1. Repeat for a square lattice the sketch of the first four Brillouin zones for a rectangular lattice (Figure 2.10).
2. The two-dimensional square crystal contains two electrons per unit cell. If they can be described as free electrons, they occupy a Fermi circle in k-space whose area is equal to that of the first Brillouin zone. Superpose this circle on the reciprocal lattice and show that parts of the first two zones are occupied. Sketch the Fermi line on these zones after they have been reduced to the square around the origin.
Repeat this for the case of four electrons per unit cell, and show that the first four zones contain electrons. This sort of construction is used to explain the properties of some metals that can be described with the nearly free electron model.