Magnetization

This article is about magnetization as it appears in Maxwell's equations of classical electrodynamics. For a microscopic description of how magnetic materials react to a magnetic field, seemagnetism.For mathematical description of fields surrounding magnets and currents, seemagnetic field.

Common symbols	M
SI unit	Ampere-meter-1
InSI base units	m-1⋅A
Dimension	L-1I

Inclassical electromagnetism,magnetizationis thevector fieldthat expresses thedensityof permanent or inducedmagnetic dipole momentsin a magnetic material. Accordingly, physicists and engineers usually define magnetization as the quantity ofmagnetic momentper unit volume. It is represented by apseudovectorM.Magnetization can be compared toelectric polarization,which is the measure of the corresponding response of a material to anelectric fieldinelectrostatics.

Magnetization also describes how a material responds to an appliedmagnetic fieldas well as the way the material changes the magnetic field, and can be used to calculate theforcesthat result from those interactions.

The origin of the magnetic moments responsible for magnetization can be either microscopicelectric currentsresulting from the motion ofelectronsinatoms,or thespinof the electrons or the nuclei. Net magnetization results from the response of a material to an externalmagnetic field.

Paramagneticmaterials have a weak induced magnetization in a magnetic field, which disappears when the magnetic field is removed.Ferromagneticandferrimagneticmaterials have strong magnetization in a magnetic field, and can bemagnetizedto have magnetization in the absence of an external field, becoming apermanent magnet.Magnetization is not necessarily uniform within a material, but may vary between different points.

Definition

The magnetization field orM-field can be defined according to the following equation: $$\mathbf {M} ={\frac {\mathrm {d} \mathbf {m} }{\mathrm {d} V}}$$

Where$\mathrm {d} \mathbf {m}$is the elementarymagnetic momentand$\mathrm {d} V$is thevolume element;in other words, theM-field is the distribution of magnetic moments in the region ormanifoldconcerned. This is better illustrated through the following relation: $$\mathbf {m} =\iiint \mathbf {M} \,\mathrm {d} V$$ wheremis an ordinary magnetic moment and the triple integral denotes integration over a volume. This makes theM-field completely analogous to theelectric polarisation field,orP-field, used to determine theelectric dipole momentpgenerated by a similar region or manifold with such a polarization: $$\mathbf {P} ={\mathrm {d} \mathbf {p} \over \mathrm {d} V},\quad \mathbf {p} =\iiint \mathbf {P} \,\mathrm {d} V$$

Where$\mathrm {d} \mathbf {p}$is the elementary electric dipole moment.

Those definitions ofPandMas a "moments per unit volume" are widely adopted, though in some cases they can lead to ambiguities and paradoxes.

TheM-field is measured inamperespermeter(A/m) inSIunits.

In Maxwell's equations

The behavior ofmagnetic fields(B,H),electric fields(E,D),charge density(ρ), andcurrent density(J) is described byMaxwell's equations.The role of the magnetization is described below.

Relations between B, H, and M

Main article:Magnetic field

The magnetization defines the auxiliary magnetic fieldHas

$\mathbf {B} =\mu _{0}(\mathbf {H+M} )$(SI units)

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

which is convenient for various calculations. Thevacuum permeabilityμ₀is, approximately,4π×10⁻⁷V·s/(A·m) (in SI units).

A relation betweenMandHexists in many materials. Indiamagnetsandparamagnets,the relation is usually linear:

$\mathbf {M} =\chi \mathbf {H},\,\mathbf {B} =\mu \mathbf {H} =\mu _{0}(1+\chi )\mathbf {H},$

whereχis called thevolume magnetic susceptibility,and μ is called themagnetic permeabilityof the material. Themagnetic potential energyper unit volume (i.e. magneticenergy density) of the paramagnet (or diamagnet) in the magnetic field is:

$-\mathbf {M} \cdot \mathbf {B} =-\chi \mathbf {H} \cdot \mathbf {B} =-{\frac {\chi }{1+\chi }}{\frac {\mathbf {B} ^{2}}{\mu _{0}}},$

the negative gradient of which is themagnetic forceon the paramagnet (or diamagnet) per unit volume (i.e. force density).

In diamagnets ($\chi <0$) and paramagnets ($\chi >0$), usually$|\chi |\ll 1$,and therefore$\mathbf {M} \approx \chi {\frac {\mathbf {B} }{\mu _{0}}}$.

Inferromagnetsthere is no one-to-one correspondence betweenMandHbecause ofmagnetic hysteresis.

Magnetic polarization

Alternatively to the magnetization, one can define themagnetic polarization,I(often the symbolJis used, not to be confused with current density).

$\mathbf {B} =\mu _{0}\mathbf {H} +\mathbf {I}$(SI units).

This is by direct analogy to theelectric polarization,$\mathbf {D} =\varepsilon _{0}\mathbf {E} +\mathbf {P}$. The magnetic polarization thus differs from the magnetization by a factor ofμ₀:

$\mathbf {I} =\mu _{0}\mathbf {M}$(SI units).

Whereas magnetization is measured typically in amperes/meter, the magnetic polarization is measured in teslas.

Magnetization current

When the microscopic currents induced by the magnetization (black arrows) do not balance out, bound volume currents (blue arrows) and bound surface currents (red arrows) appear in the medium.

The magnetizationMmakes a contribution to thecurrent densityJ,known as themagnetization current.

$\mathbf {J} _{\mathrm {m} }=\nabla \times \mathbf {M}$

and for thebound surface current:

$\mathbf {K} _{\mathrm {m} }=\mathbf {M} \times \mathbf {\hat {n}}$

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

$\mathbf {J} =\mathbf {J} _{\mathrm {f} }+\nabla \times \mathbf {M} +{\frac {\partial \mathbf {P} }{\partial t}}$

whereJ_fis the electric current density of free charges (also called thefree current), the second term is the contribution from the magnetization, and the last term is related to theelectric polarizationP.

Magnetostatics

Main article:Magnetostatics

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

${\begin{aligned}\mathbf {\nabla \times H} &=\mathbf {0} \\\mathbf {\nabla \cdot H} &=-\nabla \cdot \mathbf {M} \end{aligned}}$

These equations can be solved in analogy withelectrostaticproblems where

${\begin{aligned}\mathbf {\nabla \times E} &=\mathbf {0} \\\mathbf {\nabla \cdot E} &={\frac {\rho }{\epsilon _{0}}}\end{aligned}}$

In this sense −∇⋅Mplays the role of a fictitious "magnetic charge density" analogous to theelectric charge densityρ;(see alsodemagnetizing field).

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.

Reversal

Magnetization reversal, also known as switching, refers to the process that leads to a 180° (arc) re-orientation of the magnetizationvectorwith respect to its initial direction, from one stable orientation to the opposite one. Technologically, this is one of the most important processes inmagnetismthat is linked to the magneticdata storageprocess such as used in modernhard disk drives.As it is known today, there are only a few possible ways to reverse the magnetization of a metallic magnet:

1. an appliedmagnetic field
2. spin injectionvia a beam of particles withspin
3. magnetization reversal by circularly polarized light;i.e., incident electromagnetic radiation that iscircularly polarized

Demagnetization

Main article:Degaussing

Demagnetization is the reduction or elimination of magnetization.One way to do this is to heat the object above itsCurie temperature,where thermal fluctuations have enough energy to overcomeexchange interactions,the source of ferromagnetic order, and destroy that order. Another way is to pull it out of an electric coil with alternating current running through it, giving rise to fields that oppose the magnetization.

One application of demagnetization is to eliminate unwanted magnetic fields. For example, magnetic fields can interfere with electronic devices such as cell phones or computers, and with machining by making cuttings cling to their parent.

See also

• Magnetometer
• Orbital magnetization