Magnetization

Magnetization, M, is the magnetic moment per unit volume, V of a material, defined in terms of the magnetic moments of its constituents by:
 * $$\mathbf{M}=\frac{1}{V}\sum_{j=1}^N \mathbf{m_j}$$

where the magnetic moment mj of the j-th constituent is a vector property that determines the torque the object experiences in a magnetic field tending to align its moment with the field. Here, N is the number of magnetic moments in the volume V. The M-field is measured in amperes per meter (A/m) in SI units. Usually the magnetization is referred to a particular location r by imagining the volume V to be a microscopic region enclosing point r, and is anticipated to change with time t in the general case (perhaps because the moments are moving), defining a magnetization field, M(r, t).

At a microscopic level, the origin of the magnetic moments responsible for magnetization is traced to angular momentum, such as due to motion of electrons in atoms, or to spin, such as the intrinsic spin of electrons or atomic nuclei. According to the Bohr-van Leeuwen theorem. magnetism is a quantum mechanical effect; on the basis of classical mechanics, there cannot be diamagnetism, paramagnetism or collective magnetism (that is, most notably ferromagnetism or antiferromagnetism, but also ferrimagnetism and metamagnetism). As stated by Van Vleck: "At any finite temperature, and in all finite applied electrical or magnetic fields, the net magnetization of a collection of nonrelativistic classical electrons in thermal equilibrium vanishes identically."

Magnetic moments
As magnetization is related to magnetic moments, its understanding requires a notion of where magnetic moments originate. As a general statement, magnetic moments are related to either angular momentum or to spin, both of which at a microscopic level are related to rotational phenomena. The connection is made via the gyromagnetic ratio, the proportionality factor between magnetic moment and spin or angular momentum for a given object.

Although these ideas apply to nucleii and other particles, here attention is focused on electrons in atoms. The magnetic moment mS of a system of electrons with spin S is:
 * $$\mathbf{m_S} = 2m_B \mathbf S \, $$

and the magnetic moment mL of an electronic orbital momentum L is:
 * $$\mathbf{m_L} = m_B \mathbf{L} \ . $$

Here the factor mB refers to the Bohr magneton, defined by:
 * $$m_B = \frac{e \hbar}{2 m_e} \, $$

with e = the electron charge, ℏ = Planck's constant divided by 2π, and me = the electron mass. These relations are generalized using the g-factor:
 * $$\mathbf{m_J} = g m_B \ \mathbf J \, $$

with g=2 for spin (J = S) and g=1 for orbital motion (J = L). The resultant total spin S of an ensemble of electrons in an atom is the vector sum of the constituent spins sj:


 * $$ \mathbf {S} = \sum_{j=1}^N \ \mathbf{s_j} \ . $$

Likewise, the orbital momenta of an ensemble of electrons in an atom add as vectors.

Where both spin and orbital motion are present, they combine by vector addition:


 * $$\mathbf{J = L +S} \ ,$$

and the magnetic moment is
 * $$\mathbf {m_J} = g m_B \mathbf J \, $$

with g now the Lande g-factor or spectroscopic splitting factor:


 * $$ g = \frac {3}{2} + \frac{S(S+1)-L(L+1)}{2J(J+1)} \ . $$

If a collection of atoms with these associated magnetic moments are now subject to a magnetic flux, all the atoms will experience a torque, in part due to the applied field and in part as a response to the magnetic flux they create among themselves. The calculation of the magnetization thus involves determination of the orientation of these moments taking into account their influence upon each other and also the influence of the external magnetic flux.

Types of magnetization
The major magnetic forms are paramagnetism, ferromagnetism and diamagnetism.

Paramagnetism
When an assembly of atoms is placed in a magnetic flux, a torque is exerted upon the atoms because of their magnetic moment. The result is that atoms that are aligned with the magnetic flux have lower energy than those that are at an angle to the field. The difference in energies is proportional to the magnetic flux density and to the component of magnetic moment along the field.

According to the Boltzmann factor, higher energy configurations are less probable than lower ones, and as temperature is lowered, the population of the lower energy configurations grows at the expense of the higher energy configurations. Also, as the magnetic flux density is increased the separation of the configurations increases and the lower energies become more populated.

These observations can be made quantitative.

Ferromagnetism
In the above argument the influence of atoms upon each other was neglected. For ferromagnetic materials that approximation is invalid. The self-interaction of the atoms tends to align them even when no external magnetic flux is present. As a result, ferromagnetic materials create a net magnetic flux density in the space surrounding the material, and can form permanent magnets at temperatures below the Curie temperature of the material. At higher temperatures, the aligning interaction is inadequate to overcome the randomness introduced by thermal motions.

A simplified "toy" example of ferromagnetism helps to understand the approach. The example follows Kittel. We suppose that each atom is subject to a magnetic flux density due to the other atoms that is proportional to their magnetization:


 * $$\mathbf{B_E} = \lambda \mathbf M \, $$

where the subscript E stands for exchange field, a technical term referring to the mechanism causing cooperation between the atoms. We now bootstrap the calculation of the magnetization by assuming the total flux density is the applied plus the exchange flux, and their sum induces the magnetization. Thus,


 * $$ \mathbf M = \chi \left( \mathbf {B_a + B_E} \right) \, $$

where the approximation is made that the magnetization is simply a linear response to the flux density, according to the simple scalar susceptibility χ. Then with the previous expression for M,


 * $$ \mathbf M = \chi \left( \mathbf {B_a} + \lambda \mathbf {M} \right) \, $$

or:


 * $$ \mathbf{M} = \frac { \chi}{1-\chi \lambda } \mathbf{B_a} \ . $$

The point of interest is that the denominator can be zero if χλ can be as large as one, which means we have an infinite magnetization with even a very small applied flux density. The implication is that the self-generated field can result in a magnetization with zero applied flux density. That is the phenomenon of ferromagnetism.

Today it is still impossible to predict from first principles that iron is ferromagnetic. However, some guidance can be obtained as to which metals are candidates, and which are not, based upon estimates of how exchange energy varies with atomic radii and spacing.

Diamagnetism
Diamagnetism is the reduction within a material of an external magnetic flux by the reaction of the material. The perfect diamagnetic material is the superconductor, which completely suppresses the magnetic field inside itself in what is called the Meissner effect.

All materials exhibit diamagnetism, but it ordinarily is so weak that it is apparent only in materials where no other form of magnetic behavior is evident, like the noble gases or some diatomic gases.

Magnetization and magnetic field
In principle, the magnetic flux density B can be found by observing the force on a test charge using the formula:


 * $$ \mathbf F = q (\mathbf{v \times B } + \mathbf E) \ . $$

With B in hand, the magnetization could then be determined from Maxwell's equations using:


 * $$\mathrm{curl} \left(\frac{\mathbf B}{\mu_0} - \mathbf M \right ) = \mathbf j ,$$

where j is the sum of conduction and polarization current densities. In practice, things are done differently. The magnetic field or magnetic intensity H is introduced with the definition:


 * $$\mathbf H = \frac{\mathbf B}{\mu_0} - \mathbf M \, $$

and the magnetization M is calculated in terms of H using a so-called constitutive relation calculated by the techniques of condensed matter physics. The simplest example calculates the magnetic susceptibility χm defined as:


 * $$\mathbf M = \chi_m \mathbf H \, $$

and a magnetic permeability by:


 * $$\mu = \mu_0 \left( 1 + \chi_m \right) \ . $$

Here, χm < 0 corresponds to diamagnetism, and χm > 0 to paramagnetism. In ferromagnetic materials χm is a function of H.