So this wasn't really mentioned in the notes, but it popped up a few times in my paper, and is in chapter 25 'anharmonic effects in crystals' in the book. It's a bit of a step back to Brillouin Zones and interactions, and whilst a simple concept it comes across as quite counter intuitive!
So say we have two phonons that lie within the first brillouin zone, and they scatter off each other to produce a third phonon with a different wavevector k3. If this third wavevector lies within the first brillouin zone, then this is just a simple scattering situation. Umklapp scattering occurs when the wavevector lies outside of this zone. However, as any wavevector lying outside the FBZ can be equally well described by a wavevector lying inside of it, it is possible to have two phonons interact, and have a third phonon that appears almost reflected from the scattering. This is a bit clearer with this diagram here
http://en.wikipedia.org/wiki/File:Phonon_nu_process.png
Obviously this has implications about the phonons momenta, although at first glance I can't find anything related to conservation of it. Anyway, a little aside concerning phonons and scattering!
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That's pretty cool (although I did have to read your post, then look at the diagram, then read the post again :P ).
ReplyDeleteAnyway, with regards to the momentum, would it perhaps be that the momentum of the phonon would be the crystal momentum, and thus not need to be conserved?
Alternatively, for two phonons to scatter out of the FBZ, then they would have to have originated somewhere in the previous zone. refer to my absolutely horrible paint diagram...
https://www.dropbox.com/s/qxhew0ho6gs0f7f/scattering.png
or have I misunderstood something?
The way I understand it is that in the real world, with real crystals of finite size then the phonons will come from previous zones, or what not, and you could trace them back and have momentum conserved.
ReplyDeleteHowever, if we imagined idealised crystals that go off to infinity, perhaps then it might have some implications.
Also, I think crystal momentum is still conserved, even though it isn't classical mechanical momentum. It's defined in exactly the same way, except it's wave vector k relates to the lattice rather than in free space.