Posted by peeterjoot on January 20, 2014

Here is what will likely be the final update of my class notes from Winter 2013, University of Toronto Condensed Matter Physics course (PHY487H1F), taught by Prof. Stephen Julian.

Official course description: “Introduction to the concepts used in the modern treatment of solids. The student is assumed to be familiar with elementary quantum mechanics. Topics include: bonding in solids, crystal structures, lattice vibrations, free electron model of metals, band structure, thermal properties, magnetism and superconductivity (time permitting)”

This document contains:

• Plain old lecture notes. These mirror what was covered in class, possibly augmented with additional details.

• Personal notes exploring details that were not clear to me from the lectures, or from the texts associated with the lecture material.

• Assigned problems. Like anything else take these as is.

• Some worked problems attempted as course prep, for fun, or for test preparation, or post test reflection.

• Links to Mathematica workbooks associated with this course.

My thanks go to Professor Julian for teaching this course.

NOTE: This v.5 update of these notes is still really big (~18M). Some of my mathematica generated 3D images result in very large pdfs.

Changelog for this update (relative to the first, and second, and third, and the last pre-exam Changelogs).

January 19, 2014 Quadratic Deybe

January 19, 2014 One atom basis phonons in 2D

January 07, 2014 Two body harmonic oscillator in 3D

Figure out a general solution for two interacting harmonic oscillators, then use the result to calculate the matrix required for a 2D two atom diamond lattice with horizontal, vertical and diagonal nearest neighbour coupling.

December 04, 2013 Lecture 24: Superconductivity (cont.)

December 04, 2013 Problem Set 10: Drude conductivity and doped semiconductors.

Posted in Math and Physics Learning. | Tagged: 1 atom basis, acoustic dispersion, alkali earth metals, alkali metal, anharmonic oscillator, atomic scattering factor, band Structure, BCC, BCS theory, Bloch’s theorem, Body centered cubic, Boltzman distribution, Boltzmann-Gibbs distribution, Bose distribution, Bragg condition, Bravais, Bravais lattice, Brillouin zones, chemical bonding, conduction band, conventional unit cell, Cooper pairing, covalent bonding, crystal structure, crystal structures, Debye frequency, Debye model, Debye temperature, density of states, Deybe model, Diamond lattice, diffraction, Dirac delta function, Discrete Fourier transform, doped semiconductors, Drude formula for conductivity, Dulong-Petit law, dynamical matrix, effective mass, effective mass tensor, elastic scattering, electric current, electrical transport, electron pockets, electron-phonon interaction, entropy, Ewald sphere, Face centered cubic, FCC, Fermi energy, Fermi liquid theory, Fermi surface, Fermi velocity, Fermi wavevector, Fermi-Dirac distribution, filling factor, Fourier coefficient, fourier series, free electron, free electron gas, free electron model, freeze out, Fresnel diffraction, Germanium, group velocity, HCP, Heaviside function, hexagonal close packed, hole, hole pockets, holes, Huygens principle, Huygens-Fresnel principle, hybridization, insulator, interference, ionic bonding, isotropic model, jellium model, lattice plane, lattice structure, Laue condition, linear harmonic chain, London equations, Madelung constant, mean scattering time, melting point, metal, metallic bond, metallic bonding, Miller indices, nearly free electron, nearly free electron model, nn, noble gas, normal mode, normal modes, one atom basis, optical dispersion, perfect conductors, perfect diamagnet, periodic harmonic oscillator, periodic lattice, periodic table, periodicity, phonon, phonon mode, Phonons, promotion, reciprocal lattice, reciprocal vectors, reduced zone scheme, scattering, scattering density, semiconductors, Simple cubic, specific heat, structure factor, superconductivity, thermal energy, Thomas-Fermi screening, tight binding model, transition metal, transition metals, valence band, valence conduction, Van der Waals, van Hove singularity, wedge product | Leave a Comment »

Posted by peeterjoot on March 27, 2013

Here’s my second update of my notes compilation for this course, including all of the following:

March 27, 2013 Fermi gas

March 26, 2013 Fermi gas thermodynamics

March 26, 2013 Fermi gas thermodynamics

March 23, 2013 Relativisitic generalization of statistical mechanics

March 21, 2013 Kittel Zipper problem

March 18, 2013 Pathria chapter 4 diatomic molecule problem

March 17, 2013 Gibbs sum for a two level system

March 16, 2013 open system variance of N

March 16, 2013 probability forms of entropy

March 14, 2013 Grand Canonical/Fermion-Bosons

March 13, 2013 Quantum anharmonic oscillator

March 12, 2013 Grand canonical ensemble

March 11, 2013 Heat capacity of perturbed harmonic oscillator

March 10, 2013 Langevin small approximation

March 10, 2013 Addition of two one half spins

March 10, 2013 Midterm II reflection

March 07, 2013 Thermodynamic identities

March 06, 2013 Temperature

March 05, 2013 Interacting spin

plus everything detailed in the description of my first update and before.

Posted in Math and Physics Learning. | Tagged: addition of angular momentum, addition of spin, angular momentum, anharmonic oscillator, average, average diatomic separation, average dipole moment, average energy, average number of particles, average occupancy, binomial distribution, Boltzmann distribution, Boltzmann factor, Boson, canonical ensemble, Central limit theorem, chemical potential, classical harmonic oscillator, degeneracy pressure, delta function, density, density of states, diatomic molecule gas, differential form, eigenvalue, eigenvector, electric dipole, electric field interaction, electron, energy, energy eigenstate, energy eigenvalue, entropic force, entropy, equilibrium, Fermi distribution, Fermi energy, Fermi gas, Fermi temperature, Fermion, four momentum, four vector, free energy, fugacity, Gaussian approximation, Gibbs sum, grand canonical ensemble, grand canonical partition, grand partition function, graphene, hamiltonian, harmonic oscillator perturbation, heat capacity, high temperature limit, hole, ideal gas, integral approximation to sum, low temperature limit, magnetic field, magnetization, mean energy, microstate, moment of inertia, momentum, multiple paired spin, nuclear spin interaction, number of particles, number operator, occupancy, occupation number, occupation numbers, one form, orthonormal basis, partial derivative, particle in a box, Partition function, Pathria, pauli matrix, perturbation, PHY452H1S, Planck's constant, polymer, position mean value, pressure, probability, quantum anharmonic oscillator, random walk, relativistic gas, reservoir, singlet state, special relativity, specific heat, spherical harmonic, spin, spin hamiltonian, spin one half, spring constant, Statistics mechanics, subsystem, temperature, thermal average energy, thermal de Broglie wavelength, thermodynamic identity, trace, triplet states, two form, two variable Taylor expansion, variance, volume, zipper DNA model | 1 Comment »

Posted by peeterjoot on March 26, 2013

[Click here for a PDF of this post with nicer formatting (especially if my latex to wordpress script has left FORMULA DOES NOT PARSE errors.)]

# Disclaimer

Peeter’s lecture notes from class. May not be entirely coherent.

**Review**

Last time we found that the low temperature behaviour or the chemical potential was quadratic as in fig. 1.1.

Fig 1.1: Fermi gas chemical potential

**Specific heat**

where

**Low temperature **

The only change in the distribution fig. 1.2, that is of interest is over the step portion of the distribution, and over this range of interest is approximately constant as in fig. 1.3.

Fig 1.2: Fermi distribution

Fig 1.3: Fermi gas density of states

so that

Here we’ve made a change of variables , so that we have near cancelation of the factor

Here we’ve extended the integration range to since this doesn’t change much. FIXME: justify this to myself? Taking derivatives with respect to temperature we have

With , we have for

Using eq. 1.1.4 at the Fermi energy and

we have

Giving

or

This is illustrated in fig. 1.4.

Fig 1.4: Specific heat per Fermion

**Relativisitic gas**

- Relativisitic gas
- graphene
- massless Dirac Fermion
Fig 1.5: Relativisitic gas energy distribution

We can think of this state distribution in a condensed matter view, where we can have a hole to electron state transition by supplying energy to the system (i.e. shining light on the substrate). This can also be thought of in a relativisitic particle view where the same state transition can be thought of as a positron electron pair transition. A round trip transition will have to supply energy like as illustrated in fig. 1.6.

Fig 1.6: Hole to electron round trip transition energy requirement

**Graphene**

Consider graphene, a 2D system. We want to determine the density of states ,

We’ll find a density of states distribution like fig. 1.7.

Fig 1.7: Density of states for 2D linear energy momentum distribution

so that

Posted in Math and Physics Learning. | Tagged: chemical potential, density of states, electron, energy, Fermi distribution, Fermi gas, graphene, hole, PHY452H1S, relativisitic gas, specific heat, statistical mechanics | Leave a Comment »