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Suppose I have a ferromagnetic material. It is known that the magnetization $\textbf{M}$ can be determined from a magnetic field $\textbf{H}$ using hysteresis curves as exemplified in the figure below.

Hysteresis curves

Here, $\textbf{M} = [M_x ~ M_y~ M_z]^{T}$ and $\textbf{H} = [H_x~ H_y~ H_z]^{T}$.

P.S.: The curves are not necessarily the same (it is just an illustration).

For some magnetic field $\textbf{H}$, we can determined the differential magnetic susceptibility given by

$$\mathbf{\chi}^{(d)} = \begin{bmatrix} \chi_{11}^{(d)} & 0 & 0 \\ 0 & \chi_{22}^{(d)} & 0 \\ 0 & 0 & \chi_{33}^{(d)} \\ \end{bmatrix}, \tag{1} $$

where $\chi_{11}^{(d)}, \chi_{22}^{(d)}$ and $\chi_{33}^{(d)}$ are obtained by computing $dM/dH$ for a given $\textbf{H}$ (In the figure, the points where the derivatives are computed seem to be the same, but, again, it is just an example. As a matter of fact, the points could be everywhere on the hysteresis loop).

If I am not wrong, the tensor susceptibility of some ferromagnetic crystals can have no-zero off-diagonal elements. For instance:

$$\mathbf{\chi}^{(d)} = \begin{bmatrix} \chi_{11}^{(d)} & \chi_{12}^{(d)} & 0 \\ \chi_{21}^{(d)} & \chi_{22}^{(d)} & 0 \\ 0 & 0 & \chi_{33}^{(d)} \\ \end{bmatrix}. \tag{2} $$

In this case, I cannot figure out a hysteresis curve $M_x$ vs $H_y$ (or $M_y$ vs $H_x$) to have an idea of $\chi_{12}^{(d)}$ (or $\chi_{21}^{(d)}$). Indeed, if the hysteresis curve $M_x$ vs $H_x$ already gives the magnetization component $M_x$ for some $H_x$, a hysteresis curve $M_x$ vs $H_y$ would not necessarily give the same $M_x$ for the component $H_y$ of the same magnetic field, which seems to be contradictory.

My question is:

What would be the hysteresis curves for a ferromagnetic material whose tensor susceptibility has no-zero off-diagonal elements?

Thanks for any help!

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1 Answer 1

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You need to consider that the hysteresis curves are the outer limits of possible magnetizations, not the only possibilities. All the area within the curves are possible too.

In the case of different magnetizations in different directions, the status of the object would be represented by different positions in each area for the three dimensions.

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  • $\begingroup$ Sorry, but I didn't understand your answer. What do you mean by " area within the curves are possible too"? "Status of the object"? "Different positions in each area"? It is not clear to me. I can't even link your answer to my question. $\endgroup$
    – Alex Silva
    May 22, 2020 at 15:35
  • $\begingroup$ The actual status of the material is reflected by a point on each of your three graphs. Those points are not required to be on the line, just within their included area, nor do they need to be the same in each three graphs. $\endgroup$
    – Aganju
    May 22, 2020 at 15:38
  • $\begingroup$ Ok, I know they don't need to be the same in the three graphs. It is just an illustration. And I am considering only a hysteresis loop. Anyway, where the point is or not does not matter (or does it?). It is not the point of my question.. $\endgroup$
    – Alex Silva
    May 22, 2020 at 15:45

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