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
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Motivation
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 grouptheoretical methods in specific problems of solidstate 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 solidstate problems as molecular dynamics, spectroscopy, electronic bands, phonon spectra, Landau theory of phase transitions.
The applications of grouptheoretical 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 spectraltransition 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 symmetrymode analysis of structural phase transitions results in the decomposition of the symmetrybreaking 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 grouptheoretical procedures in practice for the analysis of phasetransition 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 online tool for the study of crystallographic symmetry and its applications. It gives access to databases with symmetry information on crystallographic groups, their groupsubgroup 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 solidstate 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.
Program
The two schools run one after the other, with a preschool optional day where the basic concepts necessary to attend the schools have been presented. Participants to the preschool 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 prerequisite to attend the second school.
Preschool day
21 June: Introduction to crystal symmetry; space groups, HermannMauguin 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
 THEORY
 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 (MetalOrganic 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 hydrogenbonded 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, selfcatenation and polythreading
 Module 4. Analysis of microporous materials and fastion 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 fastion conductors
 Module 5. Crystal design and topological relations between crystal structures
 Groupsubgroup relations in periodic nets. Subnets and supernets
 Maximumsymmetry embedding of the periodic net, working with the Systre program
 Mappings between spacegroup 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.
Weekend intermission
2627 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 nonsymmorphic space groups. Generation of space groups by composition series.
 Groupsubgroup 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 pointgroup 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 nonsymmorphic groups. Online tools for the derivation of spacegroup representations.
 Applications of representations theory in solidstate 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. Symmetryadapted 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 spacegroup 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. Grouptheoretical formulation of the necessary conditions for secondorder phase transitions.
 Symmetrymode analysis of structural phase transitions. Hierarchy of modes. Symmetrymodes applications in structure refinement. Online tools for symmetrymode analysis.
 Vibrations in molecules and solids
Schedule
 9.0010.30  morning session I
 10:3011:00  coffee break
 11.0012.30  morning session II
 12:3014:00  lunch
 14.0016.00  afternoon session I
 16:0016:30  coffee break
 16.3019.00  afternoon session II
Language
The official language of the schools was English. No simultaneous interpretation was provided.
Lecturers
 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 PerezMato, Universidad del Pays Vasco (Spain)
 Prof. Boriana Mihailova, University of Hamburg (Germany)
 Dr. Bernd Souvignier, Radboud University Nijmegen (The Netherlands)
 Prof. Massimo Nespolo, NancyUniversité (France)
Local organizing committee
 Prof. Massimo Nespolo, CRM2, Institut Jean Barriol, NancyUniversité
 Ms Anne Clausse, CRM2, Institut Jean Barriol, NancyUniversité
Online documents
 Program and abstracts of the poster presentations
 Introductory notes (M. Nespolo)
 Stereographic projection (M. Nespolo)
 Screw axes and glide planes (M. Nespolo)
 Polar and reciprocal lattices (M. Nespolo)
 Crystallographic orbits (M. Nespolo)
 Graphs, nets and tilings (D. Proserpio)
 EntanglementsI&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. PerezMato)
 Exercises on structural phase transitions (J.M. PerezMato)
 Symmetry considerations in structural phase transitions  a brief guide on internet tools (J.M. PerezMato)
 Tutorial on the Bilbao Crystallographic Server in the study of groupsubgroup phase transitions (J.M. PerezMato)
 Exercises on internet tools for structural phase transitions (J.M. PerezMato)
 The program AMPLIMODESof the Bilbao Crystallographic Server (J.M. PerezMato)
 Tutorial on the program AMPLIMODES of the Bilbao Crystallographic Server (J.M. PerezMato)
Venue
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 VandoeuvrelesNancy, in the immediate suburb of Nancy, and can
be reached from Nancy railway station in about 1520 minutes.
Google Map of the campus.
List of participants
Name  Country  Topological school  Irreps school  

1  Erik Arroyabe  Austria  X  
2  Volker Kahlenberg  Austria  X  
3  Liliana Dobrzanska  Belgium  X  
4  Jian Lu  China  X  X  
5  Yang Tao  China  X  X  
6  GuoPing Yang  China  X  X  
7  Neven Krajina  Croatia  X  
8  Frantisek Laufek  Czech Republic  X  
9  Abdellatif Bensegueni  France  X  
10  Mariya Brezgunova  France  X  X  
11  Slawomir Domagala  France  X  X  
12  Charlotte Martineau  France  X  
13  Narjes Beigom Mortazavi  France  X  
14  Agnieszka Paul  France  X  X  
15  Isabella Pignatelli  France  X  X  
16  Pascalita Prosper  France  X  X  
17  Romain Sibille  France  X  X  
18  Michael Bodensteiner  Germany  X  X  
19  Tatiana Gorelik  Germany  X  
20  Daniel Lassig  Germany  X  
21  Jörg Lincke  Germany  X  
22  Axel Pelka  Germany  X  X  
23  Guntram Schmidt  Germany  X  X  
24  Barbara Szafranowska  Germany  X  X  
25  Jagan Rajamony  India  X  X  
26  Mattia Allietta  Italy  X  
27  Giulio Giulio  Italy  X  
28  Pavlo Solokha  Italy  X  X  
29  Adrian Mermer  Poland  X  X  
30  Agnieszka Plutecka  Poland  X  
31  Magdalena Wilk  Poland  X  X  
32  Eugenia Peresypkina  Russia  X  X  
33  Alexander Virovets  Russia  X  X  
34  Anjana Chanthapally  Singapore  X  X  
35  Goutam Kumar Kole  Singapore  X  X  
36  Maria Celeste Bernini  Spain  X  
37  Ainhoa Calderon Casado  Spain  X  
38  Richard Dvries  Spain  X  
39  Roberto Fernandez de Luis  Spain  X  
40  Manuela Eloïsa Medina Munoz  Spain  X  
41  Josefina Perles  Spain  X  
42  Ana Platero  Spain  X  
43  Jimmy Retrepo Guisao  Spain  X  
44  Edurne Serrano Larrea  Spain  X  
45  Emre Tasci  Spain  X  
46  Julia Dshemuchadse  Switzerland  X  
47  Arkadiy Simonov  Switzerland  X  
48  Asli Ozturk  Turkey  X  X  
49  Thirumurugan Alagarsamy  UK  X  
50  Vladimir Bon  Ukraine  X  X  
51  Amartya Sankar Banerjee  USA  X  
52  Maw Lin Foo  USA  X  
53  Maciej Haranczyk  USA  X  
54  Vincent Jusuf  USA  X  
55  Lusann Wreng Yang  USA  X  X  
56  Elliott S. Ryan  USA  X  
49  31 
Participants to the school on Topological Crystal Chemistry
Contact
Inquiries should be sent to .
The Organizers of the Nancy 2010 MaThCryst schools have observed the basic policy of nondiscrimination 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.