# PlanetPhysics/Magnetic Susceptibility

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

${\displaystyle {\vec {M}}=\chi _{v}{\vec {H}},}$ where in SI units ${\displaystyle {\vec {M}}}$ 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 ${\displaystyle {\vec {H}}}$, also measured in amperes per meter.

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

${\displaystyle {\vec {B}}\ =\ \mu _{0}({\vec {H}}+{\vec {M}})\ =\ \mu _{0}(1+\chi _{v}){\vec {H}}\ =\ \mu {\vec {H}},}$ where ${\displaystyle \mu _{0}}$ is the magnetic constant, and ${\displaystyle \ (1+\chi _{v})}$ is the relative permeability of the material.

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

${\displaystyle \mu =\mu _{0}(1+\chi _{v})\,.}$

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

${\displaystyle \chi _{=mass=}=\chi _{v}/\rho ,}$ ${\displaystyle \chi _{mol}\,=\,M\chi _{m}=M\chi _{v}/\rho ,}$

where ${\displaystyle \rho }$ is the density and M is the molar mass.

#### Susceptibility Sign convention

If ${\displaystyle \chi }$ is positive, then ${\displaystyle (1+\chi _{v})>1}$ (or, in cgs units, ${\displaystyle (1+4\pi \chi _{v})>1)}$ 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 ${\displaystyle \chi }$ negative, and thus $\displaystyle (1+Ï‡v) < 1$ (in SI units).

### Magnetic Susceptibility Tensor, ${\displaystyle \chi }$

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

${\displaystyle M_{i}=\chi _{ij}H_{j},}$

where ${\displaystyle i}$ and ${\displaystyle j}$ 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 ${\displaystyle i}$-th direction, ${\displaystyle M_{i}}$ to the component ${\displaystyle H_{j}}$ of the external magnetic field applied along the ${\displaystyle j}$-th direction.

## References

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