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We report a comprehensive experimental investigation of the magneticstructure of the cycloidal phase in Ca$_3$Ru$_2$O$_7$, which mediates the spinreorientation transition, and establishes its magnetic phase diagram. In zeroapplied field, single-crystal neutron diffraction data confirms the scenariodeduced from an earlier resonant x-ray scattering study: between $46.7$~K $< T< 49.0$~K the magnetic moments form a cycloid in the $a-b$ plane with apropagation wavevector of $(\delta,0,1)$ with $\delta \simeq 0.025$ and anordered moment of about 1 $\mu_{\rm{B}}$, with the eccentricity of the cycloidevolving with temperature. In an applied magnetic field applied parallel to the$b$-axis, the intensity of the $(\delta,0,1)$ satellite peaks decreasescontinuously up to about $\mu_0 H \simeq 5$ T, above which field the systembecomes field polarised. Both the eccentricity of the cycloid and thewavevector increase with field, the latter suggesting an enhancement of theanti$-$symmetric Dzyaloshinskii$-$Moriya interaction via magnetostrictioneffects. Transitions between the various low-temperature magnetic phases havebeen carefully mapped out using magnetometry and resistivity. The resultingphase diagram reveals that the cycloid phase exists in a temperature windowthat expands rapidly with increasing field, before transitioning to a polarisedparamagnetic state at 5 T. High-field magnetoresistance measurements show thatbelow $T\simeq 70$ K the resistivity increases continuously with decreasingtemperature, indicating the inherent insulating nature at low temperatures ofour high-quality, untwinned, single-crystals. We discuss our results withreference to previous reports of the magnetic phase diagram ofCa$_3$Ru$_2$O$_7$ that utilised samples which were more metallic and/orpoly-domain.

The magnetic structure and field dependence of the cycloid phase mediating the spin reorientation transition in Ca$_3$Ru$_2$O$_7$