I Is there a crystal angular momentum?

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1. Apr 16, 2018

fluidistic

I've learned that in a crystal, the crystal momentum is conserved. When one considers the electrons as Bloch waves, they have a momentum that doesn't commute with the Hamiltonian and they have well definite energies, hence they cannot have a well definite momentum, because there is no basis in which both the Hamiltonian responsible for the energies and the momentum are diagonalized. Instead, the crystal momentum is what is conserved.
This conserved quantity does not come from a Noether-like theorem, because of the broken translational symmetry in a crystal. Only particular translations (by a Bravais lattice vector) yield an invariant system. However I note that some very particular rotations (only for some angles) would also keep the system invariant, while an arbitrary rotation would not, in general, leave the system invariant. This leads me to think that this property should be responsible for an angular crystal momentum. Is there such a thing? If so, why is it not commonly mentioned in solid state physics books? Why would the crystal momentum be more important than the angular crystal momentum, assuming that the latter exist?

2. Apr 17, 2018

DrDu

Ordinary momentum is still conserved, while crystal momentum is conserved only up to a multiple of a reciprocal lattice vector. Crystal momentum is a continuous quantity while rotations are quantized in discrete levels. The "conservation" of crystal angular momentum leads to a selection rule. The group theory behind this selection rules can get quite involved as rotations do not commute like translations and spin further complicates matters.

Edit: Personally I find crystal momentum to be quite an interesting concept. I don't think it is related to normal momentum. Rather, it is a consequence of the nuclear permutation group. However, I have never seen this discussed clearly in the literature.