# Why Do Ferro- magnetic/electric Domains Exist?

In short, what I am asking is what is the reason/cause of the formation of ferro- magnetic/electric domains, and how would one calculate the 'driving energy' in order to predict domain formation?

If we consider a dipole system that experience “ferromagnetic” coupling (my question does not just apply to magnetic systems, it could equally well apply to ferroelectric systems), then the ground state of this system will be the one with all the dipoles aligned in the same direction. This state minimizes the energy due to the dipole-dipole interactions, which are minimized when neighboring dipoles align with each other.

However, the dipole-dipole interaction is not the only contributor to the overall energy of the system. There are two other possibilities to consider: (1) the energy of the field around the system that is setup due to the net magnetization of the material, and (2) the favorable contribution to the free energy, via the entropy at T > 0, due formation of defects.

(1) If you align many dipoles in the same direction then the material will have a net magnetization and thus in the space around the material there will be a magnetic field. This magnetic field has energy associated with it. The more dipoles that are aligned, the stronger the field, the stronger the energy contained within the magnetic field. This energy can be reduced by creating defects in the dipole alignment, like domains, that will result in no net magnetization of the material. Of course, creating the defects costs energy, so there will be some threshold energy/temperature that will need to be met before this becomes energetically favorable to occur: For small systems of dipoles (N is small) it will be preferable to have all the dipoles aligned, but for larger systems (N is larger), it will be favorable to create domains within the material.

If I wanted to calculate the energy due to the magnetic field, however, I face the problem of having to integrate the field energy density over all space. In doing so, I find that the energy diverges for points very close to the dipole source (obviously), and thus difficult to use to calculate the field energy's contribution to the overall system's energy.

(2) Defects in the system increase the entropy and thus lower the free energy when T > 0. A perfectly ordered system will have an entropy of zero, but creating just a single spin-flip defect will decrease the free energy by a factor of k*log(N), in the case of a 1D chain of aligned dipoles, for example. Again, creating that defect increase the dipole-dipole interaction energy, so it will only occur at a high enough temperature.

In a discrete spin system, where spins can only be up or down, for example, calculating this energy is straight forward enough, in principle. But what about in continuous systems, where the spins can be in any direction? There are an infinite number of possible states, each being equally probable, and defining exactly what a 'defect' is becomes trickier. I am not sure how I might go about doing this.

So,
Which of these explanations is the 'correct' way to explain the formation of domains? Or is there something that I had not considered?

Simon Bridge
Homework Helper
In a nutshell...
Magnetism comes, ultimately, from the atomic dipole.
Magnetic domains come from clusters of them - due to the structure of the material. Ferromagnets ore usually crystalline. You can get resonances between adjacent atoms which mean their dipoles either line up or not - it turns out the ferric metals tend to line up more than not.

You don't normally get a single magnetic domain in one bulk material because of imperfections in the crystal structure. Either the structure is part-fractured or there are impurities in it. Usually both.

Calculating how this all turns out for a particular material is very hard to do since the process of manufacture affects the outcome. It's not the sort of thing you do in an undergrad course for eg.

Neither of the models you propose are correct - the actual situation is much more complicated.
However, I think you can get a good idea from stuff like this:
http://magician.ucsd.edu/essentials/WebBookse21.html [Broken]

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nasu
Gold Member
The domain structure in a ferromagnetic material minimizes the magneto-static energy of the system.
It happens in "perfect" crystals too. The specific configuration of domains in a specific sample is obviously influenced by defects and crystal size, as pointed by Simon Bridge.
Your (1) is in principle close to the accepted model. The domain structure reduces the internal energy of the system.
http://en.wikipedia.org/wiki/Magnetic_domain
or
http://www.gitam.edu/eresource/Engg_Phys/semester_2/magnetic/domain.htm [Broken]

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To be sure, as Simon Bridge said, the specific domain structure of a particular material is extremely complex and is influenced by the numerous defects and crystal structure of the material as well as many other things.

But I am interested in it in a more general sense; I am interested in the different fields or energies that go into making the domain structure energetically favorable.

As discussed in the “Landau-Lifshitz energy equation” subsection of the wikipedia article that nasu linked to, there are several contributing parts to the overall free energy of a magnetic system including the exchange energy, magnetostatic energy, anisotropy energies, etc..

I think that the answer I am looking for here is the magnetostatic energy, which is essentially my #1 from the original post. It is calculating this energy that I am curious about now.

Am I correct in amusing that the magnetic static energy could equivalently be called the demagnetizing(polarizing) field energy? And this demagnetizing field energy is only present in finite systems?

This second part might be obvious but also has an effect that I had not previously considered on domain formation....
If the magnetostatic field energy, which is (one of) the biggest drivers of domain formation is the demagnetizing field energy, and this demagnetizing field is only present in finite systems (near discontinuities of magnetization), then do domains form in infinite systems? If you had an infinite ferromagnetic medium, would it have domains? I guess not.