Domain wall ordering and motion in Mn$_3$O$_4$

Mn$_3$O$_4$ is an orbitally ordered, magnetically frustrated spinel with strong spin-lattice coupling, which exhibits a series of low temperature magnetic and structural transitions. Transverse field $\mu$SR has shown that ordered and disordered volumes coexist within this material, while MFM measurements have further shown that the magnetic domain walls themselves order in specific crystallographic directions, with a typical length scale of 100's of nm. In order to directly study these phenomena, we have performed small angle neutron scattering (SANS) and ultra-small angle neutron scattering (USANS) measurements at both zero and applied magnetic field. We will present the results of these measurements and discuss what they show as far as the formation of domains, as well as the motion of the domain walls. We will also discuss the effects of internal disorder on the behavior of the material.