International Union of Crystallography
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
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.
The two schools run one after the other, with a pre-school optional day where the basic concepts necessary to attend the schools have been presented. Participants to the pre-school day were required doing some concrete exercises allowing them to get familiar with the bases that are assumed understood during the school. The weekend between the two schools was devoted to presenting additional concepts that are pre-requisite to attend the second school.
21 June: Introduction to crystal symmetry; space groups, Hermann-Mauguin symbols, exercises on the International Tables for Crystallography
Topological Crystal Chemistry: Theory and Practice
The first school will run on four days, from 22 to 25 June
- Periodic Structures and Crystal Chemistry... aka the Topological Approach to Crystal Chemistry
- Graph, Nets & Tilings (Quotient Graphs & Natural Tilings)
- Topological Analysis of Entanglement : interpenetration, polycatenation & more
- Computer crystallochemical analysis: an overview
- Applied computer crystallochemical analysis
PRACTICE WITH PROGRAMS TOPOS, Systre, 3dt
- Module 1. Standard topological analysis and classification of nets in MOFs (Metal-Organic Frameworks), organic and inorganic crystals
- Creating a database from CIF, SHELX or Systre formats
- Computing adjacency matrix (complete set of interatomic bonds) for chemical compounds with different chemical bonding (valence, H bonding, specific interactions, intermetallic compounds)
- Visualizing 0D, 1D, 2D and 3D structures
- Standard simplified representations of MOFs or hydrogen-bonded organic crystals
- Computing topological indices (coordination sequences, point, Schläfli and vertex symbols)
- Topological identification of nets. Working with TTD collection and Systre
- Taxonomy of nets. Working with TTO collection
- Module 2. Special topological methods of searching for building units in crystal structures
- Special methods of simplification. Edge nets and ring nets. Analysis of synthons
- Standard cluster representation of MOFs
- Nanocluster representation of intermetallic compounds
- Module 3. Analysis of entanglements in MOFs and molecular crystals
- Visualization, topological analysis and classification of interpenetrating MOFs
- Detection and description of other types of entanglement in MOFs: polycatenation, self-catenation and polythreading
- Module 4. Analysis of microporous materials and fast-ion conductors with natural tilings
- Computing natural tilings and their parameters. Visualizing tiles and tilings (TOPOS & 3dt). Simple and isohedral tilings. Constructing dual nets
- Analysis of zeolites and other microporous materials, constructing migration paths in fast-ion conductors
- Module 5. Crystal design and topological relations between crystal structures
- Group-subgroup relations in periodic nets. Subnets and supernets
- Maximum-symmetry embedding of the periodic net, working with the Systre program
- Mappings between space-group symmetry and topology of the periodic net
- Searching for topological relations between nets and working with net relation graph
- Applications of net relations to crystal design, reconstructive phase transitions, taxonomy of crystal structures
Participants are invited to bring their own data/structures to be analyzed as well as personal computers to install the software.
26-27 June: preparation to the second school
- Basic facts on crystallographic groups
- Point groups. Elements of point symmetry. Groups, subgroups and theorem of Lagrange. Generators. Classes of conjugation. Abelian groups and cyclic groups. Crystallographic point groups and abstract groups. Generation of point groups by composition series. Classification of crystallographic point groups.
- Crystallographic symmetry operations and their presentation by matrices. Space groups. Translation groups and coset decompositions of space groups. Symmorphic and non-symmorphic space groups. Generation of space groups by composition series.
- Group-subgroup relations of point and space groups.
Irreducible representations of space groups
The second school will run on five days, from 28 June to 2 July
- Representations of crystallographic groups (3 days)
- General remarks on representations. Representations of discrete groups. Equivalence of representations. Unitary representations. Invariant subspaces and reducibility. Theorem of orthogonality. Characters of representations and character tables.
- Representations of point groups. Representations of Abelian groups: cyclic groups and direct products of cyclic groups. Character tables of representations of point groups. Online databases for point-group representations.
- Induction procedure for the derivation of the representations of crystallographic groups. Subduced and induced representations. Conjugate representations and orbits. Little groups, allowed representations and induction theorem. Induction procedure for indices 2 and 3. Representations of some point groups by the induction procedure.
- Representations of space groups Representation of the translation group. Star of a representation. Little groups and small representations. Representations of symmorphic and non-symmorphic groups. Online tools for the derivation of space-group representations.
- Applications of representations theory in solid-state physics and chemistry (2 days)
- Vibrations in molecules and solids
- Molecular dynamics. Small oscillations and normal modes. Zero modes and vibrational modes. Mechanical and vibrational representations. Dynamical matrix in symmetry adapted coordinates. Degeneracy.
- Electronic energy bands and phonon spectra. Assignment of small representations. Compatibility relations. Symmetry-adapted bases. Partial diagonalization of the dynamical matrix. Anticrossing.
- Direct products of irreducible representations and selection rules - general formulation. Selection rules in molecular spectroscopy: rotational and vibrational absorption, infrared and Raman effect. Direct products of space-group representations and selection rules. Online tools for infrared and Raman selection rules.
- Structural phase transitions
- Representation theory tools in the analysis of phase transitions. Primary and secondary order parameters; couplings and faintness index. Order parameter direction and isotropy subgroups. Group-theoretical formulation of the necessary conditions for second-order phase transitions.
- Symmetry-mode analysis of structural phase transitions. Hierarchy of modes. Symmetry-modes applications in structure refinement. Online tools for symmetry-mode analysis.
- Vibrations in molecules and solids
- 9.00-10.30 - morning session I
- 10:30-11:00 - coffee break
- 11.00-12.30 - morning session II
- 12:30-14:00 - lunch
- 14.00-16.00 - afternoon session I
- 16:00-16:30 - coffee break
- 16.30-19.00 - afternoon session II
The official language of the schools was English. No simultaneous interpretation was provided.
- Prof. Vladislav Blatov, Samara State University (Russia)
- Prof. Davide Proserpio, Department CSSI - University of Milan (Italy)
- Prof. Mois Aroyo, Universidad del Pays Vasco (Spain)
- Prof. Juan Manuel Perez-Mato, Universidad del Pays Vasco (Spain)
- Prof. Boriana Mihailova, University of Hamburg (Germany)
- Dr. Bernd Souvignier, Radboud University Nijmegen (The Netherlands)
- Prof. Massimo Nespolo, Nancy-Université (France)
Local organizing committee
- Prof. Massimo Nespolo, CRM2, Institut Jean Barriol, Nancy-Université
- Ms Anne Clausse, CRM2, Institut Jean Barriol, Nancy-Université
See also the list of didactic material for the MaThCryst schools.
- Program and abstracts of the poster presentations
- Graphs, nets and tilings (D. Proserpio)
- Entanglements-I&II (D. Proserpio)
- Periodic structures and crystal chemistry (D. Proserpio)
- Computer crystallochemical analysis: an overview (V. Blatov)
- TOPOS Manual (V. Blatov, D. Proserpio)
- Representation of crystallographic groups - syllabus (B. Souvignier)
- Representation of crystallographic groups - presentation (B. Souvignier)
- Representation of crystallographic groups - text (M. Aroyo)
- Vibrations in molecules and solides (B. Mihailova)
- Symmetry considerations in structural phase transitions (J.-M. Perez-Mato)
- Exercises on structural phase transitions (J.-M. Perez-Mato)
- Symmetry considerations in structural phase transitions - a brief guide on internet tools (J.-M. Perez-Mato)
- Tutorial on the Bilbao Crystallographic Server in the study of group-subgroup phase transitions (J.-M. Perez-Mato)
- Exercises on internet tools for structural phase transitions (J.-M. Perez-Mato)
- The program AMPLIMODESof the Bilbao Crystallographic Server (J.-M. Perez-Mato)
- Tutorial on the program AMPLIMODES of the Bilbao Crystallographic Server (J.-M. Perez-Mato)
The Schools were held at the Amphitheatre No. 8 of the Faculty of Sciences and Technologies of the Université Henri Poincaré Nancy I
(GPS coordinates: Latitude 48.6653088, Longitude 6.1589755).
The Faculty campus is located at Vandoeuvre-les-Nancy, in the immediate suburb of Nancy, and can
be reached from Nancy railway station in about 15-20 minutes.
Google Map of the campus.
List of participants
|Name||Country||Topological school||Irreps school|
|8||Frantisek Laufek||Czech Republic||X|
|13||Narjes Beigom Mortazavi||France||X|
|35||Goutam Kumar Kole||Singapore||X||X|
|36||Maria Celeste Bernini||Spain||X|
|37||Ainhoa Calderon Casado||Spain||X|
|39||Roberto Fernandez de Luis||Spain||X|
|40||Manuela Eloïsa Medina Munoz||Spain||X|
|43||Jimmy Retrepo Guisao||Spain||X|
|44||Edurne Serrano Larrea||Spain||X|
|51||Amartya Sankar Banerjee||USA||X|
|52||Maw Lin Foo||USA||X|
|55||Lusann Wreng Yang||USA||X||X|
|56||Elliott S. Ryan||USA||X|
Participants to the school on Topological Crystal Chemistry
Inquiries should be sent to .
The Organizers of the Nancy 2010 MaThCryst schools have observed the basic policy of non-discrimination and affirms the right and freedom of scientists to associate in international scientific activity without regard to such factors as citizenship, religion, creed, political stance, ethnic origin, race, colour, language, age or sex, in accordance with the Statutes of the International Council for Science. At these schools no barriers existed which would have prevented the participation of bona fide scientists.