84535 - SYMMETRIES, ELECTRONS AND PHONONS

Learning outcomes

At the end of the course the student will learn the basic notions regarding: symmetries of the atomic structure of molecules and crystals and their description using group theory; electronic states in crystals in the independent electron approximation (band theory) and lattice vibrations in classical and quantum approaches.

Course contents

1. Atomic Structure

1.1. Introduction to symmetry

1.1.1. Symmetry in crystallography. Isometric transformations. Direct and opposite congruence. Symmetry operations. Matrix description.

1.1.2. Elements of group theory for crystallography. Definition of a group. Abelian groups. Multiplication tables. Cyclic groups. Sub groups. Conjugate elements. Classes. Molecular symmetry. Point groups. Crystallographic point groups. Classes of molecular symmetry operations. Reducible and irreducible representations. Great Orthogonality Theorem. Character tables. BAsis functions. Wave functions as basis functions for irreducible representations. Application: splitting of atomic d levels in an octahedral field.

1.2. Crystal structures in 2 D.

Bravais lattices in 2 D. Primitive vectors. The five 2 D bravais lattices. 2D crystal structures. Unit cells: primitive, conventional and Wigner & Seitz. Point groups. The seventeen space groups in 2 D (plane groups).

1.3. Crystal structures in 3 D.

Crystal structures in 3 D. Coordination number. The fourteen Bravais lattices and seven crystal systems. Stereograms, the thirty two crystallographic point groups and two hundred and forty space groups. Relation between atomic structure and physical properties.

1.4. Structural determination by particle and wave scattering

X-rays, electrons and neutrons. Scattering by single atoms, by an ensemble of atoms and by a lattice. Laue conditions. Reciprocal lattice. Lattice planes and Miller indeces. Diffraction from a lattice with a basis, geometrical structure factor.

2. Electronic structure

2.1. The free electron gas

2.1.1. Hamiltonian for condensed matter.

2.1.2. Free electron gas. Born – von Karman boundary conditions. Fermi sphere. Fermi Dirac distrubutions. Sommerfeld expansione. Specific heat.

2.2. Non interacting electrons in a periodic potential.

2.2.1. Consequences of translational symmetry. Bloch theorem. Schroedinger equation in reciprocal space. Band index, lattice momentum. Energy bands. Energy gap. The ground state and the difference between metals and insulators. Density of states. Van Hove singularities. Velocity of Bloch electrons.

2.2.2. Nearly free electrons. Independent electrons in a weak periodic potential. Bragg planes. Representation of energy bands in three schemes. Fermi surfaces. Higher order Brillouin zone.

2.2.3. Tight binding. The atomic limit, Bloch functions built on the basis of atomic orbitals. Wannier functions. Linear combination of nearly degenerate atomic orbitals. s band.

3. Cohesion, vibrations and phonons.

3.1. Classification and cohesion in solids. Molecular, ionic, covalent and metallic solids. Cohesion energies.

3.2. Adiabatic approximation. Separation of the Schroedinger equation in two equations describing nuclear and electronic motions. Harmonic approximation.

3.3. Lattice vibrations in the harmonic approximation: classical description.

Atomic chain, dispersion relation. Vibrations in 3 D. Equation of motion and dynamical matrix; eigenvalues and eigen vectors. Acoustic and optical branches.

3.4. Lattice vibrations in the harmonic approximation: quantum description.

Hamiltonian of the harmonic crystal in terms of creation and destruction operators. Phonons. Bose – Einstein statistics. Lattice specific heat. High temperature limit and the Dulong and Petit law. Low temperature limit, T^3 dependence. Einstein and Debye Models.

3.5. Inelastic neutron scattering.

Thermal neutrons. Conservation of energy and momentum in inelastic scattering.

Readings/Bibliography

Neil W. Ashcroft and N. David Mermin, Solid State Physics, Saunders College Publishing (1976)

F. Albert Cotton, Chemical Applications of Group Theory, Third Edition, Wiley, In biblioteca DIFA c’è prima edizione

Carmelo Giacovazzo (editor), Fundamentals of Crystallography, Third Edition, Oxford University Press (2011). In biblioteca DIFA c’è questa edizione

Charles Kittel, Introduction to Solid State Physics, Eighth Edition, Wiley (2005);

Michael P. Marder, Condensed Matter Physics, Second Edition, Wiley (2010)

Lecture notes available on campus.unibo.it

Teaching methods

Lectures. Group discussion on selected topics.

Assessment methods

Oral exam

Teaching tools

Presentations, available to registered students on line

Office hours

See the website of Federico Boscherini