Number density of LO and LA phonons as a function of temperature? I'd like to know the how the number density of longitudinal optical (LO) and longitudinal acoustic (LA) phonons varies as a function of temperature of the material. Is there a simple expression for these two cases?
I'm guessing that this would work,
$N_{LO} = \int g_{LO}(E) f(E, T) dE$
$N_{LA} = \int g_{LA}(E) f(E, T) dE$
where $g(E)$ is the density of states for LO and LA phonons and $f$ is the Bose-Einstein distribution. What would be appropriate limits for the integrals. Does anybody know a reference where the density of states for these two modes is given?
EDIT
To improve the question, I'm interested in semiconductor 3D crystals. But maybe I left this too long, sorry.
Best regards,
 A: Different types of phonons can not be considered as separate systems. They are oscillations of the same crystal and interact with each-other.
For example, LO phonon lifetime is about $10^{-12}$ - $10^{-11}$ seconds while the period of the oscillations is about $10^{-13}$ seconds (GaAs). At the end it turns into two LA phonons that run opposite directions.
Bose-Einstein distribution describes thermodynamical equilibrium of the whole phonon system. You should integrate over all the modes.
The density of states can be estimated numerically or measured experimentally.  Both usually give similar results that can be found e.g. in chapter 3 of "Fundamentals of Semiconductors" by Peter Y. Yu and Manuel Cardona.
The main experimental techniques are


*

*neutron scattering

*scattering of "hard" X-rays

*Raman spectroscopy.

A: LA phonons have $$E=\hbar\omega=\hbar c k$$ where $c$ is the speed of (longitudinal) sound, and so have a density of states exactly like that of photons (with a different value of she speed, and a factor of 1/2 as there is only one polarization states) e.g. $$g(E)=V(\hbar c)^{-3}2^{-1}\pi^{-2} E^2$$ and this is only rigorously true for low values of $k$ or $E$.  And, there are only $N$ modes.  A common model is to assume that there are only the number of $k$'s so that there are $N$ models but that the form for $g$ is otherwise exact.  This "Debye" model is explained reasonably well on wikipedia.
LO phonons are a different story.  Here the simple model is that the have a single frequency and there are $N$ of them, that is a more-or-less Einstein model (also explained on wikipedia.)
