84583 - Physics of Condensed Matter and Radiation

Learning outcomes

At the end of the teaching activities, all students will be able to understand and analyze: - the phenomenology of the interaction between radiation and matter, and its description in a classical context; - the quantum physics related to the interaction between radiation and matter when exposed to semi-classical approximation (photoelectric absorption and the diffusion of atoms; the absorption and diffusion related to vibrational and rotational stimulation of molecules); - the description of direct and reciprocal crystal lattices, as well as the description of their elements of symmetry; - the fundamentals on defects both on point and on extended basis; - the basics of physics of electron states in crystalline solids; - the description of elastic waves in lattices, the knowledge related to physics of phonons and its relation with the models of both Einstein and Debye; - all introductory notions of the physics of magnetism of solid state; - all introductory notions of the thermodynamics statistics in the phases of transition.

Course contents

1. Atomic Structure

1.1. Introduction to symmetry

Symmetry in crystallography.

1.2. Crystal structures in 2D.

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

1.3. Crystal structures in 3D.

Crystal structures in 3D. Coordination number. The fourteen Bravais lattices and seven crystal systems. 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 distributions. Sommerfeld expansion. 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.

 

3. Vibrations and phonons.

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

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

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

3.3. 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. Einstein and Debye Models.

3.4. Inelastic neutron scattering.

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

 

4. Interaction between radiation and matter
General aspects and classical description. Attenuation coefficient and cross section. The dielectric response of matter, phenomenology. Permittivity and susceptibility. Index of refraction: dispersion and attenuation. Weak interaction limit. Model dielectric function for static polarization. Kramers – Heisenberg dielectric function. Kramers Kronig relations. Scattering from a damped harmonic oscillator. High and low frequency limits. Resonant scattering.
Quantum description. Fermi’s golden rule. Semiclassical approximation, Coulomb gauge. Vector potential for a polarized plane wave. Interaction Hamiltonian: linear and quadratic terms in the vector potential. Photoelectric absorption cross section in the dipole approximation. Selection rules. Scattering of radiation. Density of states for photons and derivation of the scattering cross section. Elastic scattering of linearly polarized radiation from a hydrogen – like atom in the ground states. High and low energy limits.

5. Introduction to magnetism in solids
Classical description of magnetic phenomena. Orbital and spin magnetic moment. Magnetization. Magnetostatics in ponderable matter. Magnetic materials: properties, operational definition of magnetic field; diamagnetic, paramagnetic and ferromagnetic materials. Microscopic interpretation of magnetization in ponderable matter: Larmor diamagnetism, Langevin paramagnetism; Curie – Weiss law for ferromagnetism. Hysterisis curves and Weiss domains in ferromagnetic materials.
Quantum interpretation. Quantum theory of paramagnetism and diamagnetism: Brillouin function. Quantum magnetic susceptibility. Larmor diamagnetism. Magnetization density as a function of the Helmholtz free energy. Hund’s rule for the atomic ground states. Van Vleck paramagnetism. Susceptibility of metals: Pauli paramagnetism and Landau diamagnetism. Quantum theory of ferromagnetism. Mean field theory. Curie – Weiss law. Exchange interaction and Heisenberg model. Magnons. Magnetic neutron scattering. Antiferromagnetic ordering. Ferromagnetic domains and Bloch walls.

Readings/Bibliography

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)

Other textbooks:

• 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)
• 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)
• Michael Tinkham, Group Theory and Quantum Mechanics, Dover (2003).

Copy of the slides on iol.unibo.it

Teaching methods

Front lectures

Assessment methods

Oral exam

Typically the student is asked to main questions, that is to discuss two main different arguments, among the ones in the content of the course and selected by the teacher during the exam. For each argument the student will be asked to develop a theory/calculations to obtein a physical law, by illustrating the main conceptual steps, including considerations and approssimations used, and to explain its physical meaning and its use to study some physical property of an ideal and/or real physical system, paying attention to the order of magnitude of the physical quantities at play. The typical duration of the exam is 30-50 minutes.

The purpose of the oral exam is to verify the student's knowledge and his/her ability to apply it and to make the necessary logical-deductive connections.

Graduation of the final grade: Preparation on a very limited number of topics covered in the course and analytical skills that emerge only with the help of the teacher, expression in overall correct language → 18-19;

Preparation on a limited number of topics covered in the course and autonomous analysis skills only on purely executive issues, expression in correct language → 20-24;

Knowledge of a large number of topics addressed in the course, ability to make independent choices of critical analysis, proper use of specific terminology → 25-29;

Substantially exhaustive preparation on the topics covered in the course, ability to make independent choices of critical analysis and connections, full mastery of the specific terminology and ability to argue and critical thinking → 30-30L

Teaching tools

Blackboard, Slides

Links to further information

https://www.unibo.it/sitoweb/s.sanna/contenuti-utili/6575e556

Office hours

See the website of Samuele Sanna