Potential energy in the diamond structure

(a) Show that for the diamond structure the Fourier component UG of the crystal potential seen by an electron is equal to zero for G = 2A, where A is a basis vector in the reciprocal lattice referred to the conventional cubic cell.

(b) Show that in the usual first-order approximation to the solutions of the wave equation in a periodic lattice the energy gap vanishes at the zone boundary plane normal to the end of time vector A.