Structure Determination, Building Principles, and Synthesis of Crystalline Materials in Two and Three Dimensions

Many materials in nature and technology are crystalline, i.e. their atomic structure is periodic. For a microscopic understanding of physical properties and their anisotropy and as a prerequisite for simulations, detailed knowledge of the atomic structure is essential. The lecture aims at providing fundamentals for structure determination by means of diffraction, i.e. determination of the unit cell as well as type and coordinates of the atoms contained. Since structure determination takes advantage of symmetries, an overview will be given over symmetry operations, the difference between coupling and combination, as well as the classification in point and space groups. Moreover, important methods for the growth of inorganic crystals as well as synthesis of novel organic crystalline materials as Metal-Organic-Frameworks und Covalent-Organic-Frameworks will be introduced. Finally, in relation to nano-materials, we discuss both synthesis and structure determination of two-dimensional materials.

Topics are:

• Principles of crystallography
• Bravais lattices in 2D and 3D
• Point symmetry groups and space groups in 2D and 3D
• Physical basics of structure determination by diffraction experiments
• Systematic extinctions
• Experimental realizations of diffraction experiments
• Approaches for solving the phase problem
• 2D diffraction experiments
• Elementary processes of crystal growth

After successful completion of the module the students are able to

• understand the introduction of crystal systems through lattice-compatible symmetries.
• identify and apply single and coupled symmetry operations in 2D and 3D.
• determine the point symmetry groups / crystal classes of single objects in 2D and 3D.
• determine the space groups of periodic 2D structures.
• comprehend the descriptions of 3D space groups in the International Tables for Crystallography.
• derive the Laue-equations and the Bragg-equation.
• calculate reciprocal lattice parameters and understand their physical meaning.
• calculate and interpret structure factors for a given structure.
• oversee the parameters influencing the experimentally observable reflex intensities.
• derive the rules for systematic extinctions for a given centering, screw axis or glide plane.
• analyze powder diffractograms by means of the Bragg-equation under consideration of the extinction rules.
• assess the emergence of reflexes by means of the Ewald construction in 2D and 3D for monochromatic and polychromatic x-rays and construct the respective diffraction angle.
• comprehend principal approaches to solve the phase problem.
• understand the differences for x-ray, neutron, and electron diffraction.
• evaluate the role of nucleation and growth processes in crystal growth.
• classify and describe the most important crystal growth methods.

Keine Vorkenntnisse nötig, die über die Zulassungsvoraussetzungen zum Masterstudium hinausgehen

The content is elucidated by lecturing and derivation of the physical foundations with manifold references to original work. The experimental realization of structure determination by diffraction is thoroughly discussed.  The lecture content is made intelligible through illustrative examples. Therefore macroscopic illustrative materials (minerals, crystal and molecular models, etc.) are used to elucidate the described effects and phenomena. Special emphasis is put on stimulating the interactive discussion with the students and amongst them on the just learned. The lecture is accompanied by exercises that are adapted to the lecture and further reaching. These exercises are worked out by the students themselves and require own research and literature studies.

lecturing, projected presentations, blackboard, exercise sheets, PDF-script for download, papercuts for models

• W. Massa: Kristallstrukturbestimmung, Vieweg + Teubner, (2011)
• W. Borchardt-Ott und H. Sowa: Kristallographie: Eine Einführung für Naturwissenschaftler, Springer, (2013)

There will be an oral exam of 25 minutes duration. Therein the achievement of the competencies given in section learning outcome is tested exemplarily at least to the given cognition level using comprehension questions, symmetry determinations and sample calculations.

For example an assignment in the exam might be:

• Determine the 2D space group of periodic structure
• Determine the structure factor for a given unit cell
• Determine the coordinatioes when a given symmetry element (e.g. glide plane, screw axis) acts on a point (x,y,z)
• Interpret a given powder diffractogramm
• Explain what Laue classes are