Crystallography of quasicrystals ; application to icosahedral symmetry

Abstract

2014 Les concepts de la cristallographie sont étendus aux structures quasicristallines et appliqués aux quasicristaux icosaédriques. On montre que les symétries de rotation d'ordre N bidimensionnelles sont compatibles avec les réseaux de Bravais en dimension ~ (N ) (au moins), où ~ (N) est le nombre d'Euler, alors que pour la symétrie de l'icosaèdre tridimensionnelle, la dimension minimale est 6. La cristallographie de l'icosaèdre est traitée en detail. Une classification complète des structures périodiques en six dimensions avec symétrie icosaédrique est dérivée. 11 est surprenant de voir qu'il n'y a que quelques types d'« objets cristallographiques » en 6 dimensions avec la symétrie de l'icosaèdre, en fait trois structures type réseaux de Bravais, deux groupes ponctuels et onze groupes d'espace inequivalents. Le probleme de l'équivalence des groupes d'espace icosaédriques est étudié en détail. Comme dans le cas des cristaux ordinaires à trois dimensions, les symétries de groupes d'espace non « symmorphes » conduisent à l'extinction des pics de Bragg. Ces extinctions sont calculées systématiquement. Abstract. 2014 Crystallographic concepts are extended to quasicrystalline structures and applied to icosahedral quasicrystals. 2-dimensional N fold rotational symmetries are shown to be compatible with Bravais lattices in (at least) ~ (N) dimensions, where ~ (N) is the Euler number, while for 3-dimensional icosahedral symmetry the minimal dimension is 6. The case of icosahedral crystallography is worked out in detail. A complete classification of six-dimensional periodic structures with icosahedral symmetry is derived. There are surprisingly few types of 6-dimensional « crystallographic objects » with icosahedral symmetry, namely 3 Bravais lattice types, 2 point groups, and 11 inequivalent space groups. The problem of equivalence of icosahedral space groups is studied in detail. Similar to the case of ordinary 3-dimensional crystals, nonsymmorphic space group symmetries lead to extinction of Bragg peaks. These extinctions are calculated systematically.

Affine extensions of non-crystallographic Coxeter groups induced by projection

“…They include the groups H 2 , H 3 , and the largest non-crystallographic group H 4 ; the icosahedral group H 3 and its rotational subgroup I are of particular practical importance as H 3 is the largest discrete symmetry group of physical space. Thus, many 3-dimensional systems with high symmetry, such as viruses in biology, [5][6][7][8][9] fullerenes in chemistry, [10][11][12][13] and quasicrystals in physics, [14][15][16][17] can be modeled using Coxeter groups.…”

The Most Simple Icosahedral Quasi-Crystal with Continuous Phasons

“…So in general the quasi-crystal consists of 60 different interpenetrating lattices [5,6]. This number reduces, however, if JA is left invariant as a whole by a subgroup of Y up to a shift vector out of the lattice G…”

“…But"Kalugin and Levitov reported a counterexample [5], . and soon afterwards Levitov constructed an infinite family of icosahedral quasi-crystals with continuous phason modes [6]. His model consists of 60 perfect periodic lattices related through the symmetry operations of the icosahedral group.…”

“…a :.simp1e cubic lattice G 6 in six-dimensional space R 6 , endow it with a basis of threedimensional atomic surfaces and cut it with a three-dimensional plane to arrive at the quasiperiodic three-dimensional structure [7]. The six-dimensional space group is supposed to be icosahedral and symmorphic (for a full classification of the icosahedral space groups in R 6 see [8] All these points together form a three-dimensional Bravais lattice.…”

“…Thus the requirement that the number of lattices is minimal and that the minimal interatomic distance is maximal singles out one model of the countable infinite family constructed by Levitov [6].…”