Classical magnetic dipole-dipole interaction in iron

[rogdal]

Homework Statement

A magnetic dipole μ, which is located at the origin of a coordinate system, generates a magnetic field in its environment given by:

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

Calculate the magnetic field B generated by an atom with a magnetic moment μ ≃ μB (Bohr magneton) at the position of a neighbouring atom in iron. The typical first-neighbour distance r0 for Fe ferromagnets can be calculated knowing that Fe has a bcc lattice with a = 2.866 ̊A.

Relevant Equations

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

In a BCC lattice, $r_0 = \frac{\sqrt{3}}{2}a$

I'm having a bit of trouble with this exercise because, even if I understand the physics of the dipole-dipole interaction in an ideal classical system, I don't get to know how to approach this problem. I've got a few doubts about how this system would work.

First of all, what would be the direction of the magnetic moment $\vec{\mu}$ of iron with respect to its lattice? I guess that by simplicity the direction [100] should be chosen, but I don't know if each atom's magnetic moment has the same direction.

Besides, now assuming $\vec{\mu}$ is fixed, the value of B wouldn't be the same in all the first neighbours positions, right? Since it depends on the dot product between the magnetic moment and the first neighbour's position, and this wouldn't be the same for the atom in the upper left corner and the one in the bottom right.

Do you know how to approach this exercise?

Thank you!

 

[anuttarasammyak]

You seem to worry about dipole-dipole interaction. But the problem states magnetic field caused by a dipole at the place of another dipole. You do not have to worry about the direction of the dipole at that place.

 

[rogdal]

You seem to worry about dipole-dipole interaction. But the problem states magnetic field caused by a dipole at the place of another dipole. You do not have to worry about the direction of the dipole at that place.

Really? Well, in reality I worry about the direction of the dipole that is causing the magnetic field, i.e., the one in the origin of coordinates. Wouldn't its direction influence the product $\vec{\mu}·\vec{r}$?

 

[anuttarasammyak]

Wouldn't its direction influence the product μ→·r→?

Yes, the inner product matters. For an examle you may set r coordinates according to crystal structure and place a dipole in its origin and you can rotate dipole as you like which changes B at (1,0,0) for an example.

After you got it you may be able to proceed to next story of B$\mu_2$ energy at (1,0,0) to estimate $\mu_1$ dipole-$\mu_2$ dipole interaction.

 

[rogdal]

Yes, the inner product matters. For an examle you may set r coordinates according to crystal structure and place a dipole in its origin and you can rotate dipole as you like which changes B at (1,0,0) for an example.

But if I did what you say I'd have, for instance, $\vec{r}=(1,0,0)$. Then the total magnetic field would be:

$$\vec{B}=\frac{\mu_0}{4\pi}\left(3(\mu\cos{\theta})\hat{u_x}-\vec{\mu}\right)$$

Since $|\vec{r}|=1$. But this expression cannot be simplified because we don't know what the direction of $\vec{\mu}$ is.

Is that what you mean?

And then, I could get the dipole-dipole energy. But, assuming $\vec{\mu_1}$ and $\vec{\mu_2}$ are parallel, then their specific directions wouldn't be needed, right?

Thanks for the help!

 

[anuttarasammyak]

It is also written as
$$\vec{B}(1,0,0)=\frac{\mu_0}{4\pi}(3\mu_x\hat{u_x}- \vec{\mu})$$
$\vec{\mu}=(\mu_x,\mu_y,\mu_z)$ is a parameter. It depends on direction of $\vec{\mu}$ as you suspect. Interaction energy with another dipole at (1,0,0) is
$$\vec{\mu_2}\cdot\vec{B}(1,0,0)=\frac{\mu_0}{4\pi}(3\mu_{1x}\mu_{2x}- \vec{\mu_1}\cdot\vec{\mu_2})$$

 

[rogdal]

Ah, I see. I thought the exercise was meant to obtain a well defined expression of $\vec{B}$ in terms of the Bohr magneton.

Thanks!

 

[hutchphd]

It is interesting to look at the geometry and the minimum and maximum values (although usually one looks at energies rather than forces)
You do need to specify the relative orientation of the electron magnetic moments. Of course in a real chunk of iron the interaction of an electron spin with all of its neighbors is a very interesting problem that results in creation of self organizing magnetic domains. These magnetic domains are are also influenced by the geometry of the crystaline structure, not to mention the temperature (see Curie Temp) among other thinngs . Lotsa physics here.