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This article is about a mathematical description of how magnetized a region of magnetic material is. For a microscopic description of how magnetic materials react to a magnetic field, see magnetism. For how magnets become magnetized, see magnetize. For mathematical description of fields surrounding magnets and currents, see magnetic field.

Magnetization, M, is defined as the quantity of magnetic moment per unit volume, V:

$\mathbf{M}=\frac{N}{V}\mathbf{m}=n\mathbf{m}$

Here, N is the number of magnetic moments in the sample. The quantity N/V is usually written as n, the density of magnetic moments. The M-field is measured in amperes per meter (A/m) in SI units. ^[1]

The origin of the magnetic moments responsible for magnetization can be either microscopic electric currents resulting from the motion of electrons in atoms, or the spin of the electrons or the nuclei. Net magnetization results from the response of a material to an external magnetic field, together with any unbalanced magnetic dipole moments that may be inherent in the material itself; for example, in ferromagnets. Magnetization is not always homogeneous within a body, but rather a function of position.

[edit] Magnetization in Maxwell's equations

The behavior of magnetic fields (B, H), electric fields (E, D), charge density (ρ), and current density (J) is described by Maxwell's equations. The role of the magnetization is described below.

[edit] Relations between B, H, and M

Main article: Magnetic field

The magnetization defines the auxiliary magnetic field H as

$\mathbf{B}=\mu_0\mathbf{(H + M)}$ (SI units)

$\mathbf{B} = \mathbf{(H + 4 \pi M)}$ (Gaussian units)

which is convenient for various calculations.

A relation between M and H exists in many materials. In diamagnets and paramagnets, the relation is usually linear:

$\mathbf{M} = \chi_m\mathbf{H}$

where χ_m is called the volume magnetic susceptibility.

In ferromagnets there is no one-to-one correspondence between M and H because of hysteresis.

[edit] Magnetization current

The magnetization M makes a contribution to the current density J, known as the magnetization current or bound current:

$\mathbf{J_m} = \nabla\times\mathbf{M}$

so that the total current density that enters Maxwell's equations is given by

$\mathbf{J} = \mathbf{J_f} + \nabla\times\mathbf{M} + \frac{\partial\mathbf{P}}{\partial t}$

where J_f is the electric current density of free charges (also called the free current), the second term is the contribution from the magnetization, and the last term is related to the electric polarization P.

[edit] Magnetostatics

In the absence of free electric currents and time-dependent effects, Maxwell's equations describing the magnetic quantities reduce to

$\mathbf{\nabla\cdot H} = - \nabla\cdot\mathbf{M} \qquad \mathbf{\nabla\times H} = 0$

These equations can be easily solved in analogy with electrostatic problems where

$\mathbf{\nabla\cdot E} = \frac{\rho}\epsilon_0 \qquad \qquad \qquad \mathbf{\nabla\times E} = 0$

In this sense −μ₀∇·M plays the role of a "magnetic charge density" analogous to the electric charge density ρ.

Magnetization is volume density of magnetic moment. That is: if a certain volume has magnetization M then the volume element dV has a magnetic moment of dm = MdV.

[edit] Magnetization Dynamics

Main article: Magnetization dynamics

The time-dependent behavior of magnetization becomes important when considering nanoscale and nanosecond timescale magnetization. Rather than simply aligning with an applied field, the individual magnetic moments in a material begin to precess around the applied field and come into alignment through relaxation as energy is transferred into the lattice.

[edit] Types of magnetism

Hierarchy of types of magnetism. See Myers.^[2]

[edit] Diamagnetism

Main article: Diamagnetism

This is the most common magnetic behavior. The diamagnetic magnetization is proportional and opposing to the applied magnetic field. All materials present a diamagnetic response, although it may be overshadowed by stronger magnetic behaviors. Diamagnetism can be explained by the normal response of the orbiting electrons considering Lenz's law.

This is a weak form of magnetism that is nonpermanent and persists only while external field is applied. The magnitude of induced magnetic moment is very small and in a direction opposite to that of applied field. Therefore, relative permeability is less than 1 and magnetic susceptibility is negative. When placed between the poles of a strong electromagnet, diamagnetic materials are pushed out towards the region where the field is weaker.

[edit] Paramagnetism

Main article: Paramagnetism

Paramagnetic materials present a magnetization that is proportional to the applied field and reinforces it. This arises from the existence of magnetic dipoles in the material. Paramagnetism varies inversely with temperature and is characterized by the material's saturation magnetization. When placed between the poles of a strong electromagnet, paramagnetic materials are pulled towards the region where the field is stronger.

[edit] Superparamagnetism

Superparamagnetic materials are paramagnetic materials whose magnetization saturates at very strong fields. They are obtained using magnetic nanoparticle aggregates with great net magnetic moments. Each particle is a single magnetic domain. Consequently, the alignment of spins under applied field is no longer impeded by domain walls. Above a certain temperature (called the "blocking temperature"), thermal vibrations randomly fluctuate the net spins, canceling one another and the net moment of the collective particles is zero at zero field (no coercive field). If a magnetic field is applied, the particles will align producing a net moment. This behavior is characteristic of paramagnetic materials, but the difference is that each nanoparticle has a great net moment, so the saturation of magnetization occurs at very strong fields of several teslas.

[edit] Ferromagnetism

Ferromagnetic materials present a magnetization much more than other materials. Ferromagnetism arises from the strong coupling between the neighboring magnetic dipoles in the material. Ferromagnetic materials can present spontaneous magnetization, and this gives rise to the hysteresis loops. Ferromagnetic materials can be characterized by their permeability, Curie temperature (temperature of the phase change to paramagnetic behavior), coercive field (field strength needed to demagnetize the material), and remnant magnetization (magnetization at zero external field).

[edit] See also

Permeability (electromagnetism)
Magnetic susceptibility
Earth's magnetic field
Geomagnetic reversal
Geomagnetic excursion
Magnetic constant
The Wiktionary definition of magnetization

[edit] Sources

^ "Units for Magnetic Properties". Lake Shore Cryotronics, Inc.. http://www.magneticmicrosphere.com/resources/Units_for_Magnetic_Properties.pdf. Retrieved 2009-10-24.
^ HP Meyers (1997). Introductory solid state physics (2 ed.). CRC Press. p. 322; Figure 11.1. ISBN 0748406603. http://books.google.com/books?id=Uc1pCo5TrYUC&pg=PA322.

[0] "Units for Magnetic Properties". Lake Shore Cryotronics, Inc.. http://www.magneticmicrosphere.com/resources/Units_for_Magnetic_Properties.pdf. Retrieved 2009-10-24.

[Meyers1-1] HP Meyers (1997). Introductory solid state physics (2 ed.). CRC Press. p. 322; Figure 11.1. ISBN 0748406603. http://books.google.com/books?id=Uc1pCo5TrYUC&pg=PA322.

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