International Union of Crystallography	European Crystallographic Association	Association Française de Cristallographie	Société Française de Minéralogie et Cristallographie	Crystallography Laboratory Nancy	Doctoral School of Chemistry, Materials and Physics	Pôle scientifique CPM	Métropole Grand Nancy	Malvern Panalytical

 

International Union of Crystallography Commission on Mathematical and Theoretical Crystallography

Summer School on Mathematical Crystallography

Nancy, France, 12 – 16 June 2023

Lecturers

• Massimo Nespolo, Université de Lorraine, France
• Mois Ilia Aroyo, Universidad del País Vasco, Spain
• Michal Dušek, Czech Academy of Sciences, Czech Republic
• Andrey Zameshin, Malvern Panalytical, The Netherlands
• Tutor: Isabella Pignatelli, Géoressources laboratory, Université de Lorraine, France

Local Organizing Committee

• Prof. Massimo Nespolo, CRM2 laboratory, chair
• Mrs Valérie Richalet, CRM2 laboratory, secretary

Timetable

Morning session (3:00 hours):
Lectures 9:00 – 10:30
Coffee break: 10:30 11:00
Lectures 11:00 – 13:00
Lunch: 13:00 14:30
Afternoon session (4:00 hours):
Lectures 14:30 16:30
Coffee break: 16:30 17:00
Lectures 17:00 – 19:00

Lunches were served at the Restaurant Universitaire Vélodrome.

Scientific Program

1st Day Monday, 12 June, morning session: 9:00 – 9:10 Opening of the School

General introduction to crystallographic symmetry and application to two-dimensional space

Lecturers: Massimo Nespolo, Mois Ilia Aroyo

• Symmetry groups and their classification with in terms of periodicity (MN).
• Symmetry groups and types of symmetry in the two-dimensional space (MN).
• Bravais lattices, conventional and non-conventional unit cells (MN).
• Point symmetry of lattices and symmetry directions. Hermann-Mauguin symbols for point groups. Symmetry of crystallographic patterns: holohedries and merohedries (MN).
• Subgroups, index, coset decompositions. Lagrange theorem (MIA).
• Normal and conjugate subgroups of two-dimensional point groups (MIA).
• General and special Wyckoff positions, site symmetry (MIA). 
• Supergroups and normalizers (MIA).

1st Day Monday, afternoon session

Crystallographic symmetry in the three-dimensional space

Lecturer: Massimo Nespolo

• Crystal families, lattice systems and crystal systems.
• Extension of the results in two dimensions to the three-dimensional space.
• Miller indices. Morphological symmetry, forms and zone. Stereographic projection and its use for the derivation of crystallographic point groups. Wyckoff positions for point groups. Subgroups and supergroups of point groups. Bravais-Miller indices. Diffraction and Laue indices.

2nd Day, Tuesday, 13 June, morning session

Presentation of symmetry operations

Lecturer: Mois Ilia Aroyo

• Symmetry elements: geometric elements and element sets.
• Symmetry operations and their matrix-column presentation; geometric interpretation.
• Transformations of the coordinate systems: change of origin and orientation.
• Metric tensor.

2nd Day, afternoon session

Space groups: theory and exercises

Lecturer: Massimo Nespolo

• Space groups, general introduction: screw axes and glide planes.
• Orthogonal projections of space groups. Exercises on space-group diagrams from Volume A of the International Tables for Crystallography. General and special positions, site-symmetry groups.

3rd Day, Wednesday, 14 June, morning session

Lecturer: Mois Ilia Aroyo

(i) Description of space-group symmetry in ITA

• Conventional and non-conventional descriptions of space groups; ITA settings.
• Coset decomposition. Factor group. Symmorphic and non-symmorphic types of space groups.

(ii) Computer tools for space-group symmetry

• General and special Wyckoff positions, site symmetry.