Momentum conservation in phonon-photon scattering event

Question

I am following a discussion from Kittel´s Introduction to Solid State Physics in the subchapter ‚phonon momentum‘:

We have two different conservation laws of momentum in a crystal:

1. Total momentum is conserved (if we include all processes i.e. also the center of mass motion).
2. Crystal momentum (pseudo momentum) is conserved up until a reciprocal lattice vector $K_{AB}=K_A+K_B+G$, with $K$ the crystal momentum and $G$ a reciprocal lattice vector.

If a photon hits a crystal and gets reflected, the momentum change of the photon is according to bragg’s law (or Laue equation): $$k’=k +G$$ where $k$ is the initial momentum and $k‘$ is the outgoing momentum of the photon. $G$ is a reciprocal lattice vector. This means that the photon gives some momentum to the crystal, whose center of mass starts to move (since a momentum $\hbar G$ is transferred).

In the case that a phonon (lattice vibration) gets produced or absorbed in this scattering process, the conservation of momentum looks the following: $$ k‘=k+G+K$$ where K is the crystal momentum of the phonon.

My question is:

We said crystal momentum $K$ is not conserved. By above equation, then also $k‘$ (the real momentum of the photon) is not conserved. How can this be?

As an example: Let’s say we have two photons interacting with the crystal producing two phonons A and B. $$ k^‘_A=k_A+G+K_A$$ $$ k_B^‘=k_A+G+K_B$$

Now let’s say both phonons meet at a later point in time and a scattering event happens. Since crystal momentum is not a conserved quantity, the conservation of crystal momentum looks like: $$K_{AB}=K_A+K_B+G$$ so for example, if $K_A$ and $K_B$ would be large enough, this would be an umklapp process and the total crystal momentum $K_{AB}$ would be smaller then $K_A+K_B$. So where is the real momentum of the initial photon going? It cannot (again) change the center of mass motion, because phonon-phonon interactions are not external ‚forces‘, which are the only ones that can change the center of mass motion.

Comment: I think the problem is that some of the real momentum of the photon is converted into crystal momentum of the phonon. But why is this even possible if it does not conserve total momentum?

Answer 1

Crystal momentum is a quantum number, characterizing states in a specific crystal band (whether we speak of electrons/holes or phonons). That is $k$ and $k+G$ describe the same quantum state/excitation. While the analogy between crystal momentum and real momentum is useful and not completely unfounded, we have to keep in mind that the photon actually interacts with the whole crystal - the difference between the change in the crystal momentum and the momentum of the photon is the momentum transferred to the lattice as a whole, which is not accounted in the Block waves solutions.

Another way to think of its is that Bloch waves (and phonon modes) are excitations in the frame of reference where the crystal is at rest. Crystal speed as a whole changes when it is hit by a photon, but this does not change the energy spectrum of the internal excitations of the crystal.