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UNIVERSITY OF BUCHAREST
FACULTY OF PHYSICS Guest
2024-11-22 2:26 |
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Conference: Bucharest University Faculty of Physics 2001 Meeting
Section: Theoretical Physics and Applied Mathematics
Title:
A group-theoretic approach to quasicrystals
Authors:
Nicolae Cotfas
Affiliation:
Faculty of Physics, University of Bucharest, Bucharest, Romania
E-mail
Keywords:
Abstract:
We present some new contributions to the group-theoretic approach to quasicrystals initiated by Kramer & Neri , Katz & Duneau and Elser. A mathematical model of quasicrystal is an aperiodic Delone set Q with a diffraction spectrum containing a pure point component invariant under a finite group G.
In the theoretic-group approach, the model is obtained by projection starting from a decomposition of an Euclidean space into an orthogonal sum of two subspaces. The decomposition used in the Katz-Duneau-Elser model is obtained by starting from the vectors corresponding to the vertices of a regular icosahedron, that is, from an orbit of the icosahedral group Y. The arithmetic neighbours of each point belonging to this model are distributed on the sites of a regular icosahedron having as center the considered point.
The same construction done by starting from another orbit of Y, namely, from the vectors corresponding to the vertices of a regular dodecahedron leads to a model in which the neighbours of each point are distributed on the sites of a regular dodecahedron [ N. Cotfas, Z. Kristallogr. 213 (1998) 311]. More than that, the construction used in Katz-Duneau-Elser model works if we start from a union of two or more orbits [ N. Cotfas, J. Phys. A: Math . Gen. 32 (1999) 8079]. Starting from an icosahedron and a dodecahedron we get a model in which the arithmetic neighbours of each point are distributed on two shells, namely, the vertices of an icosahedron and a dodecahedron. The self-similarities and the rational approximants of these models can be obtained in the same way as in Katz-Duneau-Elser model.
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