The members of the sartorite homologous series are complex sulfidesMe²⁺_{8N^–16–2X}Me³⁺_1⁶⁺XMe⁺_XS_8N+8whereMe²⁺is Pb andMe³⁺is As and Sb, whereasMe⁺is Ag and/or Tl. This paper presents calculation formulae for the homologue order N and for the separate substitution percentages for Tl + (As,Sb) ↔ 2Pb and Ag + (As,Sb) ↔ 2Pb substitutions. This enables one to evaluate the crystal chemistry and build a systematic classification of the sartorite homologues.

The current state of the lillianite homologous series is presented, with its two branches − the lillianite branch of the predominantly Pb-Bi-Ag sulfosalts and the andorite branch of predominantly Pb-Sb-Ag sulfosalts. Both the natural and synthetic members are discussed, especially from the structural and compositional point of view and the related, chemically distinct and structurally more complicated members of the series are described. A number of new published, or hitherto unpublished observations is given, together with fairly exhaustive tables of data. Relationships between the complex structures of different ‘andorite’ species and principal structural features of ‘oversubstituted’ As-Sb and Bi-Sb species are discussed. Contrary to many studies of this subject, synthetic phases form an integral part of the paper. It is concluded that research in this long-known homologous series still supplies new interesting phases, especially in the fields of synthetic products and in its andorite branch.

Complex sulfides of thallium with As, Sb, or Bi and with other cations (‘thallium sulfosalts’) are a large group of crystal structures with extreme variability. Incorporation of the large Tl⁺ cation in them is solved in several different ways: housing of Tl in columns of capped trigonal coordination prisms, which form separate walls in the structure (in different combinations with Pb and/or Sb), regular alternation of large Tl with small cations (As), presence of structural arrays of Tl coordination polyhedra paralleled by arrays of As coordination pyramids with a frequency ratio 1:2, omission derivatives with cavities for Tl accommodation and formation of layer structures with thallium concentrated into separate (inter)layers of different types. The first principle leads to a large family of sartorite homologues and rare lillianite homologues, as well as to the chabournéite group. The second one to the hutchinsonite family, omission derivatives form the routhierite and galkhaite groups, and the 1:2 periodicity ratio principle results in several outstanding structures from different groups. Layer structures consist of two-component and three-component layer combinations. Close cation-cation interactions are present but rare.

The polytypic structures of vaterite, CaCO3, described in recent literature, have been reinterpreted in terms of the order-disorder (OD) theory, which allows us to explain and systematize all the observed and predicted polytypes of the mineral in a unified fashion. In terms of this theory, the structure consists of OD layers that comprise a layer of calcium coordination polyhedra and the attached halves of the standing CO3 groups. The two-sided layer group of symmetry of the OD layer is c 2/ m , whereas the interlayer symmetry operations are three twofold rotation axes at 120° to one another, as well as a mirror plane in the common layer boundaries and partial c glide planes perpendicular to the boundary. Depending on the orientation of the active twofold rotation axis with respect to the above-defined layer mesh, performed independently in each stacking step, maximally ordered simple stacking sequences P 6122, P 6522, C 2/ c , C 2/ c 2/ m 21/ m , and a more complicated sequence P 312 or P 322, as well as several complicated or disordered sequences is obtained (before eventual relaxation to a subgroup of a particular space group). A perfect copy of the vaterite OD layer occurs in the structures of the bastnäsite-synchysite polysomatic series of fluorocarbonates. In these structures, the REE layers, configurationally analogous to the Ca-based OD layer, have layer symmetry p 32 and their stacking does not lead to polytypism and OD phenomena; these are generated on the Ca-based OD layers.