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Applied Geminography - Symmetry Analysis of Twinned Crystals and Definition of Twinning by Reticular Polyholohedry Massimo Nespolo, Giovanni Ferraris

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Applied Geminography - Symmetry Analysis of Twinned Crystals and Definition of Twinning by Reticular Polyholohedry Massimo Nespolo, Giovanni Ferraris Applied geminography - symmetry analysis of twinned crystals and definition of twinning by reticular polyholohedry Massimo Nespolo, Giovanni Ferraris To cite this version: Massimo Nespolo, Giovanni Ferraris. Applied geminography - symmetry analysis of twinned crystals and definition of twinning by reticular polyholohedry. Acta Crystallographica Sec- tion A Foundations and Advances, International Union of Crystallography, 2004, 60, pp.89-95. 10.1107/S0108767303025625. hal-00130548 HAL Id: hal-00130548 https://hal.archives-ouvertes.fr/hal-00130548 Submitted on 12 Feb 2007 HAL is a multi-disciplinary open access L’archive ouverte pluridisciplinaire HAL, est archive for the deposit and dissemination of sci- destinée au dépôt et à la diffusion de documents entific research documents, whether they are pub- scientifiques de niveau recherche, publiés ou non, lished or not. The documents may come from émanant des établissements d’enseignement et de teaching and research institutions in France or recherche français ou étrangers, des laboratoires abroad, or from public or private research centers. publics ou privés. electronic reprint Acta Crystallographica Section A Foundations of Crystallography ISSN 0108-7673 Editor: D. Schwarzenbach Applied geminography – symmetry analysis of twinned crystals and definition of twinning by reticular polyholohedry Massimo Nespolo and Giovanni Ferraris Copyright © International Union of Crystallography Author(s) of this paper may load this reprint on their own web site provided that this cover page is retained. Republication of this article or its storage in electronic databases or the like is not permitted without prior permission in writing from the IUCr. Acta Cryst. (2004). A60, 89–95 Nespolo and Ferraris ¯ Applied geminography research papers Acta Crystallographica Section A Foundations of Applied geminography ± symmetry analysis of Crystallography twinned crystals and definition of twinning by ISSN 0108-7673 reticular polyholohedry Received 30 June 2003 a b Accepted 5 November 2003 Massimo Nespolo * and Giovanni Ferraris aUniversite Henri Poincare Nancy 1, Laboratoire de Cristallographie et de ModeÂlisation des MateÂriaux MineÂraux et Biologiques (LCM3B), UMR ± CNRS 7036, Faculte de Sciences et Techniques, Boulevard des Aiguillettes, BP 239, F-54506 Vandúuvre-leÁs-Nancy CEDEX, France, and bUniversitaÁ di Torino, Dipartimento di Scienze Mineralogiche e Petrologiche, Istituto di Geoscienze e Georisorse, CNR, Via Valperga Caluso 35, 10125 Torino, Italy. Correspondence e-mail: [email protected] The common classi®cation of twinning into the four categories of twinning by merohedry (complete and exact overlap of the lattices of the twinned crystals), pseudomerohedry (complete but approximate overlap), reticular merohedry (partial but exact overlap) and reticular pseudomerohedry (partial and approximate overlap) is revised in terms of the complete (translational and point) lattice symmetry of the twin and of the individual. The new category of reticular polyholohedry is introduced for twins where the twin lattice has the same point symmetry but a different orientation of the individual lattice. It is shown that the degeneration to twin index 1 relates, in a parallel way, reticular merohedry to metric merohedry and reticular polyholohedry to syngonic # 2004 International Union of Crystallography merohedry. Some examples from the recent literature are analysed in terms of Printed in Great Britain ± all rights reserved this revised classi®cation. 1. Introduction The complete geminographical analysis of a twinned crystal Twinned crystals are often considered an obstacle to the should take into account three aspects: solution and re®nement of crystal structures, and the tendency 1. the symmetry relations in direct space relating the indi- towards entrusting their treatment to black-box software viduals, the individual lattice and the twin lattice; packages is increasing. Besides the risks connected to an 2. the symmetry relations in reciprocal space and the effect uncritical acceptance of the solution provided by the machine, of twinning on the diffraction pattern (Buerger, 1954, later one of the consequences of this trend is the increasing republished in Buerger, 1962; Ferraris et al., 2004); unawareness by some users of the symmetry and lattice 3. the morphology of the twin, as it appears in its forms (see, peculiarities of twins, without which a real comprehension of for example, Shafranovskii, 1973). the phenomenon of twinning is impossible. In this article, we deal with the ®rst aspect, extending some Twins are such particular `crystallographic objects' that of the classical categories introduced long ago and described J. D. H. Donnay called geminography the branch of crystal- in standard textbooks of crystallography. lography dealing with twinning (Nespolo & Ferraris, 2003). A Donnay & Donnay (1974) classi®ed twinning in two twin is an oriented association of individual crystals of the categories: TLS (twin lattice symmetry) and TLQS (twin same chemical and crystallographic species. The individuals in lattice quasisymmetry) on the basis of the value of the a twin are related by one or more geometrical laws (the twin obliquity !, which is the angle measuring the metrical laws) expressed through the point symmetry of the twin versus symmetry deviation of the crystal lattice with respect to the the point symmetry of the individual. The twin element (twin twin lattice. A plane that is a (pseudo)symmetry plane for the centre, twin axis, twin plane) is the geometric element about lattice has a lattice row (quasi)perpendicular, which is a which twin operations (operations relating different indivi- (pseudo)symmetry axis for the lattice. The obliquity ! is the duals) are performed.1 complement to 90 of the angle between these (pseudo)sym- metry axis and plane; it is zero for TLS and non-zero for 1 Twins are heterogeneous edi®ces formed by two or more homogeneous TLQS. In the diffraction pattern, TLS and TLQS differ by the structurally three-periodic individuals (modules) related by a point-group absence or presence of split re¯ections, respectively (see Koch, operation. Consequently, the twin itself does not possess a homogeneous 1992). crystal structure. Homogeneous edi®ces built by structurally less-than-three- periodic modules are termed cell twins: the cell-twin operation may have a Well before, the so-called `French school' (see Friedel, 1904, translational component (TakeÂuchi, 1997; Ferraris et al., 2004). 1926) had introduced a ®ner classi®cation into the four cate- Acta Cryst. (2004). A60, 89±95 DOI: 10.1107/S0108767303025625 89 electronic reprint research papers gories of twinning by merohedry (complete and exact overlap non-specialized metric: L 2a D DH of the lattices of the twinned crystals), pseudomerohedry specialized higher metric: L : 2b (complete but approximate overlap), reticular merohedry D DH (partial but exact overlap) and reticular pseudomerohedry (L) and can be decomposed in cosets (cf. Giacovazzo, D D (partial and approximate overlap). An alternative classi®ca- 2002a) tion, based on the morphological ± rather than reticular ± L 1D 2D ... nD 1 3a (pseudo)symmetry elements was introduced by the so-called D D[C [C [ [C `German school' (see Hahn & Klapper, 2003). H H ... H ; 3b 1 2 n0 1 Buerger (1945) also discussed the in¯uence of the structure DH[C [C [ [C where D and H are the cosets, n and n0 the indices of (L)in on twinning and reinforced the concept of necessary condi- Cj Cj D tions for twinning by adding the occurrence of structure and of in , respectively. Without an accidentally specializedD metric,D Hn = 1 and (3a) reduces to (2a). When is (pseudo)symmetry besides that of reticular (pseudo)sym- H metry. Actually, a possible existence of `suf®cient' conditions holohedral, n0 =1. for twinning appears contradicted by at least one fact: in the Friedel's geometrical crystallography was based on the same batch of crystallization, the same compound may give at lattice symmetry, but his classi®cation of twinning took into the same time twinned and untwinned crystals. The structural account only the point symmetry of lattices. To describe some conditions added by Buerger for the occurrence of a given special cases in the general framework of Friedel's classi®ca- twin are thus themselves necessary but this necessity has a tion (Friedel, 1904, 1926), we need to consider also the wider meaning than Friedel's reticular conditions. However, translational symmetry. whereas the reticular conditions can be foreseen in a very The coordinate systems of the individual and of the twin are general and simple way from the metric of the lattice, the related by a transformation matrix P that implies, in general, a structure must be already known in order to determine the change of the basis vectors: structural conditions. In this paper, we apply Friedel's lattice abc IP abc T: 4 treatment and introduce some generalizations that depend on h j h j the metric of the lattice only. P can be decomposed into the product of a rotation Ru(')by' around a rational direction u in abc|I and a metric expansion X (lattice thinning): h 2. Symmetry classification of twinning in terms of the P Ru ' X: 5 lattice point group and translation group Let us indicate by W the matrix representation of a symmetry 2.1. Definitions operation for LI, and by WT the matrix representing the twin We call `individual' an untwinned crystal and denote by the operation. TLS twinning can thus be classi®ed as follows. subscript I all the related variables. The corresponding quan- 2.1.1. and L L . This case corresponds to TT TI D T 6 D I tities for a twin are indicated by the subscript T. The vector reticular merohedry in Friedel's (1904, 1926) classi®cation. lattices of the individual and of the twin are thus LI and LT. Actually, Friedel (1926, p. 30) restricted this category to the The twin index nT is the ratio of the number of lattice nodes case in which the unit cell of LT has a symmetry higher than of the individual to the number of nodes restored, exactly or that of LI [ T I and (L)T (L)I].
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