Study-unit Code
In all curricula
Andrea Orecchini
  • Andrea Orecchini
  • 63 ore - Andrea Orecchini
Course Regulation
Coorte 2020
Learning activities
Microfisico e della struttura della materia
Academic discipline
Type of study-unit
Obbligatorio (Required)
Type of learning activities
Attività formativa monodisciplinare
Language of instruction
Specific heats in solids: experimental facts. Drude theory. Indistinguishable particles. Classical and quantum statistical distributions. Sommerfeld theory and electronic specific heat. Crystal lattices and X-ray diffraction. Bloch's theorem and electron bands. Lattice vibrations and specific heat of phonons. Elementary theories of magnetism in solids.
Reference texts
The main reference textbooks are:
- Ashcroft-Mermin, Solid State Physics, Saunders College Publishing (1976).
- Kittel, Introduzione alla fisica dello stato solido, Bollati Boringhieri (1971), or the original English version.

For specific parts of the lectures program, the following textbooks can also be consulted:
- Alonso-Finn, Fundamental University Physics vol. III, Addison-Wesley (1979).
- Reif, Fundamentals of Statistical and Thermal Physics, Mc Graw-Hill (1988).
- Bozorth, Ferromagnetism, D. van Nostrand (1959).
- Bransden-Joachain, Physics of Atoms and Molecules, Prentice Hall (2001).

For reviewing concepts dealt with in previous courses, the following textbooks might be of help:
- Goldstein, Meccanica Classica, Zanichelli, Bologna (1971), or the original English version.
- Mencuccini-Silvestrini, Fisica II, Liguori Editore (1988), no English version available. Other general textbooks about classical electromagnetism might be well suited.
Educational objectives
The course represents the first contact of the students with the microscopic physics of solids.

As such, with respect to the Dublin descriptor 1 (knowledge), the first aim of the course is to introduce the students to the simplest microscopic models to explain the mechanisms that govern some typical macroscopic phenomena of the solid state, such as electrical conductivity, lattice vibrations, behaviour of specific heats, magnetism.

Given the variety and complexity of the solid state, with respect to the necessarily limited number of credits of the course, the presented phenomena represent only a minority of those actually met with in the real physical world. As a consequence, with respect to the Dublin descriptor 2 (applying knowledge), the second and most important aim of the course consists in helping the students to acquire a critical and autonomous method for approaching solid state physics in general. This will allow the students to apply the acquired bases and method to problems of larger complexity, that they will face either in subsequent advanced courses or in their later professional experience, be it in the field of research or not.
In order to fully benefit from the lectures, the student needs to possess base knowlegde of all the main fields of classical physics, such as mechanics, thermodynamics and electromagnetism, with particular regards to:
- kinematics and mechanics of point mass and rigid body
- base notions in thermodynamics of perfect gases and solid systems
- physics of mechanical waves
- base concepts in electrostatics and electrical conductivity
- magnetic field and magnetization
- Maxwell equations
- physics of electromagnetic waves
- conditions for constructive and destructive interference

The student has to be at ease in handling the following mathematical tools:
- multi-dimensional derivatives and integrals
- differential equations
- series expansion
- Fourier transforms and Fourier analysis
- Dirac delta function
- elementary algebra of matrices and operators
- eigenvalue equations

Finally, the student has to be aware of a few base concepts of classical and quantum mechanics, such as:
- hamiltonian of a physical system
- observables in quantum physics
- quantization rules for a single particle in a potential of simple form
- relationship between symmetry properties of a hamiltonian, conserved physical quantities and commutation properties of observable operators

All the above prerequisites are to be considered as mandatory.
Teaching methods
The course is fully made up of face-to-face lectures.
Other information

Learning verification modality
The exam consists of a single oral proof, lasting up to one hour per candidate. During the examination, the candidate is asked typically three questions about three among the main topics of the program. The candidate is allowed a maximum of 20 minutes to answer each question. For each of the three answers, the candidate is evaluated on both the skill acquired in technical details (such as setting up of the calculations required to achieve a given result) and the depth of her/his critical understanding of the cause-effect relationships among the various physical elements into play. The latter is of particular importance for the final evaluation and is considered as an essential requirement for the successful candidate. In this regard, particular attention is given to the candidate's ability to span across the different topics of the course's program, by establishing the proper relationships among them. At the end of the examination, each of the three answers is assigned a score between 0 and 10. The sum of the three scores will determine the final mark out of thirty. Depending on how the oral examination evolves as a whole, the candidate might be asked a fourth question for considering the possible assignment of the "cum laude" honor.
Extended program
Introduction and review of phenomena and experiments that opened the crisis of classical physics. Specific heats of classical systems. Specific heat of solids and the law of Dulong and Petit. Comparison with low-temperature experimental results in conductors and insulators.

Drude theory of metals: electrical conductivity, dielectric function, plasma frequency and specific heat of the electron gas.

Indistinguishable particles. Exchange operators and their properties. Symmetrization and antisymmetrization operators. Wave functions of indistinguishable particles. Slater determinants. Spin and its connection with the general properties of fermions and bosons.

Introduction to statistical mechanics. Maxwell-Boltzmann distribution. Hints about the kinetic theory of ideal gases. Boltzmann constant. Quantum statistical distributions: Fermi-Dirac and Bose-Einstein distributions. Remarkable features of the three statistical distributions.

Sommerfeld theory of metals: eigenstates, eigenvalues and quantization rules for the electron gas. Ground state of the Sommerfeld electron gas. Fermi wavevector, speed, energy and temperature. Average energy and compressibility modulus. Density of states. Properties of the electron gas for T>0. Sommerfeld expansion. Chemical potential and Fermi energy. Specific heat of the Sommerfeld electron gas. Electrical conductivity.

Bravais lattices. Definitions and examples of some important lattices: sc, bcc, fcc. Primitive and conventional unit cell. Crystal structure: lattice with a basis. Examples: sc, bcc and fcc. Noticeable examples of real crystal lattices. Families of crystallographic planes. Reciprocal lattice definition. General properties and primitive vectors of the reciprocal lattice. Notable examples: sc, fcc and bcc. Brillouin zone. Relationship between families of lattice planes and reciprocal lattice vectors. Miller indices.

X-ray diffraction: von Laue formulation and Bragg's law. Examples of experimental techniques: single-crystal diffraction with a monochromatic (rotating crystal method) or polychromatic (Laue method) beam, diffraction from polycrystalline samples (Debye-Scherrer or powder method). Non-Bravais lattices: geometrical structure factor and atomic form factor.

Bloch's theorem. Consequences of the Bloch's theorem: crystal wavevector, reduction to the first Brillouin zone, energy bands. Electronic configurations of bands: conductors, insulators and intrinsic semiconductors. Electrons in a weak periodic potential in the case of second order degeneration: mechanisms for energy gap opening. Direct and indirect gaps. Electrical conductivity of Bloch electrons. Role of defects and lattice vibrations.

Small oscillations of a monatomic linear chain: dispersion curve. Diatomic linear chain: acoustic and optic modes. Hints about the monatomic linear chain with two alternating elastic constants. Vibrational normal modes in three-dimensional polyatomic crystals. Hints about the dynamical matrix.

Quantization of normal modes and the phonon gas. Specific heat of the phonon gas. Debye approximation.

Introduction to magnetism in solids. Elementary theory of the Larmor-Langevin diamagnetism. Langevin theory of paramagnetism and Curie susceptibility. Weiss theory of ferromagnetism and Curie-Weiss susceptibility. Determination of gyromagnetic factors by Einstein-de Haas experiments: the role of the electron spin. Gyromagnetic factors of electron, proton and neutron. Total atomic magnetic moment and quantum theory of paramagnetism. Role of Coulomb interactions and Hund's rules in paramagnetism and ferromagnetism. Hints about Landau diamagnetism. Pauli paramagnetism. Stoner susceptibility.
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