A Allowed k vectors in crystals


I am having troubles with understanding the allowed k vectors in a crystal.
Bloch's theorem gives us discrete energy bands for each wave vector k.

However, only discrete k vectors are allowed. Using periodic boundary conditions, the discreteness is easy to show.
But I am having a hard time in understanding how the discreteness follows from the condition that the wave amplitude must be zero at the boundaries of a crystal. Is the latter even correct? The electrons are confined to a Lx*Ly*Lz cube so that their amplitude should drop to zero at the boundaries. Can anyone offer a proof in how that translates to discrete k vectors? The volume boundaries are a continuum, so I don't really know how that would work.



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I think many things related to crystalline nature of the solid will break down at the boundary. The reason one usually ingores it is because there is a more bulk than surface. So I would not worry about the surface - its far away. The electrons in solids have finite coherence length, so in practice no electron actually 'feels' the whole crystal. Only its immediate neighbourhood. In some cases, such as mesoscopic superconductivity, this is no longer true and there you do see surafce and finite size effects affecting the material properties, e.g. the critical temperature of superconductors begins to drop once they get smaller than their coherence length.
There are two issues here that are essentially unrelated.:

1) the limitation to discrete values of k just means you have a finite piece of material. Only an infinitely large box allow truly continuous k. This usually just provides a convenient way to normalize things and does not affect the physics although for long wavelengths or small objects care must be taken. Not really the issue.

2)band structure arises because of the periodic nature of solids. It is relatively easy to show using Bloch's theorem that solutions for electrons in a periodic potential will propagate well only for a limited subset of k. (look up Dirac's Comb potential). This arises because of resonant reflection and transmission of different wavelengths (much like multi-layer optical filters for light). Of course in 3D with real atoms life is more complicated. but the fundamental physics is not .

So do not conflate the two issues......

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