# Homework Help: Thermal phonons @ low temperatures

1. Nov 13, 2012

### Liquidxlax

1. The problem statement, all variables and given/known data

a) estimate the number of phonons in a 1mm cube of a crystal at 10K, taking a sound speed of 1000m/s.

b) estimate the frequency ω for which you will find the highest phonon nrg content in the range of ω to ω+dω. (Again, assume low temps). Give your solution in the form hω = αkBT, finding the value of α.

c) show that the dominant phonon wavelength λ, assuming simple cubic lattice, can be written λ = βaθD/T with a being the lattice constant. Find β.

Problem i'm having is am I supposed to find an exact value as he has not stated. (Haven't been able to see my prof)

3. The attempt at a solution

Only thing i can think of for part a is

Nph = ∫dωD(ω)n

where n is the planck distribution and D is the density of states. Only problem is that this equation only uses the temperature

debeye frequency is ω3 = 6πv3N/V

I'm assuming N is the number of phonons which is no help.
1. The problem statement, all variables and given/known data

2. Relevant equations

3. The attempt at a solution