Magnetic moment

The magnetic moment is the magnetic strength and orientation of a magnet or other object that produces a magnetic field. Examples of objects that have magnetic moments include: loops of electric current (such as electromagnets), permanent magnets, elementary particles (such as electrons), various molecules, and many astronomical objects (such as many planets, some moons, stars, etc).
More precisely, the term "magnetic moment" normally refers to a system's magnetic dipole moment, the component of the magnetic moment that can be represented by an equivalent magnetic dipole: a magnetic north and south pole separated by a very small distance. The magnetic dipole component is sufficient for small enough magnets or for large enough distances. Higher-order terms (such as the magnetic quadrupole moment) may be needed in addition to the dipole moment for extended objects.
The magnetic dipole moment of an object is readily defined in terms of the torque that object experiences in a given magnetic field. The same applied magnetic field creates larger torques on objects with larger magnetic moments. The strength (and direction) of this torque depends not only on the magnitude of the magnetic moment but also on its orientation relative to the direction of the magnetic field. The magnetic moment may be considered, therefore, to be a vector. The direction of the magnetic moment points from the south to north pole of the magnet (inside the magnet).
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 measurement.
Definition.
The magnetic moment can be defined as a vector relating the aligning torque on the object from an externally applied magnetic field to the field vector itself. The relationship is given by:
where is the torque acting on the dipole, is the external magnetic field, and 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, , with respect to external magnetic field:
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.
Units.
The unit for magnetic moment in International System of Units (SI) base units is A⋅m2, 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:
where N is newton (SI derived unit of force), T is tesla (SI derived unit of magnetic flux density), and J is joule (SI derived unit of energy). Although torque (N·m) and energy (J) are dimensionally equivalent, torques are never expressed in units of energy.
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:
where statA is statamperes, cm is centimeters, erg is ergs, and G is gauss. The ratio of these two non-equivalent CGS units (EMU/ESU) is equal to the speed of light in free space, expressed in cm⋅s−1.
All formulae in this article are correct in SI units; they may need to be changed for use in other unit systems. For example, in SI units, a loop of current with current and area has magnetic moment (see below), but in Gaussian units the magnetic moment is .
Other units for measuring the magnetic dipole moment include the Bohr magneton and the nuclear magneton.
Measurement.
The magnetic moments of objects are typically measured with devices called magnetometers, 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.
Relation to magnetization.
The magnetic moment is a quantity that describes the magnetic strength of an entire object. Sometimes, though, it is useful or necessary to know how much of the net magnetic moment of the object is produced by a particular portion of that magnet. Therefore, it is useful to define the magnetization field as:
where and are the magnetic dipole moment and volume of a sufficiently small portion of the magnet This equation is often represented using derivative notation such that
where is the elementary magnetic moment and is the volume element. The net magnetic moment of the magnet therefore is
where the triple integral denotes integration over the volume of the magnet. For uniform magnetization (where both the magnitude and the direction of is the same for the entire magnet (such as a straight bar magnet) the last equation simplifies to:
where is the volume of the bar magnet.
The magnetization is often not listed as a material parameter for commercially available ferromagnetic materials, though. Instead the parameter that is listed is residual flux density (or remanence), denoted The formula needed in this case to calculate in (units of A⋅m2) is:
where:
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. In magnetic materials, the cause of the magnetic moment are the spin and orbital angular momentum states of the electrons, and varies depending on whether atoms in one region are aligned with atoms in another.
Magnetic pole model.
The sources of magnetic moments in materials can be represented by poles in analogy to electrostatics. This is sometimes known as the Gilbert model. In this model, a small magnet is modeled by a pair of magnetic poles of equal magnitude but opposite polarity. Each pole is the source of magnetic force which weakens with distance. Since 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 of its poles ("magnetic pole strength"), and the vector formula_11 separating them. The magnetic dipole moment is related to the fictitious poles as
It points in the direction from South to North pole. The analogy with electric dipoles should not be taken too far because magnetic dipoles are associated with angular momentum (see Relation to angular momentum). Nevertheless, magnetic poles are very useful for magnetostatic calculations, particularly in applications to ferromagnets. Practitioners using the magnetic pole approach generally represent the magnetic field by the irrotational field , in analogy to the electric field .
Amperian loop model.
After Hans Christian Ørsted discovered that electric currents produce a magnetic field and André-Marie Ampère discovered that electric currents attract and repel each other similar to magnets, it was natural to hypothesize that all magnetic fields are due to electric current loops. In this model developed by Ampère, the elementary magnetic dipole that makes up all magnets is a sufficiently small amperian loop of current I. The dipole moment of this loop is
where 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.
The magnetic dipole moment can be calculated for a localized (does not extend to infinity) current distribution assuming that we know all of the currents involved. Conventionally, the derivation starts from a multipole expansion of the vector potential. This leads to the definition of the magnetic dipole moment as:
where × is the vector cross product, is the position vector, and is the electric current density and the integral is a volume integral. When the current density in the integral is replaced by a loop of current I in a plane enclosing an area S then the volume integral becomes a line integral and the resulting dipole moment becomes
which is how the magnetic dipole moment for an Amperian loop is derived.
Practitioners using the current loop model generally represent the magnetic field by the solenoidal field , analogous to the electrostatic field .
Magnetic moment 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 identical turns (single-layer winding) and vector area ,
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 straight forward to develop for macroscopic currents using the amperian loop model (see below), neither the magnetic pole model nor the amperian loop model truly represents what is occurring at the atomic and molecular levels. At that level quantum mechanics must be used. Fortunately, the linear relationship between the magnetic dipole moment of a particle and its angular momentum still holds; although it is different for each particle. Further, care must be used to distinguish between the intrinsic angular momentum (or spin) of the particle and the particle's orbital angular momentum. See below for more details.
Effects of an external magnetic field.
Torque on a moment.
The torque on an object having a magnetic dipole moment in a uniform magnetic field is:
This is valid for the moment due to any localized current distribution provided that the magnetic field is uniform. For non-uniform B the equation is also valid for the torque about the center of the magnetic dipole provided that the magnetic dipole is small enough.
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.
A magnetic moment in an externally produced magnetic field has a potential energy :
In a case when the external magnetic field is non-uniform, there will be a
force, proportional to the magnetic field gradient, acting on the magnetic moment itself. There are two expressions for the force acting on a magnetic dipole, depending on whether the model used for the dipole is a current loop or two monopoles (analogous to the electric dipole). The force obtained in the case of a current loop model is
In the case of a pair of monopoles being used (i.e. electric dipole model), the force is
And one can be put in terms of the other via the relation
In all these expressions is the dipole and is the magnetic field at its position. Note that if there are no currents or time-varying electrical fields, 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 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.
Any system possessing a net magnetic dipole moment will produce a dipolar magnetic field (described below) in the space surrounding the system. While the net magnetic field produced by the system can also have higher-order multipole components, those will drop off with distance more rapidly, so that only the dipole component will dominate the magnetic field of the system at distances far away from it.
The magnetic field of a magnetic dipole depends on the strength and direction of a magnet's magnetic moment formula_22 but drops off as the cube of the distance such that:
where formula_24 is the current density in the amperian loop model, formula_25 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 formula_26. The denominators of these equation can be expanded using the multipole expansion to give a series of terms that have larger of power of distances in the denominator. The first nonzero term, therefore, will dominate for large distances. The first non-zero term for the vector potential is:
where formula_22 is:
where × is the vector cross product, is the position vector, and is the electric current density and the integral is a volume integral.
In the magnetic pole perspective, the first non-zero term of the scalar potential is
Here formula_22 may be represented in terms of the magnetic pole strength density but is more usefully expressed in terms of the magnetization field as:
The same symbol formula_22 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
Further, the magnetic field strength formula_35 is
Internal magnetic field of a dipole.
The two models for a dipole (current loop and magnetic poles) give the same predictions for the magnetic field far from the source. However, inside the source region, they give different predictions. The magnetic field between poles (see figure for Magnetic pole definition) is in the opposite direction to the magnetic moment (which points from the negative charge to the positive charge), while inside a current loop it is in the same direction (see the figure to the right). The limits of these fields must also be different as the sources shrink to zero size. This distinction only matters if the dipole limit is used to calculate fields inside a magnetic material.
If a magnetic dipole is formed by making a current loop smaller and smaller, but keeping the product of current and area constant, the limiting field is
Unlike the expressions in the previous section, this limit is correct for the internal field of the dipole.
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
These fields are related by , where is the magnetization.
Relation to 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." Further, a torque applied to a relatively isolated magnetic dipole such as an atomic nucleus can cause it to precess (rotate about the axis of the applied field). This phenomenon is used in nuclear magnetic resonance.
Viewing a magnetic dipole as current loop brings out the close connection between magnetic moment and angular momentum. Since the particles creating the current (by rotating around the loop) have charge and mass, both the magnetic moment and the angular momentum increase with the rate of rotation. The ratio of the two is called the gyromagnetic ratio or formula_39 so that:
where formula_41 is the angular momentum of the particle or particles that are creating the magnetic moment.
In the amperian loop model, which applies for macroscopic currents, the gyromagnetic ratio is one half of the charge-to-mass ratio. This can be shown as follows. The angular momentum of a moving charged particle is defined as:
where is the mass of the particle and is the particle's velocity. The angular momentum of the very large number of charged particles that make up a current therefore is:
where is the mass density of the moving particles. By convention the direction of the cross product is given by the right-hand rule.
This is similar to the magnetic moment created by the very large number of charged particles that make up that current:
where formula_45 and formula_46 is the charge density of the moving charged particles.
Comparing the two equations results in:
where formula_48 is the charge of the particle and formula_49 is the mass of the particle.
Even though atomic particles cannot be accurately described as orbiting (and spinning) charge distributions of uniform charge-to-mass ratio, this general trend can be observed in the atomic world so that:
where the -factor depends on the particle and configuration. For example the -factor for the magnetic moment due to an electron orbiting a nucleus is one while the -factor for the magnetic moment of electron due to its intrinsic angular momentum (spin) is a little larger than 2. The -factor of atoms and molecules must account for the orbital and intrinsic moments of its electrons and possibly the intrinsic moment of its nuclei as well.
In the atomic world the angular momentum (spin) of a particle is an integer (or half-integer in the case of spin) multiple of the reduced Planck constant . 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: 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 . 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:
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. For an atom with no nuclear magnetic moment, the magnitude of the atomic dipole moment, formula_51, is then
where is the total angular momentum quantum number, is the Landé -factor, and is the Bohr magneton. The component of this magnetic moment along the direction of the magnetic field is then
The negative sign occurs because electrons have negative charge.
The integer (not to be confused with the moment, formula_54) is called the magnetic quantum number or the "equatorial" quantum number, which can take on any of values:
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: