Magnetic moment

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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.

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 $τ$ is the torque acting on the dipole and $B$ is the external magnetic field, and $μ$ 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:^[2]^[3]

N·m/T = A·m² = J/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 statA·cm² = 3.33564095×10⁻¹⁴ A·m² (ESU)

1 erg/G = 1 abA·cm² = 10⁻³ A·m² (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.^[4] 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 $l$ separating them. The moment is related to the fictitious poles as^[4]

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

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.^[4] 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 }}={\tfrac {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. ^[5] 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 {\mathrm {M} }}={\tfrac {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 {\mathrm {M} }}={\tfrac {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 {\mathrm {M} }}={\tfrac {1}{2}}\iint _{S}\mathbf {r} \times \mathbf {j} \,{\rm {d}}S,

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

{\boldsymbol {\mathrm {M} }}={\tfrac {1}{2}}I\int _{\partial S}\mathbf {r} \times {\rm {d}}\mathbf {r} .

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

{\boldsymbol {\mathrm{M} }}=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 {\mathrm{M} }}=2I{\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.^[6] ^[7]

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

Magnetic dipoles

Main article: Magnetic dipole

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}}}

and magnetic flux density is

{\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).

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}}},

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 {m}}}}{|{\mathbf r}|^{3}}}\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.^[4]

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.^[4]^[9]

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^[4]

\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 $B = μ 0 (H + M)$ , where $M (x) = m δ (x)$ is the magnetization.

Forces between two magnetic dipoles

As discussed earlier, the force exerted by a dipole loop with moment $m 1$ on another with moment $m 2$ is

{\mathbf F}=\nabla \left({\mathbf m}_{2}\cdot {\mathbf B}_{1}\right),

where $B 1$ is the magnetic field due to moment $m 1$ . The result of calculating the gradient is^[10]^[11]

{\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),

where $r̂$ is the unit vector pointing from magnet 1 to magnet 2 and $r$ is the distance. An equivalent expression is^[11]

{\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).

The force acting on $m 1$ is in the opposite direction.

The torque of magnet 1 on magnet 2 is

{\boldsymbol \tau }={\mathbf m}_{2}\times {\mathbf 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: 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 3023071523600000000♠−9.284764×10⁻²⁴ J/T.^[12] 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.

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^[13]

m_{\text{Atom}}=g_{J}\mu _{\mathrm {B} }{\sqrt {j(j+1)}}

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^[14]

m_{\text{Atom}}(z)=-mg_{J}\mu _{\mathrm {B} }

where $m$ is called the magnetic quantum number or the equatorial quantum number, which can take on any of $2 j + 1$ values:^[15]

-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:^[16]^[17]

\frac{1}{\gamma} \frac{{\rm d}\mathbf{m}}{{\rm d}t} = \mathbf{m \times H_\text{eff}} - \frac{\lambda}{\gamma m}\mathbf{m} \times \frac{{\rm d}\mathbf{m}}{{\rm d}t}

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.

Magnetic moment of an electron

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

\mathbf {m} _{\text{S}}=-{\frac {g_{S}\mu _{\text{B}}\mathbf {S} }{\hbar }},

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: 7000200231930435999♠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.

Magnetic moment of a nucleus

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 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.

Examples of molecular magnetism

The dioxygen molecule, O₂, exhibits strong paramagnetism, due to unpaired spins of its outermost two electrons.
The 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.
The dihydrogen 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.^[18]

Number of unpaired electrons	Spin-only moment ( $μ B$ )
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 $μ$ 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^[19]
Particle	Magnetic dipole moment (10⁻²⁷ J·T⁻¹)	Spin quantum number (dimensionless)
electron (e⁻)	2996071523600000000♠−9284.764	1/2
proton (H⁺)	–0 07001141060670000000♠14.106067	1/2
neutron (n)	0 002999033764000000000♠−9.66236	1/2
muon (μ⁻)	0 02998550955220000000♠−44.904478	1/2
deuteron (²H⁺)	–0 007000433073460000000♠4.3307346	1
triton (³H⁺)	–0 07001150460940000000♠15.046094	1/2
helion (³He²⁺)	0 02998892538260000000♠−10.746174	1/2
alpha particle (⁴He²⁺)	–0 000	0

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

References and notes

1 2 Cullity, B. D.; Graham, C. D. (2008). Introduction to Magnetic Materials (2nd ed.). Wiley-IEEE Press. p. 103. ISBN 0-471-47741-9.
↑ "Magnetic units". IEEE Magnetics. Retrieved 19 February 2016.
↑ Mohr, Peter J.; Newell, David B.; Taylor, Barry N. (21 Jul 2015). "CODATA Recommended Values of the Fundamental Physical Constants: 2014". arXiv:1507.07956v1 [physics.atom-ph].
1 2 3 4 5 6 Brown, William Fuller, Jr. (1962). Magnetostatic Principles in Ferromagnetism. North-Holland.
↑ Feynman, Richard P.; Leighton, Robert B.; Sands, Matthew (2006). The Feynman Lectures on Physics. 2. ISBN 0-8053-9045-6.
↑ Krey, Uwe; Owen, Anthony (2007). Basic Theoretical Physics. Springer. pp. 151–152. ISBN 3-540-36804-3.
↑ Buxton, Richard B. (2002). Introduction to functional magnetic resonance imaging. Cambridge University Press. p. 136. ISBN 0-521-58113-3.
↑ Boyer, Timothy H. (1988). "The Force on a Magnetic Dipole". Am. J. Phys. 56 (8): 688–692. Bibcode:1988AmJPh..56..688B. doi:10.1119/1.15501.
↑ Jackson, John David (1975). Classical electrodynamics (2nd ed.). New York: Wiley. p. 184. ISBN 0-471-43132-X.
↑ Furlani, Edward P. (2001). Permanent Magnet and Electromechanical Devices: Materials, Analysis, and Applications. Academic Press. p. 140. ISBN 0-12-269951-3.
1 2 Yung, K. W.; Landecker, P. B.; Villani, D. D. (1998). "An Analytic Solution for the Force between Two Magnetic Dipoles" (PDF). Magn. Elec. Separation. 9: 39–52. Retrieved November 24, 2012.
↑ "CODATA value: electron magnetic moment".
↑ Tilley, R. J. D. (2004). Understanding Solids. John Wiley and Sons. p. 368. ISBN 0-470-85275-5.
↑ Tipler, Paul Allen; Llewellyn, Ralph A. (2002). Modern Physics (4th ed.). Macmillan. p. 310. ISBN 0-7167-4345-0.
↑ Crowther, J. A. (2007). Ions, Electrons and Ionizing Radiations (reprinted ed.). Rene Press. p. 277. ISBN 1-4067-2039-9.
↑ Rice, Stuart Alan (2004). Advances in chemical physics. Wiley. pp. 208ff. ISBN 0-471-44528-2.
↑ Steiner, Marcus (2004). Micromagnetism and Electrical Resistance of Ferromagnetic Electrodes for Spin Injection Devices. Cuvillier Verlag. p. 6. ISBN 3-86537-176-0.
↑ Figgis&Lewis, p. 406
↑ "Search results matching 'magnetic moment'". CODATA internationally recommended values of the Fundamental Physical Constants. National Institute of Standards and Technology. Retrieved 11 May 2012.

External links

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

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