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I often see pictures like this in physics, this one for Silicon band structure.

enter image description here

(source, NB: it's the German page for Silicon).

There you see the plot of the energy in terms of the momentum $k$. Now, the problem is that we generally assume that we have periodic boundary conditions in a crystal, which means that $k$ is a discrete quantity.

On the other hand, in these drawings we always seem to have continuous values of $k$.

Now, either this is due to some problems in resolving the discreteness of $k$ in these plots or the theoretical idea of periodic boundary conditions is wrong.

Could anybody comment on this issue?

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The bigger the crystal is, the closer-together those discrete values of k are. A real crystal might be 1cm long, or 100 million atoms. Then there would be 100 million little dots between the vertical bars on your figure. Those dots would be equally spaced with respect to the horizontal axis. It's too many dots to see, they blend together to look just like a series of continuous smooth curves. So that's how people draw it. :-D

The values of k are discrete regardless of the boundary conditions (periodic vs realistic).

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