We present a study of the magnetic and crystallographic structure of TbMnO$_3$ in the presence of crossed electric and magnetic fields using circularly polarised X-ray non-resonant scattering. A comprehensive account is presented of the scattering theory and data analysis methods used in our earlier studies, and in addition we present new high magnetic field data and its analysis. We discuss in detail how polarisation analysis was used to reveal structural information, including the arrangement of Tb moments which we proposed for $H = 0$ T, and how the diffraction data for $H<H_C$ can be used to determine specific magnetostrictively induced atomic displacements with femto-metre accuracy. The connection between the electric polarisation and magnetostrictive mechanisms is discussed. Similar magnetostrictive displacements have been observed for $H > H_C$ as for $H < H_C$. Finally some observations regarding the kinetics and the conservation of domain population at the transition are described.
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