# Why does dipole-dipole interaction not lead to ferromagnetic ordering?

Why does the magnetic dipole-dipole interaction between the magnetic moments of the atoms in a ferromagnetic material such as iron not lead to ferromagnetism?

• Please try to make questions accessible to users who don't already know exactly what you are talking about: Dipole-dipole interaction between what? Where does the claim that it "does not lead to ferromagnetism" come from? Cite your sources! You are not limited to single sentence in your questions. – ACuriousMind Aug 30 '17 at 22:36

The magnetic dipole dipole interaction is just by far too weak. In Stephen Blundell: "Magnetism in Condensed Matter" this is estimated to be on an energy scale of about $10^{-23}$J which may play a role at temperatures of about 1K or so. But when we are talking about magnetism we typically talk about much higher temperatures.

In detail the energy due to this interaction is

$E=\frac{\mu_0}{4\pi r^3} \left[ \boldsymbol{\mu}_1 \cdot\boldsymbol{\mu}_2 - \frac{3}{r^2}(\boldsymbol{\mu}_1\cdot \textbf{r})(\boldsymbol{\mu}_2\cdot \textbf{r}) \right]$

where $\boldsymbol{\mu}_i$ are the magnetic moments and $\textbf{r}$ is the separation vector. Assuming the magnetic moments to be on the order of $1 \mu_\text{B}$ and the separation to be on the order of 1 Angstrom one obtains the mentioned energy scale.

Ferromagnetism requires parallel alignment of magnetic moments. However, the dipole -dipole interaction $$E(r)=\frac{\mu_0}{4\pi r^3}[\boldsymbol{\mu}_1\cdot\boldsymbol{\mu}_2-\frac{3(\boldsymbol{\mu}_1\cdot \textbf{r})(\boldsymbol{\mu}_2\cdot \textbf{r})}{r^2}]$$ favours anti-parallel alignment of the dipoles because that would have lower energy compared to a parallel configuration. This rules out dipole-dipole interaction as far as the origin of ferromagnetism is concerned. However, one might think that it might lead to anti-ferromagnetism. But it's too weak to produce even anti-ferromagnetic order at room temperature.

• Are you sure about this statement? I mean if you assume a simple cubic crystal structure with a single atom per lattice site I could imagine this. For such a geometry this statement sounds reasonable. But there are also many other crystal structures. Many materials are based on an fcc or bcc lattice. There I am not so sure whether this statement holds. – Gregor Michalicek Aug 31 '17 at 19:27
• @GregorMichalicek It's a mathematical fact following from this expression of energy that the dipole-dipole interaction favours anti-parallel alignment. No, I'm not sure but I don't find anything wrong in the way I argued. – SRS Aug 31 '17 at 20:04
• @SRS Gregor is trying to point your attention to the fact that in crystals there are more than two dipoles interacting, in a variety of relative geometries. For any individual pair, the interaction might minimize energy in an antiparallel configuration, but if both dipoles are interacting with a third one, it's entirely possible that the global minimum will have a nonzero magnetization. – Emilio Pisanty Aug 31 '17 at 20:20
• @EmilioPisanty Hmm...agreed. I'll edit or remove the answer. – SRS Aug 31 '17 at 20:25