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# Magnetic moment

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### Magnetic moment

The magnetic moment of a magnet is a quantity that determines the torque it will experience in an external magnetic field. A loop of electric current, a bar magnet, an electron (revolving around a nucleus), a molecule, and a planet all have magnetic moments.

The magnetic moment may be considered to be a vector having a magnitude and direction. The direction of the magnetic moment points from the south to north pole of the magnet. The magnetic field produced by the magnet is proportional to its magnetic moment. More precisely, the term magnetic moment normally refers to a system's magnetic dipole moment, which produces the first term in the multipole expansion of a general magnetic field. 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.

## Contents

• Definition 1
• Units 2
• Two representations of the cause of the magnetic moment 3
• Magnetic pole representation 3.1
• Integral representation 3.2
• Magnetic moment of a solenoid 3.2.1
• Magnetic moment and angular momentum 4
• Effects of an external magnetic field on a magnetic moment 5
• Force on a moment 5.1
• Magnetic dipoles 6
• External magnetic field produced by a magnetic dipole moment 6.1
• Internal magnetic field of a dipole 6.2
• Forces between two magnetic dipoles 6.3
• Examples of magnetic moments 7
• Two kinds of magnetic sources 7.1
• Magnetic moment of an atom 7.2
• Magnetic moment of an electron 7.3
• Magnetic moment of a nucleus 7.4
• Magnetic moment of a molecule 7.5
• Examples of molecular magnetism 7.5.1
• 7.6 Elementary particles
• References and notes 9
• Further reading 10
• External links 11

## Definition

The magnetic moment is 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]

\boldsymbol{\tau} = \boldsymbol{\mu} \times\mathbf{B}

where \boldsymbol{\tau} is the torque acting on the dipole and \mathbf{B} is the external magnetic field, and \boldsymbol{\mu} is the magnetic moment.

This definition is based on how one would measure the magnetic moment, in principle, of an unknown sample.

## Units

The unit for magnetic moment is not a base unit in the International System of Units (SI). As the torque is measured in newton-meters (N·m) and the magnetic field in teslas (T), the magnetic moment is measured in newton-meters per tesla. This has equivalents in other base units:

\text{N}{\cdot}\text{m}/\text{T} = \text{A}{\cdot}\text{m}^2 = \text{J}/\text{T} ,

where A is amperes and J is joules.

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:

1~\text{statA}{\cdot}\text{cm}^2 = 3.33564095\cdot10^{-14}~\text{A}{\cdot}\text{m}^2 (ESU)
1~\text{erg}/\text{G} = 1~\text{abA}{\cdot}\text{cm}^2 = 10^{-3}~\text{A}{\cdot}\text{m}^2 (Gaussian and EMU),

where statA is statamperes, cm is centimeters, erg is ergs, G is gauss and abA is abamperes. 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.

## Two representations of the cause of the magnetic moment

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.[2] In magnetic materials, the cause of the magnetic moment are the spin and orbital angular momentum states of the electrons, and whether atoms in one region are aligned with atoms in another.

### Magnetic pole representation

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. Consider a bar magnet which has 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 separating them. The moment is related to the fictitious poles as[2]

\mathbf{\mu}=p\boldsymbol{\ell}.

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 Magnetic moment and angular momentum). Nevertheless, magnetic poles are very useful for magnetostatic calculations, particularly in applications to ferromagnets.[2] Practitioners using the magnetic pole approach generally represent the magnetic field by the irrotational field H, in analogy to the electric field E.

### Integral representation

Moment μ of a planar current having magnitude I and enclosing an area S

We start from the definition of the differential magnetic moment pseudovector :

\boldsymbol{\mu} = \frac{1}{2} \mathbf{r}\times\mathbf{j}

where × is the vector cross product, r is the position vector, and j is the electric current density. It is very similar to the differential angular momentum, defined as:

\boldsymbol{l}=\mathbf{r}\times (\rho \mathbf{v})

where ρ is the mass density and v is the velocity vector. Like in every pseudovector, by convention the direction of the cross product is given by the right hand grip rule. [3] Practitioners using the current loop model generally represent the magnetic field by the solenoidal field B, analogous to the electrostatic field D.

The integral magnetic moment of a charge distribution is therefore:

\boldsymbol{\Mu}=\frac{1}{2}\iiint_V \mathbf{r}\times\mathbf{j}\,{\rm d}V,

Let us start with a point particle; in this simple situation the magnetic moment is:

\boldsymbol{\Mu} =\frac{1}{2} q\, \mathbf{r}\times\mathbf{v},

where r is the position of the electric charge q relative to the center of the circle and v is the instantaneous velocity of the charge, giving an electric current density j.

On the other hand for a point particle the angular momentum is defined as:

\boldsymbol{L}=\mathbf{r}\times\mathbf{p} = m \mathbf{r}\times\mathbf{v},

and in the planar case:

\boldsymbol{\Mu}=\frac{1}{2}\iint_S \mathbf{r}\times\mathbf{j}\,{\rm d}S,

by defining the electric current with a vector area S (x, y, and z coordinates of this vector are the areas of projections of the loop onto the yz, zx, and xy planes):

\boldsymbol{\Mu}=\frac{I}{2}\int_{\partial S} \mathbf{r}\times{\rm d}\mathbf{r}.

Then by Stokes' theorem, integral magnetic moment then becomes expressible as:

\boldsymbol{\Mu}=I \mathbf{S}.

The factor 1/2 in our definition above is only due to historical reason: the old definition of the magnetic moment was this last integral equation. If one had started from a differential definition:

\boldsymbol{\mu} = \mathbf{r}\times\mathbf{j}

then the coherent integral expression would have been:

\boldsymbol{\Mu}=2 I \mathbf{S}.

#### Magnetic moment of a solenoid

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 N identical turns (single-layer winding) and vector area S,

\boldsymbol{\mu}=N I \mathbf{S}.

## Magnetic moment and angular momentum

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] In particular, when a magnetic moment is subject to a torque in a magnetic field that tends to align it with the applied magnetic field, the moment precesses (rotates about the axis of the applied field). This is a consequence of the concomitance of magnetic moment and angular momentum, that in case of charged massive particles corresponds to the concomitance of charge and mass in a particle.

Viewing a magnetic dipole as a rotating charged particle brings out the close connection between magnetic moment and angular momentum. Both the magnetic moment and the angular momentum increase with the rate of rotation. The ratio of the two is called the gyromagnetic ratio and is simply the half of the charge-to-mass ratio.[4] [5]

For a spinning charged solid with a uniform charge density to mass density ratio, the gyromagnetic ratio is equal to half the charge-to-mass ratio. This implies that a more massive assembly of charges spinning with the same angular momentum will have a proportionately weaker magnetic moment, compared to its lighter counterpart. Even though atomic particles cannot be accurately described as spinning charge distributions of uniform charge-to-mass ratio, this general trend can be observed in the atomic world, where the intrinsic angular momentum (spin) of each type of particle is a constant: a small half-integer times 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).

## Effects of an external magnetic field on a magnetic moment

### Force on a moment

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

U=-\boldsymbol{\mu}\cdot\mathbf{B}

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 has been some discussion on how to calculate the force acting on a magnetic dipole. 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).[6] The force obtained in the case of a current loop model is

\mathbf{F}_\text{loop}=\nabla \left(\boldsymbol{\mu}\cdot\mathbf{B}\right)

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

\mathbf{F}_\text{dipole}=\left(\boldsymbol{\mu}\cdot \nabla \right) \mathbf{B}

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

\mathbf{F}_\text{loop}=\mathbf{F}_\text{dipole} + \boldsymbol{\mu}\times \left(\nabla \times \mathbf{B} \right)

In all these expressions μ 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.

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

## Magnetic dipoles

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).

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

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 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 dipolar component will dominate the magnetic field of the system at distances far away from it.

The vector potential of magnetic field produced by magnetic moment m is

\mathbf{A}(\mathbf r)=\frac{\mu_{0}}{4\pi}\frac{\mathbf m\times\mathbf r}{|\mathbf r|^3}=\frac{\mu_{0}}{4\pi}\frac{\mathbf m\times \hat\mathbf r}{|\mathbf r|^2}
\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 r|^3}\right)=\frac{\mu_0}{4\pi|\bold r|^3}\left(3\left(\bold m \cdot \hat\bold r\right)\hat\bold r - \bold m\right).

Alternatively one can obtain the scalar potential first from the magnetic pole perspective,

\psi(\mathbf r)=\frac{\mathbf m\cdot\mathbf r}{4\pi|\mathbf r|^3}=\frac{\mathbf m\cdot\hat\mathbf r}{4\pi|\mathbf r|^2},

and hence magnetic field strength is

{\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 r|^3}\right)=\frac{1}{4\pi|\bold r|^3}\left(3\left(\bold m \cdot \hat\bold r\right)\hat\bold r - \bold m\right).

The magnetic field of an ideal magnetic dipole is depicted on the right.

### Internal magnetic field of a dipole

The magnetic field of a current loop

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). Clearly, 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.[2]

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

\mathbf{B}(\mathbf{x})=\frac{\mu_0}{4\pi}\left[\frac{3\mathbf{n}(\mathbf{n}\cdot \mathbf{m})-\mathbf{m}}{|\mathbf{x}|^3} + \frac{8\pi}{3}\mathbf{m}\delta(\mathbf{x})\right].

Unlike the expressions in the previous section, this limit is correct for the internal field of the dipole.[2][7]

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[2]

\mathbf{H}(\mathbf{x}) =\frac{1}{4\pi}\left[\frac{3\mathbf{n}(\mathbf{n}\cdot \mathbf{m})-\mathbf{m}}{|\mathbf{x}|^3} - \frac{4\pi}{3}\mathbf{m}\delta(\mathbf{x})\right].

These fields are related by \mathbf{B}= \mu_0\left(\mathbf{H}+\mathbf{M}\right), where

\mathbf{M}(\mathbf{x}) = \mathbf{m}\delta(\mathbf{x})

is the magnetization.

### Forces between two magnetic dipoles

As discussed earlier, the force exerted by a dipole loop with moment m1 on another with moment m2 is

\bold F =\nabla \left(\bold m_2\cdot\bold B_1\right),

where B1 is the magnetic field due to moment m1. The result of calculating the gradient is[8][9]

\bold F(\bold r,\bold m_1,\bold m_2) = \frac{3\mu_0}{4\pi|\bold r|^4}\left(\bold m_2 (\bold m_1\cdot\hat\bold r) + \bold m_1(\bold m_2\cdot\hat\bold r) + \hat\bold r(\bold m_1\cdot\bold m_2) - 5\hat\bold r(\bold m_1\cdot\hat\bold r)(\bold m_2\cdot\hat\bold r)\right),

where is the unit vector pointing from magnet 1 to magnet 2 and r is the distance. An equivalent expression is[9]

\bold F = \frac{3\mu_0}{4\pi|\bold r|^4} \left((\hat\bold r\times\bold m_1)\times\bold m _2 + (\hat\bold r\times\bold m_2)\times\bold m_1 - 2\hat\bold r(\bold m_1\cdot\bold m_2) + 5\hat\bold r(\hat\bold r\times\bold m_1)\cdot(\hat\bold r\times\bold m_2)\right).

The force acting on m1 is in the opposite direction.

The torque of magnet 1 on magnet 2 is

\boldsymbol\tau = \bold m_2\times\bold B_1.

## Examples of magnetic moments

### Two kinds of magnetic sources

Fundamentally, contributions to any system's magnetic moment may come from sources of two kinds: (1) motion of electric charges, such as electric currents, and (2) 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.[10] 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:

1. the intrinsic moment of the electron,
2. the orbital motion of the electron around the proton,
3. 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.

### 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. The magnitude of the atomic dipole moment is then[11]

m_\text{Atom} = g_J \mu_B \sqrt{j(j+1)}

where j is the total angular momentum quantum number, gJ is the Landé g-factor, and μB is the Bohr magneton. The component of this magnetic moment along the direction of the magnetic field is then[12]

m_\text{Atom}(z) = -m g_J \mu_B

where m is called the magnetic quantum number or the equatorial quantum number, which can take on any of 2j+1 values:[13]

-j, -(j-1) \cdots 0 \cdots +(j-1), +j.

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:[14][15]

\frac{1}{\gamma} \frac - \frac{\lambda}{\gamma m}\mathbf{m} \times \frac{\hbar},

where μB is the Bohr magneton, S is electron spin, and the g-factor gS is 2 according to Dirac's theory, but due to quantum electrodynamic effects it is slightly larger in reality: 2.002 319 304 36. 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.

### Magnetic moment of a nucleus

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.

### 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:

#### Examples of molecular magnetism

• Oxygen molecule, O2, exhibits strong paramagnetism, due to unpaired spins of its outermost two electrons.
• Carbon dioxide molecule, CO2, 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 13C or 17O will contribute to the molecule's magnetic moment.
• Hydrogen molecule, H2, 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.[16]
Number of
unpaired
electrons
Spin-only
moment /μB
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 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[17]
Particle Magnetic dipole moment
in SI units (JT−1)
Spin quantum number
(dimensionless)
electron −9284.764 × 10−27 1/2
proton 14.106067 × 10−27 1/2
neutron −9.66236 × 10−27 1/2
muon −44.904478 × 10−27 1/2
deuteron 4.3307346 × 10−27 1
triton 15.046094 × 10−27 1/2
helion −10.746174 × 10−27 1/2
alpha particle 0 0

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

## References and notes

1. ^ a b B. D. Cullity, C. D. Graham (2008). Introduction to Magnetic Materials (2 ed.).
2. ^ a b c d e f Brown, Jr. 1962
3. ^
4. ^ Uwe Krey, Anthony Owen (2007). Basic Theoretical Physics.
5. ^ Richard B. Buxton (2002). Introduction to functional magnetic resonance imaging.
6. ^ Boyer, Timothy H. (1988). "The Force on a Magnetic Dipole".
7. ^ Jackson 1975, p. 184
8. ^ Furlani, Edward P. (2001). Permanent Magnet and Electromechanical Devices: Materials, Analysis, and Applications.
9. ^ a b K.W. Yung, P.B. Landecker, D.D. Villani (1998). "An Analytic Solution for the Force between Two Magnetic Dipoles" (PDF). Retrieved November 24, 2012.
10. ^ "CODATA value: electron magnetic moment".
11. ^ RJD Tilley (2004). Understanding Solids.
12. ^ Paul Allen Tipler, Ralph A. Llewellyn (2002). Modern Physics (4 ed.).
13. ^ JA Crowther (2007). Ions, Electrons and Ionizing Radiations (reprinted Cambridge (1934) 6 ed.). Rene Press. p. 277.
14. ^ Stuart Alan Rice (2004). Advances in chemical physics.
15. ^ Marcus Steiner (2004). Micromagnetism and Electrical Resistance of Ferromagnetic Electrodes for Spin Injection Devices. Cuvillier Verlag. p. 6.
16. ^ Figgis&Lewis, p. 406
17. ^ "Search results matching ' magnetic moment '...". CODATA internationally recommended values of the Fundamental Physical Constants. National Institute of Standards and Technology. Retrieved 11 May 2012.