see the article Landé -factor. ' and vectors are not collinear, so only their magnitudes can be compared.

Muon -factor

The muon, like the electron, has a -factor associated with its spin, given by the equation
where ' is the magnetic moment resulting from the muon's spin, ' is the spin angular momentum, and _μ is the muon mass.
That the muon -factor is not quite the same as the electron -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 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 -factors for electrons and muons is exactly explained by the Standard Model. The muon -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, compared to the theoretical prediction of. This is a difference of suggesting that beyond-the-Standard-Model physics may be a contributory factor. The Brookhaven muon storage ring was transported to Fermilab where the Muon –2 experiment used it to make more precise measurements of muon -factor. On April 7, 2021, the Fermilab Muon −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

Measured -factor values

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