I have a hazy idea that comes from a few video lectures (see below for example) where the following is stated:

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.

Thanks for the post! This is an automated courtesy bump. Sorry you aren't generating responses at the moment. Do you have any further information, come to any new conclusions or is it possible to reword the post?