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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), permanent magnets, elementary particles (such as electrons), various molecules, and many astronomical objects (such as many planets, some moons, stars, etc).

A **magnet** is a material or object that produces a magnetic field. This magnetic field is invisible but is responsible for the most notable property of a magnet: a force that pulls on other ferromagnetic materials, such as iron, and attracts or repels other magnets.

A **magnetic field** is a vector field that describes the magnetic influence of electric charges in relative motion and magnetized materials. The effects of magnetic fields are commonly seen in permanent magnets, which pull on magnetic materials and attract or repel other magnets. Magnetic fields surround and are created by magnetized material and by moving electric charges such as those used in electromagnets. They exert forces on nearby moving electrical charges and torques on nearby magnets. In addition, a magnetic field that varies with location exerts a force on magnetic materials. Both the strength and direction of a magnetic field vary with location. As such, it is described mathematically as a vector field.

An **electric current** is the rate of flow of electric charge past a point or region. An electric current is said to exist when there is a net flow of electric charge through a region. In electric circuits this charge is often carried by electrons moving through a wire. It can also be carried by ions in an electrolyte, or by both ions and electrons such as in an ionized gas (plasma).

- Definition, units, and measurement
- Definition
- Units
- Measurement
- Relation to magnetization
- Models
- Magnetic pole model
- Amperian loop model
- Quantum mechanical model
- Effects of an external magnetic field
- Torque on a moment
- Force on a moment
- Magnetism
- Effects on environment
- Magnetic field of a magnetic moment
- Forces between two magnetic dipoles
- Torque of one magnetic dipole on another
- Theory underlying magnetic dipoles
- Magnetic potentials
- External magnetic field produced by a magnetic dipole moment
- Internal magnetic field of a dipole
- Relation to angular momentum
- Atoms, molecules, and elementary particles
- Magnetic moment of an atom
- Magnetic moment of an electron
- Magnetic moment of a nucleus
- Magnetic moment of a molecule
- Elementary particles
- See also
- References and notes
- External links

More precisely, the term *magnetic moment* normally refers to a system's **magnetic dipole moment**, the component of the magnetic moment that can be represented by an equivalent magnetic dipole: a magnetic north and south pole separated by a very small distance. The magnetic dipole component is sufficient for small enough magnets or for large enough distances. Higher order terms (such as the magnetic quadrupole moment) may be needed in addition to the dipole moment for extended objects.

A **magnetic dipole** is the limit of either a closed loop of electric current or a pair of poles as the size of the source is reduced to zero while keeping the magnetic moment constant. It is a magnetic analogue of the electric dipole, but the analogy is not perfect. In particular, a magnetic monopole, the magnetic analogue of an electric charge, has never been observed. Moreover, one form of magnetic dipole moment is associated with a fundamental quantum property—the spin of elementary particles.

The magnetic dipole moment of an object is readily defined in terms of the torque that object experiences in a given magnetic field. The same applied magnetic field creates larger torques on objects with larger magnetic moments. The strength (and direction) of this torque depends not only on the magnitude of the magnetic moment but also on its orientation relative to the direction of the magnetic field. The magnetic moment may be considered, therefore, to be a vector. The direction of the magnetic moment points from the south to north pole of the magnet (inside the magnet).

In mathematics and physics, a **vector** is an element of a vector space.

The magnetic field of a magnetic dipole is proportional to its magnetic dipole moment. The dipole component of an object's magnetic field is symmetric about the direction of its magnetic dipole moment, and decreases as the inverse cube of the distance from the object.

The magnetic moment can be defined as a vector relating the aligning torque on the object from an externally applied magnetic field to the field vector itself. The relationship is given by:^{ [1] }

In mathematics, physics, and engineering, a **Euclidean vector** is a geometric object that has magnitude and direction. Vectors can be added to other vectors according to vector algebra. A Euclidean vector is frequently represented by a line segment with a definite direction, or graphically as an arrow, connecting an *initial point**A* with a *terminal point**B*, and denoted by

**Torque**, **moment**, **moment of force** or "turning effect" is the rotational equivalent of linear force. The concept originated with the studies of Archimedes on the usage of levers. Just as a linear force is a push or a pull, a torque can be thought of as a twist to an object. Another definition of torque is the product of the magnitude of the force and the perpendicular distance of the line of action of force from the axis of rotation. The symbol for torque is typically , the lowercase Greek letter *tau*. When being referred to as moment of force, it is commonly denoted by *M*.

where * τ* is the torque acting on the dipole,

This definition is based on how one could, in principle, measure the magnetic moment of an unknown sample. For a current loop, this definition leads to the magnitude of the magnetic dipole moment equaling the product of the current times the area of the loop. Further, this definition allows the calculation of the expected magnetic moment for any known macroscopic current distribution.

An alternative definition is useful for thermodynamics calculations of the magnetic moment. In this definition, the magnetic dipole moment of a system is the negative gradient of its intrinsic energy, *U*_{int}, with respect to external magnetic field:

**Thermodynamics** is the branch of physics that deals with heat and temperature, and their relation to energy, work, radiation, and properties of matter. The behavior of these quantities is governed by the four laws of thermodynamics which convey a quantitative description using measurable macroscopic physical quantities, but may be explained in terms of microscopic constituents by statistical mechanics. Thermodynamics applies to a wide variety of topics in science and engineering, especially physical chemistry, chemical engineering and mechanical engineering, but also in fields as complex as meteorology.

Generically, the intrinsic energy includes the self-field energy of the system plus the energy of the internal workings of the system. For example, for a hydrogen atom in a 2p state in an external field, the self-field energy is negligible, so the internal energy is essentially the eigenenergy of the 2p state, which includes Coulomb potential energy and the kinetic energy of the electron. The interaction-field energy between the internal dipoles and external fields is not part of this internal energy.^{ [2] }

The unit for magnetic moment in International System of Units (SI) base units is A⋅m^{2}, where A is ampere (SI base unit of current) and m is meter (SI base unit of distance). This unit has equivalents in other SI derived units including:^{ [3] }^{ [4] }

where N is newton (SI derived unit of force), T is tesla (SI derived unit of magnetic flux density), and J is joule (SI derived unit of energy).^{ [5] } Although torque (N·m) and energy (J) are dimensionally equivalent, torques are never expressed in units of energy.^{ [6] }

In the CGS system, there are several different sets of electromagnetism units, of which the main ones are ESU, Gaussian, and EMU. Among these, there are two alternative (non-equivalent) units of magnetic dipole moment:

- (ESU)
- (Gaussian and EMU),

where statA is statamperes, cm is centimeters, erg is ergs, and G is gauss. The ratio of these two non-equivalent CGS units (EMU/ESU) is equal to the speed of light in free space, expressed in cm⋅s ^{−1}.

All formulae in this article are correct in SI units; they may need to be changed for use in other unit systems. For example, in SI units, a loop of current with current I and area A has magnetic moment IA (see below), but in Gaussian units the magnetic moment is *IA*/ *c* .

Other units for measuring the magnetic dipole moment include the Bohr magneton and the nuclear magneton.

The magnetic moments of objects are typically measured with devices called magnetometers. Not all magnetometers measure magnetic moment though. Some are configured to measure magnetic field instead. If the magnetic field surrounding an object is known well enough, though, then the magnetic moment can be calculated from that magnetic field.

The magnetic moment is a quantity that describes the magnetic strength of an entire object. Sometimes, though, it is useful or necessary to know how much of the net magnetic moment of the object is produced by a particular portion of that magnet. Therefore, it is useful to define the magnetization field as:

where and are the magnetic dipole moment and volume of a sufficiently small portion of the magnet . This equation is often represented using derivative notation such that

where *d***m** is the elementary magnetic moment and *dV* is the volume element. The net magnetic moment of the magnet therefore is

where the triple integral denotes integration over the volume of the magnet. For uniform magnetization (where both the magnitude and the direction of is the same for the entire magnet (such as a straight bar magnet) the last equation simplifies to:

where is the volume of the bar magnet.

The magnetization is often not listed as a material parameter for commercially available ferromagnetic materials, though. Instead the parameter that is listed is residual flux density (or remanence), denoted . The formula needed in this case to calculate in (units of A⋅m^{2}) is:

- ,

where:

- is the residual flux density, expressed in teslas.
- is the volume (m
^{3}) of the magnet. - is the permeability of vacuum.
^{ [7] }

The preferred classical explanation of a magnetic moment has changed over time. Before the 1930s, textbooks explained the moment using hypothetical magnetic point charges. Since then, most have defined it in terms of Ampèrian currents.^{ [8] } In magnetic materials, the cause of the magnetic moment are the spin and orbital angular momentum states of the electrons, and varies depending on whether atoms in one region are aligned with atoms in another.

The sources of magnetic moments in materials can be represented by poles in analogy to electrostatics. This is sometimes known as the Gilbert model.^{ [9] } In this model, a small magnet is modeled by a pair of magnetic poles of equal magnitude but opposite polarity. Each pole is the source of magnetic force which weakens with distance. Since magnetic poles always come in pairs, their forces partially cancel each other because while one pole pulls, the other repels. This cancellation is greatest when the poles are close to each other i.e. when the bar magnet is short. The magnetic force produced by a bar magnet, at a given point in space, therefore depends on two factors: the strength p of its poles (*magnetic pole strength*), and the vector **l** separating them. The magnetic dipole moment **m** is related to the fictitious poles as^{ [8] }

It points in the direction from South to North pole. The analogy with electric dipoles should not be taken too far because magnetic dipoles are associated with angular momentum (see Relation to angular momentum). Nevertheless, magnetic poles are very useful for magnetostatic calculations, particularly in applications to ferromagnets.^{ [8] } Practitioners using the magnetic pole approach generally represent the magnetic field by the irrotational field **H**, in analogy to the electric field **E**.

After Hans Christian Ørsted discovered that electric currents produce a magnetic field and André-Marie Ampère discovered that electric currents attract and repel each other similar to magnets, it was natural to hypothesize that all magnetic fields are due to electric current loops. In this model developed by Ampère, the elementary magnetic dipole that makes up all magnets is a sufficiently small amperian loop of current I. The dipole moment of this loop is

where *S* is the area of the loop. The direction of the magnetic moment is in a direction normal to the area enclosed by the current consistent with the direction of the current using the right hand rule.

The magnetic dipole moment can be calculated for a localized (does not extend to infinity) current distribution assuming that we know all of the currents involved. Conventionally, the derivation starts from a multipole expansion of the vector potential. This leads to the definition of the magnetic dipole moment as:

where × is the vector cross product, **r** is the position vector, and **j** is the electric current density and the integral is a volume integral.^{ [10] } When the current density in the integral is replaced by a loop of current I in a plane enclosing an area S then the volume integral becomes a line integral and the resulting dipole moment becomes

which is how the magnetic dipole moment for an Amperian loop is derived.

Practitioners using the current loop model generally represent the magnetic field by the solenoidal field **B**, analogous to the electrostatic field **D**.

A generalization of the above current loop is a coil, or solenoid. Its moment is the vector sum of the moments of individual turns. If the solenoid has N identical turns (single-layer winding) and vector area **S**,

When calculating the magnetic moments of materials or molecules on the microscopic level it is often convenient to use a third model for the magnetic moment that exploits the linear relationship between the angular momentum and the magnetic moment of a particle. While this relation is straight forward to develop for macroscopic currents using the amperian loop model (see below), neither the magnetic pole model nor the amperian loop model truly represents what is occurring at the atomic and molecular levels. At that level quantum mechanics must be used. Fortunately, the linear relationship between the magnetic dipole moment of a particle and its angular momentum still holds; although it is different for each particle. Further, care must be used to distinguish between the intrinsic angular momentum (or spin) of the particle and the particle's orbital angular momentum. See below for more details.

The torque on an object having a magnetic dipole moment in a uniform magnetic field is:

- .

This is valid for the moment due to any localized current distribution provided that the magnetic field is uniform. For non-uniform B the equation is also valid for the torque about the center of the magnetic dipole provided that the magnetic dipole is small enough.^{ [11] }

An electron, nucleus, or atom placed in a uniform magnetic field will precess with a frequency known as the Larmor frequency. See Resonance.

A magnetic moment in an externally produced magnetic field has a potential energy U:

In a case when the external magnetic field is non-uniform, there will be a force, proportional to the magnetic field gradient, acting on the magnetic moment itself. There are two expressions for the force acting on a magnetic dipole, depending on whether the model used for the dipole is a current loop or two monopoles (analogous to the electric dipole).^{ [12] } The force obtained in the case of a current loop model is

- .

In the case of a pair of monopoles being used (i.e. electric dipole model), the force is

- .

And one can be put in terms of the other via the relation

- .

In all these expressions **m** is the dipole and **B** is the magnetic field at its position. Note that if there are no currents or time-varying electrical fields ∇ × **B** = 0 and the two expressions agree.

In addition, an applied magnetic field can change the magnetic moment of the object itself; for example by magnetizing it. This phenomenon is known as magnetism. An applied magnetic field can flip the magnetic dipoles that make up the material causing both paramagnetism and ferromagnetism. Additionally, the magnetic field can affect the currents that create the magnetic fields (such as the atomic orbits) which causes diamagnetism.

Any system possessing a net magnetic dipole moment **m** will produce a dipolar magnetic field (described below) in the space surrounding the system. While the net magnetic field produced by the system can also have higher-order multipole components, those will drop off with distance more rapidly, so that only the dipole component will dominate the magnetic field of the system at distances far away from it.

The magnetic field of a magnetic dipole depends on the strength and direction of a magnet's magnetic moment but drops off as the cube of the distance such that:

where is the magnetic field produced by the magnet and is a vector from the center of the magnetic dipole to the location where the magnetic field is measured. The inverse cube nature of this equation is more readily seen by expressing the location vector as the product of its magnitude times the unit vector in its direction () so that:

The equivalent equations for the magnetic -field are the same except for a multiplicative factor of *μ*_{0} = 4*π*×10^{−7} H/m , where *μ*_{0} is known as the vacuum permeability. For example:

As discussed earlier, the force exerted by a dipole loop with moment **m**_{1} on another with moment **m**_{2} is

where **B**_{1} is the magnetic field due to moment **m**_{1}. The result of calculating the gradient is^{ [13] }^{ [14] }

where **r̂** is the unit vector pointing from magnet 1 to magnet 2 and *r* is the distance. An equivalent expression is^{ [14] }

The force acting on **m**_{1} is in the opposite direction.

The torque of magnet 1 on magnet 2 is

The magnetic field of any magnet can be modeled by a series of terms for which each term is more complicated (having finer angular detail) than the one before it. The first three terms of that series are called the monopole (represented by an isolated magnetic north or south pole) the dipole (represented by two equal and opposite magnetic poles), and the quadrupole (represented by four poles that together form two equal and opposite dipoles). The magnitude of the magnetic field for each term decreases progressively faster with distance than the previous term, so that at large enough distances the first non-zero term will dominate.

For many magnets the first non-zero term is the magnetic dipole moment. (To date, no isolated magnetic monopoles have been experimentally detected.) A magnetic dipole is the limit of either a current loop or a pair of poles as the dimensions of the source are reduced to zero while keeping the moment constant. As long as these limits only apply to fields far from the sources, they are equivalent. However, the two models give different predictions for the internal field (see below).

Traditionally, the equations for the magnetic dipole moment (and higher order terms) are derived from theoretical quantities called magnetic potentials ^{ [15] } which are simpler to deal with mathematically then the magnetic fields.

In the magnetic pole model, the relevant magnetic field is the demagnetizing field . Since the demagnetizing portion of does not include, by definition, the part of due to free currents, there exists a magnetic scalar potential such that

- .

In the amperian loop model, the relevant magnetic field is the magnetic induction . Since magnetic monopoles do not exist, there exists a magnetic vector potential such that

Both of these potentials can be calculated for any arbitrary current distribution (for the amperian loop model) or magnetic charge distribution (for the magnetic charge model) provided that these are limited to a small enough region to give:

where is the current density in the amperian loop model, is the magnetic pole strength density in analogy to the electric charge density that leads to the electric potential, and the integrals are the volume (triple) integrals over the coordinates that make up . The denominators of these equation can be expanded using the multipole expansion to give a series of terms that have larger of power of distances in the denominator. The first nonzero term, therefore, will dominate for large distances. The first non-zero term for the vector potential is:

where is:

where × is the vector cross product, **r** is the position vector, and **j** is the electric current density and the integral is a volume integral.

In the magnetic pole perspective, the first non-zero term of the scalar potential is

Here may be represented in terms of the magnetic pole strength density but is more usefully expressed in terms of the magnetization field as:

The same symbol is used for both equations since they produce equivalent results outside of the magnet.

The magnetic flux density for a magnetic dipole in the amperian loop model, therefore, is

Further, the magnetic field strength is

The two models for a dipole (current loop and magnetic poles) give the same predictions for the magnetic field far from the source. However, inside the source region, they give different predictions. The magnetic field between poles (see figure for Magnetic pole definition) is in the opposite direction to the magnetic moment (which points from the negative charge to the positive charge), while inside a current loop it is in the same direction (see the figure to the right). The limits of these fields must also be different as the sources shrink to zero size. This distinction only matters if the dipole limit is used to calculate fields inside a magnetic material.^{ [8] }

If a magnetic dipole is formed by making a current loop smaller and smaller, but keeping the product of current and area constant, the limiting field is

Unlike the expressions in the previous section, this limit is correct for the internal field of the dipole.^{ [8] }^{ [16] }

If a magnetic dipole is formed by taking a "north pole" and a "south pole", bringing them closer and closer together but keeping the product of magnetic pole charge and distance constant, the limiting field is^{ [8] }

These fields are related by **B** = *μ*_{0}(**H** + **M**), where **M**(**r**) = **m***δ*(**r**) is the magnetization.

The magnetic moment has a close connection with angular momentum called the *gyromagnetic effect*. This effect is expressed on a macroscopic scale in the Einstein–de Haas effect, or "rotation by magnetization," and its inverse, the Barnett effect, or "magnetization by rotation."^{ [1] } Further, a torque applied to a relatively isolated magnetic dipole such as an atomic nucleus can cause it to precess (rotate about the axis of the applied field). This phenomenon is used in nuclear magnetic resonance.

Viewing a magnetic dipole as current loop brings out the close connection between magnetic moment and angular momentum. Since the particles creating the current (by rotating around the loop) have charge and mass, both the magnetic moment and the angular momentum increase with the rate of rotation. The ratio of the two is called the gyromagnetic ratio or so that:^{ [17] }^{ [18] }

where is the angular momentum of the particle or particles that are creating the magnetic moment.

In the amperian loop model, which applies for macroscopic currents, the gyromagnetic ratio is one half of the charge-to-mass ratio. This can be shown as follows. The angular momentum of a moving charged particle is defined as:

where μ is the mass of the particle and **v** is the particle's velocity. The angular momentum of the very large number of charged particles that make up a current therefore is:

where ρ is the mass density of the moving particles. By convention the direction of the cross product is given by the right-hand rule.^{ [19] }

This is similar to the magnetic moment created by the very large number of charged particles that make up that current:

where and is the charge density of the moving charged particles.

Comparing the two equations results in:

where is the charge of the particle and is the mass of the particle.

Even though atomic particles cannot be accurately described as orbiting (and spinning) charge distributions of uniform charge-to-mass ratio, this general trend can be observed in the atomic world so that:

where the *g*-factor depends on the particle and configuration. For example the *g*-factor for the magnetic moment due to an electron orbiting a nucleus is one while the *g*-factor for the magnetic moment of electron due to its intrinsic angular momentum (spin) is a little larger than 2. The *g*-factor of atoms and molecules must account for the orbital and intrinsic moments of its electrons and possibly the intrinsic moment of its nuclei as well.

In the atomic world the angular momentum (spin) of a particle is an integer (or half-integer in the case of spin) multiple of the reduced Planck constant ħ. This is the basis for defining the magnetic moment units of Bohr magneton (assuming charge-to-mass ratio of the electron) and nuclear magneton (assuming charge-to-mass ratio of the proton). See electron magnetic moment and Bohr magneton for more details.

Fundamentally, contributions to any system's magnetic moment may come from sources of two kinds: motion of electric charges, such as electric currents; and the intrinsic magnetism of elementary particles, such as the electron.

Contributions due to the sources of the first kind can be calculated from knowing the distribution of all the electric currents (or, alternatively, of all the electric charges and their velocities) inside the system, by using the formulas below. On the other hand, the magnitude of each elementary particle's intrinsic magnetic moment is a fixed number, often measured experimentally to a great precision. For example, any electron's magnetic moment is measured to be −9.284764×10^{−24} J/T.^{ [20] } The direction of the magnetic moment of any elementary particle is entirely determined by the direction of its spin, with the negative value indicating that any electron's magnetic moment is antiparallel to its spin.

The net magnetic moment of any system is a vector sum of contributions from one or both types of sources. For example, the magnetic moment of an atom of hydrogen-1 (the lightest hydrogen isotope, consisting of a proton and an electron) is a vector sum of the following contributions:

- the intrinsic moment of the electron,
- the orbital motion of the electron around the proton,
- the intrinsic moment of the proton.

Similarly, the magnetic moment of a bar magnet is the sum of the contributing magnetic moments, which include the intrinsic and orbital magnetic moments of the unpaired electrons of the magnet's material and the nuclear magnetic moments.

For an atom, individual electron spins are added to get a total spin, and individual orbital angular momenta are added to get a total orbital angular momentum. These two then are added using angular momentum coupling to get a total angular momentum. For an atom with no nuclear magnetic moment, the magnitude of the atomic dipole moment is then^{ [21] }

where j is the total angular momentum quantum number, g_{J} is the Landé *g*-factor, and *μ*_{B} is the Bohr magneton. The component of this magnetic moment along the direction of the magnetic field is then^{ [22] }

where m is called the magnetic quantum number or the *equatorial* quantum number, which can take on any of 2*j* + 1 values:^{ [23] }

- .
^{[ dubious – discuss ]}

The negative sign occurs because electrons have negative charge.

Due to the angular momentum, the dynamics of a magnetic dipole in a magnetic field differs from that of an electric dipole in an electric field. The field does exert a torque on the magnetic dipole tending to align it with the field. However, torque is proportional to rate of change of angular momentum, so precession occurs: the direction of spin changes. This behavior is described by the Landau–Lifshitz–Gilbert equation:^{ [24] }^{ [25] }

where γ is the gyromagnetic ratio, **m** is the magnetic moment, λ is the damping coefficient and **H**_{eff} is the effective magnetic field (the external field plus any self-induced field). The first term describes precession of the moment about the effective field, while the second is a damping term related to dissipation of energy caused by interaction with the surroundings.

Electrons and many elementary particles also have intrinsic magnetic moments, an explanation of which requires a quantum mechanical treatment and relates to the intrinsic angular momentum of the particles as discussed in the article Electron magnetic moment. It is these intrinsic magnetic moments that give rise to the macroscopic effects of magnetism, and other phenomena, such as electron paramagnetic resonance.

The magnetic moment of the electron is

where *μ*_{B} is the Bohr magneton, **S** is electron spin, and the *g*-factor g_{S} is 2 according to Dirac's theory, but due to quantum electrodynamic effects it is slightly larger in reality: 2.00231930436. The deviation from 2 is known as the anomalous magnetic dipole moment.

Again it is important to notice that **m** is a negative constant multiplied by the spin, so the magnetic moment of the electron is antiparallel to the spin. This can be understood with the following classical picture: if we imagine that the spin angular momentum is created by the electron mass spinning around some axis, the electric current that this rotation creates circulates in the opposite direction, because of the negative charge of the electron; such current loops produce a magnetic moment which is antiparallel to the spin. Hence, for a positron (the anti-particle of the electron) the magnetic moment is parallel to its spin.

The nuclear system is a complex physical system consisting of nucleons, i.e., protons and neutrons. The quantum mechanical properties of the nucleons include the spin among others. Since the electromagnetic moments of the nucleus depend on the spin of the individual nucleons, one can look at these properties with measurements of nuclear moments, and more specifically the nuclear magnetic dipole moment.

Most common nuclei exist in their ground state, although nuclei of some isotopes have long-lived excited states. Each energy state of a nucleus of a given isotope is characterized by a well-defined magnetic dipole moment, the magnitude of which is a fixed number, often measured experimentally to a great precision. This number is very sensitive to the individual contributions from nucleons, and a measurement or prediction of its value can reveal important information about the content of the nuclear wave function. There are several theoretical models that predict the value of the magnetic dipole moment and a number of experimental techniques aiming to carry out measurements in nuclei along the nuclear chart.

Any molecule has a well-defined magnitude of magnetic moment, which may depend on the molecule's energy state. Typically, the overall magnetic moment of a molecule is a combination of the following contributions, in the order of their typical strength:

- magnetic moments due to its unpaired electron spins (paramagnetic contribution), if any
- orbital motion of its electrons, which in the ground state is often proportional to the external magnetic field (diamagnetic contribution)
- the combined magnetic moment of its nuclear spins, which depends on the nuclear spin configuration.

- The dioxygen molecule, O
_{2}, exhibits strong paramagnetism, due to unpaired spins of its outermost two electrons. - The carbon dioxide molecule, CO
_{2}, mostly exhibits diamagnetism, a much weaker magnetic moment of the electron orbitals that is proportional to the external magnetic field. The nuclear magnetism of a magnetic isotope such as^{13}C or^{17}O will contribute to the molecule's magnetic moment. - The dihydrogen molecule, H
_{2}, in a weak (or zero) magnetic field exhibits nuclear magnetism, and can be in a para- or an ortho- nuclear spin configuration. - Many transition metal complexes are magnetic. The spin-only formula is a good first approximation for high-spin complexes of first-row transition metals.
^{ [26] }

Number of

unpaired

electronsSpin-only

moment (μ_{B})1 1.73 2 2.83 3 3.87 4 4.90 5 5.92

In atomic and nuclear physics, the Greek symbol μ represents the magnitude of the magnetic moment, often measured in Bohr magnetons or nuclear magnetons, associated with the intrinsic spin of the particle and/or with the orbital motion of the particle in a system. Values of the intrinsic magnetic moments of some particles are given in the table below:

Intrinsic magnetic moments and spins of some elementary particles ^{ [27] }Particle Magnetic dipole moment

(10^{−27}J⋅T^{−1})Spin quantum number

(dimensionless)electron (e ^{−})−9284.764 1/2 proton (H ^{+})14.106067 1/2 neutron (n) −9.66236 1/2 muon (μ ^{−})−44.904478 1/2 deuteron ( ^{2}H^{+})4.3307346 1 triton ( ^{3}H^{+})15.046094 1/2 helion ( ^{3}He^{2+})−10.746174 1/2 alpha particle ( ^{4}He^{2+})0 0

For relation between the notions of magnetic moment and magnetization see magnetization.

- 1 2 Cullity, B. D.; Graham, C. D. (2008).
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*Classical electrodynamics*(2nd ed.). New York: Wiley. ISBN 9780471431329. - ↑ Jackson, John David (1975).
*Classical electrodynamics*(2nd ed.). New York: Wiley. p. 184. ISBN 978-0-471-43132-9. - ↑ Krey, Uwe; Owen, Anthony (2007).
*Basic Theoretical Physics*. Springer. pp. 151–152. ISBN 978-3-540-36804-5. - ↑ Buxton, Richard B. (2002).
*Introduction to functional magnetic resonance imaging*. Cambridge University Press. p. 136. ISBN 978-0-521-58113-4. - ↑ Feynman, Richard P.; Leighton, Robert B.; Sands, Matthew (2006).
*The Feynman Lectures on Physics*.**2**. p. 13–12. ISBN 978-0-8053-9045-2. - ↑ "CODATA Value: electron magnetic moment".
*physics.nist.gov*. - ↑ Tilley, R. J. D. (2004).
*Understanding Solids*. John Wiley and Sons. p. 368. ISBN 978-0-470-85275-0. - ↑ Tipler, Paul Allen; Llewellyn, Ralph A. (2002).
*Modern Physics*(4th ed.). Macmillan. p. 310. ISBN 978-0-7167-4345-3. - ↑ Crowther, J. A. (1949).
*Ions, Electrons and Ionizing Radiations*(8th ed.). London: Edward Arnold. p. 270. - ↑ Rice, Stuart Alan (2004).
*Advances in chemical physics*. Wiley. pp. 208ff. ISBN 978-0-471-44528-9. - ↑ Steiner, Marcus (2004).
*Micromagnetism and Electrical Resistance of Ferromagnetic Electrodes for Spin Injection Devices*. Cuvillier Verlag. p. 6. ISBN 978-3-86537-176-8. - ↑ Figgis, B. N.; Lewis, J. (1960). "The magnetochemistry of complex compounds". In Lewis, J.; Wilkins, R. G. (eds.).
*Modern coordination chemistry: Principles and methods*. New York: Interscience. pp. 405–407. - ↑ "Search results matching 'magnetic moment'".
*CODATA internationally recommended values of the Fundamental Physical Constants*. National Institute of Standards and Technology. Retrieved 11 May 2012.

- Bowtell, Richard (2009). "
*μ*– Magnetic Moment".*Sixty Symbols*. Brady Haran for the University of Nottingham.

In electromagnetism, there are two kinds of **dipoles**:

In quantum mechanics, a **Hamiltonian** is an operator corresponding to the sum of the kinetic energies plus the potential energies for all the particles in the system. It is usually denoted by , but also or to highlight its function as an operator. Its spectrum is the set of possible outcomes when one measures the total energy of a system. Because of its close relation to the time-evolution of a system, it is of fundamental importance in most formulations of quantum theory.

**Magnetism** is a class of physical phenomena that are mediated by magnetic fields. Electric currents and the magnetic moments of elementary particles give rise to a magnetic field, which acts on other currents and magnetic moments. The most familiar effects occur in ferromagnetic materials, which are strongly attracted by magnetic fields and can be magnetized to become permanent magnets, producing magnetic fields themselves. Only a few substances are ferromagnetic; the most common ones are iron, cobalt and nickel and their alloys. The prefix *ferro-* refers to iron, because permanent magnetism was first observed in lodestone, a form of natural iron ore called magnetite, Fe_{3}O_{4}.

**Paramagnetism** is a form of magnetism whereby some materials are weakly attracted by an externally applied magnetic field, and form internal, induced magnetic fields in the direction of the applied magnetic field. In contrast with this behavior, diamagnetic materials are repelled by magnetic fields and form induced magnetic fields in the direction opposite to that of the applied magnetic field. Paramagnetic materials include most chemical elements and some compounds; they have a relative magnetic permeability slightly greater than 1 and hence are attracted to magnetic fields. The magnetic moment induced by the applied field is linear in the field strength and rather weak. It typically requires a sensitive analytical balance to detect the effect and modern measurements on paramagnetic materials are often conducted with a SQUID magnetometer.

In atomic physics, **hyperfine structure** refers to small shifts and splittings in the energy levels of atoms, molecules, and ions, due to interaction between the state of the nucleus and the state of the electron clouds.

In atomic physics, the **spin quantum number** is a quantum number that parameterizes the intrinsic angular momentum of a given particle. The **spin quantum number** is the fourth of a set of quantum numbers, which completely describe the quantum state of an electron. It is designated by the letter s. It describes the energy, shape and orientation of orbitals. The name comes from a physical spinning about an axis that was proposed by Uhlenbeck and Goudsmit. However this simplistic picture was quickly realized to be physically impossible and replaced by a more abstract quantum-mechanical description.

In physics, the **gyromagnetic ratio** of a particle or system is the ratio of its magnetic moment to its angular momentum, and it is often denoted by the symbol *γ*, gamma. Its SI unit is the radian per second per tesla (rad⋅s^{−1}⋅T^{−1}) or, equivalently, the coulomb per kilogram (C⋅kg^{−1}).

In atomic physics, the **electron magnetic moment**, or more specifically the **electron magnetic dipole moment**, is the magnetic moment of an electron caused by its intrinsic properties of spin and electric charge. The value of the electron magnetic moment is approximately −9.284764×10^{−24} J/T. The electron magnetic moment has been measured to an accuracy of 7.6 parts in 10^{13}.

In quantum physics, the **spin–orbit interaction** is a relativistic interaction of a particle's spin with its motion inside a potential. A key example of this phenomenon is the spin–orbit interaction leading to shifts in an electron's atomic energy levels, due to electromagnetic interaction between the electron's magnetic dipole, its orbital motion, and the electrostatic field of the positively charged nucleus. This phenomenon is detectable as a splitting of spectral lines, which can be thought of as a Zeeman effect product of two relativistic effects: the apparent magnetic field seen from the electron perspective and the magnetic moment of the electron associated with its intrinsic spin. A similar effect, due to the relationship between angular momentum and the strong nuclear force, occurs for protons and neutrons moving inside the nucleus, leading to a shift in their energy levels in the nucleus shell model. In the field of spintronics, spin–orbit effects for electrons in semiconductors and other materials are explored for technological applications. The spin–orbit interaction is one cause of magnetocrystalline anisotropy and the spin Hall effect.

The **Breit equation** is a relativistic wave equation derived by Gregory Breit in 1929 based on the Dirac equation, which formally describes two or more massive spin-1/2 particles interacting electromagnetically to the first order in perturbation theory. It accounts for magnetic interactions and retardation effects to the order of *1/c ^{2}*. When other quantum electrodynamic effects are negligible, this equation has been shown to give results in good agreement with experiment. It was originally derived from the Darwin Lagrangian but later vindicated by the Wheeler–Feynman absorber theory and eventually quantum electrodynamics.

In classical electromagnetism, **magnetization** or **magnetic polarization** is the vector field that expresses the density of permanent or induced magnetic dipole moments in a magnetic material. The origin of the magnetic moments responsible for magnetization can be either microscopic electric currents resulting from the motion of electrons in atoms, or the spin of the electrons or the nuclei. Net magnetization results from the response of a material to an external magnetic field, together with any unbalanced magnetic dipole moments that may be inherent in the material itself; for example, in ferromagnets. Magnetization is not always uniform within a body, but rather varies between different points. Magnetization also describes how a material responds to an applied magnetic field as well as the way the material changes the magnetic field, and can be used to calculate the forces that result from those interactions. It can be compared to electric polarization, which is the measure of the corresponding response of a material to an electric field in electrostatics. Physicists and engineers usually define magnetization as the quantity of magnetic moment per unit volume. It is represented by a pseudovector **M**.

In physics, **Larmor precession** is the precession of the magnetic moment of an object about an external magnetic field. Objects with a magnetic moment also have angular momentum and effective internal electric current proportional to their angular momentum; these include electrons, protons, other fermions, many atomic and nuclear systems, as well as classical macroscopic systems. The external magnetic field exerts a torque on the magnetic moment,

A ** g-factor** is a dimensionless quantity that characterizes the magnetic moment and angular momentum of an atom, a particle or nucleus. It is essentially a proportionality constant that relates the observed magnetic moment

The **Einstein-de Haas** effect is a physical phenomenon in which a change in the magnetic moment of a free body causes this body to rotate. The effect is a consequence of the conservation of angular momentum. It is strong enough to be observable in ferromagnetic materials. The experimental observation and accurate measurement of the effect demonstrated that the phenomenon of magnetization is caused by the alignment (polarization) of the angular momenta of the electrons in the material along the axis of magnetization. These measurements also allow the separation of the two contributions to the magnetization: that which is associated with the spin and with the orbital motion of the electrons. The effect also demonstrated the close relation between the notions of angular momentum in classical and in quantum physics.

In physics, **relativistic quantum mechanics (RQM)** is any Poincaré covariant formulation of quantum mechanics (QM). This theory is applicable to massive particles propagating at all velocities up to those comparable to the speed of light *c*, and can accommodate massless particles. The theory has application in high energy physics, particle physics and accelerator physics, as well as atomic physics, chemistry and condensed matter physics. *Non-relativistic quantum mechanics* refers to the mathematical formulation of quantum mechanics applied in the context of Galilean relativity, more specifically quantizing the equations of classical mechanics by replacing dynamical variables by operators. *Relativistic quantum mechanics* (RQM) is quantum mechanics applied with special relativity. Although the earlier formulations, like the Schrödinger picture and Heisenberg picture were originally formulated in a non-relativistic background, a few of them also works with special relativity.

The interaction of an electromagnetic wave with an electron bound in an atom or molecule can be described by time-dependent perturbation theory. Magnetic dipole transitions describe the dominant effect of the coupling to the magnetic part of the electromagnetic wave. They can be divided into two groups by the frequency at which they are observed: optical magnetic dipole transitions can occur at frequencies in the infrared, optical or ultraviolet between sublevels of two different electronic levels, while magnetic Resonance transitions can occur at microwave or radio frequencies between angular momentum sublevels within a single electronic level. The latter are called Electron Paramagnetic Resonance (EPR) transitions if they are associated with the electronic angular momentum of the atom or molecule and Nuclear Magnetic Resonance (NMR) transitions if they are associated with the nuclear angular momentum.

Magnets exert forces and torques on each other due to the rules of electromagnetism. The forces of attraction field of magnets are due to microscopic currents of electrically charged electrons orbiting nuclei and the intrinsic magnetism of fundamental particles that make up the material. Both of these are modeled quite well as tiny loops of current called magnetic dipoles that produce their own magnetic field and are affected by external magnetic fields. The most elementary **force between magnets**, therefore, is the magnetic dipole–dipole interaction. If all of the magnetic dipoles that make up two magnets are known then the net force on both magnets can be determined by summing up all these interactions between the dipoles of the first magnet and that of the second.

**Magnetic resonance** is a phenomenon that affects a Magnetic dipole when placed in a uniform static magnetic field. Its energy is split into a finite number of energy levels, depending on the value of quantum number of angular momentum. This is similar to energy quantization for atoms, say ^{}e^{−}_{} in H atom; in this case the atom, in interaction to an external electric field, transitions between different energy levels by absorbing or emitting photons. Similarly if a magnetic dipole is perturbed with electromagnetic field of proper frequency( ), it can transit between its energy eigenstates, but as the separation between energy eigenvalues is small, the frequency of the photon will be the microwave or radio frequency range. If the dipole is tickled with a field of another frequency, it is unlikely to transition. This phenomenon is similar to what occurs when a system is acted on by a periodic force of frequency equal to its natural frequency.

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