Event Name

Summer Schools on Mathematical and Theoretical Crystallography

Start Date 21st Jun 2010
End Date 2nd Jul 2010

Summer Schools on Mathematical Crystallography

Nancy, France, 21 June - 2 July 2010

On the occasion of the fifth anniversary of its foundation, the Commission on Mathematical and Theoretical Crystallography organised two summer schools devoted to the topology of crystal structures and to the irreducible representations of space groups

Topological Crystal Chemistry: Theory and Practice

The explosive growth in inorganic and organic materials chemistry has seen a great upsurge in the synthesis of crystalline materials with extended framework structures (zeolites, coordination polymers/coordination networks, Metal Organic Frameworks MOFs, supramolecular architectures formed by Hydrogen bonds and/or Halogen bonds etc.). There is a concomitant interest in simulating such materials and in designing new ones. However, it is a truism that before one can embark on systematic design of materials, one must know what the possibilities are. Indeed, in the last two decades there have been many parallel outcomes in the theoretical aspects of description and analysis of periodic structures (nets, tilings, surfaces, etc.), in the elaboration of databases, and in the development of software for analyzing and describing (illustrating) topological aspects of both real crystal structures and theoretical extended architectures. With these achievements, materials science and crystal chemistry comes up to a new level of their development that is characterized by deeper integration of mathematical methods, computer algorithms and programs into modeling and interpretation of periodic systems of chemical bonds in crystals.

The goal of this school is to give an introduction to this whole new area that we call Topological Crystal Chemistry. There will be large time dedicated to hands-on session on the use of the novel and still not widespread computer methods/software/databases so the student at the end of the course should be able to analyze any kind of extended structure through the eye of the topology and describe it in term of nets, entanglements, catenation etc.

The target audience is young scientists (graduate students and postdoctoral associates) actively engaged in materials research, (experimental and/or theoretical) but also crystallographers who want to look at familiar structure types with a different eye. Some basic knowledge in chemistry and crystallography will be assumed which will provided during the pre-school day, for those needing a basic introduction .

Irreducible representations of space groups

Group Theory is an indispensable mathematical tool in many branches of chemistry and physics. The school aims at giving the necessary background and practical skills for an efficient use of the group-theoretical methods in specific problems of solid-state physics, structural chemistry and material sciences. After a revision of the basic concepts of spatial symmetry and its description by crystallographic point and space groups according to International Tables of Crystallography, the principal results of the theory of group representations will be introduced with an emphasis on the practical aspects of the subject. Irreducible representations of crystallographic point and space groups and their derivation will be discussed in details. The abstract theory is limited to a reduced set of fundamental facts and statements. More attention is paid to different tools and techniques necessary for practical applications of the symmetry methods in solid-state problems as molecular dynamics, spectroscopy, electronic bands, phonon spectra, Landau theory of phase transitions.

The applications of group-theoretical methods to molecular vibrations including the concept of normal modes of vibrations will be discussed in details. The students will learn how, starting from symmetry requirements, to determine the spectral-transition selection rules with special attention to infrared and Raman spectra. The important role of representations of crystallographic groups in the classification, labeling and the analysis of the degeneracies of the lattice vibrations and electronic energy bands of crystalline solids will be reviewed. The applications related to phase transition studies will include the introduction of efficient techniques allowing the determination of the principle characteristics of a system undergoing a phase transition. For example, the determination of the order parameter from the knowledge of the initial and final phases, or the enumeration of all symmetry allowed phases that can result from a continuous phase transition. The symmetry-mode analysis of structural phase transitions results in the decomposition of the symmetry-breaking distortion, present in the distorted structure into contributions from different symmetry modes. The exposition of the general theory and methods will be illustrated with number of examples of typical phase transitions of different nature so that the participant can learn to apply the group-theoretical procedures in practice for the analysis of phase-transition mechanisms and in the search for new functional materials.

A tutorial and practical guide to the Bilbao Crystallographic server (www.cryst.ehu.es) forms an essential part of the course. The server provides an excellent on-line tool for the study of crystallographic symmetry and its applications. It gives access to databases with symmetry information on crystallographic groups, their group-subgroup relations and irreducible representations. The school aims at giving the necessary background and practical skills for an efficient use of the computer databases and programs on the Bilbao Crystallographic Server focused on solid-state physics and chemistry applications.

The participants of the school will benefit from the practical training in the application of advanced symmetry methods in solid state physics and crystal chemistry problems. The minimal mathematical prerequisites for the school widen the participation audience to students and researchers from chemistry, physics, geological sciences and engineering.
Location Nancy
URL http://www.crystallography.fr/mathcryst/nancy2010.php

Category IUCr sponsored meetings | Schools
Topics Mathematical and Theoretical Crystallography