Electron magnetic dipole moment
Encyclopedia
In atomic physics
, the electron magnetic dipole moment is the magnetic moment
of an electron
caused by its intrinsic property of spin
.
comes from two types of rotation: spin
and orbital motion. From classical electrodynamics, a rotating electrically charged
body creates a magnetic dipole
with magnetic poles of equal magnitude but opposite polarity. This analogy holds as an electron indeed behaves like a tiny bar magnet. One consequence is that an external magnetic field exerts a torque
on the electron magnetic moment
depending on its orientation with respect to the field.
If the electron is visualized as a classical charged particle literally rotating about an axis with angular momentum
, its magnetic dipole moment 
is given by:
Here the charge is −
, where 
is the elementary charge
. The mass is the electron
rest mass
. Note that the angular momentum
in this equation may be the spin angular momentum, the orbital angular momentum, or the total angular momentum. It turns out the classical result is off by a proportional factor for the spin magnetic moment. As a result, the classical result is corrected by multiplying it with a correction factor.
The dimensionless
correction factor g is known as the g-factor. Finally, it is customary to express the magnetic moment in terms of the Planck constant
and the Bohr magneton:
Here
is the Bohr magneton and 
is the reduced Planck constant.
is intrinsic for an electron. It is:
Here
is the electron spin angular momentum. The spin g-factor is approximately two: 
. The magnetic moment of an electron is approximately twice what it should be in classical mechanics. The factor of two difference implies that the electron appears to be twice as effective in producing a magnetic moment as the corresponding classical charged body.
The spin magnetic dipole moment is approximately one
because 
and the electron is a spin one-half particle: 
.
The z component of the electron magnetic moment is:
where mS is the spin quantum number. Note that
is a negative constant multiplied by the spin
, so the magnetic moment is antiparallel to the spin angular momentum.
The spin g-factor
comes from the Dirac equation
, a fundamental equation connecting the electron's spin with its electromagnetic properties. Reduction of the Dirac equation for an electron in a magnetic field to its non-relativistic limit yields the Schrödinger equation with a correction term which takes account of the interaction of the electron's intrinsic magnetic moment with the magnetic field giving the correct energy.
For the electron spin, the most accurate value for the spin g-factor has been experimentally determined to have the value 2.00231930419922 ± (1.5 × 10−12). Note that it is only two thousandths larger than the value from Dirac equation. The small correction is known as the anomalous magnetic dipole moment
of the electron; it arises from the electron's interaction with virtual photons in quantum electrodynamics
. In fact, one famous triumph of the Quantum Electrodynamics
theory is the accurate prediction of the electron g-factor. The most accurate value for the electron magnetic moment is -928.476377 × 10−26 ± 0.000023 × 10−26 J/T.
. Then the orbital magnetic dipole moment is:
Here
is the electron orbital g-factor and 
is the Bohr magneton. The value of 
is exactly equal to one, by a quantum-mechanical argument analogous to the derivation of the classical gyromagnetic ratio.
The g-factor
is known as the Landé g-factor
, which can be related to
and 
by quantum mechanics. See the article on Landé g-factor
.
atom, an electron
occupying the atomic orbital
, the magnetic dipole moment is given by:
Here
is the orbital angular momentum
,
and 
are the principal, azimuthal and magnetic quantum numbers.
The z-component of the orbital magnetic dipole moment for an electron with a magnetic quantum number
ml is given by:
Atomic physics
Atomic physics is the field of physics that studies atoms as an isolated system of electrons and an atomic nucleus. It is primarily concerned with the arrangement of electrons around the nucleus and...
, the electron magnetic dipole moment is the magnetic moment
Magnetic moment
The magnetic moment of a magnet is a quantity that determines the force that the magnet can exert on electric currents and the torque that a magnetic field will exert on it...
of an electron
Electron
The electron is a subatomic particle with a negative elementary electric charge. It has no known components or substructure; in other words, it is generally thought to be an elementary particle. An electron has a mass that is approximately 1/1836 that of the proton...
caused by its intrinsic property of spin
Spin (physics)
In quantum mechanics and particle physics, spin is a fundamental characteristic property of elementary particles, composite particles , and atomic nuclei.It is worth noting that the intrinsic property of subatomic particles called spin and discussed in this article, is related in some small ways,...
.
Magnetic moment of an electron
The electron is a charged particle. Its angular momentumAngular momentum
In physics, angular momentum, moment of momentum, or rotational momentum is a conserved vector quantity that can be used to describe the overall state of a physical system...
comes from two types of rotation: spin
Spin (physics)
In quantum mechanics and particle physics, spin is a fundamental characteristic property of elementary particles, composite particles , and atomic nuclei.It is worth noting that the intrinsic property of subatomic particles called spin and discussed in this article, is related in some small ways,...
and orbital motion. From classical electrodynamics, a rotating electrically charged
Electric charge
Electric charge is a physical property of matter that causes it to experience a force when near other electrically charged matter. Electric charge comes in two types, called positive and negative. Two positively charged substances, or objects, experience a mutual repulsive force, as do two...
body creates a magnetic dipole
Magnetic dipole
A magnetic dipole is the limit of either a closed loop of electric current or a pair of poles as the dimensions of the source are reduced to zero while keeping the magnetic moment constant. It is a magnetic analogue of the electric dipole, but the analogy is not complete. In particular, a magnetic...
with magnetic poles of equal magnitude but opposite polarity. This analogy holds as an electron indeed behaves like a tiny bar magnet. One consequence is that an external magnetic field exerts a torque
Torque
Torque, moment or moment of force , is the tendency of a force to rotate an object about an axis, fulcrum, or pivot. Just as a force is a push or a pull, a torque can be thought of as a twist....
on the electron magnetic moment
Magnetic moment
The magnetic moment of a magnet is a quantity that determines the force that the magnet can exert on electric currents and the torque that a magnetic field will exert on it...
depending on its orientation with respect to the field.
If the electron is visualized as a classical charged particle literally rotating about an axis with angular momentum
Angular momentum
In physics, angular momentum, moment of momentum, or rotational momentum is a conserved vector quantity that can be used to describe the overall state of a physical system...
, its magnetic dipole moment is given by:Here the charge is −
, where is the elementary chargeElementary charge
The elementary charge, usually denoted as e, is the electric charge carried by a single proton, or equivalently, the absolute value of the electric charge carried by a single electron. This elementary charge is a fundamental physical constant. To avoid confusion over its sign, e is sometimes called...
. The mass is the electron
Electron
The electron is a subatomic particle with a negative elementary electric charge. It has no known components or substructure; in other words, it is generally thought to be an elementary particle. An electron has a mass that is approximately 1/1836 that of the proton...
rest mass
. Note that the angular momentumAngular momentum
In physics, angular momentum, moment of momentum, or rotational momentum is a conserved vector quantity that can be used to describe the overall state of a physical system...
in this equation may be the spin angular momentum, the orbital angular momentum, or the total angular momentum. It turns out the classical result is off by a proportional factor for the spin magnetic moment. As a result, the classical result is corrected by multiplying it with a correction factor.The dimensionless
Dimensionless quantity
In dimensional analysis, a dimensionless quantity or quantity of dimension one is a quantity without an associated physical dimension. It is thus a "pure" number, and as such always has a dimension of 1. Dimensionless quantities are widely used in mathematics, physics, engineering, economics, and...
correction factor g is known as the g-factor. Finally, it is customary to express the magnetic moment in terms of the Planck constant
Planck constant
The Planck constant , also called Planck's constant, is a physical constant reflecting the sizes of energy quanta in quantum mechanics. It is named after Max Planck, one of the founders of quantum theory, who discovered it in 1899...
and the Bohr magneton:
Here
is the Bohr magneton and is the reduced Planck constant.Spin magnetic dipole moment
The spin magnetic momentSpin magnetic moment
-Basis for spin magnetic moments:A spin magnetic moment is induced by all charged particles. The electron is an example of one such charged particle. A spin magnetic moment is created because a particle has physical properties known as spin and electric charge. The spin within classical physics...
is intrinsic for an electron. It is:
Here
is the electron spin angular momentum. The spin g-factor is approximately two: . The magnetic moment of an electron is approximately twice what it should be in classical mechanics. The factor of two difference implies that the electron appears to be twice as effective in producing a magnetic moment as the corresponding classical charged body.The spin magnetic dipole moment is approximately one
because and the electron is a spin one-half particle: .The z component of the electron magnetic moment is:
where mS is the spin quantum number. Note that
is a negative constant multiplied by the spinSpin (physics)
In quantum mechanics and particle physics, spin is a fundamental characteristic property of elementary particles, composite particles , and atomic nuclei.It is worth noting that the intrinsic property of subatomic particles called spin and discussed in this article, is related in some small ways,...
, so the magnetic moment is antiparallel to the spin angular momentum.
The spin g-factor
comes from the Dirac equationDirac equation
The Dirac equation is a relativistic quantum mechanical wave equation formulated by British physicist Paul Dirac in 1928. It provided a description of elementary spin-½ particles, such as electrons, consistent with both the principles of quantum mechanics and the theory of special relativity, and...
, a fundamental equation connecting the electron's spin with its electromagnetic properties. Reduction of the Dirac equation for an electron in a magnetic field to its non-relativistic limit yields the Schrödinger equation with a correction term which takes account of the interaction of the electron's intrinsic magnetic moment with the magnetic field giving the correct energy.
For the electron spin, the most accurate value for the spin g-factor has been experimentally determined to have the value 2.00231930419922 ± (1.5 × 10−12). Note that it is only two thousandths larger than the value from Dirac equation. The small correction is known as the anomalous magnetic dipole moment
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...
of the electron; it arises from the electron's interaction with virtual photons in quantum electrodynamics
Quantum electrodynamics
Quantum electrodynamics 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 special relativity is achieved...
. In fact, one famous triumph of the Quantum Electrodynamics
Quantum electrodynamics
Quantum electrodynamics 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 special relativity is achieved...
theory is the accurate prediction of the electron g-factor. The most accurate value for the electron magnetic moment is -928.476377 × 10−26 ± 0.000023 × 10−26 J/T.
Orbital magnetic dipole moment
The revolution of an electron around an axis through another object, such as the nucleus, gives rise to the orbital magnetic dipole moment. Suppose that the angular momentum for the orbital motion is. Then the orbital magnetic dipole moment is:Here
is the electron orbital g-factor and is the Bohr magneton. The value of is exactly equal to one, by a quantum-mechanical argument analogous to the derivation of the classical gyromagnetic ratio.Total magnetic dipole moment
The total magnetic dipole moment resulting from both spin and orbital angular momenta of an electron is related to the total angular momentum J = L+S by:The g-factor
is known as the Landé g-factorLandé 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....
, which can be related to
and by quantum mechanics. See the article on Landé g-factorLandé 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....
.
Example: Hydrogen atom
For a hydrogenHydrogen
Hydrogen is the chemical element with atomic number 1. It is represented by the symbol H. With an average atomic weight of , hydrogen is the lightest and most abundant chemical element, constituting roughly 75% of the Universe's chemical elemental mass. Stars in the main sequence are mainly...
atom, an electron
Electron
The electron is a subatomic particle with a negative elementary electric charge. It has no known components or substructure; in other words, it is generally thought to be an elementary particle. An electron has a mass that is approximately 1/1836 that of the proton...
occupying the atomic orbital
, the magnetic dipole moment is given by:Here
is the orbital angular momentumAngular momentum
In physics, angular momentum, moment of momentum, or rotational momentum is a conserved vector quantity that can be used to describe the overall state of a physical system...
,
and are the principal, azimuthal and magnetic quantum numbers.The z-component of the orbital magnetic dipole moment for an electron with a magnetic quantum number
Magnetic quantum number
In atomic physics, the magnetic quantum number is the third of a set of quantum numbers which describe the unique quantum state of an electron and is designated by the letter m...
ml is given by:
See also
- Anomalous magnetic dipole momentAnomalous magnetic dipole momentIn 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...
- Nuclear magnetic momentNuclear magnetic momentThe nuclear magnetic moment is the magnetic moment of an atomic nucleus and arises from the spin of the protons and neutrons. It is mainly a magnetic dipole moment; the quadrupole moment does cause some small shifts in the hyperfine structure as well....
- Fine structureFine structureIn atomic physics, the fine structure describes the splitting of the spectral lines of atoms due to first order relativistic corrections.The gross structure of line spectra is the line spectra predicted by non-relativistic electrons with no spin. For a hydrogenic atom, the gross structure energy...
- Hyperfine structureHyperfine structureThe term hyperfine structure refers to a collection of different effects leading to small shifts and splittings in the energy levels of atoms, molecules and ions. The name is a reference to the fine structure which results from the interaction between the magnetic moments associated with electron...
- g-factor