Relationship between Debye Temperature and Speed of Sound in Metals

[Hemmer]

Homework Statement

I'm struggling to understand the relationship between the Debye temperature and the speed of sound in a substance. An example problem given is:

Estimate the Debye Temperature of Silicon and Lead, given that their respective speeds of sound are 9150 m/s and 1320 m/s. (not sure if its relevant to this part of the question but also given graph of C_v vs T for Argon from which you can read a Debye temp of about ~80K).

Homework Equations

\Theta_D = \hbar \omega_D / k_b
where \omega_D = c k_D
and k_D is the radius of the "Debye Sphere"

The Attempt at a Solution

Not sure how to attempt this to be honest, there seems like there are too many unknowns. Presumably there is some simplifying assumption, but I'm not sure where to begin...

 

[captainjack2000]

you first need to find the cut off frequency wd by using the linear dispersion relation. For the Debye model w is proportional to k. At the long wavelength limit k=pi/a where a is the dimension of the unit cell. Having found w you can then substitute it back into your first equation for temperature.

 

[Hemmer]

you first need to find the cut off frequency wd by using the linear dispersion relation. For the Debye model w is proportional to k. At the long wavelength limit k=pi/a where a is the dimension of the unit cell. Having found w you can then substitute it back into your first equation for temperature.

Thanks yes this makes sense. I had funnily enough just worked it out 5 mins ago, I forgot that k could be found fairly easily by estimating a.