# PlanetPhysics/Magnetic Susceptibility

In Electromagnetism, the volume magnetic susceptibility , represented by the symbol $\chi _{v}$ is defined by the following equation

${\vec {M}}=\chi _{v}{\vec {H}},$ where in SI units ${\vec {M}}$ is the magnetization of the material (defined as the magnetic dipole moment per unit volume, measured in amperes per meter), and H is the strength of the magnetic field ${\vec {H}}$ , also measured in amperes per meter.

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

${\vec {B}}\ =\ \mu _{0}({\vec {H}}+{\vec {M}})\ =\ \mu _{0}(1+\chi _{v}){\vec {H}}\ =\ \mu {\vec {H}},$ 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 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 $\displaystyle (1+Ï‡v) < 1$ (in SI units).

### Magnetic Susceptibility Tensor, $\chi$ The magnetic susceptibility of most crystals (that are anisotropic) cannot be represented only by a scalar, but it is instead representable by a tensor $\chi$ . Then, the crystal magnetization ${\vec {M}}$ is dependent upon the orientation of the sample and can have non-zero values along directions other than that of the applied magnetic field ${\vec {H}}$ . 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.