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 Neel 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.
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