PlanetPhysics/Magnetic Susceptibility

In [[../Electromagnetism/|Electromagnetism]], the volume magnetic susceptibility , represented by the symbol ${\chi }_v$ is defined by the following equation

$\overrightarrow{M}={\chi }_v\overrightarrow{H},$ where in SI units $\overrightarrow{M}$ is the magnetization of the material (defined as the magnetic dipole moment per unit [[../Volume/|volume]], measured in amperes per meter), and H is the strength of the [[../NeutrinoRestMass/|magnetic field]] $\overrightarrow{H}$, also measured in amperes per meter.

On the other hand, the magnetic induction $\overrightarrow{B}$ is related to $\overrightarrow{H}$ by the equation

$\overrightarrow{B}={\mu }_0(\overrightarrow{H}+\overrightarrow{M})={\mu }_0(1+{\chi }_v)\overrightarrow{H}=\mu \overrightarrow{H},$ where ${\mu }_0$ is the magnetic constant, and $(1+{\chi }_v)$ is the relative permeability of the material.

Note that the magnetic susceptibility ${\chi }_v$ and the magnetic permeability $\mu$ of a material are related as follows:

$\mu ={\mu }_0(1+{\chi }_v).$

There are two other measures of susceptibility, the mass magnetic susceptibility , ${\chi }_g$ or ${\chi }_m$, and the molar magnetic susceptibility , ${\chi }_{mol}:$

${\chi }_{=mass=}={\chi }_v/\rho ,$ ${\chi }_{mol}=M{\chi }_m=M{\chi }_v/\rho ,$

where $\rho$ is the density and M is the molar [[../CosmologicalConstant/|mass]].

Susceptibility Sign convention

If $\chi$ is positive, then $(1+{\chi }_v)>1$ (or, in cgs units, $(1+4\pi {\chi }_v)>1)$ and the material can be paramagnetic, ferromagnetic, ferrimagnetic, or anti-ferromagnetic; then, the magnetic field inside the material is strengthened by the presence of the material, that is, the magnetization value is greater than the external H-value.

On the other hand there are certain materials--called diamagnetic -- for which $\chi$ negative, and thus Failed to parse (syntax error): {\displaystyle (1+χv) < 1} (in SI units).

Magnetic Susceptibility Tensor, χ

The magnetic susceptibility of most crystals (that are anisotropic) cannot be represented only by a [[../Vectors/|scalar]], but it is instead representable by a [[../Tensor/|tensor]] $\chi$ . Then, the crystal magnetization $\overrightarrow{M}$ is dependent upon the orientation of the sample and can have non-zero values along directions other than that of the applied magnetic field $\overrightarrow{H}$. Note that even non-crystalline materials may have a residual anisotropy, and thus require a similar treatment.

In all such magnetically anisotropic materials, the volume magnetic susceptibility tensor is then defined as follows:

$M_i={\chi }_{ij}{H}_j,$

where $i$ and $j$ refer to the directions (such as, for example, x, y, z in Cartesian coordinates) of, respectively, the applied magnetic field and the magnetization of the material. This rank 2 tensor (of dimension (3,3)) relates the component of the magnetization in the $i$-th direction, $M_i$ to the component $H_j$ of the external magnetic field applied along the $j$-th direction.