The layer-like crystal structure of franckeite from the mine of San José, Bolivia, exhibits a pronounced one-dimensional transversal wave-like modulation and a non-commensurate layer match in two dimensions. It consists of alternating pseudohexagonal (H) layers and pseudotetragonal (Q) slabs and forms a homologous pair with cylindrite, which has thinner Q slabs. The Q slabs in franckeite are four atomic layers thick. The two components have their own lattices and a common modulation. The Q slab of the refined franckeite structure, Pb21.74Sn9.34Fe3.95Sb8.08S56.87, is an MS layer (M = Pb2+, Sn2+, Sb3+) four atomic planes thick, with a = 5.805(8), b = 5.856(16) Å, and the layer-stacking vector c = 17.338(5) Å. The lattice angles are α = 94.97(2)°, β = 88.45(2)°, γ = 89.94(2)°; the modulation vector q = −0.00129(8) a* + 0.128436(10) b* − 0.0299(3) c*. The H layer is a single-octahedron MS2 layer (M = Sn4+, Fe2+) with a = 3.665(8), b = 6.2575(16), c = 17.419(5) Å, α = 95.25(2)°, β = 95.45(2)°, γ = 89.97(2)°; the modulation vector is q = −0.00087(8) a* + 0.13725(16) b* − 0.0314(4) c*. The a and b vectors of both subsystems are parallel; the c vectors diverge. (3+2)D superspace refinement was performed in the superspace group C1̄, using 7397 observed reflections. It resulted in the overall R(obs) value equal to 0.094. The Q slabs are composed of two tightly bonded double-layers, separated by an interspace hosting non-bonding electron pairs. Average composition of cations on the outer surface was refined as Pb0.74(Sn,Sb)0.26, whereas that of cations, which are adjacent to the interspace with lone electron pairs, with a configuration analogous to that observed in orthorhombic SnS, corresponds to (Sn,Sb)0.73Pb0.27. Iron is dispersed over the octahedral Sn4+ sites in the H layer. Transversal modulation of the Q slab is achieved by local variations in the Pb:(Sn,Sb) ratios at its surface and interior. Its purpose is to re-establish a one-dimensional commensurate contact along [010] between the curved Q and H surfaces to the greatest extent possible. Layer-stacking disorder and divergence of the Q and H stacking directions, and the divergence between modulation wave-front and these stacking directions are typical for the composite structures of franckeite and cylindrite. Because of the increased rigidity of the Q component, franckeite usually forms masses of curved crystals rather than cylindrical aggregates. The existence of this family depends critically on the radius ratios of the cations involved, especially those involving (Pb2+, Sn2+) and Sn4+. Their replacement by a Pb2+:Bi3+ combination leads to misfit layer structures of a very different type, typified by cannizzarite.

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