Opening crystallography
- PDF / 3,016,708 Bytes
- 14 Pages / 595.276 x 790.866 pts Page_size
- 105 Downloads / 198 Views
REVIEW ARTICLE
Opening crystallography Marjorie Senechal 1 & Jean E. Taylor 2 Received: 15 July 2020 / Accepted: 22 July 2020 / Published online: 7 August 2020 # Springer Science+Business Media, LLC, part of Springer Nature 2020
Abstract This essay surveys results of the past 4 years, starting with a week-long workshop on “Soft Packings, Nested Clusters, and Condensed Matter” held in September, 2016, at the American Institute of Mathematics; our subgroup focused on geometric frustration in the self-assembly of complex crystals. We hope that our geometric study will shed light on actual materials that can be grown in laboratories, such as intermetallics, and patterns that emerge from computer simulations of interactions between “particles” of various sorts. We start with 13 spheres packed with icosahedral symmetry and by the end literally link clusters of the same or related symmetry to global structures that fill space. In the process, we begin the extension of the classical theory of parallelohedra (i.e., polyhedra that tile space face-to-face by translation) from its requirement of convexity and the necessity of central symmetry to more general polyhedra. Keywords Crystallography . Parallelohedra . Icosahedral clusters
Introduction This essay is rooted in a week-long workshop on “Soft Packings, Nested Clusters, and Condensed Matter” held in September, 2016, at the American Institute of Mathematics in San Jose, California. We are grateful to AIM, both for the workshop and for supporting our 3-year follow-up SQuaRE1 with our colleagues in that program, Pablo Damasceno, Yoav Kallus, Davide Proserpio, and Erin Teich. The title of the workshop attests to the revolution in mathematical crystallography sparked by Dan Shechtman’s discovery of quasicrystals in 1982. Long since accepted as real crystals, the unfortunate modifier “quasi” no longer questions them, and it signifies symmetries impossible for the formerly paradigmatic lattices. 1 SQuaRE stands for Structured Quartet Research Ensemble, a small focused research program. Our SQuaRE was, however, hexagonal.
* Marjorie Senechal [email protected] Jean E. Taylor [email protected] 1
Department of Mathematics, Smith College, Northampton, MA, USA
2
Department of Mathematics, University of California at Berkeley, Berkeley, CA, USA
In awarding Shechtman its 2011 prize in chemistry, the Nobel committee noted that his discovery had changed the way chemists think about solid matter, including the very definition of a crystal. “Clearly, the old definition of crystallinity was insufficient to cover this new class of ordered solids, and as a consequence, the definition of ‘crystal’ given by the International Union of Crystallography was changed,” wrote Sven Lidin, a member of the prize committee [1]. “While formal definitions may be more or less important to science, this one is interesting because it makes no attempt to define the concept of ‘crystal’ directly, but rather provides an operative definition based on the diffraction pattern of the material: By ‘Crysta
Data Loading...
