Repulsive interaction of Neel walls, and the internal length scale of the cross-tie wall

Neel 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 Neel 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 Neel walls. This analogy, combined with our results on Neel 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.