Trying to understand the Brillouin zone

[phrygian]

I am having a really hard time understanding how the Brillouin zone relates to electron states and have a couple of questions that might help clear it up for me.

For a band structure like this:

https://wiki.fysik.dtu.dk/gpaw/_images/silicon_banddiagram.png

I know that the different symbols on the bottom correspond to different points in the Brillouin zone. Gamma is the center of the Brillouin zone with K=0, so how can an electron with that wavevector have any energy? And why does Gamma appear on the bottom twice?

Do these diagrams mean that when an electrons wavevector increases in magnitude it also must point in a different direction? And what do these states mean, that the electrons are plane waves traveling in one direction?

Thanks a lot for the help, it would really be great to finally understand these concepts.

 

[DrDu]

The wavevector only describes the variation of the phase going from elementary cell to elementary cell. But this does not exclude a variation of phase or more generally wavefunction (and accompanying kinetic energy) within an elementary cell. E.g. think about a crystal made up of helium atoms. The energy of the lowest band is approximately equal to that in a free helium atom irrespective of the phase (k vector) with which the wavefunctions of different atoms are superposed.