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!
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!
]. 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 
