Are Phonon Collisions Responsible for Quantized Lattice Vibrations?

[aaaa202]

Phonons are quantized lattice vibrations. They are traveling waves of the form
exp(ikr) taken to satisfy periodic boundary conditions. I'm guessing periodic boundary conditions are equivalent to demanding the waves to be zero at the boundary. which is what characterizes standing waves (right? :S)
So are all phonons really just the number of possible standing waves you can form from the solutions exp(ikr).
And if so, how is one to interpret phonon-phonon collisions? What happens physically when two standing wave vibrations collide and why can we speak of conservation of the crystal momentum?

 

[dauto]

No. Periodic boundary conditions are the unphysical conditions of requiring the wave function at opposite sides of the box to match each other. It is used for mathematical convenience.

Yes. the phonons are the normal modes of vibration.

Phonons collision happen because of non linear behavior of the medium. One phonon changes the configuration of atoms as it passes by them and the changed configuration affects the propagation of the other phonon.

 

[aaaa202]

So when you say: Yes the phonons are normal modes of vibrations? They are not standing waves in general? (frankly normal mode is not common nomenclature for me)

 

[dauto]

Pretty much the same thing. In a standing wave all atoms move in phase with each other. A normal mode is a more general concept which allows for other possibilities where different atoms are oscillating out of phase with each other, but still at the same frequency.

 

[aaaa202]

But surely that kind of exotic behaviour does not happen for plane waves exp(ikr) satisfying the boundary conditions?
And it seems that according to my book these are the only possible phonon modes you can have. At least for calculating the heat capacity these are the only ones considered as contributing - so if you can have all kinds of weird waves which satisfy periodic boundary conditions why is it only the eigenmodes of the system that contribute to the heat capacity?

 

[DrDu]

The collision of phonons is best considered in terms of wavepackets as any other scattering process.
The energetic width of these wavepackets is inversely proportional to the width of the packets in real space.
As long as this localization length is much smaller than the crystall dimension, the energetic splitting of the sin kx and cos kx standing wave solutions, which is inversely proportional to the crystal dimension L, is negligible and the wavepackets can be considered as being made up of traveling waves exp ikx instead.
This approximation will also break down for very long times during which the wavepackets travel to the crystal boundary and get reflected. But usually, scattering times are much shorter, so that this isn't important either.

 

[aaaa202]

but you too agree that phonons are in general normal modes right? It just seems there is much confusion about the term but this is what I came to conclude.
Also: The possible number of phonon modes are either found by imposing periodic or fixed end boundary conditions on the crystal - why are these procedures equivalent?

 

[DrDu]

Bulk properties do usually not depend on the boundary conditions chosen in the infinite volume limit. That is why usually periodic boundary conditions (Born - von Karman boundary conditions are chosen). Ashcroft and Mermin has some references on that topic.