1
$\begingroup$

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?

$\endgroup$
1
$\begingroup$

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".

| cite | improve this answer | |
$\endgroup$
  • 1
    $\begingroup$ 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. $\endgroup$ – dgwp May 14 '17 at 9:13

Your Answer

By clicking “Post Your Answer”, you agree to our terms of service, privacy policy and cookie policy

Not the answer you're looking for? Browse other questions tagged or ask your own question.