International School on Mathematical and Theoretical Crystallography

Program

Monday April 28, morning. Introduction to crystallographic symmetry I. Massimo Nespolo, Université Henri Poincaré Nancy I, France

• Basics of group theory. Space groups and space group types. Hermann-Mauguin symbols. Wyckoff positions, site symmetry. Unconventional settings of space groups. Extracting information from Volume A of the International Tables for Crystallography

Monday April 28, afternoon. Introduction to crystallographic symmetry II. Mois I. Aroyo, Universidad del Pais Vasco, Bilbao, Spain

• Mappings, symmetry operations, 4x4 matrices. Basis and coordinate transformations, origin shifts. Subgroups and splitting of Wyckoff positions.

Tuesday April 29. Group theory applied to crystallography. Bernd Souvignier, Radboud University Nijmegen, The Netherlands

• Dimension-independent fundamental notions of crystallographic groups (lattices, point groups, vector systems); classification of crystallographic groups (arithmetic/geometric classes, crystal systems, lattice systems, crystal families). Equivalent descriptions for a crystal structure (use of normalizers).

Wednesday April 30, morning. Subgroups of space groups. Mois I. Aroyo, Universidad del Pais Vasco, Bilbao, Spain

• Exercises on the Volume A1 of the International Tables for Crystallography

Wednesday April 30, afternoon. The symmetry principle in crystal chemistry - I. Ulrich Müller, University of Marburg, Germany

• Instructions how to build trees of group-subgroup relations between space groups. Examples for translationengleiche, klassengleiche and isomorphic subgroups; space groups related by a common supergroup; large structural families. Pitfalls and possible sources of errors.

Thursday May 1, morning. The symmetry principle in crystal chemistry - II. Ulrich Müller, University of Marburg, Germany

• Applications: Structures with close-packed atoms and occupied interstices; symmetry aspects at phase transitions and topotactic reactions, twinning, domains; wrong structure determinations due to twinning; symmetry relations among molecular structures

Thursday May 1, afternoon. Role of symmetry and energy in structural phase transitions. Michele Catti, University of Milano-Bicocca, Italy

• Elements of Landau theory of structural phase transitions. Symmetry group-subgroup relationships. Reconstructive, displacive and order-disorder solid-solid transformations. Evaluation of the crystal energy by periodic quantum-mechanical techniques. Applications to thermodynamic and kinetic aspects of displacive and reconstructive phase transitions of inorganic crystals.

Friday May 2, morning. Computer tools in mathematical crystallography. Mois I. Aroyo, Universidad del Pais Vasco, Bilbao, Spain

• Introduction to the Bilbao Crystallographic Server

Friday May 2, afternoon. Understanding organic crystal packing: how intermolecular energies determine crystal symmetry and intermolecular geometry. Angelo Gavezzotti, University of Milano, Italy

• Fundamentals of intermolecular potentials. Correlation-dispersion and exchange-repulsion. Close packing symmetry elements. Restrictions on space groups for organic crystals. Use of crystallographic Databases. Models for the calculation of intermolecular energies: recognition in gas-phase dimers versus pairwise molecular interactions in crystals. Quantitative ranking of approach modes: the hydrogen bond versus other bonding interactions. Quantitative ranking of lattice energies among polymorphs; crystals structure 'prediction'.

Conference venue

The school took place at Palazzo Feltrinelli, on the Garda Lake. This beautiful historical Villa is now owned by the University of Milan and is dedicated to Workshop and Meeting activities. Gargnano is a small resort village on Garda Lake, about 150 km east of Milan.