Applications of Group Theory in Solid State Physics

Course guarantor:

Zdeněk Potůček

Lecturer:

Zdeněk Bryknar, Zdeněk Potůček

Tutor:

Zdeněk Potůček

Supervisor:

Department of Solid State Engineering

Synopsis:

Consideration of atomic system symmetry allows, without any quantitative calculations, rigorously and precisely determine how many energy states there are and what interactions and transitions between them may occur. Therefore, the main purpose of this course is to describe the methods by which we can extract the information on the object that symmetry alone will provide. The application of these methods is illustrated by an example of molecular orbitals, inner orbitals of ions in the crystal field environment, normal modes of molecular vibrations, and selection rules for optical absorption and emission transitions.

Requirements:

Oral exam in the field of topics listed in the lectures syllabus including solution of typical exercise.

Syllabus of lectures:

1. Definitions and theorems of group theory.

2. Molecular symmetry, the symmetry point groups, classes of symmetry operations.

3. Reducible and irreducible representations of groups, properties of irreducible representations, decomposition of reducible representations, character tables.

4. Wave functions as bases for irreducible representations.

5. The direct product and its use.

6. Molecular orbital theory - transformation properties of atomic orbitals, complete and incomplete projection operator, molecular orbitals for sigma and pi bonding.

7. Ligand field theory - splitting of the terms of free ions in a crystalline environment, application of the method of descending symmetry for construction of energy level correlation diagrams for ions in a crystalline environment.

8. Selection rules and polarization of optical transitions.

9. Normal modes of molecular vibrations, selection rules of their activity in infrared absorption and Raman spectra.

Syllabus of tutorials:

Study Objective:

Knowledge:

Theory of group representations and methods of its application to determination of molecular orbitals, energy levels of ions in a crystalline environment, normal modes of molecular vibrations, and probability and polarization of optical transitions.

Skills:

Ability to obtain attainable information about physical properties of atomic system (molecule, crystal) arising from its symmetry alone.

Study materials:

Key references:

[1] F. A. Cotton: Chemical applications of group theory, 1990, John Willey & Sons, New York.

[2] R. Ameta, S. C. Ameta: Chemical applications of symmetry and group theory, 2016, Apple Academic Press.

Recommended references:

[3] M. S. Dresselhaus, G. Dresselhaus, A. Jorio: Group theory: application to the physics of condensed matter, 2008, Springer-Verlag, Berlin.

[4] G. Burns: Introduction to group theory with applications, 2014, Academic Press.

[5] R. C. Powell: Symmetry, group theory, and the physical properties of crystals, 2010, Springer Verlag.

Note:

Time-table for winter semester 2025/2026:

Time-table is not available yet

Time-table for summer semester 2025/2026:

Time-table is not available yet

The course is a part of the following study plans: