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## Main Question or Discussion Point

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?

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?