A ''g''-factor (also called ''g'' value or dimensionless magnetic moment) is a dimensionless quantity that characterizes the
magnetic moment
In electromagnetism, the magnetic moment is the magnetic strength and orientation of a magnet or other object that produces a magnetic field. Examples of objects that have magnetic moments include loops of electric current (such as electromagnets ...
and angular momentum of an atom, a particle or the
nucleus
Nucleus ( : nuclei) is a Latin word for the seed inside a fruit. It most often refers to:
*Atomic nucleus, the very dense central region of an atom
* Cell nucleus, a central organelle of a eukaryotic cell, containing most of the cell's DNA
Nucl ...
. It is essentially a proportionality constant that relates the different observed magnetic moments ''μ'' of a particle to their
angular momentum
In physics, angular momentum (rarely, moment of momentum or rotational momentum) is the rotational analog of linear momentum. It is an important physical quantity because it is a conserved quantity—the total angular momentum of a closed syst ...
quantum numbers
In quantum physics and chemistry, quantum numbers describe values of conserved quantities in the dynamics of a quantum system. Quantum numbers correspond to eigenvalues of operators that commute with the Hamiltonian—quantities that can be kno ...
and a unit of magnetic moment (to make it dimensionless), usually the
Bohr magneton
In atomic physics, the Bohr magneton (symbol ) is a physical constant and the natural unit for expressing the magnetic moment of an electron caused by its orbital or spin angular momentum.
The Bohr magneton, in SI units is defined as
\mu_\mat ...
or
nuclear magneton
The nuclear magneton (symbol ''μ'') is a physical constant of magnetic moment, defined in SI units by:
:\mu_\text =
and in Gaussian CGS units by:
:\mu_\text =
where:
:''e'' is the elementary charge,
:''ħ'' is the reduced Planck constant ...
.
Definition
Dirac particle
The spin magnetic moment of a charged, spin-1/2 particle that does not possess any internal structure (a Dirac particle) is given by
where ''μ'' is the spin magnetic moment of the particle, ''g'' is the ''g''-factor of the particle, ''e'' is the
elementary charge
The elementary charge, usually denoted by is the electric charge carried by a single proton or, equivalently, the magnitude of the negative electric charge carried by a single electron, which has charge −1 . This elementary charge is a fundame ...
, ''m'' is the mass of the particle, and S is the spin angular momentum of the particle (with magnitude ''ħ''/2 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 ''g''-factor is undefined). Conventionally, the associated ''g''-factors are defined using the
nuclear magneton
The nuclear magneton (symbol ''μ'') is a physical constant of magnetic moment, defined in SI units by:
:\mu_\text =
and in Gaussian CGS units by:
:\mu_\text =
where:
:''e'' is the elementary charge,
:''ħ'' is the reduced Planck constant ...
, 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 ''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
The nuclear magneton (symbol ''μ'') is a physical constant of magnetic moment, defined in SI units by:
:\mu_\text =
and in Gaussian CGS units by:
:\mu_\text =
where:
:''e'' is the elementary charge,
:''ħ'' is the reduced Planck constant ...
, ''e'' is the elementary charge and ''m''
p is the proton rest mass.
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 most known of these is the ''electron spin g-factor'' (more often called simply the ''electron g-factor''), ''g''
e, defined by
:
where μ
s is the magnetic moment resulting from the spin of an electron, S is its
spin angular momentum, and
is the
Bohr magneton
In atomic physics, the Bohr magneton (symbol ) is a physical constant and the natural unit for expressing the magnetic moment of an electron caused by its orbital or spin angular momentum.
The Bohr magneton, in SI units is defined as
\mu_\mat ...
. In atomic physics, the electron spin ''g''-factor is often defined as the ''absolute value'' or ''negative'' of ''g''
e:
:
The ''z''-component of the magnetic moment then becomes
:
The value ''g''
s is roughly equal to 2.002318, and is known to extraordinary precision. The reason it is not ''precisely'' two is explained by
quantum electrodynamics
In particle physics, quantum electrodynamics (QED) is the relativistic quantum field theory of electrodynamics. In essence, it describes how light and matter interact and is the first theory where full agreement between quantum mechanics and spec ...
calculation of the
anomalous magnetic dipole moment
In quantum electrodynamics, the anomalous magnetic moment of a particle is a contribution of effects of quantum mechanics, expressed by Feynman diagrams with loops, to the magnetic moment of that particle. (The ''magnetic moment'', also called '' ...
.
The spin ''g''-factor is related to spin frequency for a free electron in a magnetic field of a cyclotron:
:
Electron orbital ''g''-factor
Secondly, the ''electron orbital g-factor'', ''g''
''L'', is defined by
:
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
In atomic physics, the Bohr magneton (symbol ) is a physical constant and the natural unit for expressing the magnetic moment of an electron caused by its orbital or spin angular momentum.
The Bohr magneton, in SI units is defined as
\mu_\mat ...
. 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
In atomic physics, the magnetic quantum number () is one of the four quantum numbers (the other three being the principal, azimuthal, and spin) which describe the unique quantum state of an electron. The magnetic quantum number distinguishes the ...
''m''
l, the ''z''-component of the orbital angular momentum is
:
which, since ''g''
''L'' = 1, is −''μ''
B''m''
l
For a finite-mass nucleus, there is an effective ''g'' value
:
where ''M'' is the ratio of the nuclear mass to the electron mass.
Total angular momentum (Landé) ''g''-factor
Thirdly, the ''
Landé g-factor
In physics, the Landé ''g''-factor is a particular example of a ''g''-factor, namely for an electron with both spin and orbital angular momenta. It is named after Alfred Landé, who first described it in 1921.
In atomic physics, the Landé '' ...
'', ''g''
''J'', is defined by
:
where μ
J is the total magnetic moment resulting from both spin and orbital angular momentum of an electron, is its total angular momentum, and ''μ''
B is the
Bohr magneton
In atomic physics, the Bohr magneton (symbol ) is a physical constant and the natural unit for expressing the magnetic moment of an electron caused by its orbital or spin angular momentum.
The Bohr magneton, in SI units is defined as
\mu_\mat ...
. 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 colinear, 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
:
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
In quantum electrodynamics, the anomalous magnetic moment of a particle is a contribution of effects of quantum mechanics, expressed by Feynman diagrams with loops, to the magnetic moment of that particle. (The ''magnetic moment'', also called '' ...
. 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 Standard Model of particle physics is the theory describing three of the four known fundamental forces (electromagnetism, electromagnetic, weak interaction, weak and strong interactions - excluding gravity) in the universe and classifying a ...
. 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
Brookhaven National Laboratory (BNL) is a United States Department of Energy national laboratory located in Upton, Long Island, and was formally established in 1947 at the site of Camp Upton, a former U.S. Army base and Japanese internment c ...
. 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 3.4
standard deviation
In statistics, the standard deviation is a measure of the amount of variation or dispersion of a set of values. A low standard deviation indicates that the values tend to be close to the mean (also called the expected value) of the set, while ...
s, suggesting that beyond-the-Standard-Model physics may be having an effect. The Brookhaven muon storage ring was transported to
Fermilab
Fermi National Accelerator Laboratory (Fermilab), located just outside Batavia, Illinois, near Chicago, is a United States Department of Energy national laboratory specializing in high-energy particle physics. Since 2007, Fermilab has been operat ...
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.
Measured ''g''-factor values
The electron ''g''-factor is one of the most precisely measured values in physics.
See also
*
Anomalous magnetic dipole moment
In quantum electrodynamics, the anomalous magnetic moment of a particle is a contribution of effects of quantum mechanics, expressed by Feynman diagrams with loops, to the magnetic moment of that particle. (The ''magnetic moment'', also called '' ...
*
Electron magnetic moment
In atomic physics, the electron magnetic moment, or more specifically the electron magnetic dipole moment, is the magnetic moment of an electron resulting from its intrinsic properties of spin and electric charge. The value of the electron magnet ...
*
Landé ''g''-factor
Notes and references
Further reading
CODATA recommendations 2006
External links
*
*
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Atomic physics
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Particle physics
Physical constants