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 $$(1+Ï‡v) < 1$$ (in SI units).

Magnetic Susceptibility Tensor, χ
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.