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# Magnetization

Magnetization is a property of some materials (e.g. magnets) that describes to what extent they are affected by magnetic fields, and also determines the magnetic field that the material itself creates. Magnetization is defined as the amount of magnetic moment per unit volume. The origin of the magnetic moments that create the magnetization can be either microscopic electric currents corresponding to the motion of electrons in atoms, or the spin of the electrons.

In some materials (e.g., ferromagnets) magnetization can exist even without an external magnetic field (spontaneous magnetization). In other types of materials, magnetization is induced only when an external magnetic field is present. Magnetization is not always homogeneous within a body, but rather a function of position.

## Magnetization in Maxwell's equations

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

#### Relations between B, H and M

The magnetization defines the auxiliary magnetic field $\mathbf{H}$ as

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

which is convenient for various calculations.

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

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

where $\chi_m\,$ is called the volume magnetic susceptibility.

In ferromagnets there is no one-to-one correspondence between $\mathbf{M}$ and $\mathbf{H}$ because of hysteresis.

#### Magnetization current

The magnetization $\mathbf{M}$ makes a contribution to the current density $\mathbf{J}$, 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 $\mathbf{J_f}$ is the electric current density of free charges, the second term is the contribution from the magnetization, and the last term is related to the electric polarization $\mathbf{P}$.

#### 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 $-\epsilon_0\nabla\cdot\mathbf{M}$ plays the role of a "magnetic charge density" analogous to the electric charge density $\rho\,$.

Magnetization is volume density of magnetic moment. That is: if a certain volume has magnetization $\mathbf{M}$ then volume element dV has magnetic moment of $d \mathbf{m} = \mathbf{M} dV$.

## Types of magnetism

### 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 shadowed by stronger magnetic behaviors. Diamagnetism can be explained by the normal response of the orbiting electrons considering the Lenz's law. This is a weak form of magnetism that is non permanent 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.

### 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.

### Superparamagnetism

Superparamagnetic materials are paramagnetic materials whose magnetization saturates at very large fields. They are obtained using magnetic nanoparticle aggregates with large 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 the blocking temperature, thermal vibrations randomly fluctuate the net spins, cancelling 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 large net moment, so the saturation of magnetization occurs at very large fields of several teslas.

### Ferromagnetism

Ferromagnetic materials present a magnetization much larger 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).