Geometric models for continuous transitions from quasicrystals to crystals (original) (raw)

Starting from variable p-veetors half-stars whíeh verify Hadwiger's theorem, the cut-projeetion method is used here. The strip ofprojeetion is projeeted on a rotatory subspaee and a variable tiling is obtained. Two out standing examples are developed. The first, a eontinuous evolution from a two-dímensional octagonal quasilattiee to two square lattiees 45° rotated in between. The seeond is a eontinuous evolution from a three-dimensional Penrose tiling to an f.e.e. vertex lattiee. Physieal applieations to quasierystal-<:rystal transitions are poínted out. After quasicrystalline phases were discovered (Shechtman, Blech, Gratias and Cahn 1984), some theoretical (El ser and Henley 1985, Kramer 1987) and experimental works (Guyot and Audier 1985, Urban, Moser and Kronmüller 1985, Audier and Guyot 1986a, b, Guyot, Audier and Lequette 1986) began to pay attention to the close and systematic relationship between quasicrystals and crystals. RecentIy, many works have pointed in the same direction (Poon, Dmowski, Egami, Shen and Shiflet 1987, Zhou, Li, Ye and Kuo 1987, Yamamoto and Hiraga 1988, Zhang, Wang and Kuo 1988, Sadananda, Singh and Imam 1988, Yu-Zhang, Bigot, Chevalier, Gratias, Martin and Portier 1988, Fitz Gerald, Withers, Stewart and Calka 1988, Yang 1988, Henley 1988, Chandra and Suryanarayana 1988, Cahn, Gratias and Mozer 1988). Some authors even state that the transition from quasicrystalline to crystalline phases is continuous over a range ofintermediate phases (Reyes-Gasga, Avalos-Borja and José-Yacamán 1988, Zhou, Ye, Li and Kuo 1988). We present here a geometric model to describe simple and plausible continuous evolutions from quasilattices to lattices. Our method is a version ofthe well known cutprojection method (Kramer and Neri 1984, Duneau and Katz 1985, EIser 1986). In the above mentioned work, EIser and Henley (1985) modified the cut-projection method to allow study of the connection between crystal and quasicrystal structures. These authors tilted the strip of projection with respect to the hypercubic lattice (defined in the hyperspace EP) but they fixed the projection hyperplane (or projection subspace P, p> n). So, different hypercubic roofs were projected in such a way that the quasicrystal structure was the limit of a discontinuous sequence of periodic structures. In this work, we develop the contrary strategy and we describe a lattice as an atrophical quasilattice. We fix the particular strip (in the p-dimensional hypercubic lattice of EP) which generates the standard quasiperiodic tiling but we rotate the projection hyperplane (or 0950--0839/