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The condition for constructive interference (Bragg condition) in reciprocal space is that the wave vector should touch a Brillouin zone boundary. In this case, the standing wave that is created opens a band gap. Is this a correct way of understanding opening of band gaps and if yes, can someone explain or point me to a reference that shows

*why this constructive interference condition results in a band gap*? I'm actually looking for intuition more than a proof to be honest.

The other way I've seen it done is using perturbation theory where the math is fine but there is no intuition on why a band gap occurs only near the edge of the Brillouin zone.

Here is a lecture that explains the connection between Bragg diffraction and the opening of a band gap but I still can't see why.