This semester I am teaching half of a solid state physics course at the level of Ashcroft and Mermin. This week I introduced crystal structures and started on the reciprocal lattice. I have taught this many times before, but this time I realised something basic that I have overlooked before.
In the past I motivated the reciprocal lattice as a way to understand how one determines crystal structures, i.e. one shines an external beam of waves (x-rays, electrons, or neutrons) on a crystal and one sees how their wavevector is changed by a reciprocal lattice vector of the crystal.
However, similar physics applies to electron waves within the crystal. They can also undergo Bragg diffraction. This is particularly relevant for band structures and understanding how band gaps open up at the zone boundaries.
In the past, I never mentioned the latter motivation when introducing the reciprocal lattice. It only came up much later when discussing band structures...