| ||||

| ||||

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.

**Reading**

Read the Normal Modes and Phonons section of the outline.

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

- 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 TiO
_{2}. There are always three acoustic branches. The acoustic branches are linear near*k*= 0. There are 3*p*- 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

International Tables for Crystallography: Phonons