  The high-field magnetic properties and magnetic order of the gem mineral
green dioptase Cu$_6[$Si$_6$O$_{18}]\cdot 6$H$_2$O have been studied by means
of single-crystal neutron diffraction in magnetic fields up to $21~$T and
magnetization measurements up to $30~$T. In zero field, the Cu$^{2+}$-moments
in the antiferromagnetic chains are oriented along the $c$-axis with a small
off-axis tilt. For a field applied parallel to the $c$-axis, the magnetization
shows a spin-flop-like transition at $B^*=12.2~$T at $1.5~$K. Neutron
diffraction experiments show a smooth behavior in the intensities of the
magnetic reflections without any change in the periodicity of the magnetic
structure. Bulk and microscopic observations are well described by a model of
ferromagnetically coupled antiferromagnetic $XXZ$ spin-$\frac{1}{2}$ chains,
taking into account a change of the local easy-axis direction. We demonstrate
that the magnetic structure evolves smoothly from a deformed N\'eel state at
low fields to a deformed spin-flop state in a high field via a strong crossover
around $B^*$. The results are generalized for different values of interchain
coupling and spin anisotropy.
