Question

Due to Umklapp phonon scattering, the conservation of momentum is broken. Why does the conservation of momentum break for Umklapp scattering of phonons?

One assumption could be that phonons are quasi-particles, and hence the conservation of momentum does not matter and does not apply. If this is true, what are the further explanations/implications of this assumption. If not, what are other possible explanations to describe the loss of the extra momentum that is not accounted for in Umklapp scattering?

Answer 1

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The crystal has a center of mass which can also move. lhe center of mass of the crystal can also have a velocity. Then the total momentum of the moving lattice is the total mass of the lattice times the velocity of the center of mass. The total momentum of the lattice is included in the "true momentum". Although the momenta of the phonons alone are not conserved, the momenta of the phonons plus the momenta of the crystal is conserved. In Umklapp scattering, the crystal lattice gains the negative of the reciprocal lattice added during the scattering event. So the "true momentum" is conserved. Almost always, the translation of the entire crystal is ignored. The displacement of the entire crystal due to phonon scattering is generally too small to measure anyway. However, the displacement of the crystal is theoretically real. The translation of the crystal is unimportant for calculating thermal resistivity. When calculating resistivity, one only has to acknowledge that Umklapp scattering takes place. It does make sense for a time averaged momentum and a time averaged lattice. If there isa phonon in the crystal, then the atoms are vibrating. However, the lattice vectors are not portrayed as vibrating. Each lattice point is the time averaged position of an atom or molecule, not an instantaneous position. The position has to be averaged over an integration time much larger than the period of the vibration. Similarly, the momentum of each atom in a vibrating crystal changes rapidly. However. the momentum of a phonon is constant between collisions. This only makes sense if the momentum of a phonon is time averaged over a very long time. Similarly, the center of mass has to be averaged over a long time compared to the periods of vibration. Therefore, the pseudomomentum of a phonon is the time averaged momentum of the vibrational mode modulo a reciprocal vector. In Umklapp scattering, the time averaged momentum of the phonons is not conserved. However, the time averaged center of mass changes momentum enough to compensate for the momentum of the phonons. Making the vibrational mode into a quasipaticle is like time averaging. The energy of a phonon is precisely determinable. So the uncertainty in time of the phonon has to be very long. Therefore, just in order to use the concept of quasiparticle one has to assume a very long integration time. So consider the lattice points as time averages of the position of averages. If the lattice moves at a constant velocity, it means that the time averaged position of the center of mass has that velocity. That is where the "real momentum" is going.