Materials Science and Engineering, A 134 (1991 ) 893-895 893 Quasicrystals: local structure versus global structure Sven Lidin Max-Planck Institute fur Festk6rperforschung, Heisenbergstrasse 1, D- 7000 Stuttgart 80 (F.R.G.) Abstract The pros and cons of different quasicrystal structure descriptions are discussed. A novel model using tiles of different sizes is suggested. 1. Introduction We are told that truth is eternal. In science, supposedly the pursuit of truth, we find however that some truths are more eternal than others. In solid state science we have experienced the chal- lenging of some very deeply rooted truths lately, with varying results. Ten years ago most of the workers in the field agreed that superconduction was impossible at temperatures higher than 25 K, that fusion of hydrogen into helium only takes place when catalyzed by murons or at very high temperatures and pressures and that fivefold symmetry was forbidden in ordered crystals (cf. Fig. 1). While the claims about cold fusion seem to be refuted, although a conclusive verdict is still lacking, today superconductors with a Tc around 100 K are made at laboratories around the globe and icosahedral quasicrystals seem to defy the laws of geometry. How could yesterday's truths about supercon- duction and fivefold symmetry be refuted? What was wrong with the original reasoning? The answer is that nothing was wrong. Superconduc- tion is possible at temperatures far above 25 K, but not superconduction according to the classi- cal mechanism. The mechanism for high T c superconduction is still not elucidated, but it is certain that it differs from the classical mechan- isms. Analogously, quasicrystals do not violate the geometrical rule forbidding an ordered crystal to have fivefold symmetry. First of all quasi- crystals do not have fivefold symmetry and secondly, they are not even ordered crystals. The reason why we think the rules are broken is that we change the basic definitions to which the rules apply. The recognition of this semantic shift is essential for the understanding of the nature of quasicrystals. Quasicrystals differ pro- foundly from ordinary crystals, and should be treated accordingly. 2. What are quasicrystals? Quasicrystals are not ordered in the same sense that a classical crystal is ordered, but they do have one important property in common with these in that they are both Poisson combs. A Pois- son comb is a discrete distribution of Dirac delta- functions whose Fourier transform is also a discrete distribution of Dirac delta-functions. The geometrical criterion sufficient for a structure to be a Poisson comb is still an open problem of mathematics. There are two mainstreams in the description of quasicrystals, both of which aim at forcing (~). I p Q • t "''~) ¢, i i • i • t x 0• 'b i' Fig. 1. A simple proof that fivefold symmetry is incompat- ible with translation. Two fivefold centra at a distance, a, will always create new centra at a shorter distance, b, and so on ad infinitum, filling the plane densely with fivefold centra. 0921-5093/91/$3.50