magnetic susceptibility

Definition 0.1   In Electromagnetism, the volume magnetic susceptibility, represented by the symbol $ \chi_{v} $ is defined by the following equation

$\displaystyle \vec{M} = \chi_{v} \vec{H},$

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

$\displaystyle \vec{B} \ = \ \mu_0(\vec{H} + \vec{M}) \ = \ \mu_0(1+\chi_{v}) \vec{H} \ = \ \mu \vec{H},$

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:

$\displaystyle \mu = \mu_0(1+\chi_v) \, .$

Remark 0.1   There are two other measures of susceptibility, the mass magnetic susceptibility, $ \chi_g$ or $ \chi_m$ , and the molar magnetic susceptibility, $ \chi_{mol}:$

$\displaystyle \chi_{\text{mass}}= \chi_v/\rho ,$

$\displaystyle \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+χv) < 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:

$\displaystyle 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.

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