Experimental Physics V (Solid State Physics)
WS 2021/2022
Sheet 8
Exercise 1:
The phonon density of states per branch is given for small k-vectors with: 𝑫(𝝎) =
𝑽∙𝒌𝟐
𝟐𝝅𝟐 ∙
. Calculate the
𝒅𝝎
𝒅𝒌
phonon state-density per volume V in aluminum with the following sound velocities: longitudinal vl =
6.32×103 m/s and transversal: vt = 3.1×103 m/s. Determine the total density of states up to a frequency limit of  = 1013 Hz. Consider that there are two transversal and one longitudinal mode. (4P)
Exercise 2:
The dispersion relation of an optical phonon branch in 3D near the -point can be written as: 𝜔(𝑘) = 𝜔0 −
𝐴𝑘 2 . Show that the phonon density of states is then given by: 𝐷(𝜔) = {
𝐿
3
( ) (
2𝜋
2𝜋
𝐴3/2
) √(𝜔 − 𝜔0 )
0
At what conditions can a singularity in the phonon density of states be expected? (4P)
𝑓ü𝑟 𝜔 < 𝜔0
𝑓ü𝑟 𝜔 > 𝜔0
.
Exercise 3:
The Webpage: http://lampx.tugraz.at/~hadley/ss1/phonons/table/dos2cv.html provides useful insights into the interplay between the phonon density of states, the dispersion relation and the heat capacity. Explore the possibility of the
program to better understand the physics behind. Calculate the Debye temperature for Ag, Al and Ta by fitting Debye’s T3 law at low temperature to the provided tables. Use the density of the materials to find an expression for the
particle number N (details follow in the next lecture). (3P)
Return to your tutors until 20. December 2021