1. Confirm that the first Brillouin zone of a face-centred cubic crystal contains one state per spin per atom of the lattice.

2. Show that the angle between any two of the bonds between adjacent atoms in the diamond lattice is arccos

3. Show that the hexagonal faces of the Brillouin zone of a face-centred cubic crystal are regular hexagons; this is not obvious, as pointed out in Section 2.6.1, because there is only a threefold rotational symmetry about the [111]-axis. This also shows that all edges of the zone have the same length.