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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2$\begingroup$ Related (and possible duplicates): Is crystal momentum really momentum?, Crystal Momentum in a Periodic Potential. $\endgroup$– lemonCommented Mar 17, 2016 at 8:45
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$\begingroup$ 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. $\endgroup$– CoffeeIsLifeCommented Mar 17, 2016 at 9:02
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$\begingroup$ 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? $\endgroup$– garypCommented Mar 17, 2016 at 12:28
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$\begingroup$ @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. $\endgroup$– CoffeeIsLifeCommented Mar 17, 2016 at 12:55
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$\begingroup$ 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. $\endgroup$– garypCommented Mar 17, 2016 at 13:16
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2 Answers
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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.
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1$\begingroup$ This is also how Nyquist frequency works. $\endgroup$ Commented Mar 27, 2016 at 20:03
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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.
