We give some intuition today about when you should expect the Wiedemann-Franz ratio (which relates the electrical to the thermal conductivity in a metal) to hold, and when you should expect a deviation from the ratio we calculated for free electrons. (The ratio holds at low and high temperatures, but deviates from the free electron picture in between.) We also introduce a crystalline lattice into our free electron box today. We'll do this through adding the lattice in perturbatively (that is, pretend the Coulomb attraction between electron and lattice is very weak compared to the Fermi energy of the free electron system). We find this causes band gaps to open in the electronic energies at the Brillouin zone boundaries. These are caused by the formation of standing electronic waves due to reflections in the crystal. Lecture Audio

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