... one point x' K such that|| xx'||= e. The open ball Bex> does not i... more ... one point x' K such that|| xx'||= e. The open ball Bex> does not intersect S/t (e) so that for r small enough 5/c (e) D Br, XC Br, i\ Bex'. ... References [BM] M. Baake and RV Moody, Self-similar measures for quasicrystals, Directions in Math-ematical Quasicrystals, CRM Monogr. ...
Quasiperiodic packings of ®bres with icosahedral symmetry
Different icosahedral packings of ®bres have been experimentally realized. A packing construction... more Different icosahedral packings of ®bres have been experimentally realized. A packing construction with straight ®bres of the same circular cross section, only parallel to ®vefold icosahedral axes and respecting the closest packing condition, is reported. Its characteristics of point-group symmetry and related two-dimensional tilings are analysed. But for determining unambiguously all the ®bre positions it appears that a mathematical construction has to be made from the cut and projection of a ®ve-dimensional space. Through such a method, the volume fraction of ®brous reinforcement in a composite material can be calculated. The related two-dimensional tiling can be proved to be different from a Penrose tiling. Finally, the characteristics of other icosahedral packings where ®bres are parallel to threefold axes or to both threefold and ®vefold axes are brie ̄y discussed and a few further experiments on their elasticity properties and photonic band-gap structure are suggested.
We determine the ground state of the two-dimensional, fully polarized electron gas within the Har... more We determine the ground state of the two-dimensional, fully polarized electron gas within the Hartree-Fock approximation without imposing any particular symmetries on the solutions. At low electronic densities, the Wigner crystal solution is stable, but for higher densities ($r_s$ less than $\sim 3$) we obtain a ground state of different symmetry: the charge density forms a triangular lattice with about
This lecture deals with the theory of quasiperiodic tilings, and more generally with quasiperiodi... more This lecture deals with the theory of quasiperiodic tilings, and more generally with quasiperiodic patterns. We present the ideas introduced in [9] and developed in [10].
L'accès aux archives de la revue « Annales de l'I. H. P., section A » implique l'accord avec les ... more L'accès aux archives de la revue « Annales de l'I. H. P., section A » implique l'accord avec les conditions générales d'utilisation (). Toute utilisation commerciale ou impression systématique est constitutive d'une infraction pénale. Toute copie ou impression de ce fichier doit contenir la présente mention de copyright. Article numérisé dans le cadre du programme Numérisation de documents anciens mathématiques 249
L'accès aux archives de la revue « Annales de l'I. H. P., section A » implique l'accord avec les ... more L'accès aux archives de la revue « Annales de l'I. H. P., section A » implique l'accord avec les conditions générales d'utilisation (). Toute utilisation commerciale ou impression systématique est constitutive d'une infraction pénale. Toute copie ou impression de ce fichier doit contenir la présente mention de copyright. Article numérisé dans le cadre du programme Numérisation de documents anciens mathématiques 269
Structural characteristics of pentagonal Al—Fe—Cu phases
Philosophical Magazine A, 1998
ABSTRACT The vector basis and super spacegroup of the pentagonal Al-Fe-Cu phases, observed as a t... more ABSTRACT The vector basis and super spacegroup of the pentagonal Al-Fe-Cu phases, observed as a transient state of reversible icosahedral-to-rhombohedral transformation, are precised in the six-dimensional representation. The proposed theoretical description of the pentagonal structures is shown to be in a good agreement with experimental observations. As a result of this study, a scheme for the next step of transformation, pentagonal-to-rhombohedral, can be suggested.
The structural analysis of various approximant phases of icosahedral quasicrystals shows local en... more The structural analysis of various approximant phases of icosahedral quasicrystals shows local environments with icosahedral symmetry: icosahedra, Mackay clusters (M) and Bergman clusters (B). For the icosahedral phases i-AlCuFe and i-AlPdMn, these clusters have been proposed as complementary building blocks centered on particular nodes. However, computations showed that these genuine 2-shells or 3-shells clusters don't cover all atomic positions given by 6D models. One the other hand, the recent concept of a unique covering cluster was shown to apply to 2D Penrose tilings and Amman-Beenker tilings. In this paper we examine the local environments in i-AlCuFe and i-AlPdMn models about Wyckoff positions of the 6D lattice. We consider extended Bergman clusters of 6 shells that appear naturally in the Katz-Gratias model. We discuss the cell decomposition of the atomic surfaces and the variable occupation number of some of the shells. We show that a fixed extended Bergman cluster of 6 shells and 106 atoms covers about 98% of atomic positions. We also prove that a variable extended Bergman cluster of 6 shells, which contains the previous fixed cluster, covers all atomic positions of the theoretical model.
We consider quasiperiodic structures obtained by the cut and project method and show that wheneve... more We consider quasiperiodic structures obtained by the cut and project method and show that whenever the window satisfies a non-generic tiling property the associated structure can be transformed into a periodic lattice by bounded displacements. The case of the octagonal tiling is examined in detail.
Structural defects, observed by TEM, in Al-Pd-Mn approximant phases are identified as phason defe... more Structural defects, observed by TEM, in Al-Pd-Mn approximant phases are identified as phason defects. Assuming their motion under an applied shear strain, it is shown that two approximant phases can be related between them by a phase transformation occurring through plastic deformations. Then, other approximant phases, experimentally observed, can be interpreted as intermediate states of such a phase transformation. Atomic motions corresponding to these phason shifts are partly examined on the basis of HREM observations and crystallographic data, obtained from a single-crystal X-ray study on one of these approximants. The description of this transformation in six-dimensional hyperspace is proposed in order to establish a relationship with the icosahedral quasicrystalline state. The possibility that a plastic deformation of the icosahedral phase, occurring without the strain-hardening effect, could only result in a motion of phason defects is briefly discussed.
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