Ordering and symmetry in condensed media

P. V. Dolganov

Abstract

This lecture course gives an introduction to physics of crystalline and non-crystalline condensed media. The course comprises two introductory chapters, which are devoted to description of ordering in different types of materials and basics of symmetry theory, and ten main chapters, where an overview of different physical phenomena in condensed media is given and their relation to symmetry is discussed. Description of various structures is given on the basis of general physical notions.

1. Types of condensed media. Crystals, liquids, amorphous state, liquid crystals, plastic (rotational) crystals. Incommensurate phases. Dynamics of ordered media formed by atoms and molecules. Types of interatomic and intermolecular bonds.
2. Symmetry. Symmetry operations and elements. Point and space symmetry groups. Continuous point groups. Crystalline classes and systems (syngonies).
3. Electrical and magnetic properties of crystals. Electric polarization. Ferroelectrics, ferrielectrics, antiferroelectrics. Crystals in electric field. Physical properties near the ferroelectric-paraelectric transition. Susceptibility.
4. Piezoelectricity. Direct and inverse piezoelectric effect. Transport processes. Electric conductivity and thermal conductivity. Deformations and elasticity. Thermal expansion.
5. Optical properties of crystals. Birefringence. Optical indicatrix. Relation of optical properties of crystals to their symmetry. Optical methods for investigations of condensed media.
6. Transitions with change of structure in crystalline and non-crystalline media. Relation of symmetry with possible transition types. Order parameter space. Spontaneous symmetry breaking. Goldstone modes in various systems.
7. Liquid-crystalline structures with orientational and translational ordering. Chirality. Structures formed as a result of competing interaction, frustration. Behavior of helical structures in external field.
8. Defects in media with broken orientational and translational symmetry. Topological defects. Topological charge. Solitons, 2?-walls, ?-walls. Phases formed by ordered structure of defects. Kosterlitz-Thouless transitions, dislocation melting.
9. Fractals. Fractal dimension of polymers. Biopolymers. Amphiphilic molecules. Micelles, lyotropic liquid crystals. Biological structures forming liquid-crystalline phases. DNA, biological macromolecules and their ordering.
10. Influence of surface on structure and phase transitions. Systems of confined geometry. Surface melting, surface crystallization. Layer by layer thinning transitions. Stability of thin films.
11. Self-organization of microparticles in anisotropic media. Formation of ordered structures. Interaction of particles in smectic films. Topological charge of a particle in ordered medium. The role of boundary conditions. Topological dipoles and quadrupoles.
12. Photonic crystals. One-dimensional, two-dimensional and three-dimensional photonic crystals. Examples of photonic crystals, photonic crystals in nature. Interaction of light with spatially modulated structure. Photon dispersion. Forbidden photonic zones.

1. J.F. Nye, Physical Properties of Crystals: Their Representation by Tensors and Matrices, Oxford University Press.
2. M. Kleman, O. D. Lavrentovich, Soft Matter Physics: An Introduction, Chapters 1-3, 7, 11, 13.
3. L. D. Landau, E. M. Lifshitz, Quantum Mechanics, зз91-93.
4. L. D. Landau, E. M. Lifshitz, Statistical Physics, зз128-132.
5. Ch. Kittel, Introduction to Solid State Physics.
6. N. V. Karlov, N. A. Kirichenko, Oscillations, Waves, Structures (in Russian), Fizmatlit, Moscow, 2001, Chapter IX