What happens when the wavevector of an electron lies on the Brillouin Zone boundary (BZB)?

Answer 1

The first Brillouin Zone encompasses all unique k-states (wavevectors) of an electron in the reciprocal space/lattice. To put it more physically, it contains all unique phonon (normal mode) vibrations of a crystal lattice. When the wavevector is at the Brillouin Zone boundary (k = ±π/a), the periodicity of the phonon in the crystal lattice will repeat, and the k-states will repeat again, but shifted by the reciprocal lattice vector π/a. At the B.Z. boundary, you will also have a bandgap/energy gap open up due to the wavevector k = ±π/a satisfying Bragg's law and leading to scattering.