Friday, March 15, 2013

Motivations for learning the reciprocal lattice

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...


  1. Is this how Sheldon works out the linear spectrum of graphene at half filling?

    How does filling factor fit in? Maybe I just need to think about it more.

  2. I have trouble imagining the motivating part of this, since usually at this point in the development of the topics, electron waves within the crystal are still some ways off. That usually click in after Bloch's theorem, no?

    1. I agree that this may not be that motivational for students.
      Kittel does crystal structures, then phonons, then the Sommerfeld model, and then the Bloch model.

      But, I like and use the Ashcroft and Mermin sequence.
      First, one does the Drude and Sommerfeld model and then motivates the need to take into account the crystal lattice (Bloch model). One then does crystal structures and the reciprocal lattice. This then leads into the Bloch model.

    2. I also prefer Ashcroft and Mermin's approach. Starting out with crystals is abstract and, in my opinion, sets the tone in the wrong note.