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{{Short description|Magnetic strength and orientation of an object that produces a magnetic field}}

[[File:Magnetic moment.PNG|thumb|Magnetic moment '''m''' of a current '''I''', enclosing an area ''a''.]]

{{Electromagnetism|cTopic=Magnetostatics}}

In [[electromagnetism]], the '''magnetic moment''' or ''' ''magnetic dipole'' moment''' is the combination of strength and orientation of a [[magnet]] or other object or system that exerts a [[magnetic field]]. The magnetic dipole moment of an object determines the magnitude of [[torque]] the object experiences in a given magnetic field. When the same magnetic field is applied, objects with larger magnetic moments experience larger torques. 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.  Its direction points from the south pole to north pole of the magnet (i.e., inside the magnet).

The magnetic moment also expresses the magnetic force effect of a magnet. 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.

Examples of objects or systems that produce magnetic moments include: permanent magnets; astronomical objects such as many [[planet]]s, including [[Earth's magnetic field|the Earth]], and some [[Natural satellite|moons]], [[star]]s, etc.; various [[molecule]]s; [[elementary particle]]s (e.g. [[electron]]s); [[List of particles#Composite particles|composites of elementary particles]] ([[proton]]s and [[neutron]]s{{mdash}}as of the nucleus of an atom); and loops of [[electric current]] such as exerted by [[electromagnet]]s.

== Definition, units, and measurement ==

=== Definition ===
The magnetic moment can be defined as a vector (really [[pseudovector]]) relating the aligning [[torque]] on the object from an externally applied [[magnetic field]] to the field vector itself. The relationship is given by:<ref name=Graham>{{cite book |title=Introduction to Magnetic Materials |first1=B. D.|last1= Cullity|first2= C. D.|last2= Graham |url=https://books.google.com/books?id=ixAe4qIGEmwC&pg=PA103 |page=103 |isbn=978-0-471-47741-9 |year=2008 |publisher=[[Wiley-IEEE Press]]|edition=2nd}}</ref>
<math display="block"> \boldsymbol{\tau} = \mathbf{m} \times\mathbf{B}</math>
where {{math|'''''τ'''''}} is the torque acting on the dipole, {{math|'''B'''}} is the external magnetic field, and {{math|'''m'''}} is the magnetic moment.

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, {{math|''U''<sub>int</sub>}}, with respect to external magnetic field:
<math display="block"> \mathbf{m} = -\hat\mathbf x \frac{\partial U_\text{int}}{\partial B_x} - \hat\mathbf y \frac{\partial U_\text{int}}{\partial B_y} - \hat\mathbf z \frac{\partial U_\text{int}}{\partial B_z}.</math>

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.<ref>See, for example, {{cite book |title=Thermodynamics and an Introduction to Thermostatistics |url=https://archive.org/details/thermodynamicsin00call_349 |url-access=limited |first1=Herbert B.|last1= Callen|page=[https://archive.org/details/thermodynamicsin00call_349/page/n215 200] |isbn=978-0-471-86256-7 |year=1985 |publisher=[[John Wiley & Sons]]|edition=2nd}} where the relevant {{math|U}} is {{math|U[B{{sub|e}}]}}.</ref>

=== Units ===
The unit for magnetic moment in [[International System of Units]] (SI) [[SI base unit|base units]] is A⋅m<sup>2</sup>, 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:<ref name=IEEEM>{{cite web | title=Magnetic units|url=http://www.ieeemagnetics.org/index.php?option=com_content&view=article&id=118&Itemid=107|publisher=IEEE Magnetics | access-date=19 February 2016}}</ref><ref>{{cite journal |first1=Peter J. |last1=Mohr |first2=David B. |last2=Newell |first3=Barry N. |last3=Taylor | arxiv=1507.07956 |title=CODATA Recommended Values of the Fundamental Physical Constants: 2014 |date=21 Jul 2015 |doi=10.1103/RevModPhys.88.035009 | volume=88 |issue=3 |journal=Reviews of Modern Physics |page=035009 |bibcode=2016RvMP...88c5009M|s2cid=1115862 }}</ref>

<math display="block">\mathrm{A \cdot m^2} = \frac{\mathrm{N \cdot m}}{\mathrm{T}} = \frac{\mathrm{J}}{\mathrm{T}},</math>

where N is [[newton (unit)|newton]] (SI derived unit of force), T is [[Tesla (unit)|tesla]] (SI derived unit of magnetic flux density), and J is [[joule]] (SI derived unit of [[energy]]).<ref name="SI2019">{{SIbrochure9th}}</ref>{{rp|pp=20–21}} Although torque (N·m) and energy (J) are dimensionally equivalent, torques are never expressed in units of energy.<ref name="SI2019"/>{{rp|p=23}}

In the [[Centimetre gram second system of units|CGS]] system, there are several different sets of electromagnetism units, of which the main ones are [[Centimetre gram second system of units#Electrostatic units (ESU)|ESU]], [[Gaussian units|Gaussian]], and [[Centimetre gram second system of units#Electromagnetic units (EMU)|EMU]]. Among these, there are two alternative (non-equivalent) units of magnetic dipole moment:
<math display="block">1 \text{ statA} {\cdot} \text{cm}^2 = 3.33564095 \times 10^{-14} \text{ A} {\cdot} \text{m}^2 ~~ \text{  (ESU)}</math>
<math display="block">1 \;\frac{\text{erg}}{\text{G}} = 10^{-3} \text{ A} {\cdot} \text{m}^2 ~~ \text{  (Gaussian and EMU),}</math>

where statA is [[statampere]]s, cm is [[centimeter]]s, erg is [[erg]]s, and G is [[Gauss (unit)|gauss]]. The ratio of these two non-equivalent CGS units (EMU/ESU) is equal to the [[speed of light|speed of light in free space]], expressed in [[Centimetre|cm]]⋅[[Second|s]]<sup>−1</sup>.

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 {{mvar|I}} and area {{mvar|A}} has magnetic moment {{mvar|IA}} (see below), but in [[Gaussian units]] the magnetic moment is {{math|{{sfrac|''IA''|[[speed of light|''c'']]}}}}.

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

=== Measurement ===
{{Main|magnetometer}}

The magnetic moments of objects are typically measured with devices called [[magnetometer]]s, though not all magnetometers measure magnetic moment: 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.{{citation needed|date=January 2024}}

== Relation to magnetization ==
{{Main|Magnetization}}
{{See also|Remanence}}

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 {{math|'''M'''}} as:
<math display="block">\mathbf{M} = \frac{\mathbf{m}_{\Delta V}}{V_{\Delta V}},</math>
where {{math|'''m'''<sub>Δ''V''</sub>}} and {{math|''V''<sub>Δ''V''</sub>}} are the magnetic dipole moment and volume of a sufficiently small portion of the magnet {{nowrap|{{math|Δ''V''}}.}} This equation is often represented using derivative notation such that
<math display="block">\mathbf M = \frac{\mathrm{d}\mathbf{m}}{\mathrm{d}V},</math>
where {{math|d'''m'''}} is the elementary magnetic moment and {{math|d''V''}} is the [[volume element]]. The net magnetic moment of the magnet {{math|'''m'''}} therefore is
<math display="block">\mathbf m = \iiint \mathbf M\,\mathrm{d}V,</math>
where the triple integral denotes integration over the volume of the [[Magnetic field|magnet]]. For uniform magnetization (where both the magnitude and the direction of {{math|'''M'''}} is the same for the entire magnet (such as a straight bar magnet) the last equation simplifies to:
<math display="block"> \mathbf{m} = \mathbf{M} V,</math>
where {{mvar|V}} 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  {{math|'''B'''<sub>r</sub>}}. The formula needed in this case to calculate {{math|'''m'''}} in (units of A⋅m<sup>2</sup>) is:
<math display="block"> \mathbf{m} = \frac{1}{\mu_0}\mathbf{B}_\text{r} V,</math>

where:
* {{math|'''B'''<sub>r</sub>}} is the residual flux density, expressed in [[Tesla (unit)|teslas]].
* {{mvar|V}} is the volume of the magnet (in m<sup>3</sup>).
* {{math|''μ''<sub>0</sub>}} is the permeability of vacuum ({{val|4|end=π|e=-7|u=H/m}}).<ref>{{cite web|url=https://www.kjmagnetics.com/glossary.asp | title=K&J Magnetics – Glossary | website=www.kjmagnetics.com}}</ref>

== Models ==
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.<ref name=Brown62>{{cite book |last=Brown| first = William Fuller Jr. |author-link = William Fuller Brown, Jr.|title = Magnetostatic Principles in Ferromagnetism|year=1962 |publisher = [[North-Holland Publishing Company|North-Holland]]}}</ref> In magnetic materials, the cause of the magnetic moment are the [[#Magnetic moment and angular momentum|spin and orbital angular momentum states]] of the [[electron]]s, and varies depending on whether atoms in one region are aligned with atoms in another.{{citation needed|date=January 2024}}

=== Magnetic pole model ===
[[File:VFPt dipole electric.svg|thumb|upright|An electrostatic analog for a magnetic moment: two opposing charges separated by a finite distance.]]
The sources of magnetic moments in materials can be represented by poles in analogy to [[electrostatics]]. This is sometimes known as the Gilbert model.<ref name="GriffithsIntroED">{{cite book
 | last=Griffiths
 | first=David J.
 | year=1999
 | title=Introduction to Electrodynamics
 | edition=3rd
 | publisher=[[Prentice Hall]]
 | page=258
 | isbn=978-0-13-805326-0
 | oclc=40251748
 }}</ref>{{rp|p=[https://archive.org/details/introductiontoel00grif_0/page/258 258]}} In this model, a small magnet is modeled by a pair of ''fictitious'' magnetic monopoles of equal magnitude but opposite [[Magnetic polarity|polarity]]. Each pole is the source of magnetic force which weakens with distance. Since [[magnet#Two models for magnets: magnetic poles and atomic currents|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 {{mvar|p}} of its poles (''magnetic pole strength''), and the vector <math>\mathrm{\boldsymbol \ell}</math><!-- {{math|'''ℓ'''}} --> separating them. The magnetic dipole moment {{math|'''m'''}} is related to the fictitious poles as<ref name="Brown62"/>
<math display="block">\mathbf{m} = p\,\mathrm{\boldsymbol \ell}\,.</math>

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|Relation to angular momentum]]). Nevertheless, magnetic poles are very useful for [[Magnetostatics|magnetostatic]] calculations, particularly in applications to [[ferromagnets]].<ref name=Brown62/> Practitioners using the magnetic pole approach generally represent the [[magnetic field]] by the [[irrotational]] field {{math|'''H'''}}, in analogy to the [[electric field]] {{math|'''E'''}}.

=== Ampèrian loop model<span class="anchor" id="Amperian loop model"></span>===

[[Image:VFPt dipole magnetic3.svg|thumbnail|200px|right|The Amperian loop model: A current loop (ring) that goes into the page at the x and comes out at the dot produces a {{math|'''B'''}}-field (lines). The north pole is to the right and the south to the left.]]

After [[Hans Christian Ørsted]] discovered that electric currents produce a magnetic field and [[André-Marie Ampère]] discovered that [[Ampère's force law|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
<math display="block">\mathbf{m} = I\boldsymbol{S},</math>
where {{mvar|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.

==== Localized current distributions ====
[[File:Magnetic moment.svg|thumb|left|200px|Moment <math>\boldsymbol{\mu}</math> of a planar current having magnitude {{math|''I''}} and enclosing an area {{mvar|S}}]]
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 [[Magnetic vector potential|vector potential]]. This leads to the definition of the magnetic dipole moment as:
<math display="block">\mathbf{m} = \tfrac{1}{2} \iiint_V \mathbf{r} \times \mathbf{j} (\mathbf{r}) \, \mathrm dV,</math>
where {{math|×}} is the [[vector cross product]], {{math|'''r'''}} is the position vector, and {{math|'''j'''}} is the [[electric current density]] and the integral is a volume integral.<ref name=JDJacksonCED>{{Cite book
| last = Jackson
| first = John David
| author-link = John David Jackson (physicist)
| title = Classical Electrodynamics
| edition = 2
| year = 1975
| isbn = 978-0-471-43132-9 <!--| title-link = Classical Electrodynamics (book) -->
| publisher = Wiley
| url = https://archive.org/details/classicalelectro00jack_0
}}</ref>{{rp|at=§5.6}} 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
<math display="block">\mathbf{m} = I\mathbf{S},</math>
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 {{math|'''B'''}}, analogous to the electrostatic field {{math|'''D'''}}.

==== Magnetic moment of a solenoid ====
[[File:Solenoid, air core, insulated, 20 turns, (shaded).svg|thumb|right|Image of a solenoid]]
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 {{mvar|N}} identical turns (single-layer winding) and vector area {{math|'''S'''}},
<math display="block">\mathbf{m} = N I \mathbf{S}.</math>

=== Quantum mechanical model ===
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 straightforward to develop for macroscopic currents using the amperian loop model (see [[#Relation to angular momentum|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 (physics)|spin]]) of the particle and the particle's orbital angular momentum. See [[#Magnetic moment of an atom|below]] for more details.

== Effects of an external magnetic field ==

=== Torque on a moment ===

The torque {{mvar|'''τ'''}} on an object having a magnetic dipole moment {{math|'''m'''}} in a uniform magnetic field {{math|'''B'''}} is:
<math display="block"> \boldsymbol{\tau} = \mathbf{m} \times\mathbf{B}.</math>

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.<ref name="GriffithsIntroED"/>{{rp|p=[https://archive.org/details/introductiontoel00grif_0/page/257 257]}}

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

=== Force on a moment ===
{{See also|Force between magnets}}
A magnetic moment in an externally produced magnetic field has a potential energy {{mvar|U}}:
<math display="block">U = -\mathbf{m}\cdot\mathbf{B} </math>

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 [[Magnet#Modelling magnets|model used for the dipole]] is a current loop or two monopoles (analogous to the electric dipole).<ref>{{cite journal
| last= Boyer|first= Timothy H.
| title = The Force on a Magnetic Dipole
| year = 1988
| journal = [[American Journal of Physics|Am. J. Phys.]]
| volume = 56
| issue = 8
| pages = 688–692
| doi = 10.1119/1.15501|bibcode = 1988AmJPh..56..688B }}
</ref> The force obtained in the case of a current loop model is
<math display="block" id="Floop">\mathbf{F}_\text{loop}=\nabla \left(\mathbf{m}\cdot\mathbf{B}\right) .</math>

Assuming existence of magnetic monopole, the force is modified as follows:
<math display="block">\begin{align}
 \mathbf{F}_\text{loop} &= \left(\mathbf{m}\times\nabla\right) \times \mathbf{B} \\[1ex]
 &= \nabla \left(\mathbf{m}\cdot\mathbf{B}\right) - \left(\nabla\cdot\mathbf{B}\right) \mathbf{m}
\end{align}</math>

In the case of a pair of monopoles being used (i.e. electric dipole model), the force is
<math display="block">\mathbf{F}_\text{dipole} = \left(\mathbf{m} \cdot \nabla \right) \mathbf{B}.</math>And one can be put in terms of the other via the relation
<math display="block" id="Feq">\mathbf{F}_\text{loop}=\mathbf{F}_\text{dipole} + \mathbf{m}\times \left(\nabla \times \mathbf{B} \right) - \left(\nabla\cdot\mathbf{B}\right) \mathbf{m}.</math>

In all these expressions {{math|'''m'''}} is the dipole and {{math|'''B'''}} is the magnetic field at its position. Note that if there are no currents or time-varying electrical fields or magnetic charge, {{math|1=∇×'''B''' = 0}}, {{math|1=∇⋅'''B''' = 0}} and the two expressions agree.

=== Relation to Free Energy ===

One can relate the magnetic moment of a system to the free energy of that system.<ref>{{cite book
| last1= Landau|first1= L. D.
| last2= Lifshitz|first2= E. M.
| last3= Pitaevskii|first3= L. P.
| title = Electrodynamics of Continuous Media: Volume 8 (Course of Theoretical Physics)
| edition = 2
| date = January 15, 1984
| publisher = Butterworth-Heinemann
| page = 130
| isbn = 978-0750626347}}
</ref> In a uniform magnetic field {{math|'''B'''}}, the [[Helmholtz free energy#Relating free energy to other variables|free energy]] {{math|F}} can be related to the magnetic moment {{math|'''M'''}} of the system as
<math display="block">\mathrm dF = -S\,\mathrm dT - \mathbf{M}\,\cdot \mathrm d\mathbf{B}</math>
where {{math|S}} is the [[entropy]] of the system and {{math|T}} is the temperature. Therefore, the magnetic moment can also be defined in terms of the free energy of a system as
<math display="block">m = \left. - \frac{\partial F}{\partial B}  \right|_{T}.</math>

=== Magnetism ===

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]].

== Effects on environment ==

=== Magnetic field of a magnetic moment ===
{{Main|Magnetic dipole}}
{{See also|Dipole}}
[[File:Magnetic dipole moment.jpg|thumbnail|right|Magnetic field lines around a "magnetostatic dipole". The magnetic dipole itself is located in the center of the figure, seen from the side, and pointing upward.]]
Any system possessing a net magnetic dipole moment {{math|'''m'''}} will produce a [[dipole|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 <math>\mathbf{m}</math> but drops off as the cube of the distance such that:
<math display="block">\mathbf{H}(\mathbf{r}) = \frac{1}{4\pi} \left(\frac{3\mathbf{r}(\mathbf{m}\cdot\mathbf{r})}{|\mathbf r|^5} - \frac{{\mathbf{m}}}{|\mathbf r|^3}\right),</math>

where <math>\mathbf{H}</math> is the [[magnetic field]] produced by the magnet and <math>\mathbf{r}</math> 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 <math>\mathbf{r}</math> as the product of its magnitude times the unit vector in its direction (<math>\mathbf{r} = |\mathbf{r}| \mathbf\hat{r}</math>) so that:
<math display="block">\mathbf{H}(\mathbf{r}) =\frac{1}{4\pi}\frac{3\mathbf\hat{r}(\mathbf\hat{r}\cdot \mathbf{m})-\mathbf{m}}{|\mathbf{r}|^3}.</math>

The equivalent equations for the magnetic <math>\mathbf{B}</math>-field are the same except for a multiplicative factor of ''μ''<sub>0</sub> = {{val|4|end=''π''|e=-7|u=[[Henry (unit)|H]]/[[Metre|m]]}}, where ''μ''<sub>0</sub> is known as the [[vacuum permeability]]. For example:
<math display="block">\mathbf{B}(\mathbf{r}) =\frac{\mu_0}{4\pi}\frac{3\mathbf\hat{r}(\mathbf\hat{r}\cdot \mathbf{m})-\mathbf{m}}{|\mathbf{r}|^3}.</math>

=== Forces between two magnetic dipoles ===
{{See also|Magnetic dipole–dipole interaction}}

As discussed earlier, the force exerted by a dipole loop with moment {{math|'''m'''<sub>1</sub>}} on another with moment {{math|'''m'''<sub>2</sub>}} is
<math display="block">\mathbf F = \nabla \left(\mathbf m_2 \cdot \mathbf B_1\right),</math>
where {{math|'''B'''<sub>1</sub>}} is the magnetic field due to moment {{math|'''m'''<sub>1</sub>}}. The result of calculating the gradient is<ref>{{cite book
 |last = Furlani
 |first = Edward P.
 |title = Permanent Magnet and Electromechanical Devices: Materials, Analysis, and Applications
 |publisher = [[Academic Press]]
 |year = 2001
 |page = 140
 |url = https://books.google.com/books?id=irsdLnC5SrsC&q=3.130
 |isbn = 978-0-12-269951-1
}}</ref><ref name=Yung>{{cite journal  |title= An Analytic Solution for the Force between Two Magnetic Dipoles|first1=K. W.|last1= Yung|first2= P. B.|last2= Landecker|first3= D. D.|last3= Villani|url=http://downloads.hindawi.com/archive/1998/079537.pdf|access-date=November 24, 2012 | date=1998 | journal= Magnetic and Electrical Separation|volume=9|pages=39–52|doi=10.1155/1998/79537|doi-access=free}}</ref>
<math display="block">\mathbf F(\mathbf r,\mathbf m_1,\mathbf m_2) = \frac{3\mu_0}{4\pi|\mathbf r|^4} \left[\mathbf m_2 (\mathbf m_1\cdot\hat\mathbf r) + \mathbf m_1(\mathbf m_2\cdot\hat\mathbf r) + \hat\mathbf r(\mathbf m_1\cdot\mathbf m_2) - 5\hat\mathbf r(\mathbf m_1\cdot\hat\mathbf r)(\mathbf m_2\cdot\hat\mathbf r)\right],
</math>
where {{math|'''r̂'''}} is the unit vector pointing from magnet 1 to magnet 2 and {{math|''r''}} is the distance. An equivalent expression is<ref name=Yung/>
<math display="block">
\mathbf F = \frac{3\mu_0}{4\pi|\mathbf r|^4} \left[(\hat\mathbf r\times\mathbf m_1)\times\mathbf m _2 + (\hat\mathbf r\times\mathbf m_2)\times\mathbf m_1 - 2 \hat\mathbf r(\mathbf m_1 \cdot \mathbf m_2 ) + 5\hat\mathbf r(\hat\mathbf r\times\mathbf m_1)\cdot(\hat\mathbf r\times\mathbf m_2)\right].
</math>
The force acting on {{math|'''m'''<sub>1</sub>}} is in the opposite direction.

=== Torque of one magnetic dipole on another ===
The torque of magnet 1 on magnet 2 is
<math display="block">\boldsymbol\tau = \mathbf m_2 \times \mathbf B_1.</math>

== Theory underlying magnetic dipoles ==
{{Main|Multipole expansion}}
{{See also|Magnetic dipole}}

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 [[Multipole expansion|monopole]] (represented by an isolated magnetic north or south pole) the [[Magnetic dipole|dipole]] (represented by two equal and opposite magnetic poles), and the [[Quadrupole#Magnetic quadrupole|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.{{citation needed|date=January 2024}}

For many magnets the first non-zero term is the magnetic dipole moment. (To date, no isolated [[magnetic monopole]]s 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).

=== Magnetic potentials ===

Traditionally, the equations for the magnetic dipole moment (and higher order terms) are derived from theoretical quantities called [[Magnetic vector potential|magnetic potential]]s<ref name="JDJacksonCED"/>{{rp|at=§5.6}} which are simpler to deal with mathematically than the magnetic fields.{{citation needed|date=January 2024}}

In the magnetic pole model, the relevant magnetic field is the [[demagnetizing field]] <math>\mathbf{H}</math>. Since the demagnetizing portion of <math>\mathbf{H}</math> does not include, by definition, the part of <math>\mathbf{H}</math> due to free currents, there exists a [[magnetic scalar potential]] such that
<math display="block">{\mathbf{H}}({\mathbf{r}}) = -\nabla\psi.</math>

In the amperian loop model, the relevant magnetic field is the magnetic induction <math>\mathbf{B}</math>. Since magnetic monopoles do not exist, there exists a [[magnetic vector potential]] such that
<math display="block">\mathbf{B}(\mathbf{r}) = \nabla \times \mathbf{A}.</math>

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:
<math display="block">\begin{align}
  \mathbf{A} \left(\mathbf{r}, t\right) &= \frac{\mu_0}{4\pi} \int \frac{\mathbf{j}\left(\mathbf{r}'\right)}{\left|\mathbf{r} - \mathbf{r}'\right|}\, \mathrm{d}V', \\[1ex]
        \psi \left(\mathbf{r}, t\right) &= \frac{1}{4\pi} \int \frac{\rho \left(\mathbf{r}'\right)}{\left|\mathbf{r} - \mathbf{r}'\right|}\, \mathrm{d}V',
\end{align}</math>
where <math>\mathbf{j}</math> is the [[current density]] in the amperian loop model, <math>\rho</math> 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 <math>\mathbf{r}'</math>. 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:
<math display="block">\mathbf{A}(\mathbf r) = \frac{\mu_0}{4\pi} \frac{\mathbf m\times\mathbf r}{|\mathbf r|^3},</math>
where <math>\mathbf{m}</math> is:
<math display="block">\mathbf{m} = \frac{1}{2} \iiint_V \mathbf{r}\times\mathbf{j} \, \mathrm dV,</math>
where {{math|×}} is the [[vector cross product]], {{math|'''r'''}} is the position vector, and {{math|'''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 [[Magnetic scalar potential|scalar potential]] is
<math display="block">\psi(\mathbf r) = \frac{\mathbf m\cdot\mathbf r}{4\pi|\mathbf r|^3}.</math>

Here <math>\mathbf{m}</math> may be represented in terms of the magnetic pole strength density but is more usefully expressed in terms of the [[magnetization]] field as:
<math display="block">\mathbf m = \iiint \mathbf M\,\mathrm{d}V.</math>

The same symbol <math>\mathbf{m}</math> is used for both equations since they produce equivalent results outside of the magnet.

=== External magnetic field produced by a magnetic dipole moment ===

The [[magnetic flux density]] for a magnetic dipole in the amperian loop model, therefore, is
<math display="block">\mathbf{B}(\mathbf{r}) = \nabla\times{\mathbf{A}} = \frac{\mu_{0}}{4\pi}\left(\frac{3\mathbf{r}(\mathbf{m}\cdot\mathbf{r})}{|\mathbf r|^5}-\frac{{\mathbf{m}}}{|\mathbf r|^3}\right).</math>

Further, the [[magnetic field strength]] <math>\mathbf{H}</math> is
<math display="block">{\mathbf{H}}({\mathbf{r}}) = -\nabla\psi = \frac{1}{4\pi}\left(\frac{3\mathbf{r}(\mathbf{m}\cdot\mathbf{r})}{|\mathbf r|^5} - \frac{{\mathbf{m}}}{|\mathbf r|^3}\right).</math>

=== Internal magnetic field of a dipole ===
[[File:VFPt dipole magnetic3.svg|thumbnail|right|The magnetic field of a current loop]]
The two models for a dipole (magnetic poles or current loop) 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 the figure for [[#Magnetic pole model|Magnetic pole model]]) 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.<ref name=Brown62/>

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<ref name=Brown62/>
<math display="block">\mathbf{H}(\mathbf{r}) =\frac{1}{4\pi}\left[\frac{3\mathbf\hat{r}(\mathbf\hat{r}\cdot \mathbf{m})-\mathbf{m}}{|\mathbf{r}|^3} - \frac{4\pi}{3}\mathbf{m}\delta(\mathbf{r})\right].</math>

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
<math display="block">\mathbf{B}(\mathbf{r})=\frac{\mu_0}{4\pi}\left[\frac{3\mathbf\hat{r}(\mathbf\hat{r}\cdot \mathbf{m})-\mathbf{m}}{|\mathbf{r}|^3} + \frac{8\pi}{3}\mathbf{m}\delta(\mathbf{r})\right].</math>
Unlike the expressions in the previous section, this limit is correct for the internal field of the dipole.<ref name=Brown62/><ref name="JDJacksonCED"/>{{rp|p=[https://archive.org/details/classicalelectro00jack_0/page/184 184]}}

These fields are related by {{math|'''B''' {{=}} ''μ''<sub>0</sub>('''H''' + '''M''')}}, where {{math|'''M'''('''r''') {{=}} '''m'''''δ''('''r''')}} is the [[magnetization]].

== Relation to angular momentum ==
{{See also|gyromagnetic ratio}}
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".<ref name=Graham/> Further, a [[torque]] applied to a relatively isolated magnetic dipole such as an [[atomic nucleus]] can cause it to [[precession|precess]] (rotate about the axis of the applied field). This phenomenon is used in [[nuclear magnetic resonance]].{{citation needed|date=January 2024}}

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 <math>\gamma</math> so that:<ref name=Krey>{{cite book |title=Basic Theoretical Physics |first1=Uwe |last1=Krey |first2=Anthony |last2=Owen |pages=151–152 |url=https://books.google.com/books?id=xZ_QelBmkxYC&pg=PA151 |isbn=978-3-540-36804-5 |year=2007 |publisher=[[Springer Science+Business Media|Springer]]}}</ref><ref name=Buxton>{{cite book |first=Richard B. |last=Buxton |year=2002 |title=Introduction to functional magnetic resonance imaging |page=136 |url=https://books.google.com/books?id=6XVu0NKzgekC&pg=PA136 |isbn=978-0-521-58113-4 |publisher=[[Cambridge University Press]]}}</ref>
<math display="block">\mathbf{m} = \gamma\,\mathbf{L},</math>
where <math>\mathbf{L}</math> 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:
<math display="block"> \mathbf{L} = \mathbf{r}\times\mathbf{p} = \mu \, \mathbf{r}\times\mathbf{v},</math>
where {{mvar|μ}} is the mass of the particle and {{mvar|'''v'''}} is the particle's [[velocity]]. The angular momentum of the very large number of charged particles that make up a current therefore is:
<math display="block"> \mathbf{L} = \iiint_V \, \mathbf{r} \times (\rho \mathbf{v})\, \mathrm dV\,,</math>
where {{mvar|ρ}} is the [[mass density]] of the moving particles. By convention the direction of the cross product is given by the [[right-hand rule]].<ref name=Feynman>{{Cite book
| last1 = Feynman
| first1 = Richard P.
| author-link = Richard Feynman
| first2 = Robert B.
| last2 = Leighton
| author2-link = Robert B. Leighton
| first3 = Matthew
| last3 = Sands
| author3-link = Matthew Sands
| title = The Feynman Lectures on Physics
| volume = 2
| pages = 13–12
| year = 2006
| isbn = 978-0-8053-9045-2
| title-link = The Feynman Lectures on Physics
| publisher = Pearson/Addison-Wesley
}}</ref>

This is similar to the magnetic moment created by the very large number of charged particles that make up that current:
<math display="block">\mathbf{m} = \tfrac{1}{2}\iiint_V \mathbf{r} \times (\rho_{Q} \mathbf{v})\, \mathrm dV\,,</math>
where <math>\mathbf{j} = \rho_Q \mathbf{v}</math> and <math>\rho_Q</math> is the [[charge density]] of the moving charged particles.

Comparing the two equations results in:
<math display="block">\mathbf{m} = \frac{e}{2\mu}\,\mathbf{L}\,,</math>
where <math>e</math> is the charge of the particle and <math>\mu</math> 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:
<math display="block">\mathbf{m} = g \, \frac{e}{2\mu} \, \mathbf{L},</math>
where the [[g-factor (physics)|{{mvar|g}}-factor]] depends on the particle and configuration. For example, the {{mvar|g}}-factor for the magnetic moment due to an electron orbiting a nucleus is one while the {{mvar|g}}-factor for the magnetic moment of electron due to its intrinsic angular momentum ([[Spin (physics)|spin]]) is a little larger than 2. The {{mvar|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 (physics)|spin]]) of a particle is an [[integer]] (or [[half-integer]] in the case of spin) multiple of the reduced [[Planck constant]] {{mvar|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.

== Atoms, molecules, and elementary particles ==

Fundamentally, contributions to any system's magnetic moment may come from sources of two kinds: 1) motion of [[electric charge]]s, such as [[electric current]]s; and  2) the intrinsic magnetism due [[Spin (physics)#Magnetic moments|spin]] of [[elementary particle]]s, such as the [[electron]].{{citation needed|date=January 2024}}

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. 

Contributions due to particle spin sum the [[Euclidean vector|magnitude]] of each elementary particle's intrinsic magnetic moment, a fixed number, often measured experimentally to a great precision. For example, any electron's magnetic moment is measured to be {{val|-9.284764|e=-24|u=J/T}}.<ref>{{Cite web | url=https://physics.nist.gov/cgi-bin/cuu/Value?muem | title=CODATA Value: electron magnetic moment | website=physics.nist.gov}}</ref> The [[Euclidean vector|direction]] of the magnetic moment of any elementary particle is entirely determined by the direction of its [[Spin (physics)|spin]], with the [[minus sign|negative value]] indicating that any electron's magnetic moment is antiparallel to its spin.

The net magnetic moment of any system is a [[Euclidean vector|vector sum]] of contributions from one or both types of sources.
For example, the magnetic moment of an atom of [[Hydrogen atom|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 [[electron]]s of the magnet's material and the nuclear magnetic moments.

=== Magnetic moment of an atom ===
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, <math>\mathfrak m_\text{atom}</math>, is then<ref name=Tilley2>{{cite book |title=Understanding Solids |url=https://books.google.com/books?id=ZVgOLCXNoMoC&pg=PA368 |page=368 |isbn=978-0-470-85275-0 |first=R. J. D. |last=Tilley |year=2004 |publisher=[[John Wiley and Sons]]}}</ref>
<math display="block">\mathfrak m_\text{atom} = g_\text{J}\,\mu_\text{B}\,\sqrt{j\,(j+1)\,}</math>
where {{mvar|j}} is the [[total angular momentum quantum number]], {{mvar|g}}{{sub|J}} is the [[Landé g-factor|Landé {{mvar|g}}-factor]], and {{mvar|μ}}{{sub|B}} is the [[Bohr magneton]]. The component of this magnetic moment along the direction of the magnetic field is then<ref name=Llewellyn>{{cite book |title=Modern Physics |first1=Paul Allen|last1= Tipler |first2=Ralph A. |last2=Llewellyn |url=https://books.google.com/books?id=tpU18JqcSNkC&pg=PA310 |page=310 |isbn=978-0-7167-4345-3 |year=2002 |edition=4th |publisher=[[Macmillan Publishers (United States)|Macmillan]]}}</ref>
<math display="block">\mathfrak m_{\text{atom},z} = -m\,g_\text{J}\,\mu_\text{B} \, .</math>

The negative sign occurs because electrons have negative charge.

The [[integer]] {{mvar|m}} (not to be confused with the moment, <math>\mathfrak m</math>) is called the [[magnetic quantum number]] or the ''equatorial'' quantum number, which can take on any of {{math|2''j'' + 1}} values:<ref name=Crowther>{{cite book |title=Ions, Electrons, and Ionizing Radiations |first=J.A. |last=Crowther |page=[https://archive.org/details/IonsElectronsIonizingRadiations/page/n283 270] |url=https://archive.org/details/IonsElectronsIonizingRadiations |year=1949 |location=London |publisher=Edward Arnold |edition=8th}}</ref>
<math display="block">-j,\ -(j-1),\ \cdots,\ -1,\ 0,\ +1,\ \cdots,\ +(j-1),\ +j~.</math>

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]]:<ref name=Rice>{{cite book |title=Advances in chemical physics |first=Stuart Alan |last=Rice |url=https://books.google.com/books?id=wK3Vhq-VnBQC&pg=PA208 |pages=208ff |isbn=978-0-471-44528-9 |publisher=[[John Wiley & Sons|Wiley]] |year=2004}}</ref><ref name=Steiner>{{cite book |title=Micromagnetism and Electrical Resistance of Ferromagnetic Electrodes for Spin Injection Devices |first=Marcus |last=Steiner |publisher=Cuvillier Verlag |year=2004 |isbn=978-3-86537-176-8 |url=https://books.google.com/books?id=tnX1edkCB-wC&pg=PA6 |page=6}}</ref>
<math display="block">\frac{1}{\gamma} \frac{\mathrm d\mathbf{m}}{\mathrm dt} = \mathbf{m} \times \mathbf{H}_\text{eff} - \frac{\lambda}{\gamma m} \mathbf{m} \times \frac{\mathrm d\mathbf{m}}{\mathrm dt}</math>
where {{mvar|γ}} is the [[gyromagnetic ratio]], {{math|'''m'''}} is the magnetic moment, {{mvar|λ}} is the damping coefficient and {{math|'''H'''}}{{sub|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.

=== Magnetic moment of an electron ===
{{See also|Anomalous magnetic dipole moment|Electron magnetic moment}}
[[Electron]]s 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]].{{citation needed|date=January 2024}}

The magnetic moment of the electron is
<math display="block"> \mathbf{m}_\text{S} = -\frac{g_\text{S} \mu_\text{B} \mathbf{S}}{\hbar},</math>
where [[Bohr magneton|{{mvar|μ}}{{sub|B}}]] is the [[Bohr magneton]], {{math|'''S'''}} is electron [[Spin (physics)|spin]], and the [[g-factor (physics)|''g''-factor]] {{mvar|g}}{{sub|S}} is 2 according to [[Dirac equation|Dirac's theory]], but due to [[quantum electrodynamics|quantum electrodynamic]] effects it is slightly larger in reality: {{val|2.00231930436}}. The deviation from 2 is known as the [[anomalous magnetic dipole moment]].

Again it is important to notice that {{math|'''m'''}} is a negative constant multiplied by the [[Spin (physics)|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.

=== Magnetic moment of a nucleus ===
{{See also|Nucleon magnetic moment|Nuclear magnetic moment}}
The nuclear system is a complex physical system consisting of nucleons, i.e., [[proton]]s and [[neutron]]s. 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 [[isotope]]s have long-lived [[excited state]]s. 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.

=== Magnetic moment of a molecule ===
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 spin]]s ([[paramagnetic]] contribution), if any
* orbital motion of its electrons, which in the [[Stationary state#Ground state|ground state]] is often proportional to the external magnetic field ([[diamagnetic]] contribution)
* the combined magnetic moment of its [[nuclear spin]]s, which depends on the [[Angular momentum coupling#Spin-spin coupling|nuclear spin configuration]].

==== Examples of molecular magnetism ====
* The [[Oxygen|dioxygen]] molecule, O{{sub|2}}, exhibits strong [[paramagnetism]], due to unpaired spins of its outermost two electrons.
* The [[carbon dioxide]] molecule, CO{{sub|2}}, mostly exhibits [[diamagnetism]], a much weaker magnetic moment of the electron [[Molecular orbital|orbitals]] that is proportional to the external magnetic field. The nuclear magnetism of a magnetic [[isotope]] such as {{sup|13}}C or {{sup|17}}O will contribute to the molecule's magnetic moment.
* The [[dihydrogen]] molecule, H{{sub|2}}, in a weak (or zero) magnetic field exhibits nuclear magnetism, and can be in a [[parahydrogen|para-]] or an [[orthohydrogen|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 metal]]s.<ref>{{cite encyclopedia |last2=Lewis |first2=J. |last1=Figgis |first1=B.N. |editor1-last=Lewis |editor1-first=J. |editor2-last=Wilkins | editor2-first=R.G. |encyclopedia=Modern Coordination Chemistry: Principles and methods |title=The magnetochemistry of complex compounds |date=1960 |publisher=Interscience |location=New York |pages=405–407 |url=https://archive.org/details/moderncoordinati00lewi/page/404}}</ref>
{|class="wikitable" style="text-align:center"
! Number of<br />unpaired<br />electrons
! Spin-only<br />moment<br />({{math|[[Bohr magneton|''μ''{{sub|B}}]]}})
|-
| 1 || 1.73
|-
| 2 || 2.83
|-
| 3 || 3.87
|-
| 4 || 4.90
|-
| 5 || 5.92
|}

=== Elementary particles ===
In atomic and nuclear physics, the Greek symbol {{mvar|μ}} represents the [[Euclidean vector|magnitude]] of the magnetic moment, often measured in Bohr magnetons or [[nuclear magneton]]s, 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:

{| class="wikitable"
|+ Intrinsic magnetic moments and spins<br />of some elementary particles<ref>{{cite web |title=Search results matching 'magnetic moment' |url=http://physics.nist.gov/cgi-bin/cuu/Results?search_for=+magnetic+moment |work=CODATA internationally recommended values of the Fundamental Physical Constants |publisher=National Institute of Standards and Technology |access-date=11 May 2012}}</ref>
! Particle<br />name (symbol)
! Magnetic<br />dipole moment<br />(10{{sup|−27}}&nbsp;[[Joule (unit)|J]]⋅[[Tesla (unit)|T]]{{sup|−1}})
! [[Spin (physics)|Spin]]<br />quantum number<br />([[dimensionless]])
|-
| [[electron]] (e<sup>−</sup>)
| style="padding-left:2.5em" | {{val|−9284.764}}
| align=center | {{sfrac|1|2}}
|-
| [[proton]] (H<sup>+</sup>)
| style="padding-left:2.5em" | {{0|–0 0}}{{val|14.106067}}
| align=center | {{sfrac|1|2}}
|-
| [[neutron]] (n)
| style="padding-left:2.5em" | {{0|0 00}}{{val|−9.66236}}
| align=center | {{sfrac|1|2}}
|-
| [[muon]] (μ{{sup|−}})
| style="padding-left:2.5em" | {{0|0 0}}{{val|−44.904478}}
| align=center | {{sfrac|1|2}}
|-
| [[deuteron]] ({{sup|2}}H{{sup|+}})
| style="padding-left:2.5em" | {{0|–0 00}}{{val|4.3307346}}
| align=center | 1
|-
| [[tritium|triton]] ({{sup|3}}H{{sup|+}})
| style="padding-left:2.5em" | {{0|–0 0}}{{val|15.046094}}
| align=center | {{sfrac|1|2}}
|-
| [[helion (chemistry)|helion]] ({{sup|3}}He{{sup|++}})
| style="padding-left:2.5em" | {{0|0 0}}{{val|−10.746174}}
| align=center | {{sfrac|1|2}}
|-
| [[alpha particle]] ({{sup|4}}He{{sup|++}})
| style="padding-left:2.5em" | {{0|–0 00}}0
| align=center | 0
|-
|}

== See also ==
* [[Moment (physics)]]
* [[Electric dipole moment]]
* [[Toroidal moment|Toroidal dipole moment]]
* [[Magnetic susceptibility]]
* [[Orbital magnetization]]
* [[Magnetic dipole–dipole interaction]]

== References and notes ==
{{Reflist}}

==External links==
* {{cite web |last=Bowtell |first=Richard |title=''μ'' – Magnetic Moment |url=http://www.sixtysymbols.com/videos/magneticmoment.htm |website=Sixty Symbols |publisher=[[Brady Haran]] for the [[University of Nottingham]] |year=2009}}

{{Authority control}}

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[[Category:Moment (physics)]]
[[Category:Magnetic moment]]