Incommensurate to commensurate transition under magnetic field in TbMnO$_{3}$

We have studied the evolution of the incommensurate satellite peaks of TbMnO$_{3}$ by neutron and x-ray single crystal diffraction with field H$\| b$ and H$\| c$. For H$\| b$ and a temperature of T=3K, Tb undergoes the first order phase transition characterized by a change of the magnetic wavevector from $q_{Tb}\sim$0.44 to 0.33 r.l.u. at a field of 1.75T. X-ray diffraction measurements show that the structural counterpart for the magnetic Tb reflection follows a similar dependence with 2$q_{Tb}$=0.82 r.l.u. changing to 0.66 r.l.u.. At H=6T and T=3K, the structural superlattice reflections for both the Mn and Tb wavevectors collapse to a single $q$-commensurate structure with propagation vector $Q$=[0,1/2,l]. This transition coincides with anomalies in the polarization P$\| c$. For fields of up to 4T applied along the c-direction no new magnetic phases arise, except for the shift of the antiferromagnetism and locking transitions.