I am familiar with some of the definitions of crystal momentum and I am familiar with how it is related to Bloch's theorem. I also am familiar that crystal momentum is not the momentum of each electron. However, does crystal momentum have a physical meaning? Is it an observable quantity? Whose momentum is it? Electrons, phonons, or something else?
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2Related (and possible duplicates): Is crystal momentum really momentum?, Crystal Momentum in a Periodic Potential. – lemon Mar 17 '16 at 08:45
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I read these but I am not sure if they give a physical meaning. It appears that crystal momentum is the total momentum of a specific point inside a crystal from what I could infer from the two posts. But nowhere in the books that I have come across say something like this. Also I am not sure as to what such "point" would be. – CoffeeIsLife Mar 17 '16 at 09:02
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Why do you say "crystal momentum is the total momentum of a specific point inside a crystal". Drop that idea. You ask if it's observable. First tell me how you would observe "conventional" momentum, then I might be able to answer. That is, what kind of answer are you looking for? – garyp Mar 17 '16 at 12:28
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@garyp I am not believing that. What I am saying is that "crystal momentum...point inside a crystal" is what I think other people are saying in both posts. I do not believe that it is correct. I am looking as to where the crystal momentum is "distributed." Is it the momentum of a group of electrons, and is this group of electrons any different from other electrons in the system? I would measure conventional momentum by observing what "carries" momentum and by observing how it can be changed. – CoffeeIsLife Mar 17 '16 at 12:55
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In the simplest one-electron/periodic-potential model (the Bloch model), crystal momentum is carried by each electron individually. The electron carries the momentum. It can be changed by collisions or external force. Just like conventional momentum. It's also conserved, just like conventional momentum. – garyp Mar 17 '16 at 13:16
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@garyp - I think I'd disagree that the electron, and only the electron, carries the momentum. Although not dealing with electrons, one can make the distinction between normal and umklapp processes in phonon scattering - in normal the total initial and final crystal momenta are strictly equal, while for an umklapp scattering they differ by a reciprocal lattice vector - the rest must go to the crystal itself. – Jon Custer Mar 17 '16 at 14:26
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@JonCuster I suppose that's a point of view, but I might argue that two values of crystal momentum that differ by a reciprocal lattice vector are identical. Crystal momentum is defined only in the first Brillouin zone. Also, I don't see the need for the bulk crystal to take up any momentum. For example: $-\pi/a$ is the same reciprocal lattice vector as $+\pi/a$, so nothing special needs to be done to conserve crystal momentum when scattering one into the other. – garyp Mar 17 '16 at 14:58
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@garyp - I think that we are in agreement, but circling around a question more of language and 'intuition'. But, lets step back to a perhaps simpler problem - diffraction of a photon off a diffraction grating. The momentum of the photon has clearly changed because of the interaction with the grating (functioning as a 1D Bloch system). Does the momentum of the grating change? – Jon Custer Mar 17 '16 at 15:04
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@garyp - or put yet differently - in the crystal, the very act of defining it as a system solved by Bloch waves implicitly intertwines the momenta of the electron and the lattice such that they are conserved together. – Jon Custer Mar 17 '16 at 15:05
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@JonCuster A question remains for me. If an electron scatters from $\vec{k}$ to $\vec{k} + \vec{R}$ (a recip. lattice vector), the electron it has not made a transition to a new state. The state of the crystal is exactly the same. Does that mean that it's bulk momentum is also the same, or not? The elementary presentation considers a static external potential so the question doesn't arise. – garyp Mar 17 '16 at 15:38
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@garyp - somewhat surprisingly, Wikipedia has a good, short writeup on Crystal Momentum - one point in it to consider is that, since the crystal lattice has discrete symmetry, Noether's theorem does not apply, so crystal momentum is at best conserved as a 'quasi-momentum'. I'll have to ponder that in my spare time... – Jon Custer Mar 17 '16 at 16:02
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@garyp- The momentum of the crystal is usually ignored, but certainly in real life if an electron scatters off an ionic core the ion recoils as well, so momentum is conserved. It is only when one considers the crystal as a static external potential that translational symmetry is broken and momentum is not conserved. Basically the same story as throwing a ball against a wall and having it bounce back. – Rococo Mar 23 '16 at 02:32
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Crystal momentum is a modulo momentum. As said in the comments, momentum is somewhat difficult to observe, as we usually measure velocity and multiply by mass.
A good way to "visualize" crystal momentum is watching how spokes on a tire appear to precess when observed under 60 Hz streetlights. You could think of your wheel's "crystal momentum" by observing the apparent angular velocity and multiplying by the wheel's moment of inertia. The wheel is likely spinning much faster than it appears when under a strobe, but because of the period nature of its spokes, you only see the modulo velocity.
This effect shows up in crystals, where each crystal site looks almost the same, and your wave vector is "sampling" every spatial period.
Jonathan Wheeler
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I would only add to Jonathan's answer a picture (which I can't put as a comment):
(from Kittel, Solid State Physics)
In this image, think of the state as completely determined by the value of the wave at the solid points (which are like nuclei). Then it is clear that both waves, despite having different wavelengths, represent the same state. This is really a practical application of sampling theory.
Alternatively, if an electron in a crystal is prepared in a momentum eigenstate that is like one of these two waves, the crystal will scatter it into states like the other one, as well as other wavelengths that belong to this "class" of being the same at every black point. So the full momentum is not conserved, but this sort of "discrete momentum," which is the values of the waves at these points, is. And that's crystal momentum.
In a real crystal the nuclei are not points, and have an extended force, but as long as the potential is periodic this logic turns out to still work, as proved in Bloch's theorem.
Rococo
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