|Magnetic domains changing their size by movements of domain walls in non-oriented electrical steel
A domain wall is a term used in physics which can have similar meanings in magnetism, optics, or string theory. These phenomena can all be generically described as topological solitons which occur whenever a discrete symmetry is spontaneously broken.
In magnetism, a domain wall is an interface separating magnetic domains. It is a transition between different magnetic moments and usually undergoes an angular displacement of 90° or 180°. A domain wall is a gradual reorientation of individual moments across a finite distance. The domain wall thickness depends on the anisotropy of the material, but on average spans across around 100–150 atoms.
The energy of a domain wall is simply the difference between the magnetic moments before and after the domain wall was created. This value is usually expressed as energy per unit wall area.
The width of the domain wall varies due to the two opposing energies that create it: the magnetocrystalline anisotropy energy and the exchange energy, both of which tend to be as low as possible so as to be in a more favorable energetic state. The anisotropy energy is lowest when the individual magnetic moments are aligned with the crystal lattice axes thus reducing the width of the domain wall. Conversely, the exchange energy is reduced when the magnetic moments are aligned parallel to each other and thus makes the wall thicker, due to the repulsion between them (where anti-parallel alignment would bring them closer, working to reduce the wall thickness). In the end an equilibrium is reached between the two and the domain wall's width is set as such.
An ideal domain wall would be fully independent of position, but the structures are not ideal and so get stuck on inclusion sites within the medium, also known as crystallographic defects. These include missing or different (foreign) atoms, oxides, insulators and even stresses within the crystal. This prevents the formation of domain walls and also inhibits their propagation through the medium. Thus a greater applied magnetic field is required to overcome these sites.
Note that the magnetic domain walls are exact solutions to classical nonlinear equations of magnets (Landau–Lifshitz model, nonlinear Schrödinger equation and so on).