By Alan L. Mackay (auth.), J.-C. Tolédano (eds.)

Distinct clinical groups are typically interested by the 3 fields of quasi-crystals, of liquid crystals, and of platforms having modulated crystalline constructions. besides the fact that, in recent times, there was a growing to be feeling variety of universal difficulties have been encountered within the 3 fields. those include the necessity to recur to "exotic" areas for describing the kind of order of the atomic or molecular configurations of those structures (Euclidian "superspaces" of dimensions more than three, or four-dimensional curved spaces); the popularity that one has to accommodate geometrically pissed off platforms, and in addition the occurence of particular excitations (static or dynamic) as a result of the continual degeneracies of the good constructions thought of. within the view of discussing those difficulties, aNA TO-Advance learn Workshop has assembled in Preveza (Greece), in september 1989,50 specialists of the 3 thought of fields (with an equivalent share of theorists and experimentalists). 35 hours of meetings and discussions have ended in a extra precise review of the similarities and of the diversities within the techniques carried out within the experiences of the 3 different types of platforms. The papers contained during this NATO-series booklet give you the substance of this workshop. The reader will locate 3 different types of papers. a few very brief papers giving the most principles said on a subject matter. Papers comprising 8-10 pages which stick heavily to the contents of the talks offered. Longer papers offering extra commonly the history and effects relative to a given subject. it truly is worthy summarizing the critical outputs of the workshop.

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Lett. 61, 2774 (1988). Y. L. V. Chester, Phys. Rev. B 39, 446 (1989). P. Bak, Scripta Met. 20, 1199 (1986). B. M. Lang-;-M. Tanaka, P. Sainfort and M. Audier, Nature 324, 48 (1986). ~Jaric, Anniversary Adriatico Research Conference on Quasicrystals, Trieste, July 4-7, 1989 (to be published by Yorld Scientific, Singapore). M. de Boissieu, M. Audier, Ch. M. Dubois, P. Guyot and B. Dubost, Ibidem. P. Tsai, A. Inoue and T. Masumoto, Japan J. Appl. Phys. 26, L1505 (1987) S. Ebalard and F. Spaepen, J.

Complicating factor is that the 6D-density function describing a single atom is not known, contrary to the case for atoms in 3D, physical space. All vertices of the 3D Penrose tiling can be obtained as a 3D section of a hypercubic lattice in 6D space, when each 6D lattice point is decorated with a object which has the shape of a triacontahedron (TR) in perpendicular space and is infinitely thin in physical space. 10- 12 It immediataly follows that a decoration of the 3D Penrose tiling with an atom on each vertex is represented in 6D space by a 60-atom, which is the convolution of an ordinary density in physical space and the triacontahedron in perpendicular space.

The two coordinate axis are two-fold symmetry axis. The peaks 0, A, Band G are obtained for atoms on the vertices of a Penrose tiling. The peaks C, E and F can be explained by assuming both the vertices and middle-edge positions to be decorated by atoms. 45 ( X,X,X,D, D, D) Fig. 3. Section of the 60 Patterson function for icosahedral Al6CuLi3, obtained by Fourier transformation of the measured X-ray diffraction intensities. Contour lines are drawn at intervals of 10% of the maximum value. This section incorporates the origin.