Magnetostructural coupling in the magnetodielectric Mn$_3$O$_4$

At room temperature, the spinel Mn$_3$O$_4$ is distorted from cubic symmetry due to Jahn-Teller distortions of octahedral Mn$^{3+}$ and is described by the tetragonal spacegroup $I4_1/amd$. It undergoes ferrimagnetic ordering near 43$\,$K where anomalies in heat capacity and dielectric measurements are also observed. High-resolution variable-temperature synchrotron X-ray powder diffraction reveals a distortion of the long-range crystal structure at the onset of magnetic order. The structure of Mn$_3$O$_4$ below the magnetostructural ordering temperature is described by coexisting tetragonal and orthorhombic phases. A discontinuous change in lattice parameters at the magnetostructural ordering temperature illustrates that this system undergoes a first-order phase transition. Sharp changes in heat capacity at the magnetic ordering temperature are also consistent with a first-order phase transition. We present the complete crystallographic description of this important magnetodielectric spinel and suggest mechanisms behind the spin-driven lattice distortion.