Classical dipole-dipole interaction in iron

Question

I've been reading about the classical magnetic dipole-dipole interaction and I'm wondering how it would work in a ferromagnet element like iron (theoretically under the assumptions I will write below).

The $B$ field generated by a magnetic dipole is given by:

$$\vec{B}(\vec{r})=\frac{\mu_0}{4\pi}\frac{3(\vec{\mu}·\vec{r})\vec{r}-r^2\vec{\mu}}{r^5}.$$

In order to simplify my question, let's consider only the B field generated by a certain atom in the iron lattice and measured in the position of a first neighbour atom, neglecting further interactions with other atoms. As in this case, we are dealing with iron, and it has a BCC lattice, let's say the distance between first neighbours is $r_0$.

Now, first of all. What is the direction of the magnetic moment $\vec{\mu}$ of iron with respect to its lattice? Secondly, because $\vec{\mu}$ is fixed, the value of $B$ wouldn't be the same in all the first neighbours positions, right? Since it depends on $\vec{\mu}$'s orientation.

Answer 1

There is no simple answer to this question even for the simplest crystal as the BCC is because there are at least three competing dynamics involved. One is the tendency for a thermodynamic system to occupy its smallest energy state, the second is that a crystal lattice is inherently anisotropic, and the third is that the quantum mechanical exchange forces between the magnetic moments prefer them be parallel.

The result of these competing dynamics is a multi-domain structure in a single crystal, see the very good pictures in. The arrows represent domains that are regions of homogeneous parallel distribution of the atomic dipole moments meanwhile the total magnetic energy of the single crystal consisting of many domains is minimized. Locally, the arrows themselves point relative to the crystal lattice in the direction of so-called easy magnetization in contrast to some other possible direction. For the case of a BCC the easy direction is [100]. Notice that the walls between neighboring domains are not necessarily parallel with the easy magnetization but are determined by the requirement for minimum magnetostatic energy that demands "pole avoidance". For details, see, W.F. Brown: Magnetostatic Principles on Ferromagnetism.