Abstract
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 c2/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 P6122, P6522, C2/c, C2/c2/m21/m, and a more complicated sequence P312 or P322, 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 p32 and their stacking does not lead to polytypism and OD phenomena; these are generated on the Ca-based OD layers.
Acknowledgments
Rafaella Demichelis kindly supplied CIF files of the vaterite variants derived in their work. This paper contributed to the Nano-Chalk Venture, funded by Maersk Oil and Gas A/S and the Danish Advanced Technology foundation (HTF). Discussions with the members of the Nano-Geochemistry group, Nano-Science Center, University of Copenhagen, chaired by Susan L.S. Stipp, were appreciated. I am grateful to the associate editor, D.R. Hummer, and two anonymous referees for the helpful comments.
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Appendix: Basic Notions of the Theory of Od Phenomena
All crystal structures can be sectioned into equivalent slabs. For the majority of structures, the orientation and position of the (n+1)st slab after the nth slab is unambiguously determined. The less frequent cases when these rules are relaxed but still deterministic, are dealt with by the order-disorder (OD) theory. In the sense used in this contribution, OD phenomena describe ambiguity in the position (and orientation) of the (n+1)st slab of the structure after every nth slab. In the OD structures, the two (or more) positional/orientational choices give rise to two or several geometrically (and thus also crystal-chemically) equivalent pairs of slabs, which differ from one another only by their orientation. These slabs are purely geometrical and not always identical with crystal chemically defined layers. They are called OD layers, with their own crystallographic (pseudo)symmetry, described by one of the eighty layer groups [consisting of the so-called intralayer (or λ-) partial operations (POs); the partial character of the operations means that they are not necessarily valid for the whole structure; they may be valid only for the OD layer]. Another set of operations (the so-called interlayer partial operations or σ-POs) transforms an OD layer into the adjacent one. Together these two kinds of operations do not form a symmetry group but a weaker combination of symmetry operations, a groupoid (Dornberger-Schiff 1966).The above-mentioned positional ambiguity means that—although all layer pairs are equivalent—layer triples, quadruples, etc., are not automatically identical even in one and the same structure or in various crystals of the same substance. All OD-structures with the same set of λ-POs and σ-POs to the exclusion of translations belong to one OD-groupoid family. Our case, vaterite, is described as one of the families of OD structures composed of one kind of OD layers, but ample examples of OD structures composed of two (or more) kinds of OD layers in regular alternation exist as well (e.g., Ferraris et al. 2008).
The partial character of interlayer operations (glide-reflection planes and screw axes) in the OD structures involves unusual translation components with special notation rules. In all cases the translation component is referred to the basis vectors of the OD layer. Generalized notation rules for partial symmetry operations that interconnect an nth layer with the (n+1)st layer (σ-POs) are as follows: for a glide plane, the translation component is the value of the subscript attached to the symbol divided by two: n (≡ n1) has translation equal to ½ of the full translation vector, n½ has translation component equal to ¼ of the translation vector, n2 has a translation component equal to a full translation period (from one OD layer to the next one), etc. In a similar way, 21 has the translation component equal to ½ of the full translation period parallel to the axis, 22 is a partial operation with the translation equal to a full translation period of the lattice along the given axis, etc.
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