Let's make some single-domain magnetic nanoparticles

[etotheipi]

Might be interesting, to determine how small a spherical single-domain nanoparticle needs to be in order for it to be stable against forming more domains. The energy of such a nanoparticle (occupying a spherical region $\Omega$) will be$$E_1 = -\frac{\mu_0}{2}\int_{\Omega} \boldsymbol{M}_0 \cdot \boldsymbol{H} d\tau = \frac{\mu_0}{2} \int_{\Omega} \boldsymbol{M}_0 \cdot \nabla U d\tau$$where $\boldsymbol{H} = - \nabla U$ where $\nabla^2 U = \nabla \cdot \boldsymbol{M}_0$. [This stray field energy we expect to scale $\propto r^3$]. Let's then consider forming a second domain, and forming a domain wall [at some energy cost] along the equatorial plane. Then, in one hemisphere the magnetisation is $\mathbf{M}_0$, and in the other $-\mathbf{M}_0$. Denoting $\Omega = \Omega_1 \cup \Omega_2$,$$E_2 = \underbrace{\frac{\mu_0}{2} \sum_{i=1}^{2} \int_{\Omega_i} (-1)^i \boldsymbol{M}_0 \cdot \nabla U d\tau}_{\text{stray field}} + \underbrace{E_{\text{bloch}}}_{\text{domain wall}}$$To be stable against forming a second domain, it's required that $E_2 > E_1$. Forming more domains reduces stray field but at the cost of increasing domain wall energy, so the aim is to find that critical radius.

I would like some help to flesh out this calculation! Firstly, I expect the domain wall energy to scale $\propto r^2$ as the domain wall area, but how can I determine an explicit expression for this? Also, what approximations are justified, in order to simplify the determination of the stray field energy? Thanks!

 

[hutchphd]

How about doing something in 1D first? I am not at all expert here but aren't you going to end up with the Ising model or similar? This is a very interesting question but by no means a new one.

 

[etotheipi]

I didn't even know what the Ising model was until about 1 minute and 30 seconds ago, but it does look cool. I'm sure Professor Tong has written some notes about them, somewhere [EDIT: yes, he has, under 'statistical field theory' ]. I'm not familiar with this field so I don't know what tools are available or what models already exist. But open to suggestions

What do you mean by trying something in 1D first?

 

[hutchphd]

Check the Ising model stuff. I know the questions were usually regarding phase transitions (order-disorder). The 1D models are just a string of interacting spins with some temperature maybe? This led to much of Ken Wilsons renormalization group. Enough here to last a lifetime! Wish I knew more.

 

[etotheipi]

Okay cool, thanks, looks like I've got a bit of reading to do... better go and make a cup of tea first

 

[Dale]

I would like some help to flesh out this calculation!

This seems like an interesting calculation. In MRI we use super-paramagnetic iron oxide which consists of iron oxide particles that are small enough to consist of a single domain. So you can use the size of those as a check

 

[nasu]

Single domain particles were used in the magnetic coatings of the old magnetic tapes. You can find a lot of relating info by looking up magnetic recording. And iron oxide was the most common material too.