On the Description of Symmetry of OD Structures (I) OD Groupoid Family, Parameters, Stacking

Abstract

OD Groupoid Family, Parameters, StackingDedicated to Professor Kiite BOLL-DORNBEROER on the occasion of her 70th birthdayThe relations between the groupoid of partial symmetry operations of an OD structure and the description of the symmetry of this structure by an OD groupoid family as well as parameters and the stacking of layers are discussed. Proposals are suggested for some changes in the symbols of the OD groupoid families in order to simplify and unify these symbols.Der Zusatnmenhang zwischen dern Gruppoid der partiellen Deckoperationen einer OD Struktur und der Beschreibung der Symmetrie dieser Struktur durch OD-GruppoidFamjlie, Parameter und Stspelung dcr Schiehten wird diskutiert. Es werden Vorschliige fur Anderungen in den Syrnbolen der OD-Gruppoid-Faniilien gemacht, die eine Vereinfachung und Vereinheitlichung dieser Symbole zuni Ziel haben.

Non-Crystallographic Layer Lattice Restrictions in Order-Disorder (OD) Structures

“…A shortening and an update of the notation was proposed by Fichtner [4,23]. Until a more general notation is developed, a few ad hoc adaptations are proposed below to enable the description of OD groupoid families fulfilling only VCβ or VCβ .…”

“…One of these systems, the mineral, wollastonite (CaSiO 3 ) [1], and the isotypic inorganic compounds, Madrell's salt (NaPO 3 ) and NaAsO 3 [2], sparked the development of the order-disorder (OD) theory [3]. The OD theory can be considered as the crystallographic theory of local symmetry and has, since its inception, been developed into a versatile theory for the explanation and description of polytypism, diffuse scattering, non-crystallographic absences, twinning and as a means to classify structures by "symmetry principles" [4]. The fruitful application of OD theory to all major classes of compounds-from minerals to biological macro-molecules [5]-demonstrates the universality of the underlying concepts.…”

“…In this case, they lead to non-crystallographic layer lattice restrictions, which can be expressed as linear relationships of the scalar products of the primitive lattice basis vectors, as will be discussed in Section 2. 4.…”

Symbols for OD groupoid families referring to OD structures (polytypes) consisting of more than one kind of layer

“…In such cases five-or seven-place symbols, respectively, have to be used. These have already been introduced and used for the symbols of OD groupoid families with M = 1 (Dornberger-Schiff & Grell-Niemann, 1961;Dornberger-Schiff, 1964;Dornberger-Schiff & Fichtner, 1972;Fichtner, 1979).…”

The pseudo-inversion symmetry of 9,90′ (1,3,4-oxadiazole-2,5-diyldi(1,1′-biphenyl)-2′,4- diyl)bis[9H-carbazole] in the light of OD theory

“…OD structures are not necessarily periodic, nevertheless they adhere to strict symmetry principles [7]: They fulfill the vicinity condition (VC) stating that equivalent sides of equivalent layers join to other layers in such a way, that the resulting pairs of layers are equivalent [3]. To specify the stacking rules, OD theory introduced the concept of OD groupoid families, which are a generalization of the space group concept.…”