Is the Condition for an Energy Gap Equivalent to Bragg Refraction?

[painfive]

I'm asked to prove that the condition for the existence of an energy gap at the boudnry of the Brillouin zone on a 1D lattice is equivalent to the condition for Bragg refraction. I don't understand this question. Doesn't an energy gap arise solely when there is a potential? Meanwhile, Bragg refraction only depends on the lattice structure, the potential is irrelevant. What am I missing here?

 

[Physics Monkey]

If there isn't a potential, then the electrons can't see the positive ions hence no scattering. The point of the Bragg formula is that it allows you extract the diffraction pattern without knowing the details of the potential. It follows just from the periodicity of the lattice and the fact that the sample is very large. The details of the scattering are encoded in the structure factors. These structure factors can modify the brightness of diffraction spots (perhaps even making some special spots vanish) but they can't influence the location of the spots.