Difference between Debye frequency and highest phonon frequency

Question

In my lecture it says that the Debye frequency is the cutoff frequency. So is this also the highest phonon frequency? And if not, what is the difference?

A little background: I do have an exercise where I should say what the debye frequency of a monoatomic, cubic material with lattice constant $a$ and sound velocity $c$ is. I thought this is $$\omega_D^3 = \frac{6\pi}{a^3}\cdot v^3.$$ But in my exercise there is also given the highest phonon frequency $\omega$. So I wonder if there is a difference between those two things: debye frequency and highest phonon frequency.

Answer 1

It is easier to think of the shortest wavelength for phonons (the largest wavevector).

In the Debye model, it is assumed that there is no dispersion up to that wavevector, that the proportionality is given by the velocity of sound in the crystal. This is not true, the Debye frequency overestimates the highest frequency.

Often the Debye frequency is calculated from the experimentally determined Debye temperature.