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PHY.K02UF Molecular and Solid State Physics

## 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.

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 that solve these equations.
• 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 dispersion relation and 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.

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