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# Statistical Mechanics & Solid State II - PHY00049H

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• Department: Physics
• Module co-ordinator: Dr. Paul Davies
• Credit value: 20 credits
• Credit level: H
• Academic year of delivery: 2022-23
• See module specification for other years: 2021-22

## Module summary

The course aims to provide a formal understanding of the underlying energy distribution of systems containing many particles. The consequences of the distribution is related to classical descriptions of thermodynamics and the behaviour of electrons in solids.

• None

## Module will run

Occurrence Teaching period
A Autumn Term 2022-23 to Spring Term 2022-23

## Module aims

Statistical Mechanics

In Statistical Mechanics we will develop formalisms of equilibrium statistical mechanics from fundamental considerations of the microscopic states available to the system, and relate statistical mechanics to the classical thermodynamical descriptions of heat, work, temperature and entropy. Statistical mechanics will be used to derive formulae for the internal energy, entropy, specific heat, free energy and related properties of classical and quantum-mechanical systems, and to apply these formulae to a variety of realistic examples.

Solid State II

This is a prerequisite for the MPhys modules Nanomaterials and Light and Matter. If we want to understand physical properties such as electrical and thermal conductivity, magnetism or reflectivity and absorption, it is necessary to study the electronic structure and transport properties of electrons in solids. Starting with the classical free electron gas approximation we will develop the concepts of the Fermi gas and nearly free electron theory making use of the quantum mechanical description of electrons in a periodic potential. This leads to the band structure model, which will allow us to describe material systems such as semiconductors and metals. These concepts will then be used to obtain insight into the origin of magnetism and optical properties of materials.

## Module learning outcomes

Learning outcomes: at the end of this module successful students will be able to:

Statistical Mechanics

• Discuss, at the level of detail given in the lectures, how the results of statistical mechanics may be derived from fundamental statistical considerations and how they are related to classical thermodynamics;

• Apply the definitions and results of statistical mechanics to deduce physical properties of the systems studied in the lectures and other systems of similar complexity, drawing in part on your knowledge of the microstates of simple systems from core courses in quantum mechanics and solid state physics.

Solid State II

• Understand the different models involved describing the interaction between electrons and electrons as well between electrons and crystal lattice and the underlying physical principles.

• Explain the concept of the free electron approximation in metals.

• Describe the interaction of free electrons with a constant electric and a constant magnetic field.

• Calculate the density of states based on the Fermi statistics.

• Determine the electronic contribution to the heat capacity.

• Distinguish direct and indirect band gap semiconductors.

• Describe the different mechanisms of conductivity in semiconductors.

• Explain the principles of semiconductor devices such as diodes and transistors.

• Distinguish the different types of magnetic properties in solids.

• Understand the principles of superconductivity

## Module content

Please note, students wishing to take this module should have taken PHY00031I (Thermodynamics & Solid State II) in addition to either PHY00032I (Quantum Physics II) or PHY00036I ((NS) Quantum & Atomic Physics) as well as either PHY00030I (Mathematics II), MAT00039I (Applied Mathematics for Mathematics and Physics) or PHY00035I ((NS Mathematics II) and MAT00007C (Mathematics for the Sciences I)

Syllabus

Statistical Mechanics

Microstates: microstates (quantum states) and macrostates of a system, degeneracy W, density of states, illustration for a set of N harmonic oscillators, principle of equal equilibrium probability of an isolated system, term “microcanonical ensemble” [1 lecture]

Thermal equilibrium, temperature: statistical nature of equilibrium illustrated for 2 sets of N harmonic oscillators, definition of temperature, Boltzmann distribution, partition function Z, term “canonical ensemble” 

Entropy: general statistical definition of entropy S, law of increase of entropy, entropy of isolated system in internal equilibrium (“microcanonical ensemble”), entropy of system in thermal equilibrium with a heat bath (“canonical ensemble”), Helmholtz free energy F; equivalence of classical and statistical entropy [2.5]

Elementary applications: Vacancies in solids; two-level systems (including magnetic susceptibility of dilute paramagnetic salt), simple harmonic oscillator (partition function, heat capacity). 

Vibrational heat capacity of solids: Quantisation of phonon modes, labelling of modes using wavevector k; Einstein and Debye models 

Ideal gas: Partition function of monatomic gas, classical gas law, Maxwell-Boltzmann speed distribution, molecular gases (rotation and vibration), classical limit of occupation numbers 

Systems with variable number of particles: Grand canonical ensemble, chemical potential, Gibbs distribution [1.5]

Identical particles: Fermions and bosons, Fermi and Bose distributions, Bose-Einstein condensation, with applications to free-electron metals and nuclear physics (fermions), and liquid 4He and superconductivity (bosons) 

Black body radiation: Energy density, pressure 

The classical limit: Phase space, classical equipartition theorem 

Comprehensive lecture notes should be taken down from the blackboard during lectures, and will be supplemented by a small number of handouts. These handouts, together with audio recordings of lectures, interactive apps, a record of problems set, and similar information, will also be made available through the VLE.

Solid State II

Recap of the Fermi-gas model

• Free electron gas approximation (Drude – d.c. and a.c. response)
• Fermi-Dirac statistics, Fermi-sphere, Fermi-distribution, density of electronic states, Energy dispersion
• Heat capacity and electrical conductivity of the Fermi-gas
• Interactions with constant electric/magnetic fields.

Nearly Free electron model

• Perturbation Theory
• Electrons in a periodic potential
• Reduced Zone scheme, Extended Zone scheme
• Tight Binding Model
• Band structure and band gaps
• Fermi surfaces and Brilliouin zones
• Effective electron mass approximation
• Measuring the Fermi surface. de-Haas van Alphen effect and ARPES
• Failures of the Band-theory of Metals and Insulators

Semiconductors

• Direct and indirect band gaps
• Intrinsic and doped semiconductors
• Cyclotron Resonance
• Impact of temperature on charge density and conductivity
• The p-n junction

Dielectric and optical properties

• Optical transitions in direct and indirect semiconductors
• Plasma Frequency
• Reflectivity and absorption of metals

Magnetic properties

• Para-, dia-, ferro- and antiferromagnetism
• Ising Model of Ferromagnetism
• Hubbard Model of Itinerant Magnetism and the Stoner Criteria

Superconductivity

• London Equations
• Meissner effect
• BCS-theory

## Assessment

Task Length % of module mark
Essay/coursework
Solid State II Assignment 1
N/A 20
Essay/coursework
Solid State II Assignment 2
N/A 30
Essay/coursework
Statistical Mechanics Assignment 1
N/A 20
Essay/coursework
Statistical Mechanics Assignment 2
N/A 30

None

### Reassessment

Task Length % of module mark
Essay/coursework
Solid State II Assignment 1
N/A 20
Essay/coursework
Solid State II Assignment 2
N/A 30
Essay/coursework
Statistical Mechanics Assignment 1
N/A 20
Essay/coursework
Statistical Mechanics Assignment 2
N/A 30

## Module feedback

Our policy on how you receive feedback for formative and summative purposes is contained in our Department Handbook.

Statistical Mechanics

Waldram JR: The theory of thermodynamics (Cambridge University Press)***

Bowley R and Sánchez M: Introductory statistical mechanics (Oxford University Press)***

Glazer M and Wark J: Statistical Mechanics: A Survival Guide (Oxford University Press)***

Mandl F: Statistical physics (Wiley)**

Reif F: Fundamentals of statistical and thermal physics (McGraw-Hill)**

Blundell SJ and KM: Concepts in Thermal Physics (Oxford University Press)*

Swendsen RJ: An Introduction to Statistical Mechanics and Thermodynamics (Oxford University Press 2012)*

Solid State II

C. Kittel: Introduction to Solid State Physics (Wiley and Sons)

N.W. Ashcroft and N.D. Mermin: Solid State Physics (Saunders College Publishing)

H. Ibach and H. Lüth: Solid-State Physics – An Introduction to Principles of Materials Science (Springer-Verlag)

The information on this page is indicative of the module that is currently on offer. The University is constantly exploring ways to enhance and improve its degree programmes and therefore reserves the right to make variations to the content and method of delivery of modules, and to discontinue modules, if such action is reasonably considered to be necessary by the University. Where appropriate, the University will notify and consult with affected students in advance about any changes that are required in line with the University's policy on the Approval of Modifications to Existing Taught Programmes of Study.