Is crystal momentum really momentum? Almost every solid state physics textbook says crystal momentum is not really physical momentum. For example, phonons always carry crystal momentum but they do not cause a translation of the sample at all.
However, I learned that in indirect-band-gap semiconductors, we need phonons to provide the crystal momentum transfer to make happen electron transitions between the top of the valance band and the bottom of the conduction band. Along with absorbing or emitting photons, of course.
Photons do carry physical momentum. For the purpose of momentum conservation, it seems that phonons do carry physical momentum as well.
How can we explain this?
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To put it more specifically, I drew a graph to tell the story:

$K$ (capital) is crystal momentum.
For such transition, photon provides most of the energy transfer (and a little momentum transfer $hk$, $k$ in lower case), phonon provides most of the momentum transfer (and a little energy).
Similar graphs can be found in most solid state physics textbooks. The picture tells me, either the photon participating in the transition carries crystal momentum, which value is equal to physical moemntum hk, or the crystal momentum itself is a kind of physical momentum.
However, one can prove that a phonon does not carry physical momentum (here I quote Kittel's "Introduction to Solid State Physics"):

So, how do we explain the momentum transfer in the electron transition aforementioned?
 A: Crystal Momentum can be seen as "less than momentum", that holds a part of the information from what real momentum usually tells, whether or not the other part makes any sense. 
Once wave function values is of interest only on lattice sites, for example, as in case of phonons, what matters is how phase changes from one site to another and not elsewhere in space. In such case, 2pi jumps are meaningless, so it is known "up to addition of a reciprocal latice vector"
A: I don't see why the crystal should move as a whole. A first argument is the one given by Marek in the comments of the question. But that's also the very principle of a wave: transfer of energy/momentum and no transfer of matter; in this case the ions will vibrate around their equilibrium position in the lattice and the modes of this vibration if a phonon.
Considering this "real momentum"-ness whatever it means this looks strange to me: consider the absorption of a photon (carrying momentum and energy) by the crystal. This process can happen due to the creation of a phonon, or more generally a bunch of them, creating a wave packet. This wave packet will propagate in the crystal (seen as a medium, from the point of view of the phonon this is equivalent to the vacuum from the point of view of the photon) and conserve the momentum.
More generally I think this question is not really interesting from a physical point of view, it is the same as asking if the phonons are real particles or not. I will depend on what you call "real" and following a pragmatic physicist way we call them particles because they have properties that more "naturally real" particles have and they can be treated in the same way...
Edit
I also find that proof a little odd. A real propagating phonon (or plasmon, magnon, whatever) must be seen as a wave packet. In that case the proof given will show that this packet does carry momentum.
A: First (as already pointed out in my comments) crystal momentum is in every regard the same as usual momentum except the fact that it takes values only in Brillouin zone (as a consequence of discrete symmetry of lattice; or more precisely its continuum limit). So the answer to your question is: for most purposes crystal momentum is a real momentum.
Now, the term crystal momentum is being used here in two different meanings and that is probably where confusion arises. Your citation uses it as a total momentum of the crystal. This is obviously zero for phonons (which are just harmonic modes of the material) because on average the atoms of the crystal don't move (they just oscillate around stable positions). And that is precisely why nobody uses the term in this way (and I don't understand why your book does).
But locally energy and momentum are still being transfered (jumping from one atom to the next as they interact). So in fact, phonon is a wave that propagates in the material in some direction and carries some energy. Obviously this is a very physical wave with physical energy and physical momentum. It is this latter momentum which is usually referred to as crystal momentum.
A: I'm not an expert, but I dont see any contradiction between following statements.


*

*Crystal momentum is not really a physical momentum.

*Phonons carry crystal momentum.

*Phonons do carry physical momentum as well.

A: Cristal momentum is not a real momentum because this is not an eingevalue of the hamiltonian. Infact you define the cristal momentum according to the periodicity of the bravais lattice. Then the sistem is invariant only under discrete traslation and then k is not a good quantic number.
