# 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,

• Sure seems right to me. Of course thee Bose-Einstein distribution allows occupation numbers greater than one. Jun 9, 2011 at 20:05
• Question number one: do you want experimental or theoretical results? Question number two: what kind of materials? Pure crystals, crystals with defects, arbitrary solids? Jun 10, 2011 at 8:56
• @Marek; It seems to me that the equations should work irrespective of the material. Can you correct me on this? I'm thinking that they would simply have different $g(E)$. Jun 12, 2011 at 4:33
• @Carl: are you saying the temperature dependence for any material is only present in the Bose-Einstein part of the formula? This statement seems quite unobvious and is probably wrong too. Jun 12, 2011 at 8:08
• It's never too late to try for an answer ;-) Feb 1, 2012 at 4:51

## 2 Answers

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

• This answer is technically correct, but if the interaction is weak enough (as I believe it will be for this case), you can describe the optical phonons and the acoustic phonons both with separate Bose-Einstein distributions for free fields and not be too far off. The question is not whether they interact, but whether the average energy of an optical phonon in the thermal background of acoustic phonons is going to be changed significantly (and vice versa). My impulse is that opticals are absent at reasonable temperature, while long-wavelength acoustic phonons have a near-perfect Debye spectrum. Feb 2, 2012 at 12:24
• @Maksim, Would suggest that you include neutron scattering with appropriate application of the incoherent approximation rather than solely Raman spectroscopy, since not all optical phonon modes are Raman-active.
– Jen
Feb 2, 2012 at 16:58
• @RonMaimon, I have never heard anything about quasi-equilibrium state of optical phonons. In order to let this equilibrium exist there should be some mechanism of LO phonon - LO phonon interaction with characteristic time much shorter then the decay time. This is an interesting question. Feb 2, 2012 at 17:38

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.)