# Understanding magnetic hysteresis curves

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.

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!