Syllabus

Module: 1 [8-classes]
Basic matrix algebra: Matrix calculus applied to crystallography. Point symmetry operations.
Crystal pattern: Lattices and unit cells, Crystal Structure, Crystal Systems, symmetry of lattices symmetry of the unit cell content symmetry of crystallographic pattern, Centering of lattice, Bravais lattices.

Module: 2 [8-classes]
Crystallographic point groups: Point group, Hermann-Mauguin symbols for point groups. Basic concepts of group theory. Group actions, Development of 32 crystallographic point groups. Stereographic projections of point groups. Space groups and their description: Space groups - general introduction, Screw axes and glide planes. Structure of space groups: point groups of space groups. Coset decomposition of the space group with respect to its translation subgroup, Hermann-Mauguin symbols of space groups. Space-group diagrams. Derivation of space group, Orthogonal projections of space groups.

Module: 3 [8-classes]
Representation of Crystallographic groups: Representations of point groups: General remarks on representations. Equivalent, reducible and irreducible representations. Characters of representations and character tables. Representations of space groups: Representation of the translation group. Star of a representation. Little groups and small representations. Representations of symmorphic and non-symmorphic groups.

Module: 4 [6-classes]
Phase Transitions: General introduction to phase transitions in the solid state: Thermodynamics aspects concerning phase transition, 1st order and 2nd order phase transitions, Structural classifications of phase transitions, The Landau theory of continuous phase transition and discontinues phase transition.

Module: 5 [6-classes]
Symmetry considerations in structural phase transitions: Primary and secondary order parameters, Order parameter direction and isotropy subgroups. Group-theoretical formulation of the necessary conditions for second-order phase transitions. Group-subgroup relations between space groups for structural phase transition, Subgroups of space groups, types of subgroups of space groups.

To define the concept of lattice, symmetry operations and crystal systems.

To introduce the concept of symmetry and its mathematical treatment using group theory.

To demonstrate the importance and practical utility of point and space groups in crystallography.

To develop the understanding of different phase transitions and the Symmetry consideration in structural phase transitions.

After completion of this course, the students will have the understanding of the fundamental crystallography and their applications in the analysis of the structural phase transitions.

G. Burns and A. M. Glazer,, Space Group for Solid State Scientist,, Academic Press, USA, 2013.

Ulrich Muller, Symmetry Relationships Between Crystal Structures,, Oxford University Press, UK, 1st Edition 2013.

R. Mcweeny,, Symmetry: An Introduction to Group Theory and Its Applications,, Pergamon Press, UK, 1963.

A. W. Joshi, Elements of Group Theory for Physicist, New Age International (P) Limited Publisher, 4th Edition, 2005.

Lecture notes on “Representation of Crystallographic Groups” during the Summer School on Mathematical and Theoretical Crystallography, Nancy, June 28- July, 2, 2008.