# Magnetic susceptibility for ferromagnetic materials

Magnetic susceptibility $$\chi_m$$ is defined by following relation:

$$\vec{M}=\chi_m \vec{H}$$

where $$\vec{M}$$ is the magnetization density and $$\vec{H}$$ is the magnetizing field, defined by $$\vec{H}=\frac{1}{\mu_0} \vec{B}-\vec{M}$$. It is often said that for diamagnetic and paramagnetic materials, $$\chi_m$$ is a very small number with respect to 1. For ferromagnetic materials it is often said that $$\chi_m$$ is not small and not constant. I imagine that "not constant" here means that susceptibility depends on the modulus of $$\vec{H}$$ so that the previous equation is not linear anymore (right?).

Question: why is previous relation (definition of $$\chi_m$$) appliable to ferromagnetic materials? This relation in fact implies that magnetization and $$\vec{H}$$ field are always parallel (point by point). This should not be true for ferromagnets, that could have a permanent magnetization, independent from applied magnetizing field...

What is actually implied for ferromagnetic materials is $$\mathbf{M} = \chi_\mathrm{m}(\mathbf{H})\mathbf{H}$$, which turn the material's response to the field non-linear, as you have stated. For an anisotropic material, $$\chi_\mathrm{m}$$ is naturally a tensor, which should describe the magnetic domains. Notice that high intensity fields are, in principle, capable of magnetizing the material in a single direction -- making it isotropic, but not necessarily homogeneous.