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We present a study of the magnetic and crystallographic structure ofTbMnO$_3$ in the presence of crossed electric and magnetic fields usingcircularly polarised X-ray non-resonant scattering. A comprehensive account ispresented of the scattering theory and data analysis methods used in ourearlier studies, and in addition we present new high magnetic field data andits analysis. We discuss in detail how polarisation analysis was used to revealstructural information, including the arrangement of Tb moments which weproposed for $H = 0$ T, and how the diffraction data for $H<H_C$ can be used todetermine specific magnetostrictively induced atomic displacements withfemto-metre accuracy. The connection between the electric polarisation andmagnetostrictive mechanisms is discussed. Similar magnetostrictivedisplacements have been observed for $H > H_C$ as for $H < H_C$. Finally someobservations regarding the kinetics and the conservation of domain populationat the transition are described.

Circularly Polarised X-ray Scattering Investigation of Spin-Lattice Coupling in TbMnO$_3$ in Crossed Electric and Magnetic Fields