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I am quite confused about the different representations of the dispersion relation in a lattice. enter image description here

This image makes a lot of sense to me, since it only represents one dispersion curve and transforms it back to the 1st-Brillouin-Zone due to the periodicity of the lattice in the Bloch theorem.

But sometimes, the dispersion relation is also showed in the so called "periodic zone scheme" as follows:

enter image description here

In this one, there are a lot of dispersion curves, what would make me think there are infitenely many wave-functions with the same energy, but different reciprocal wave vectors $k=k+n \cdot \frac{\pi}{a}$, what would lead to an infinity number of possible states in each band. What am I getting wrong?

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  • $\begingroup$ Well, the next atom over sees that things look just the same as the first atom... $\endgroup$
    – Jon Custer
    Feb 5 at 13:27
  • $\begingroup$ Yes, but that is not in position space. It is in the fourier space. $\endgroup$
    – Kubrik
    Feb 5 at 16:46
  • $\begingroup$ Indeed - the picture mixes up the bases, which makes it pretty useless. $\endgroup$
    – Jon Custer
    Feb 5 at 17:02
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    $\begingroup$ The states outside of the first Brillouin zone are the same states as those in the first Brillouin zone. That is, the state labeled by $k$ and $n$ (a band index) in the periodic zone scheme is the same as the one that has a value $k+G$, where $G$ is the reciprocal lattice vector that takes $k$ back into the first Brillouin zone. So there aren't more states. Instead, this picture can be useful for processes like Umklapp scattering, in which the scattering process results in a wave vector outside the first Brillouin zone. $\endgroup$
    – march
    Feb 5 at 17:22

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The periodic zone scheme is redundant, as it shows the same states multiple times. All of the same band structure information is contained within the reduced zone scheme (fig.(a) in your first image).

Piggybacking off of the comment by march on your post: The periodic zone scheme could be useful for Umklapp processes, where scattering between different Brillouin zones occur.

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