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Balangeroite, with ideal crystal chemical for- mula (Mg,Fe)42O6(Si4O12)4(OH)40, displays two MDO polytypes, balangeroite-2M (a ¼ 19.179, b ¼ 9.601, c ¼ 19.218 A ˚ , b ¼ 90.52 � ; space group P2/n) and balan- geroite-1A (a ¼ 9.602, b ¼ 13.891, c ¼ 14.012 A ˚ , a ¼ 86.99 � , b ¼ 76.79 � , g ¼ 76.67 � ; space group P � Both polytypes are built up by octahedral walls 3 � 1, octahedral bundles 2 � 2 and four-repeat silicate chains. All these modules run parallel to the 9.6 Aaxis (b and a in the monoclinic and triclinic polytype, respectively). The distinguishing features between the two polytypes are in the different positioning of the silicate chains, which gives rise to distinct cells and space groups.

Single-crystal X-ray diffraction studies of brochantite demonstrate its Order-Disorder (OD) character. The OD structures of the brochantite family can be described as built up by equivalent OD layers with symmetry Pn2 1 m. Two MDO polytypes are possible in this family, and both have been identified in specimens of brochantite from various localities. The MDO 1 polytype corresponds to “normal” brochantite P12 1 /a 1 , a = 13.140(2) A, b = 9.863(2), c = 6.024(1), β = 103.16(3)°, whereas the newly discovered MDO 2 polytype is monoclinic, P2 1 /n11, a = 12.776(2) A, b = 9.869(2), c = 6.026(1), α = 90.15(3)°. Given their polytypic relationships, the two MDO 1 and MDO 2 polytypes can be designated as brochantite-2M 1 and brochantite-2M 2 , respectively. The refined structure (R 1 = 0.049) of brochantite-2M 1 substantially agrees with the model of Cocco & Mazzi (1959) and confirms the assignment of the OH bands due to Schmidt & Lutz (1993). On the basis of the OD relationships between the two polytypes, a structural model for the newly discovered 2M 2 polytype was derived and subsequently refined up to R 1 = 0.062.

The chemical formula of fiedlerite, a rare hydrated lead halide, has been revised. The mineral is now known to contain also fluorine, and the new, correct formula is Pb3Cl4F(OH)·H2O. X-ray diffraction studies on fiedlerite from Laurion, Greece (the type locality), and from Baratti, Italy (the second known occurrence), revealed its Order-Disorder (OD) character. All structures within this OD family can be built up by layers of the same kind. The two polytypes with Maximum Degree of Order (MDO) display triclinic and monoclinic symmetry, with one and two OD layers, respectively, in the unit cell. On these grounds the nomenclature of fiedlerite has been revised, and the mineral is designated together with the polytype suffix (i.e. fiedlerite-1A, fiedlerite-2M). The crystal structures of the two MDO polytypes of fiedlerite have been solved and refined: fiedlerite-1A: P, a = 8.574(3) Å, b = 8.045(4), c = 7.276(2), α = 89.96(4)° β = 102.05(4), γ = 103.45(4), R = 0.092; fiedlerite-2M: P21/a, a = 16.681(4) Å, b = 8.043(3), c = 7.281(2), β = 102.56(4)°, R = 0.061. In both structures Pb is eight-coordinated by different ligands [CI−, F−, (OH)−, H2O] that define bicapped trigonal prisms.

Acta Crystallographica Section A : http://journals.iucr.org/a/journalhomepage.html; International audience; Mathematical crystallography is the branch of crystallography dealing specifically with the fundamental properties of symmetry and periodicity of crystals, topological properties of crystal structures, twins, modular and modulated structures, polytypes and OD structures, as well as the symmetry aspects of phase transitions and physical properties of crystals. Mathematical crystallography has had its most evident success with the development of the theory of space groups, at the end of the XIX century; since then, it has greatly enlarged its applications, but crystallographers are not always familiar with the developments that followed, partly because the applications sometimes require some additional background that the structural crystallographer doe not always possess (it is the case, for example, of graph theory). The knowledge offered by mathematical crystallography is at present only partly mirrored in the International Tables for Crystallography and is sometimes still enshrined in more specialistic texts and publications. To cover this communication gap is one of the tasks of the IUCr Commission on Mathematical and Theoretical Crystallography (MaThCryst).

International audience; The occurrence of residues in the Fourier map of OD structures (polytypes in which pairs of building modules are geometrically equivalent) in the positions of the 'virtual atoms' of the corresponding family structure derives from the presence of stacking faults inside an otherwise ordered (periodic) matrix. These residues are commonly spurious peaks deriving from the refinement of the structure with a single scale factor for both the family and the non-family reflections, which may be instead on a different scale because of the different peak shape and background characterizing the two types of reflections, resulting in broadening and streaking of the non-family reflections. The 'virtual atoms' occur in the same positions of the atoms corresponding to the stacking faults, but the spurious peaks are in a quantitative relation with them only if the stacking faults diffract coherently. The case of mica polytypes is illustrated also with the aid of examples taken from the literature.

A review article containing in its first part a short summary of the most important terms, definitions and logical principles of the theory of OD (order-disorder) structures by Dornberger-Schiff. It is concluded that the OD theory is a symmetry theory of polytypic structures. A few selected examples in the second part : phyllosilicates, ZnS, SiC, TaS2, orpiment, and γ-Hg3S2Cl2 demonstrate the use of the OD apparatus in the investigation of polytypic substances ; in particular the high abstraction power of the OD approach, the concept of MDO polytypes (polytypes with maximum degree of order), the explanation of unexpected vibrational spectra, and the notion of desymmetrization., Article de revue contenant en première partie un bref rappel des points les plus importants, définitions et principes logiques de la théorie des structures OD (ordre-désordre) de Dornberger-Schiff. On conclut que la théorie OD est une théorie de symétrie des structures polytypiques. Dans la seconde partie quelques exemples choisis : phyllosilicates, ZnS, SiC, TaS2, orpiment et Hg3 S2Cl2 - γ démontrent l'utilisation de l'outil OD pour l'étude des substances polytypiques ; en particulier le fort pouvoir d'abstraction de l'approche OD, le concept de polytypes MDO (polytypes avec maximum degré d'ordre), l'explication de spectres de vibration inattendus et la notion de désymétrisation., Ďurovič Slavomil, Weiss Zdenĕk. OD structures and polytypes. In: Bulletin de Minéralogie, volume 109, 1-2, 1986. Polytypes.