Coupled antiferromagnetic spin-1/2 chains in green dioptase, Cu$_6[$Si$_6$O$_{18}]\cdot 6$H$_2$O

Gem crystals of natural dioptase with colors ranging from emerald-green to bluish have delighted people since ancient times and still attract attention of mineral collectors around the globe. The crystal structure of green dioptase (space group $R\bar{3}$) consists of corrugated silicate rings Si$_6$O$_{18}$ interconnected by Cu$^{2+}$ ions. Oxygen atoms form axially-elongated octahedral of CuO$_4$(H$_2$O)$_2$. The magnetic ground state of green dioptase remains controversial. We report inelastic neutron scattering measurements of the magnetic excitations of green dioptase Cu$_6[$Si$_6$O$_{18}]\cdot 6$H$_2$O. The observed spectrum contains two magnetic modes and a prominent spin gap that is consistent with the ordered ground state of Cu moments coupled antiferromagnetically in spiral chains along the $c$ axis and ferromagnetically in $ab$ planes on the hexagonal cell. The data are in excellent agreement with a spin-1/2 Hamiltonian that includes AFM nearest-neighbor intra-chain coupling $J_{c} = 10.6(1)$ meV, ferromagnetic inter-chain coupling $J_{ab} = -1.2(1)$ meV and exchange anisotropy $\Delta J_{c} = 0.14(1)$ meV. This appears compatible with reduced N\`{e}el temperature, $T_\mathrm{N}=14.5$ K $\ll J_{c}$, and can be explained by a presence of quantum spin fluctuations.