Magnetic hexamers interacting in layers in the (Na,K)$_2$Cu$_3$O(SO$_4)_3$ minerals

Magnetic properties and underlying magnetic models of the synthetic A$_2$Cu$_3$O(SO$_4)_3$ fedotovite (A = K) and puninite (A = Na) minerals, as well as the mixed euchlorine-type NaKCu$_3$O(SO$_4)_3$ are reported. We show that all these compounds contain magnetic Cu$_6$ hexamer units, which at temperatures below about 100 K carry an overall spin-1 and represent magnetic molecules. Weak interactions between such molecules lead to long-range magnetic order below $T_N$ = 3.4 K (A = Na), 4.7 K (A = NaK), and 2.9 K (A = K). The formation of the magnetic order is elucidated by ab initio calculations that reveal predominant inter-hexamer interactions forming a $S=1$ square lattice within crystallographic $bc$ planes, and challenge previous description of these minerals in terms of Haldane spin chains. We emphasize the role of the anisotropy inherent to the $S=1$ molecules in the formation of a magnetically ordered ground state with a magnon gap instead of a Haldane gap discussed in previous studies.