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In multiferroic TbMn$_2$O$_5$, the behavior of the terbium ions forms acrucial part of the magneto-electric coupling. The result is a magneticallyinduced reversal of the electric polarization at 2 T. In this article wepresent the first direct measurement of the terbium magnetic structure underapplied magnetic fields. Contrary to the current interpretation of the magneticproperties of \tmo, we show that upon the electric polarization reversal theterbium sub-lattice adopts a canted antiferromagnetic structure with a largecomponent of magnetic moment parallel to the \textit{a}-axis. Furthermore, weprovide evidence for a coupling between the manganese 3d magnetic structure andthe terbium $4f$ magnetism, which is of great significance in the elusivemagneto-electric mechanisms at play.

Magnetically induced electric polarization reversal in multiferroic TbMn$_2$O$_5$: Terbium spin reorientation studied by resonant x-ray diffraction