International Union of Crystallography
Commission on Mathematical and Theoretical Crystallography

• Introduction to group theory: the notion of symmetry and symmetry groups.
• Lattices and unit cells. Crystallographic symmetry in two dimensions

• Lattice planes, Miller indices and Bravais-Miller indices. Forms, zones and zone axes.
• Choice of the zones to look at in ED for the different crystal systems.
• Introduction to the stereographic projection and exercises on the stereographic projection to build crystallographic point groups.

• Symmetry operations with a screw or glide component.
• From point to space groups.

• Examples and exercises on space group diagrams.
• Projections of space groups and their two-dimensional symmetry.

• Group-subgroup relations. Klassengleiche, translationengleiche, isomorphic subgroups. Change of basis from group to subgroup. Definition and basic features of parent and daughter phases.
• Integral, zonal and serial diffraction conditions.
• Diffraction symbol and possible space groups corresponding to it.

• basics of TEM
• sample preparation for TEM
• experimental SAED and CBED patterns caption

• Scanning electron microscopy (SEM) - laboratory visit
• basics of SEM
• sample preparation for SEM
• Electron Back Scattered Diffraction (EBSD)

• Indexing of an ED pattern.
• Calculation / simulation of a SAED pattern for a given compound (from cell parameters and zone axis).
• Reconstruction of the unit cell from SAED along different zone axes.

• Connection of the symmetry in real space to that of the ED pattern.
• Analysis of the information one can obtain from CBED.

• Differences between the symmetry information from SAED (addition of inversion centre, can't distinguish m or 2) and CBED (can distinguish all point symmetry elements).
• Exercises on experimental SAED and CBED patterns

• From superlattices (subcells) in ED to sublattices (supercells) in real space.
• Determination of point groups and space groups from ED patterns.

• Domains (twins, antiphase domains); effect of twinning on the diffraction pattern.
• Calculation of twin index, obliquity, unit cell of the twin lattice.
• Splitting of reflections following a phase transition to a lower lattice system.

• Effect of the presence of symmetry operations with a glide component on ED patterns, effect of screw axes, Gjonnes-Moodie lines
• Determination of space groups from ED patterns

• Dynamical effects of the (multiple diffractions) and their effect on the reflection conditions and expected equivalence of reflections.
• Partial reconstruction of the space group symmetry from atomic resolution images (projection of space groups, effects of misorientation).

• Short introduction on the possibilities of extracting and using the reflections on ED patterns in a quantitative manner to solve structures ab initio and to refine structures.
• Exercises on the determination of space group from electron diffraction patterns.