Repulsive interaction of Néel Walls, & the internal length scale of the cross-tie wall

Ńeel walls and cross-tie walls are two structures commonly seen in ferromagnetic thin films. They are interesting because their internal length scales are not determined by dimensional analysis alone. This paper studies (a) the repulsive interaction of one-dimensional Ńeel walls and (b) the internal length scale of the cross-tie wall. Our analysis of (a) is mathematically rigorous; it provides, roughly speaking, the first two terms of an asymptotic expansion for the energy of a pair of interacting walls. Our analysis of (b) is heuristic, since it rests on an analogy between the cross-tie wall and an ensemble of Ńeel walls. This analogy, combined with our results on Ńeel walls and a judicious choice of parameter regime, yields a specific prediction for the internal length scale of a cross-tie wall. This prediction is consistent with the experimentally observed trends.