On the Symmetry of OD‐Structures Consisting of Equivalent Layers

Abstract

After some introductory remarks a survey is given (in part 2) of the basic notions of OD-theory, including the concept of the OD-groupoid family. In part. 3 a new procedure is presented for the determination of t.he OD-groupoid family of an OD-crystal from general features of its X-ray diagrams and applied to two examples. This proccdure makes use of a complete list of OD-groupoid families of which Table 3 contains an important part. For the rest. of this list, the changes from a similar list. which had been published earlier ist given in tJhis table. The mathematical method used to obtain this list is outlined in part 4.Nach einleitenden Bemerkungen wird (in Teil 2) ein tfberblick uber die gruiidle-gander1 Begriffe der OD-Theorie, insbesondere der OD-Gruppoid-Familie gegeben. In Teil3 wird ein neues Verfahreri zur Bestimmung der OD-Gruppoid Familie eines OD-Kristalls aus allgemeinen Eigenschaften seiner Rontgenbeugungsaufnahmen vorgest,ellt und auf zwei Beispiele angewendet,. Fur dieses Verfahren wird eine komplette Lista aller OD-Gruppoid-Familien benotigt, von der in Tabelle 3 ein wichtiger Teil vollstandig und im ubrigen die Anderurigen gegenuber einer fruher veroffentlichteri Liste wiedergegeben werden. Die mathematische Methode, mit deren Hilfe diese neue Liste gewonnen wurde, wird ein Teil 4 in groben Umrissen dargestcllt.

Twins and allotwins of basic mica polytypes: theoretical derivation and identification in the reciprocal space

“…The symmetry and basis vectors of the family structure are obtained by completing the local symmetry operations of a space groupoid of any member of the family, to global symmetry operations of a space group (Fichtner, 1977). 4 The Fourier transform of the family structure corresponds to a three-dimensional subset of re¯ections, called family re¯ections (Dornberger-Schiff & Fichtner, 1972), which are always sharp and are common to all polytypes of the same family. The remaining ones are called nonfamily re¯ec-tions and are typical of each polytype; they can be sharp or diffuse, depending whether the polytype is ordered or not (D Ï urovic Ï & Weiss, 1986;D Ï urovic Ï, 1997.…”

New chain diborate TmH[B2O5] and its relation to GdH[B2O5]: group–subgroup relation

“…Combination of the "layers" in pairs is realized by other symmetry elements which correspond to orthorhombic space group Pbna, other subgroup of Cmca. The multiplicities of both groups are equal and there are no variants in pairs determined by symmetry relation in accordance to OD-theory [3]. The origin of existence of two structural models is in the difference in ionic radii of the elements at the middle and at the end of the REE-elements row.…”

“…The structure was solved using Patterson method in CSD package in a space group C2/c = C 2h 6 and refined in SHELXL-97 package. In TmH[B 2 O 5 ] structure two independent polyhedra: BO 4 -tetrahedron and BO 3 [2] which has monoclinic symmetry with other unit cell. Both structures demonstrate equal pseudosymmetry described in common unit cell by orthorhombic supergroup Cmca.…”

Geometrical properties of MDO polytypes and procedures for their derivation. I. General concept and applications to polytype families consisting of OD layers all of the same kind

“…According to the vicinity condition (VC) (Dornberger-Schiff & Grell-Niemann, 1961: DornbergerSchiff, 1964, 1966: Dornberger-Schiff & Fichtner, 1972, there are only three kinds of sequences of letters for families of OD structures containing equivalent layers and four kinds for families containing more than one kind of layer (Fichtner, 1977: Grell & Dornberger-Schiff, 1982 …”

“…A family of OD structures (Dornberger-Schiff, 1964, 1966Dornberger-Schiff & Fichtner, 1972) is defined as the set of structures consisting of the same kind(s) of layer(s) and the same kind(s) of layer pair(s). The symmetry of any kind of layer and any kind of layer pair occurring in the structure is described by an OD groupoid family.…”