# Landau free energy for magnetic phase transition - field, susceptibility

In standard treatments the Landau (mean field) expansion for the free energy in this case includes a field-dependent term \begin{equation*} -Bm \end{equation*} where $B$ is an external field and $m=MV$, with $M$ the magnetization and $V$ the volume. However, I have seen the susceptibility defined as both \begin{equation*} \chi=\frac{\partial M}{\partial B}\quad\text{and}\quad \chi=\frac{\partial M}{\partial H} \end{equation*} but my understanding is that this should be dimensionless, so shouldn't the first of these should have a factor of $\mu_0$ (assuming $M$ is small)? Or is the field in the expression for the free energy actually $H$ rather than $B$?

Perhaps someone could clarify exactly which field we are dealing with here, and what is the correct expression for the susceptibility?

I think that the problem is that you are confusing the magnetic field $B$ with the magnetic field intensity $H$. When the material is linear (for example, in diamagnets) magnetization and field intensity are related by $M=\chi H$. When the relation is non-linear, you have the derivative:
$$\chi = \frac{\partial M}{\partial H}$$
Both the magnetization and the field strenght $H$ have the same units so you have an adimensional quantity. The problem you are having is the abuse of language that leads most people to call both $B$ and $H$ "magnetic field" when they represent slightly different things. The abuse of language is due to the relation between both fields in linear materials, $B=\mu H$.
So you should derive with respect to $H$, because the material is non-linear and the relation between $B$ and $H$ becomes more complicated. When you study the Ising model and related topics in general "magnetic field" usually refers to "magnetic field intensity".
• I am not confusing anything, and fully understand the difference between $B$ and $H$. You have not read my question properly. I would like to know whether it should be $B$ or $H$ in the free energy term, and also why some authors define the susceptibility as $\chi=\partial M/\partial B$, which is clearly NOT dimensionless. – dgwp May 14 '17 at 9:13