Field induced linear magneto-elastic coupling in multiferroic TbMnO$_{3}$

We have used in-field neutron and X-ray single crystal diffraction to measure the incommensurability $\delta$ of the crystal and magnetic structure of multiferroic TbMnO$_{3}$. We show that the flop in the electric polarization at the critical field $H_{C}$, for field $H$ along the $a-$ and $b-$axis coincides with a 1st order transition to a commensurate phase with propagation vector $\kappa=(0,\frac{1}{4},0)$. In-field X-ray diffraction measurements show that the quadratic magneto-elastic coupling breaks down with applied field as shown by the observation of the 1st harmonic lattice reflections above and below $H_{C}$. This indicates that magnetic field induces a linear magneto-elastic coupling. We argue that the commensurate phase can be described by an ordering of Mn-O-Mn bond angles.