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Abstracts ICQ12.Indd 12th International Conference on Quasicrystals BOOK OF ABSTRACTS 1-6 September 2013 Cracow, Poland Committees Distinguished Guest Dan Shechtman International Advisory Board Michael Baake (Germany) Guido Kreiner (Germany) Marc de Boissieu (France) Ron Lifshitz (Israel) Janez Dolinšek (Slovenia) Ronan McGrath (UK) Chuang Dong (China) N.K. Mukhopadhyay (India) Jean-Marie Dubois (France) Walter Steurer (Switzerland) Michael Feuerbacher (Germany) Patricia Thiel (USA) Yasushi Ishii (Japan) Hans Reiner Trebin (Germany) Tsutomu Ishimasa (Japan) An Pang Tsai (Japan) Ken Kelton (USA) Michael Widom (USA) Marian Krajci (Slovakia) Janusz Wolny (Poland) Program Committee Michael Baake (Germany) Tsutomu Ishimasa (Japan) Marc de Boissieu (France) Ron Lifshitz (Israel) Janez Dolinšek (Slovenia) Walter Steurer (Switzerland) Tomonari Dotera (Japan) Patricia Thiel (USA) Jean-Marie Dubois (France) An Pang Tsai (Japan) Yasushi Ishii (Japan) Janusz Wolny (Poland) Organizing Committee Jacek MiBkisz (University of Warsaw) Marian Surowiec (University of Silesia Katowice) Janusz Wolny - chairman (AGH Kraków) Local Organizing Committee Włodzimierz Bogdanowicz Radosław Strzałka Maciej ChodyL Marian Surowiec Iwona Idryan Roman Wawszczak Paweł Kuczera Anna WnBk Lucjan Pytlik Janusz Wolny - chairman ISBN 978-83-934620-6-3 Honorary Doctorate Lecture D-01 The Discovery of Quasi-Periodic Crystals and its consequences D. Shechtman Technion, Haifa, Israel ISU, Ames, Iowa, USA [email protected] Crystallography has been one of the mature sciences. Over the years, the modern science of crystallography that started by experimenting with x-ray diffraction from crystals in 1912, has developed a major paradigm – that all crystals are ordered and periodic. Indeed, this was the basis for the definition of “crystal” in textbooks of crystallography and x-ray diffraction. Based upon a vast number of experimental data, constantly improving research tools, and deepening theoretical understanding of the structure of crystalline materials no revolution was anticipated in our understanding the atomic order of solids. However, such revolution did happen with the discovery of the Icosahedral phase, the first quasi-periodic crystal (QC) in 1982, and its announcement in 1984 [1,2]. QCs are ordered materials, but their atomic order is quasiperiodic rather than periodic, enabling formation of crystal symmetries, such as icosahedral symmetry, which cannot exist in periodic materials. The discovery created deep cracks in this paradigm, but the acceptance by the crystallographers' community of the new class of ordered crystals did not happen in one day. In fact it took almost a decade for QC order to be accepted by most crystallographers. The official stamp of approval came in a form of a new definition of “Crystal” by the International Union of Crystallographers. The paradigm that all crystals are periodic has thus been changed. It is clear now that although most crystals are ordered and periodic, a good number of them are ordered and quasi- periodic. While believers and nonbelievers were debating, a large volume of experimental and theoretical studies was published, a result of a relentless effort of many groups around the world. Quasi-periodic materials have developed into an exciting interdisciplinary science. This talk will outline the discovery of QCs and describe the important role of electron microscopy as an enabling discovery tool. 1. D. Shechtman, I. Blech, Met. Trans. 16A (1985), p. 1005-1012. 2. D. Shechtman, I. Blech, D. Gratias, J. W. Cahn, Phys. Rev. Lett. 53 (1984), p. 1951-1953. D-01 Tutorial lectures T-01 – T-06 Aperiodic order in equilibrium systems Aernout van Enter Johann Bernoulli Institute for Mathematics and Computer Science Faculty of Mathematics and Natural Sciences University of Groningen, Nijenborgh 9, 9747 AG Groningen, The Netherlands [email protected] We discuss the nature of aperiodically ordered systems, in terms of their dynamical spectrum and their diffraction spectrum, and compare those descriptions. We discuss how aperiodic order can arise in ground states and Gibbs states of lattice models. Moreover we discuss how aperiodically ordered systems can provide examples of types of order expected in disordered systems, such as (spin) glasses. T-01 Fusion: a general framework for hierarchical tilings Natalie Frank Department of Mathematics, Vassar College, Poughkeepsie, USA [email protected] One well studied way to construct quasicrystalline tilings is via inflate-and- subdivide (a.k.a. substitution) rules. These produce self-similar tilings, of which the Penrose, octagonal, and pinwheel tilings are famous examples. We present a different model for generating hierarchical tilings we call "fusion rules". Inflate-and-subdivide rules are a special case of fusion rules, but general fusion rules allow the composition rules to change from level to level. Spectral, dynamical, and topological results will be discussed that parallel those known for self-similar tilings. T-02 Recent Progress in Mathematical Diffraction Uwe Grimm1 and Michael Baake2 1. Department of Mathematics and Statistics, The Open University, Walton Hall, Milton Keynes MK7 6AA, U.K. 2. Fakultät für Mathematik, Universität Bielefeld, Postfach 100131, 33501 Bielefeld, Germany [email protected] Keywords: diffraction, order, model sets Diffraction methods continue to provide the main tool for the structure analysis of solids. The corresponding inverse problem of determining a structure from its diffraction pattern is difficult and, in general, does not define a structure uniquely. Kinematic diffraction, which is an approximation that is reasonable for X-ray diffraction where multiple scattering effects can be neglected, is well suited for a mathematical approach via measures. Measures provide a natural mathematical concept to quantify the distribution of matter in space as well as the distribution of scattering intensity. This approach has substantially been developed since the discovery of quasicrystals required an extension of the methods used to compute the diffraction of periodic crystals. The need for further insight emerged from the question of which distributions of matter, beyond perfectly periodic crystals, lead to pure point diffraction patterns, hence to diffraction patterns comprising sharp Bragg peaks only. More recently, it has become apparent that one also has to study continuous diffraction in more detail, with a careful analysis of the different types of diffuse scattering involved. In this tutorial review, we summarise key results, putting particular emphasis on the analysis of non-periodic structures. Following the presentation in our recent review articles [1,2], general results are introduced and discussed on the basis of various characteristic examples, with minimal use of formal arguments or proofs. A systematic, and more rigorous, development of the theory (including lots of references) can be found in [3]. 1. M. Baake and U. Grimm, “Mathematical diffraction of aperiodic structures”, Chem. Soc. Rev. 41 (2012) p. 6821 – 6843; arXiv:1205.3633. 2. M. Baake and U. Grimm, “Kinematic diffraction from a mathematical viewpoint”, Z. Kristallogr. 226 (2011) p. 711 – 725; arXiv:1105.0095. 3. M. Baake and U. Grimm, “Aperiodic Order: A Mathematical Invitation” (Cambridge University Press, Cambridge), in press. T-03 Symmetry, excitations, and defects in quasicrystals Ron Lifshitz Raymond and Beverly Sackler School of Physics & Astronomy, Tel Aviv University E-mail: [email protected] Keywords: long-range order, indistinguishability, symmetry, point groups, dislocations, phasons The discovery of quasicrystals has changed our view of some of the most basic notions related to the condensed state of matter, leading to a new underlying paradigm for the notion of a crystal as a solid with long-range positional order [1]. This paradigm shift brought about a new fundamental understanding of the basic characteristics of a crystal – its symmetry, its elementary excitations, and its typical defects [2]. Before we knew about quasicrystals, it was believed that crystals break the continuous translation and rotation symmetries of the liquid-phase into a discrete lattice of translations, and a finite point group of rotations. Quasicrystals, on the other hand, possess no such symmetries – there are no translations, nor, in general, are there any rotations, leaving them invariant. Does this imply that no symmetry is left, or that the meaning of symmetry should be revised? Without an underlying lattice, can we describe the elementary excitations? Can we even talk about dislocations without a lattice? If so, what are they like? How can we characterize them? In this tutorial I shall review these and other notions, related to the symmetry- breaking transition from a liquid to a crystal, using the notion of indistinguishability, which was introduced many years ago by Rokhsar, Wright, and Mermin [3,4]. This will allow me to provide some physical and geometric intuition, while avoiding the formal use of abstract high-dimensional spaces. The author acknowledges support from the Israel Science Foundation through Grant No. 556/10. 1. R. Lifshitz, What is a crystal? Z. Kristallogr. 222 (2007), p. 313. 2. R. Lifshitz, Symmetry breaking and order in the age of quasicrystals, Isr. J. Chem. 51 (2011), p. 1156. 3. D. S. Rokhsar, N. D. Mermin, D. C. Wright, Phys. Rev. B 35 (1987), p. 5487. 4. D. S. Rokhsar, D. C. Wright, N. D. Mermin, Acta Crystallogr. A 44 (1988), p. 197. T-04 Valence Electrons and Bonding
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