In this joint experimental and theoretical work magnetic properties of the Cu$^{2+}$ mineral szenicsite Cu$_3$(MoO$_4$)(OH)$_4$ are investigated. This compound features isolated triple chains in its crystal structure, where the central chain involves an edge-sharing geometry of the CuO$_4$ plaquettes, while the two side chains feature a corner-sharing zig-zag geometry. The magnetism of the side chains can be described in terms of antiferromagnetic dimers with a coupling larger than 200 K. The central chain was found to be a realization of the frustrated antiferromagnetic $J_1$-$J_2$ chain model with $J_1simeq 68$ K and a sizable second-neighbor coupling $J_2$. The central and side chains are nearly decoupled owing to interchain frustration. Therefore, the low-temperature behavior of szenicsite should be entirely determined by the physics of the central frustrated $J_1$-$J_2$ chain. Our heat-capacity measurements reveal an accumulation of entropy at low temperatures and suggest a proximity of the system to the Majumdar-Ghosh point of the antiferromagnetic $J_1$-$J_2$ spin chain, $J_2/J_1=0.5$.
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