Title: Magnetoelectric Coupling Based on Protons in Ammonium Sulfate

Abstract: Most ferroelectric crystals have their own set of unique characteristics and ammonium sulfate (NH$_4$)$_2$SO$_4$ is no exception. We report on two previously unidentified features in ammonium sulfate: 1) that there are at least two successive transitions instead of one occurring at the Curie temperature $T$$_C$ = 223 K according to dielectric constant measurements; and 2) pronounced step-like anomalies are found in the magnetic susceptibility exactly at $T$$_C$. To explain these results, we take into account that there exists a previously unidentified linear coupling between the magnetic and electric dipole moments of the NH$_4$$^+$ tetrahedra due to their rapid reorientations and distorted geometry, respectively. The magnetic moments are small, 0.0016 $\mu$$_B$ for every $C$$_3$ reorientation which involve three protons (H$^+$) undergoing orbital motion. Nevertheless, short-range correlations exist in the paraelectric phase because the magnetic moments are restricted to only point along 14 possible orientations due to the symmetry and periodic nature of the potential wells. At $T$$_C$, $C$$_2$ reorientations (involving four protons) are no longer energetically feasible so the reduction in the degrees of freedom to 8 further enhances the effect of the magnetic interactions. This triggers long-range ordering of the orbital moments in an antiferromagnetic configuration along the $ab$-plane, which via Dzyaloshinskii-Moriya interactions, end up canting slightly toward the $c$-axis direction. Since there exists two types of inequivalent NH$_4$$^+$ groups that reorient at different frequencies with temperature and do not have the same degree of distortion, the emerging polar phase is ferrielectric.