Analytic theory for the switch from Bloch to N\'eel domain wall in nanowires with perpendicular anisotropy

Rectangular nanowires may have domain walls nucleated and moved through them to realize many devices. It has been shown that at a particular width and thickness of a nanowire with perpendicular anisotropy, there is a switch from the domain wall being of Bloch-type to it being of N\'eel-type. This critical shape can be found through micromagnetics simulations, but here we present an analytic calculation for the energies of both wall types involving two iterations for the demagnetizing energy density. The expressions developed are long, but have the advantage that by simply inputting material parameters for the magnetic material, the critical shape can be found using a calculator in a matter of seconds. We compare our results to those found using micromagnetics and those found experimentally and the agreement is good.