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Disclination Density in Atomic Structures Described in Curved Spaces J.F Disclination density in atomic structures described in curved spaces J.F. Sadoc, R. Mosseri To cite this version: J.F. Sadoc, R. Mosseri. Disclination density in atomic structures described in curved spaces. Journal de Physique, 1984, 45 (6), pp.1025-1032. 10.1051/jphys:019840045060102500. jpa-00209832 HAL Id: jpa-00209832 https://hal.archives-ouvertes.fr/jpa-00209832 Submitted on 1 Jan 1984 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 45 (1984) 1025-1032 JUIN 1984, 1025 Classification Physics Abstracts 61.00 - 61.70 Disclination density in atomic structures described in curved spaces J. F. Sadoc (*) and R. Mosseri (**) (*) Lab. de Physique des Solides, 91405 Orsay, France (**) Physique des Solides, CNRS, 1 Pl. A. Briand, 92190 Meudon Bellevue, France (Reçu le 28 octobre 1983, accepte le 8 février 1984) Résumé. 2014 La courbure de l’espace et la densité de disinclinaisons sont deux grandeurs connectées. Il y a une relation exacte à 2 dimensions. Nous montrons comment utiliser une relation approchée à 3 dimensions. Les appli- cations à la structure W-03B2 et aux phases de Laves sont présentées. La coordinance, dans les structures denses aléatoires, est expliquée à partir de la notion de densité de disinclinaison. Abstract. 2014 The curvature of a space and the density of disclinations are two related quantities. There is an exact relation in 2-D spaces. We show how an approximate solution can be used in 3-D space. Applications to the 03B2-W structure and the Laves phase are presented The coordination number in dense random structures is explained in terms of disclination density. 1. Introduction. deficit of a surface topologically equivalent to a sphere, and the sum of the Gaussian curvature on the In a 2-D space the curvature of space and the density surface. In fact this is a simple example of the Gauss- of disclinations are related to each other exactly by Bonnet theorem, and this relation is applicable to the Gauss-Bonnet theorem [1]. An attractive example all the surfaces approximated by a polyhedron occurs when disclinations are not infinitesimal, with flat faces. Consequently in 2d space, there is but correspond to a finite angular deficit (or excess) : an entire equivalence between the disclination density the surface is everywhere flat except on point defects (with a weighting factor equal to their angular deficit) to a local concentration of curvature. corresponding and the curvature. In this case the 2-D space is the surface of a polyhedron Unfortunately in 3-D space no such simple result [2] a since it is not a closed (in general sense, necessarily exists. surface). In the case of regular polyhedra simple The aim of this paper is to relate disclination density defined two Schlafli indices { p, q } the angular by and curvature in 3-D spaces. (For example polytopes deficit 6 of each vertex is 6 = 2 7C 2013 where qap, ap embedded in 4-D Euclidean space.) Application to is the vertex angle of a p-gonal face. some complex periodic structure [3] which can be So described in corrugated 3-D space (with a zero mean curvature) is presented. We show how positive and negative disclinations approximatively annul each The number of vertices for regular polyhedra is other, if positive and negative curvatures balance mutually in order to give a zero mean curvature. Coordination numbers in dense structures are explai- ned in terms of disclination density. (from the Euler relation). Consequently the total angular deficit for a poly- hedron is Lb = 4 7L Now consider a polyhedron 2. Polytopes and disclinations. to be an approximation of a sphere with a constant A polytope is an approximation by flat spaces (cells) gaussian curvature K. We can write Kda = 4n of a 3-D spherical space. The curvature is concen- ff trated on edges acting as disclinations. and conclude to the equality between the total angular Polytopes are very well described in the books Article published online by EDP Sciences and available at http://dx.doi.org/10.1051/jphys:019840045060102500 1026 by Coxeter [4, 5]. They are characterized by three Schlafli indices I p, q, r } : I p, q } is characteristic of the polyhedral cell and r is the number of cells sharing a common edge. In the following paper the word polytope is used with two different meanings : some times it is an ensemble of cells with straight edges, and some times it is a spherical space containing vertices connected by geodesic edges; in this case we call it a spherical polytope. 3. Orthoscheme tetrahedron and symmetry sphere. The orthoscheme tetrahedron is a very efficient way to describe a The faces of this tetrahedron polytope. Fig. 2. - The orthoscheme tetrahedron for the {4, 3, 4 } are mirrors. A given vertex is reproduced in these tessellation (the euclidean cubic tessellation). Point Po mirrors like in a kaleidoscope. All these images are correspond to a vertex, P1 is the centre of edge, P2 is the the vertices of the polytope. The faces of the ortho- centre of face, P3 the centre of cell. scheme tetrahedron are rectangular triangles defined in the spherical space. An Euclidean tetrahedron can also be defined by analogy. A M6bius spherical triangle (which is equivalent to the orthoscheme tetrahedron in 2-D spherical space) is presented in figure 1. All identical triangles obtained by reflection in their sides form a pattern covering the sphere just once. If a vertex is placed on one of the vertices of the M6bius triangle, its images form a polyhedron. Fig. 3. - The unfolded orthoscheme. The value of all angles as functions of p, q, r are given in the appendix. 2-sphere. These geodesic lines are the reflection lines of the polyhedron. In 3-D spherical space a face of the spherical orthoscheme is part of a 2-sphere which is Fig. 1. - A Mobius spherical triangle VCE, and other called « symmetry sphere of the polytope ». identical triangles obtained by reflection in their sides to form a pattern covering the sphere just once. The angles of this triangle are V = nlp, C = nlq, E = n/2 if the poly- 4. Disclinations and symmetry spheres. hedron is { p, q }. Consider a spherical 3-D space with an inscribed spherical polytope. This polytope is characterized by The orthoscheme tetrahedron of {4, 3, 4 } (the its symmetry spheres. If the curvature of the space is Euclidean cubic tessellation) is represented on figure 2. then concentrated on the edges of the polytope, a The relation between a polytope cell and the ortho- symmetry sphere is distorted and becomes a poly- scheme tetrahedron is clearly defined on this figure. hedron (but we always call it « symmetry sphere »). In order to define angles of the orthoscheme tetra- The basic idea of this paper is to study the effect of hedron faces, it is drawn unfolded on figure 3. The disclinations going through a symmetry sphere in vertex of the polytope { p, q, r } is called Po, the centre order to understand the combination law of these of cell is P3, the centre of face is P2 and the centre of disclinations. We take advantage of the existence of an edge is Pl. In the appendix we give the different ele- exact solution for a symmetry sphere (which is a 2-D ments of the Mobius triangle and of the orthoscheme space) and we extend results to the spherical 3-D tetrahedron following the Coxeter notation. space characterized by the symmetry spheres. Looking at figure 1 we observe that sides of the Disclination lines are edges of the polytope, conse- Mobius triangle are parts of great circles of the quently they go through a symmetry sphere with 1027 respect to the mirror symmetry. A disclination is perpendicular to this surface, if there is no vertex of the polytope corresponding to its intersection with the symmetry sphere. If there is a vertex on the symmetry sphere it is a node for disclinations. Some of these disclination segments are in the surface, others are disposed symmetrically with respect to the mirror. In this case they can be considered as a continuous discli- nation line with a cusp where it goes through the symmetry sphere. - 4.1 EXAMPLE OF THE { 5, 3, 3} POLYTOPE. - The Fig. 5. Positive disclinations relative to a pentagonal face of the of the 5, 3, { 5, 3, 3 } is a packing of dodecahedra in which an symmetry sphere { 3 } polytope. edge joins three cells. The angular deficit, leading to disclinations, is due to the deviation from 2 7r/3 of the there are five disclinations a dodecahedron dihedral angle. More formally this going through pentagonal face with a bend at the crossing point point). deficit is related to the deviation of the 7r/r (r = 3) (Po We can estimate the contribution to the curvature angle in the spherical orthoscheme, from the same of the different disclinations crossing the symmetry angle in the Euclidean orthoscheme arcos sphere. (Keeping in mind that the total curvature is (2013 2013 4 n). Disclinations to the surface = x perpendicular symmetry cos nlq The° disclination angle is 6 2 n - 6 sIn n p )) contribute with their exact angular deficit 6 to the 1.01722 or 6 - 0.1798 rd curvature.
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