Crystallography: Symmetry Groups and Group Representations B
EPJ Web of Conferences 22, 00006 (2012) DOI: 10.1051/epjconf/20122200006 C Owned by the authors, published by EDP Sciences, 2012 Crystallography: Symmetry groups and group representations B. Grenier1 and R. Ballou2 1SPSMS, UMR-E 9001, CEA-INAC / UJF-Grenoble, MDN, 38054 Grenoble, France 2Institut Néel, CNRS / UJF, 25 rue des Martyrs, BP. 166, 38042 Grenoble Cedex 9, France Abstract. This lecture is aimed at giving a sufficient background on crystallography, as a reminder to ease the reading of the forthcoming chapters. It more precisely recalls the crystallographic restrictions on the space isometries, enumerates the point groups and the crystal lattices consistent with these, examines the structure of the space group, which gathers all the spatial invariances of a crystal, and describes a few dual notions. It next attempts to familiarize us with the representation analysis of physical states and excitations of crystals. 1. INTRODUCTION Crystallography covers a wide spectrum of investigations: i- it aspires to get an insight into crystallization phenomena and develops methods of crystal growths, which generally pertains to the physics of non linear irreversible processes; ii- it geometrically describes the natural shapes and the internal structures of the crystals, which is carried out most conveniently by borrowing mathematical tools from group theory; iii- it investigates the crystallized matter at the atomic scale by means of diffraction techniques using X-rays, electrons or neutrons, which are interpreted in the dual context of the reciprocal space and transposition therein of the crystal symmetries; iv- it analyzes the imperfections of the crystals, often directly visualized in scanning electron, tunneling or force microscopies, which in some instances find a meaning by handling unfamiliar concepts from homotopy theory; v- it aims at providing means for discerning the influences of the crystal structure on the physical properties of the materials, which requires to make use of mathematical methods from representation theory.
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