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Properties of the Single and Double D4d Groups and Their Isomorphisms with D8 and C8v Groups A

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Properties of the Single and Double D4d Groups and Their Isomorphisms with D8 and C8v Groups A Properties of the single and double D4d groups and their isomorphisms with D8 and C8v groups A. Le Paillier-Malécot, L. Couture To cite this version: A. Le Paillier-Malécot, L. Couture. Properties of the single and double D4d groups and their isomorphisms with D8 and C8v groups. Journal de Physique, 1981, 42 (11), pp.1545-1552. 10.1051/jphys:0198100420110154500. jpa-00209347 HAL Id: jpa-00209347 https://hal.archives-ouvertes.fr/jpa-00209347 Submitted on 1 Jan 1981 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. J. Physique 42 (1981) 1545-1552 NOVEMBRE 1981, 1545 Classification Physics Abstracts 75.10D Properties of the single and double D4d groups and their isomorphisms with D8 and C8v groups A. Le Paillier-Malécot Laboratoire de Magnétisme et d’Optique des Solides, 1, place A.-Briand, 92190 Meudon Bellevue, France and Université de Paris-Sud XI, Centre d’Orsay, 91405 Orsay Cedex, France and L. Couture Laboratoire Aimé-Cotton (*), C.N.R.S., Campus d’Orsay, 91405 Orsay Cedex, France and Laboratoire d’Optique et de Spectroscopie Cristalline, Université Pierre-et-Marie-Curie, Paris VI, 75230 Paris Cedex 05, France (Reçu le 15 mai 1981, accepté le 9 juillet 1981) Résumé. 2014 Le groupe de symétrie D4d est intéressant car certains ions dans les cristaux peuvent présenter cette symétrie de façon approchée. Nous donnons la table de caractères pour les trois groupes isomorphes D8, C8v et D4d, simples et doubles. Le type d’isomorphisme choisi établit une correspondance entre les rotations inverses notées IC8 de D4d et les rotations directes notées C8 de D8 et de C8v. Le groupe D4d ayant seul un intérêt spectroscopique, nous présentons, uniquement pour ce groupe, la table de décomposition des représentations du groupe des rotations, la table de décomposition des représentations de D4d pour les différents sous-groupes cristallographiques, ainsi qu’une table relative aux règles de sélection. Enfin nous discutons les avantages du type d’isomorphisme choisi. Abstract. 2014 D4d group is of interest because it may be an approximate symmetry for ions in crystals. We present here the character table for the three isomorphic D8, C8v and D4d single and double groups. The chosen type of isomorphism establishes a correspondence between the inverse rotation IC8 of D4d and the direct rotation C8 of D8 and C8v. Since only D4d is of spectroscopic interest we give, for this group only, the full rotation group compatibility table, the subgroup compatibility table and the spectroscopic selection rules. The advantages of the chosen type of isomorphism are discussed. 1. Introduction. - D4d symmetry group which approximate higher symmetry if there is one. For presents an inverse eightfold axis is not one of the example, using the approximate icosahedral sym- thirty-two point groups which are allowed in crystal- metry of rare earth ions in double nitrates, Judd lography. However, in crystals, an approximate D4d was able to explain many spectral features of these symmetry may occur in the surroundings of an ion ions [2]. when the coordination number of this ion is eight In a study of the Zeeman effect of ytterbium ions in and when the coordination polyhedron is a square monoclinic YbCl3, 6 H20, Dieke and Crosswhite [3] or tetragonal Archimedean antiprism (a trigonal first observed that the spectra exhibited an approxi- antiprism is an octahedron). mate symmetry higher than monoclinic. After that, As an example of an approximate D4d site sym- many studies of Zeeman effect or paramagnetic metry in crystals we may mention the site of a tan- resonance of rare earth hexahydrated chlorides reveal- talum Ta+ 5 ion in the monoclinic crystal Na3(TaFg) [1]. ed a similar symmetry, which was sometimes inter- To understand the spectroscopic properties of preted as being a sixfold approximate symmetry. In paramagnetic ions in crystals, it is important to know a forthcoming article Couture and Le Paillier-Malécot the true symmetry of the crystal field, but also the will show that a sixfold axis cannot be found in the surroundings of rare earth ions in such crystals; the coordination polyhedron of rare earth ions will (*) Laboratoire associé à l’Université Paris-Sud. be shown to have the form of an approximate square Article published online by EDP Sciences and available at http://dx.doi.org/10.1051/jphys:0198100420110154500 1546 antiprism made of six water molecules and two (rotation of 2 03C0/8 around z axis) and C’ (rotation of chloride ions (a preliminary presentation is given 03C0 around x axis perpendicular to z). in [4]). The observations can then be interpreted by The symmetry elements of the Os group are an an approximate D4d symmetry. eightfold axis taken as the z direction and eight The surroundings of paramagnetic ions with the twofold axes perpendicular to it. The four twofold shape of a square antiprism are very interesting for axes related to C2 operations are taken as the x and spectroscopic and magnetic applications because they y axes, and the first and second bisectors of the (x, y) have the following three properties : angle ; the four twofold axes related to C2 operations bisect the of the four axes. (i) an approximate axial symmetry of high order, angles C2 The order of this is sixteen. (ii) no centre of symmetry, even approximate, and group (iii) a high anisotropy. C8, group For these reasons we think that the is D4d group All the operations of the C., group may be obtained worth in a similar that the studying way crystal- as products of the two operations C’ and a’ (reflec- have been studied Koster, lographic point groups by tion in a plane containing z and perpendicular to Wheeler and Statz and Prather Dimmock, [5] [6]. x axis which is perpendicular to z). In contrast to ions in molecules crystalline sites, The symmetry elements of the C8v group are an exhibit true and this has may D4d symmetry, group eightfold axis taken as the z direction and eight been studied by molecular spectroscopists [7-9]. The reflection planes containing this axis. The four exhibits such a puckered ring sulfur molecule, S8, reflection planes related to av operations are taken symmetry [8]. in the planes yz and xz and the first and second The D4d symmetry group is isomorphic with D8 bisecting planes of the (yz, xz) angle ; the four reflec- and and we shall the character table and C8, give tion planes related to Oj operations bisect the angles the which are for the multiplication table, common, between the planes (J" v’ three groups. But as D4d group is the only one of interest for spectroscopists we shall present, for this D4d group group only, the full rotation group compatibility All the operations of the group may be obtained table, the subgroup compatibility table and the as products of the operation IC’ and the opera- spectroscopic selection rules. tion C’ (with x axis perpendicular to the z axis). This study will be treated in part 2 and will use The symmetry elements of the D4d group are an inverse rotations as operations of the second kind inverse eightfold axis in the z direction, four twofold of D4d group. The chosen type of isomorphism will axes perpendicular to z, two of them being taken be what we call inversion isomorphism. In part 3 along the x and y axes (Cz operations), and four we shall discuss another type of isomorphism (Tisza’s reflection planes bisecting the angles between these isomorphism) and shall consider improper rotations, binary axes (ad operations). Sn, according to Schoenflies notations. 2.3 GEOMETRICAL REPRESENTATIONS OF THE D8, C8v AND D4d SYMMETRY GROUPS. - We present in 2. of and and double Properties D8, C8, D4d single 1, for each of the three and inverse rotations and inversion isomor- figure D8, C8v D4d groups groups using and with the same conventions as in the Interna- phism. - 2.1 NOTATIONS FOR OPERATIONS OF THE tional Tables for X-ray Crystallography [10], stereo- GROUPS. - The following notations are used : grams of poles of general equivalent directions and E : identity, the symmetry elements of the groups. We have in addition shown the directions of the x axes. Cn : rotation of + 2 03C0/n around an axis and y (n integral), I : inversion, 2.4 CHOICE OF ISOMORPHISM OF THE THREE SINGLE 2 around an rotation of + axis, - 1Cn : 03C0/n GROUPS OS, C8v AND D4d The choice of the corres- followed by an inversion (the inverse ponding operations in the isomorphism of the D. rotation must be considered as a IC,, and C8v groups is obvious, namely C8(D8) H C8(C8v) single operation), and C2(Dg) H u’(C8v); we must nevertheless remark Q = reflection in a to 1C2 : plane perpendicular that the choice of isomorphism is not unique. the axis of rotation of C2. In the case of the D. and D4d group isomorphism, the choice of the corresponding operations which we 2.2 DEFINITIONS OF THE THREE SINGLE POINT GROUPS have made is the following : D8, Csv AND D4d. D8 group All the operations of the D8 group may be obtained We shall call this correspondence inversion iso- as products of the two elementary operations C8z morphism.
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