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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, Pb 21.74 Sn 9.34 Fe 3.95 Sb 8.08 S 56.87 , is an MS layer (M = Pb 2+ , Sn 2+ , Sb 3+ ) 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 MS 2  layer (M = Sn 4+ , Fe 2+ ) 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 C1, 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 Pb 0.74 (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) 0.73 Pb 0.27 . Iron is dispersed over the octahedral Sn 4+  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 (Pb 2+ , Sn 2+ ) and Sn 4+ . Their replacement by a Pb 2+ :Bi 3+  combination leads to misfit layer structures of a very different type, typified by cannizzarite.

The crystal structure of franckeite, Pb 21.7 Sn 9.3 Fe 4.0 Sb 8.1 S 56.9