International Union of Crystallography European Crystallographic Association Mineral and Inorganic Crystallography Special Interest Group Association Française de Cristallographie Crystallography Laboratory Nancy
International Union of Crystallography European Crystallographic Association Mineral and Inorganic Crystallography Special Interest Group Association Française de Cristallographie Crystallography Laboratory Nancy
Université Henri Poincaré Nancy 1 EMMA Graduate School SESAMES Graduate School Région Lorraine Communauté Urbaine du Grand Nancy
Université Henri Poincaré Nancy 1 Graduate School of Physics, Nancy-Metz Graduate School of Molecular Chemistry and Physics, Nancy-Metz Région Lorraine Communauté Urbaine du Grand Nancy
logo MaThCryst

International Union of Crystallography
Commission on Mathematical and Theoretical 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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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. Voronoi-Dirichlet polyhedron of a hydrogen atom in the crystal structure of β-quinol (HYQUIN05)

Clck for details of the image.

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. β-quinol HYQUIN05

Clck for details of the image.

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 The hetero-interpenetrating nets lcy+srs

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The hetero-interpenetrating nets lcy+srs Parallel interpenetrating sql sheets and parallel polycatenation of 3-fold thick layers

Clck for details of the images.

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 ( 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.

Pre-school day

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


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.

Weekend intermission

26-27 June: preparation to the second school

  1. Basic facts on crystallographic groups
    1. 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.
    2. 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.
    3. 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

  1. Representations of crystallographic groups (3 days)
    1. 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.
    2. 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.
    3. 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.
    4. 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.
  1. Applications of representations theory in solid-state physics and chemistry (2 days)
    1. Vibrations in molecules and solids
      1. Molecular dynamics. Small oscillations and normal modes. Zero modes and vibrational modes. Mechanical and vibrational representations. Dynamical matrix in symmetry adapted coordinates. Degeneracy.
      2. Electronic energy bands and phonon spectra. Assignment of small representations. Compatibility relations. Symmetry-adapted bases. Partial diagonalization of the dynamical matrix. Anticrossing.
      3. 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.
    2. Structural phase transitions
      1. 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.
      2. Symmetry-mode analysis of structural phase transitions. Hierarchy of modes. Symmetry-modes applications in structure refinement. Online tools for symmetry-mode analysis.



The official language of the schools was English. No simultaneous interpretation was provided.


Local organizing committee

Online documents

See also the list of didactic material for the MaThCryst schools.


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

NameCountryemailTopological schoolIrreps school
1Erik ArroyabeAustria X
2Volker KahlenbergAustria X
3Liliana DobrzanskaBelgium X
4Jian LuChina XX
5Yang TaoChina XX
6Guo-Ping YangChina XX
7Neven KrajinaCroatia X
8Frantisek LaufekCzech Republic X
9Abdellatif BensegueniFrance X
10Mariya BrezgunovaFrance XX
11Slawomir DomagalaFrance XX
12Charlotte MartineauFrance X
13Narjes Beigom MortazaviFrance X
14Agnieszka PaulFrance XX
15Isabella PignatelliFrance XX
16Pascalita ProsperFrance XX
17Romain SibilleFrance XX
18Michael BodensteinerGermany XX
19Tatiana GorelikGermany X
20Daniel LassigGermany X
21Jörg LinckeGermany X
22Axel PelkaGermany XX
23Guntram SchmidtGermany XX
24Barbara SzafranowskaGermany XX
25Jagan RajamonyIndia XX
26Mattia AlliettaItaly X
27Giulio GiulioItaly X
28Pavlo SolokhaItaly XX
29Adrian MermerPoland XX
30Agnieszka PluteckaPoland X
31Magdalena WilkPoland XX
32Eugenia PeresypkinaRussia XX
33Alexander VirovetsRussia XX
34Anjana ChanthapallySingapore XX
35Goutam Kumar KoleSingapore XX
36Maria Celeste BerniniSpain X
37Ainhoa Calderon CasadoSpain X
38Richard DvriesSpain X
39Roberto Fernandez de LuisSpain X
40Manuela Eloïsa Medina MunozSpain X
41Josefina PerlesSpain X
42Ana PlateroSpain X
43Jimmy Retrepo GuisaoSpain X
44Edurne Serrano LarreaSpain X
45Emre TasciSpain X
46Julia DshemuchadseSwitzerland X
47Arkadiy SimonovSwitzerland X
48Asli OzturkTurkey XX
49Thirumurugan AlagarsamyUK X
50Vladimir BonUkraine XX
51Amartya Sankar BanerjeeUSA X
52Maw Lin FooUSA X
53Maciej HaranczykUSA X
54Vincent JusufUSA X
55Lusann Wreng YangUSA XX
56Elliott S. RyanUSA X

Topology school Nancy2010 group photo
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.

Last updated: 11th August 2010
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