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Hi,I have a problem for scattering matrix theory. Currently, the book I've read is about collision between free particles.

What if collision between non-free particles? For example, in lattice, only certain quasi-momentum of phonon satisfying Bloch equation. When two phonons collide, how to treat it?

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Yes, in principle only certain quasi-momentum are allowed. However, the spacing between allowed quasi-momenta will scale inversely with the number of atoms in the solid. Since most solids contain a lot of atoms, the spacing will be very tiny and the spectrum of quasi-momenta is treated as continuous.

The main difference between phonons and free particles is that momentum must be conserved, but quasi-momentum only has to be conserved up to a reciprocal lattice vector. This leads to umklapp scattering, which can lead to counter-intuitive results. For example two phonons can interact and scatter in the "wrong" direction (as shown in the link).

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  • $\begingroup$ What if the solids contain only very limited number of atoms? How to treat this case? $\endgroup$ – xiang sun Aug 18 '19 at 10:28
  • $\begingroup$ I mean for particles moving in finite lattice, where the momentum must be conserved $\endgroup$ – xiang sun Aug 18 '19 at 10:29

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