Mapping the magneto-structural phases of magnetodielectric Mn$_{3}$O$_{4}$*

We report temperature-dependent x-ray diffraction and temperature- and field-dependent Raman scattering studies of single crystal samples of Mn$_{3}$O$_{4}$, which allow us to elucidate the microscopic origins of the magneto-dielectric and --elastic behaviors of this material.~ Among the key results:~ we find that the T$_{2g}$ Raman mode exhibits a clear splitting below T$\sim $33K, indicating a tetragonal-to-monoclinic structural change that coincides with the commensurate magnetic transition at T$_{2}$=33K; we also show evidence for the magnetic-field induced monoclinic distortions sensitive to magnetic field direction as well as magnitude, and provide evidence for a novel field-induced phase in the intermediate field regime of H$\vert \vert [1\bar {1}0]$.