I've been reading Kittel's book on Solid state physics and while it's been mostly smooth sailing, the abrupt loss of rigour in places in unsettling. In particular, the bits about scattering seem to be just thrown in here and there without any rigourous mathematical treatment at all. He talks about electron-phonon , phonon-phonon, and x ray - phonon scattering processes. What exactly are the mechanisms behind these processes. How and why does this kind of scattering occur. Further, how does phonon and electron scattering off a lattice imperfection occur? Surely, there must be some kind of theory behind scattering which makes all this rigourous. Secondly, he says that phonon-phonon scattering processes can only occur by an anharmonic interaction. I'm not sure if i've understood this. What exactly does this mean ? An lastly, Umklapp scattering. I get how the mathematics behind the whole things works out, but if you're claiming that momentum isn't conserved in a certain process, shouldn't you identify the force in play that makes it happen? Shouldn't there be some kind of 'recoil' or 'force' associated with the addition of a reciprocal lattice vector G. The book i'm using doesn't identify any such 'force' or 'recoil'. Thanks for all the help. Greatly appreciated. :)
No. I haven't. I've heard it's a much better treatment of the subject though. Would you recommend it highly ?
For Umklapp scattering, the assumption is usually that the "missing" momentum is transferred onto the crystal as a whole. For an (near) infinite crystal the mass is (near) infinite, such that the recoil velocity is (near) zero, and also the kinetic energy transferred to the crystal as a whole is (near) zero (p^2/2m).
But then wouldn't enough of these phonon-phonon U-processes cause the crystal as a whole to recoil with an appreciable velocity? Or an appreciable increase in the kinetic energy of the crystal.
I also recommend very much Ashcroft and Mermin. A phonon is an eigenstate of the harmonic part of the hamiltonian, thus phonon phonon scattering must be due to anharmonic terms. You can check this expanding e.g. a Q^3 term in terms of phonon creation and anihiliation operators. Concerning Umklapp scattering, true momentum is conserved in these processes and is taken up by the crystal, however, for an infinitely large crystal, the change in energy is negligible. On the other hand, crystal momentum isn't conserved, but it is a completely different quality than true momentum.