Hexagonal close-packed structure. Consider the first Brillouin zone of a crystal with a simple hexagonal lattice in three dimensions with lattice constants a and c. Let Gc denote the shortest reciprocal lattice vector parallel to the c axis of the crystal lattice. (a) Show that for a hexagonal close-packed crystal structure the Fourier component U(Gc) of the crystal potential U(r) is zero. (b) Is U(2Gc) also zero? (c)Why is it possible in principle to obtain an insulator made up of divalent atoms at the lattice points of a simple hexagonal lattice? (d) Why is it not possible to obtain an insulator made up of monovalent atoms in a hexagonal-close-packed structure?