Magnetic Moment

Magnetic moment is the magnetic force and orientation of a magnet or other object that produces a magnetic field . Examples of objects with magnetic moments include: loops of electric current (such as electromagnets ), permanent magnets, elementary particles (such as electrons ), various molecules , and many celestial bodies (such as many planets , some moons , stars , etc.).

More precisely, the term magnetic moment refers in general to the magnetic dipole moment of a system, the component of the magnetic moment that can be represented by a uniform magnetic dipole: a magnetic north and south pole separated by a very small distance . . The magnetic dipole component is sufficient for small enough magnets or for large enough distances. Extended objects may require higher-order terms (such as the magnetic quadrupole moment ) in addition to the dipole moment.

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

Definition, Units and Measurements

Definition

The magnetic moment can be defined as a vector that is related to the torque aligned to the field vector from an externally applied magnetic field on the object. The relation is given by:

{\displaystyle {\boldsymbol {\tau }}=\mathbf {m} \times \mathbf {B} }

where is the torque acting on the dipole, b is the external magnetic field, and m is the magnetic moment.

This definition is based on how one can measure the magnetic moment of an unknown sample. For a current loop, this definition leads to the magnitude of the magnetic dipole moment being equal to the area of ​​the loop times the product of the current. Furthermore, 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 magnetic moment . In this definition, the magnetic dipole moment of a system is the negative gradient of its internal energy, U int , with respect to the external magnetic field :

{\displaystyle \mathbf {m} =-{\hat {\mathbf {x} }}{\frac {\partial U_{\rm {int}}}{\partial B_{x}}}-{\hat {\mathbf {y} }}{\frac {\partial U_{\rm {int}}}{\partial B_{y}}}-{\hat {\mathbf {z} }}{\frac {\partial U_{\rm {int}}}{\partial B_{z}}}.}

Generally, internal energy includes the self-field energy of the system and the energy of the internal workings of the system. For example, for a hydrogen atom in the 2p state in the outer sphere, the self-field energy is negligible, so the internal energy is essentially the eigenstate energy of the 2p state, which includes the Coulomb potential energy and the electron’s kinetic energy. The inter-field energy between the inner dipoles and the outer fields is not part of this internal energy. 

units

The unit of magnetic moment in the International System of Units (SI) base unit is A⋅m² , where A is the ampere ( the SI base unit of current) and m is the meter (the SI base unit of distance). This unit has equivalents in other SI derived units including:

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

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

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

{\displaystyle 1{\text{ statA}}{\cdot }{\text{cm}}^{2}=3.33564095\times 10^{-14}{\text{ A}}{\cdot }{\text{m}}^{2}}(es U)\\
{\displaystyle 1\;{\frac {\text{erg}}{\text{G}}}=10^{-3}{\text{ A}}{\cdot }{\text{m}}^{2}}(Gaussian~~ and ~~EMU),

where Stata is statamperes , cm is centimeter erg , erg , and g is gauss . The ratio of these two non-equal CGS units (EMU/ESU) is equal to the speed of light in free space , expressed in cms – 1 .

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

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

Measurement

The magnetic moments of objects are usually measured with instruments called magnetometers, although not all magnetometers measure magnetic moment: some are configured to measure magnetic fields instead. If the magnetic field surrounding an object is known sufficiently, the magnetic moment can be calculated from that magnetic field.

Magnetism

Magnetic moment is a quantity that describes the magnetic force of an object as a whole. Sometimes, however, it is useful or necessary to know how much of an object’s net magnetic moment is generated by a particular part of that magnet. Therefore, it is useful to define the magnetization field M as:

{\displaystyle \mathbf {M} ={\frac {\mathbf {m} _{\Delta V}}{V_{\Delta V}}},}

where M _V and _V V is the magnetic dipole moment and _{the amount} of a sufficiently small part of the magnet _V. This equation is often represented using derivative notation such as

{\displaystyle \mathbf {M} ={\frac {\mathrm {d} \mathbf {m} }{\mathrm {d} V}},}

where d m is the primary magnetic moment and d v is the quantity element. Hence the net magnetic moment of the magnet m is

{\displaystyle \mathbf {m} =\iiint \mathbf {M} \,\mathrm {d} V,}

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

{\displaystyle \mathbf {m} =\mathbf {M} V,}

Where V is the volume of the bar magnet.

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

{\displaystyle \mathbf {m} ={\frac {1}{\mu _{0}}}\mathbf {B} _{\rm {r}}V},

Where from:

• B _r is the residual flux density, expressed in Tesla.
• V is the volume of the magnet ( ^{in m 3} ).
• μ ₀ is the permittivity of the vacuum (4π × 10 ⁻⁷  h/m ).

Model

The preferred classical interpretation of the magnetic moment has changed over time. Before the 1930s, textbooks explained the moment using imaginary magnetic point charges. Since then, most have defined it in terms of Amperian currents. ^[8] In magnetic materials, the cause of the magnetic moment is the spin and orbital angular momentum of the electrons, and depends on whether atoms in one region are aligned with atoms in another.

Magnetic pole model

The sources of magnetic moments in materials can be represented by poles corresponding to electrostatics. This is sometimes referred to as the Gilbert model. ^{[9] In} this model, a small magnet is created by a pair of magnetic poles of equal magnitude but opposite polarity. Each pole is a source of magnetic force which weakens with distance. Since magnetic poles always come in pairs, their forces partially cancel each other out as 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 small. The magnetic force produced by a bar magnet at a given point in space, therefore, depends on two factors: the power of its poles p ( magnetic pole strength ), and the vector “l” separate them. Magnetic dipole moment m is related to imaginary poles 

{\displaystyle \mathbf {m} =p\,\mathrm {\boldsymbol {\ell }} \,.}

It points 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 Relationship to angular momentum). Nevertheless, magnetic poles are very useful for magnetostatic calculations, especially in applications of ferromagnets. ^[8] Practitioners using the magnetic pole approach typically represent the magnetic field by the irrotational field H , in analogy to the electric field E.

Emperian Loop Model

After Hans Kristin rsted discovered that electric current produced a magnetic field and André Marie Ampere discovered that electric currents attract and repel each other alike, it was natural to hypothesize that all magnetic fields were caused by electric current loops. are doing. In this model developed by Ampere, the primary magnetic dipole that makes up all magnets is a sufficiently small Amperian loop of current I. The dipole moment of this loop is

{\displaystyle \mathbf {m} =I{\boldsymbol {S}},}

where S is the area of ​​the loop. The direction of the magnetic moment is in the direction normal to the region surrounded by current, corresponding to the direction of the current using the right-hand rule.

localized current distribution

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

{\displaystyle \mathbf {m} ={\tfrac {1}{2}}\iiint _{V}\mathbf {r} \times \mathbf {j} \,{\rm {d}}V,}

where × is the vector product across, r is the position vector, and j is the electric flux density and the integral is a quantity integral. ^[10] When the current density in the integral is replaced by a loop of current I in the plane enclosing the S region, the volume integral becomes a line integral and the resultant dipole moment becomes

{\displaystyle \mathbf {m} =I\mathbf {S} ,}

Thus the magnetic dipole moment for the Amperian loop is obtained.

Therapists using the current loop model usually represent the magnetic field by the solenoidal field B corresponding to the electrostatic field D.

Magnetic moment of the solenoid

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

{\displaystyle \mathbf {m} =NI\mathbf {S} .}

Quantum mechanical model

When calculating the magnetic moments of materials or molecules at the microscopic level, it is often convenient to use a third model for magnetic moment that exploits the linear relationship between the angular momentum and the particle’s magnetic moment. While this relationship is straight forward for the evolution of macroscopic currents using the Emperian loop model (see below), neither the magnetic pole model nor the Emperian loop model actually represents events occurring at the atomic and molecular levels. Quantum mechanics should be used at that level. Fortunately, the linear relationship between a particle’s magnetic dipole moment and its angular momentum is still maintained; However it is different for each particle. In addition, care must be exercised to distinguish between the particle’s intrinsic angular momentum (or spin) and the particle’s orbital angular momentum. See below for more details.

Effects of external magnetic fields

Torque in a moment

Torque m in a uniform magnetic field when an object having magnetic dipole moment B is:

{\displaystyle {\boldsymbol {\tau }}=\mathbf {m} \times \mathbf {B} }.

This is valid for the time being due to any localized current distribution provided that the magnetic field is uniform. The equation for the non-uniform B is also valid for the torque about the center of the magnetic dipole provided that the magnetic dipole is small enough. ^{[11 1]}

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

The potential energy U at a magnetic moment in an externally produced magnetic field is :

{\displaystyle U=-\mathbf {m} \cdot \mathbf {B} }

In the case when the external magnetic field is unequal, there will be a force proportional to the magnetic field gradient, which will act on the magnetic moment itself. There are two expressions for the force acting on a magnetic dipole, depending on whether the model used for the dipole is a current loop or two monopoles (corresponding to an electric dipole). ^[12] The force obtained in the case of the present loop model is

{\displaystyle \mathbf {F} _{\text{loop}}=\nabla \left(\mathbf {m} \cdot \mathbf {B} \right)}.

In the case of using a pair of monopoles (ie the electric dipole model), the force is

{\displaystyle \mathbf {F} _{\text{dipole}}=\left(\mathbf {m} \cdot \nabla \right)\mathbf {B} }.

and one can be placed with reference to the other through the relationship

{\displaystyle \mathbf {F} _{\text{loop}}=\mathbf {F} _{\text{dipole}}+\mathbf {m} \times \left(\nabla \times \mathbf {B} \right)}.

In all these expressions m is the dipole and B is the magnetic field at its position. Note that if there is no current or time-varying electric field, then × B = 0 and the two expressions agree.

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 a material causing both paramagnetism and ferromagnetism. Additionally, magnetic fields can affect currents creating magnetic fields (such as nuclear orbits) that cause diamagnetism.

Impact on environment

Magnetic field of magnetic moment

Any system having a net magnetic dipole moment m will generate a dipole magnetic field (described below) in the space around the system. While the net magnetic field produced by the system may also have higher-order multipole components, they will fade away more rapidly, so that only the dipole component will dominate the distance away from the system’s magnetic field.

The magnetic field of a magnetic dipole depends on the strength and direction of the magnetic moment of the magnet but falls as a cube of the distance such that: m

{\displaystyle {\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),}

where is the magnetic field produced by the magnet and is a vector from the center of the magnetic dipole to the point where the magnetic field is measured. The inverse cubic nature of this equation can be seen more easily by expressing the space vector as the product of the magnitude of the unit vector in its direction ( ) so that:

\mathbf{H}\mathbf {r}\mathbf {r}{\displaystyle \mathbf {r} =|\mathbf {r} |\mathbf {\hat {r}} }

{\displaystyle \mathbf {H} (\mathbf {r} )={\frac {1}{4\pi }}{\frac {3\mathbf {\hat {r}} (\mathbf {\hat {r}} \cdot \mathbf {m} )-\mathbf {m} }{|\mathbf {r} |^{3}}}.}

magnetic. Equation equivalent to -field μ ₀ = . The multiples of is the same except the factors of B 4 × 10 −7  H/m , where μ ₀ is known as the vacuum permittivity ^. for example:

{\displaystyle \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}}}.}

Force between two magnetic dipoles

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

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

where B1 is the magnetic field due _to the moment _M1 . The result of calculating the gradient is

{\displaystyle \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 r is the unit vector pointing from magnet 1 to magnet 2 and r is the distance. There is a similar expression

{\displaystyle \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.}

Torque of one magnetic dipole on the other

The torque of magnet 1 on magnet 2 is

{\displaystyle {\boldsymbol {\tau }}=\mathbf {m} _{2}\times \mathbf {B} _{1}.}

Theory Built-in Magnetic Dipole

The magnetic field of any magnet can be modeled by a series of terms for which each term is more complex (with a better angular description) than its former. The first three terms of that series are called monopole (represented by an isolated magnetic north or south pole), dipole (represented by two equal and opposite magnetic poles), and quadruple (represented by four poles that together form two equal and opposite poles). are called). dipole). The magnitude of the magnetic field for each term decreases progressively faster with distance than the previous term, with the first non-zero term dominating at a sufficient distance.

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

Magnetic potential

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

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

{\displaystyle {\mathbf {H} }({\mathbf {r} })=-\nabla \psi }.

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

{\displaystyle \mathbf {B} ({\mathbf {r} })=\nabla \times {\mathbf {A} }.}

Both of these possibilities can be calculated for any arbitrary current distribution (for the Amperian loop model) or magnetic charge distribution (for the magnetic charge model), provided they are confined to a small enough area to give:

{\displaystyle {\begin{aligned}\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',\\\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{aligned}}}

where is the current density. In the amperian loop model, the magnetic pole corresponding to the electric charge density is the power density that leads to the electric potential, and there are integrals at the integral volume (triple) coordinates that make up . The denominators of these equations can be expanded using polyhedron expansion to give a series of terms in which the denominator has a greater power of distances. Therefore, the first non-zero term will dominate for larger distances. The first non-zero term for vector potential is:

\mathbf {j}\rho\mathbf {r} '

{\displaystyle \mathbf {A} (\mathbf {r} )={\frac {\mu _{0}}{4\pi }}{\frac {\mathbf {m} \times \mathbf {r} }{|\mathbf {r} |^{3}}},}

Where is it: m

{\displaystyle \mathbf {m} ={\tfrac {1}{2}}\iiint _{V}\mathbf {r} \times \mathbf {j} \,{\rm {d}}V,}

where × is the vector product across, r is the position vector, and j is the electric flux density and the integral is a quantity integral.

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

{\displaystyle \psi (\mathbf {r} )={\frac {\mathbf {m} \cdot \mathbf {r} }{4\pi |\mathbf {r} |^{3}}}.}

Here the magnetic pole can be represented in terms of power density but is more usefully expressed in terms of magnetization field: m

{\displaystyle \mathbf {m} =\iiint \mathbf {M} \,\mathrm {d} V.}

The same symbol is used for both equations because they produce similar results outside the magnet. m

External magnetic field produced by magnetic dipole moment

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

{\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).

Furthermore, the magnetic field strength is H

{{\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).

Dipole’s internal magnetic field

The two models for a dipole (current loop and magnetic pole) give similar predictions for the magnetic field away from the source. However, inside the source region, they make different predictions. The magnetic field between the poles (see figure for magnetic pole definition) is in the opposite direction of the magnetic moment (which points from negative charge to positive charge), while inside the current loop it is in the same direction (see figure to the right) And). The boundaries of these regions should also vary as the sources shrink to zero size. This difference only matters when the dipole limit is used to calculate the fields inside the magnetic material. ^[8]

If a magnetic dipole is formed by shortening and shortening a current loop, but keeping the product of current and field constant, the finite field is

{\displaystyle \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].}

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

If a “north pole” and a “south pole” form a magnetic dipole, bringing them closer together, but keeping the magnetic pole charge and distance constant, is a finite field

{\displaystyle \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].}

These fields are related to B = μ ₀ ( H + M ) , where M ( R ) = m ( R ) is magnetization .

Relation to angular momentum

Magnetic moment has a close relationship with angular momentum which is called gyromagnetic effect . This effect is expressed on a macroscopic scale in the Einstein–de Haas effect, or “rotation by magnetization,” and conversely, in the Barnett effect, or “magnetism by rotation.” ^[1] In addition, a torque applied to a relatively isolated magnetic dipole such as an atomic nucleus can propel it forward (rotate about the axis of the applied field). This phenomenon is used in nuclear magnetic resonance.

Viewing a magnetic dipole as a current loop reveals a close relationship between magnetic moment and angular momentum. Since the particles that produce the current (moving around the loop) have charge and mass, both the magnetic moment and the angular momentum increase with the rate of rotation. The ratio of the two is called the gyromagnetic ratio or . So called : Y

{\displaystyle \mathbf {m} =\gamma \,\mathbf {L} ,}

where is is the angular momentum of the particle or particles making up the magnetic moment. L

In the Ampere loop model, which is applicable for macroscopic currents, the gyromagnetic ratio is half of the charge-to-mass ratio. It can be shown like this. The angular momentum of a moving charged particle is defined as:

{\displaystyle \mathbf {L} =\mathbf {r} \times \mathbf {p} =\mu \,\mathbf {r} \times \mathbf {v} ,}

where μ is the mass of the particle and v is the velocity of the particle. The angular momentum of a large number of charged particles that form a current is therefore:

{\displaystyle \mathbf {L} =\iiint _{V}\,\mathbf {r} \times (\rho \mathbf {v} )\,{\rm {d}}V\,,}

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

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

{\displaystyle \mathbf {m} ={\tfrac {1}{2}}\iiint _{V}\,\mathbf {r} \times (\rho _{Q}\mathbf {v} )\,{\rm {d}}V\,,}

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

{\displaystyle \mathbf {j} =\rho _{Q}\mathbf {v} }{\displaystyle \rho _{Q}}

Comparing the two equations gives the result:

{\displaystyle \mathbf {m} ={\frac {e}{2\mu }}\,\mathbf {L} \,,}

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

e\mu

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

{\displaystyle \mathbf {m} =g\,{\frac {e}{2\mu }}\,\mathbf {L} ,}

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

In the atomic world the angular momentum (spin of a particle) is an integer (or in the case of half-integer spin) several of the lesser Planck constant h . This is the basis for defining the magnetic moment units of the Bohr magneton (assuming charge-to-mass ratio of electrons) and atomic magneton (assuming charge-to-mass ratio of protons). See Magnetic moment electron and Bohr magneton for more information.

Atoms, Molecules and Elementary Particles

Basically, contributions to the magnetic moment of any system can come from two types of sources: the movement of electric charges, such as electric currents; and the intrinsic magnetism of elementary particles such as electrons.

Using the formulas below, knowing the distribution of all electric currents (or, alternatively, all electric charges and their velocities) inside the system, the contribution due to the first type of sources can be calculated. On the other hand, the magnitude of the intrinsic magnetic moment of each elementary particle is a fixed number, often experimentally measured to a great precision. For example, the magnetic moment of any electron is measured as−9.284 764 × 10 ⁻²⁴  J/t . ^[20] The direction of any elementary particle’s magnetic moment is determined entirely 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, containing one proton and one electron) is a vector sum of the following contributions:

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

Similarly, the magnetic moment contribution of a bar magnet is the sum of the 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 atom

For an atom, the individual electron spins are added up to get the total spin, and the individual orbital angular momentum is added up to get the total orbital angular momentum. These two are combined using angular momentum coupling to get the total angular momentum. For an atom with no nuclear magnetic moment, the magnitude of the nuclear dipole moment, , is then 

{\displaystyle {\mathfrak {m}}_{atom}}

{\displaystyle {\mathfrak {m}}_{\text{atom}}=g_{\rm {J}}\,\mu _{\rm {B}}\,{\sqrt {j\,(j+1)\,}}}

where j is the total angular momentum quantum number, g _j is the Lande g-factor, and μ _b is the Bohr magneton. The component of this magnetic moment is along the direction of the magnetic field

{\displaystyle {\mathfrak {m}}_{{\text{atom}},z}=-m\,g_{\rm {J}}\,\mu _{\rm {B}}~}.

The negative sign occurs because electrons have a negative charge.

The integer m (not to be confused with the moment ) is called the magnetic quantum number or the equatorial quantum number, which can take any of the values ​​of 2 j + 1 : m

{\displaystyle -j,\ -(j-1),\ \cdots ,\ -1,\ 0,\ +1,\ \cdots ,\ +(j-1),\ +j~}.

Because of the angular momentum, the mobility of a magnetic dipole in a magnetic field is different from that of an electric dipole in an electric field. The field exerts a torque on the magnetic dipole which aligns it with the field. However, torque is proportional to the rate of change of angular momentum, so precedence occurs: the direction of the spin changes. This behavior is described by the Landau–Lifshitz–Gilbert equation:

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

where is the gyromagnetic ratio, m is the magnetic moment, is the indifference coefficient and h _eff is the effective magnetic field (external field plus any self-induced field). The first term describes the precession of the moment about the effective field, while the second is a damping term related to the dissipation of energy due to interaction with the surroundings.

Magnetic moment of an electron

Electrons and many elementary particles also have intrinsic magnetic moments, the interpretation of which requires quantum mechanical treatment and is related 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_\text{S} \mu_\text{B} \mathbf{S}}{\hbar},

where μ is the Bohr magneton, s is the electron spin, and the g-factor gs _is 2 according to Dirac’s principle, but because of the quantum electrochemical effect it is actually slightly larger:2.002 319 304 36 . Deviation from 2 is known as anomalous magnetic dipole moment.

Again it is important to note that m is a negative constant multiplied by the spin, so the magnetic moment of the electron is parallel to the spin. This can be understood from the following classical diagram: If we assume that the spin angular momentum is created by the electron mass rotating around an axis, then the electric current produced by this rotation is driven in the opposite direction by the negative charge of the electron. it occurs. , Such current loops produce a magnetic moment that is parallel to the spin. Therefore, the magnetic moment for a positron (the electron’s anti-particle) is parallel to its spin.

Magnetic moment of the nucleus

The nuclear system is a complex physical system consisting of nucleons, i.e. protons and neutrons. The quantum mechanical properties of nucleons include spin among others. Since the electromagnetic moments of nuclei depend on the spins of individual nucleons, these properties can be observed with measurements of nuclear moments, and in particular nuclear magnetic dipole moments.

Most normal nuclei exist in their ground state, although the nuclei of some isotopes have longer excited states. Each energy state of the nucleus of a given isotope is characterized by a well-defined magnetic dipole moment with a fixed number of magnitudes, which is often experimentally measured to a great precision. This number is very sensitive to individual contributions from nucleons, and measurement or prediction of its value can reveal important information about the content of the nuclear wave function. There are many theoretical models that predict the value of the magnetic dipole moment and several experimental techniques aimed at making measurements in nuclei with atomic charts.

Magnetic moment of the molecule

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

• Magnetic moment, if any, due to its unpaired electron spin (paramagnetic contribution)
• the orbital speed 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 cycles, which depends on the nuclear spin configuration.

Examples of molecular magnetism

• The dioxygen molecule, O ₂ , exhibits strong paramagnetism, due to the unpaired spin of its outermost two electrons.
• _The carbon dioxide molecule, CO , mostly exhibits diamagnetism, a very weak magnetic moment of the electron that orbits that is proportional to the external magnetic field. The atomic magnetization of a magnetic isotope such as ¹³ C or ¹⁷ O would contribute to the magnetic moment of the molecule.
• _The dihydrogen molecule, H2 , exhibits nuclear magnetism in a weak (or zero) magnetic field, and may 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-line transition metals. ^[26]

Elementary particles

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

For a connection between the notions of magnetic moment and magnetization, see Magnetism.