  BaCo2V2O8 is a nice example of a quasi-one-dimensional quantum spin system
that can be described in terms of Tomonaga-Luttinger liquid physics. This is
explored in the present study where the magnetic field-temperature phase
diagram is thoroughly established up to 12 T using single-crystal neutron
diffraction. The transition from the N\'eel phase to the incommensurate
longitudinal spin density wave (LSDW) phase through a first-order transition,
as well as the critical exponents associated with the paramagnetic to ordered
phase transitions, and the magnetic order both in the N\'eel and in the LSDW
phase are determined, thus providing a stringent test for the theory.
