84535 - Symmetries, Electrons and Phonons

Academic Year 2023/2024

  • Teaching Mode: Traditional lectures
  • Campus: Bologna
  • Corso: Second cycle degree programme (LM) in Physics (cod. 9245)

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.

2.3 Complements on electronic structure

2.3.1 Overview of band structures. Metals, semiconducturs and insulators. Nearly free electron metals (Al), noble gases (Kr), semiconductors (Si, Ge, GaAs), graphene, transition metals (V).

2.3.2 Photoemission. The work function. Experimental scheme. Initial and final states in many body and one electron schemes. Angle integrated photoemission, density of states. Angle resolved photoemission, band structure determination; example: graphite.

2.4 Beyond the indepedent electron approximation. Hartree and Hartree-Fock equations. Hartree – Fock for the free electron gas. Screening; Thomas – Fermi approach to screening.

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.


Main textbooks

  • F. Albert Cotton, Chemical Applications of Group Theory, Third Edition, Wiley (1990)
  • Michael P. Marder, Condensed Matter Physics, Second Edition, Wiley (2010)
  • Neil W. Ashcroft and N. David Mermin, Solid State Physics, Saunders College Publishing (1976)
  • Charles Kittel, Introduction to Solid State Physics, Eighth Edition, Wiley (2005)

Complementary textbooks

  • General condensed matter
  • Efthimios Kaxiras and John D. Joannopoulos, Quantum Theory of Materials, Cambridge University Press (2019)
  • Marvin Cohen and Steven Louie, Fundamentals of Condensed Matter Physics, Cambridge University Press (2016)
  • Attilio Rigamonti and Piero Carretta, Structure of Matter, Springer (2015)
  • Gian Franco Bassani e Umberto Grassano Fisica dello Stato Solido, Bollati Boringhieri (2000), only in Italian.
  • Feng Duan and Jin Guojon, Introduction to Condensed Matter Physics, Volume 1, World Scientific (2005).
  • Giuseppe Grosso and Giuseppe Pastori Parravicini, Solid State Physics, Second Edition, Academic Press (2014)
  • Peter Y. Yu and Manuel Cardona, Fundamentals of Semiconductors, Fourth Edition, Springer (2010)
  • G.L. Squires, Introduction to the theory of thermal neutron scattering, Cambridge University Press, 1978
    • Crystallography
  • Carmelo Giacovazzo (editor), Fundamentals of Crystallography, Third Edition, Oxford University Press (2011).
  • Boris K. Vainshtein, Fundamentals of Crystals: Symmetry, and Methods of Structural Crystallography, Second Enlarged Edition, Springer (1994).
  • Boris K. Vainshtein, Vladimir M. Fridkin and Vladimir L. Indenbom, Modern Crystallography II: Structure of Crystals, Springer (1979)
  • Gerald Burns and Anthony M. Glazer, Space groups for solid state scientists, Academic Press (2013)
    • Group theory
  • Michael Tinkham, Group Theory and Quantum Mechanics, Dover (2003).

Teaching methods

Formal lectures

Assessment methods

Oral exam, approximately 30 minutes long. The objective of the exam is to assess the understanding of the basic physical meaning of the topics described in the course. Typically at least two different topics will be discussed. The exam must be booked via almaesami, 3 students every hour and a half slot, 12 students a day.

Teaching tools

Presentations, available to registered students on line

Office hours

See the website of Federico Boscherini


Affordable and clean energy

This teaching activity contributes to the achievement of the Sustainable Development Goals of the UN 2030 Agenda.