# Confusion with Magnetic Susceptibility

Could someone point out the error made in the following two simple calculations? I am trying to understand how to find magnetic and electric susceptibility using thermodynamics. I know something is not correct, I just don't know why what I'm doing is wrong. Consider a system of $$N$$ magnetic moments where each magnetic moment can be in two states, $$\pm \mu$$. Then the partition function of one moment us, when they are applied to an external field $$B$$, $$Z_1 = 2\cosh\left(\frac{\mu B}{T}\right)\implies Z = Z_1^N$$ Then the magnetization is given by $$M = -\frac{\partial (-T\ln Z)}{\partial B} = NT\frac{\partial}{\partial B}\ln\left(2\cosh\left(\frac{\mu B}{T}\right)\right) = N\mu \tanh\left(\frac{\mu B}{T}\right)$$ Then the magnetic susceptibility is $$\chi_1 = \lim_{B\to 0}\frac{\partial M}{\partial B} = \frac{\mu^2 N}{T}$$ Since $$B$$ is an external magnetic field, so I assume in a vacuum, I replace $$B$$ by $$\mu_0 H$$. Repeating that whole calculation then shows that $$\chi_2 = \frac{\mu^2\mu_0^2 N}{T}$$. Here is my confusion: In electrodynamics one says that $$B = \mu_0(H + M)$$. One of the $$\chi_1 B$$ one or $$\chi_2 H$$ should replace $$M$$. It seems to me that if $$M = \chi_m H$$ then it should be $$\chi_2 H$$. However, it is clear that $$M$$ cannot be both $$\chi_1 B$$ and $$\chi_2 H$$, as if we had $$M = \chi B$$ then $$\chi$$ is related to $$\chi_m$$ through a rational function, but $$\chi_2 = \mu_0^2 \chi_1$$. Which is correct, $$\chi_1 = \chi_m$$ or $$\chi_2 = \chi_m$$ and why is that one chosen over the other, when it seems to me arbitrary which one is chosen.

The quantities $$\mathbf B,\mathbf H$$ in the relation

$$\mathbf B = \mu_0 \mathbf H + \mu_0 \mathbf M\tag{*}$$ are total macroscopic fields in material medium. They include contributions (in the sense of the superposition principle) due to sources ("magnetic moments") in the material medium. None of these fields is just an external field due to some laboratory source.

In order to find susceptibility (as defined in EM theory) from some model, we have to express magnetization as function of total field $$\mathbf H$$ in the medium in that model:

$$\mathbf M = \chi \mathbf H.\tag{**}$$ $$\mathbf H$$ is used here instead of $$\mathbf B$$ because historical reasons and also in practice $$\mathbf H$$ is directly relatable to electric current which is easier to measure and control than $$\mathbf B$$ field.

The quantity $$B$$ in the model of paramagnetism from statistical physics you describe, on the other hand, is just the external field due to some other source than the magnetic moments in the system. This simplest statistical model ignores interaction between magnetic elements in the material medium. This is justified in special cases where the magnetic moment are few and far between, such as in 2D systems with inter-particle distance that is large enough. In that case, we can make the simplifying assumption that magnetic moments are acted upon by microscopic field $$\mathbf b$$ that is the same as the macroscopic field $$\mathbf B$$: $$\mathbf b = \mathbf B.$$ We are thus ignoring the interaction with the magnetic field due to the system itself.

Since the magnetic particles are few and far between, we can neglect magnetization term in (*) and write $$\mathbf B = \mu_0 \mathbf H$$ both outside and inside the magnetic medium.

With these simplifying assumptions, it does not matter whether partition function is expressed using $$B$$ or $$H$$, they differ just by constant factor $$\mu_0$$.

However, susceptibility should be defined based on (**) in any case.