  Single crystal neutron diffraction studies have been performed on the S = 3/2
kagome staircase compound Co3V2O8 with a magnetic field applied along the
magnetization easy-axis (H || a). Previous zero field measurements [Y. Chen, et
al., Phys. Rev. B 74, 014430 (2006)] reported a rich variety of magnetic
phases, with a ferromagnetic ground state as well as incommensurate,
transversely polarized spin density wave (SDW) phases (with a propagation
vector of k = (0 delta 0)) interspersed with multiple commensurate lock-in
transitions. The magnetic phase diagram with H || a adds further complexity.
For small applied fields, H = 0.05 T, the commensurate lock-in phases are
destabilized in favor of the incommensurate SDW, while slightly larger applied
fields restore the commensurate lock-in phase with delta = 1/2 and yield a new
commensurate phase with delta = 2/5. For all measurements in an applied field,
higher-order scattering is observed that corresponds to the second-harmonic.
