Phonons

Phonons are quantized particles of sound. Similar to photons, the phonon energy is related to the frequency of the sound waves E = hf and the phonon momentum is related to the wavelength of the sound waves p = h/λ. Sound waves with wavelengths much longer than the lattice constant of a crystal, are described by the wave equation. The wave equation was quantizied in the section on the quantization of the electromagnetic field and the quantization of sound proceeds similarly. A table summarizing the results for phonons is given below.

Outline
• Calculating phonon dispersion relations by assumming the atoms are connected by linear springs     W
  • Linear chain
  • Linear chain with two different masses
  • Einstein model      W
  • Debye model      W
  • fcc with linear springs to nearest neighbors
  • bcc with linear springs to nearest neighbors
• Thermodynamic properties of phonons
• Dispersion relation
• Density of states
• Internal energy density
• Specific heat
• Helmholtz free energy density
• Entropy
• Table summarizing the thermodynamic properties of phonons
• Kinetic theory
• Thermal conductivity
• Fermi's golden rule     W

Reading
Kittel chapter 4: Crystal vibrations or R. Gross und A. Marx: Dynamik des Kristallgitters
Kittel chapter 5: Phonons and lattice vibrations R. Gross und A. Marx: Thermische Eigenschaften des Kristallgitters

For the exam
• Be able to write down Newton's law for a periodic arrangement of atoms connected by linear springs. Know the form of the eigenfunction solutions.
• Be able to draw the dispersion relation for crystals such as Ag, NaCl, or TiO₂. There are always three acoustic branches. The acoustic branches are linear near k = 0. There are 3p - 3 optical branches where p is the number of atoms in the primitive unit cell. All acoustic and optical branches meet the Brillouin zone boundary at 90°.
• Know how to calculate the density states and from the density of states how to calculate the internal energy, Helmholtz free energy, specific heat, and entropy.
• Be able to define: Einstein model, Debye model, and Umklapp scattering.
• Be able to describe how the phonon dispersion relation can be measured with neutron scattering.
• Be able to describe how kinetic theory can be used to describe the phonon contribution to the thermal conductivity.

Student Presentations
• Berechnung der Phononen Dispersionsrelation am Beispiel eines FCC Gitters - Richard Romirer-Maierhofer, 2009
• Die Quantisierung einer Gitterschwingung - Patrick Kraus, 2008
• Phononen: Thermische Eigenschaften - Markus Neuschitzer und Harald Spreitzer, 2008
• Phonon dispersion relations for fcc crytals with one atom per unit cell - Philipp Thaler, 2009
• Thermische Eigenaschaften 2 - Patrick Mayrhofer und Hanno Buchner, 2009

Resources
Table summarizing the properties of phonons
Phonon software
Using complex numbers to describe oscillations
International Tables for Crystallography: Phonons
Phonon dispersion
Java Phonon Applet
Program to calculate phonon dispersion of fcc crystals The solid state simulation project
Ashcroft and Mermin: Chapters 22-26
A recent article on phonons: Phonon spectrum and specific heat of silicon nanowires, Y. Zhang, J. X. Cao, Y. Xiao, and X. H. Yan, Journal of Applied Physics 102,104303 (2007).
Weißmantel und Hamann: 5.2 Gitterdynamik des Festkörpern, 5.3 Spezifische Wärmekapazität von Festkörpern