(a) Alkali metals adopt a BCC structure. Its first Brillioun zone is sketched in the figure below. Reproduce the sketch and mark in the special points, r(0,0,0), H(1,0,0), N(0.5,0.5,0), and P(0.5,0.5,0.5) [3 marks] Z (b) The closest BZ surface passes through one of the three points above, H, N or P. Show that the closest BZ surface is at a distance of 0.707 x 2r/a away from the zone centre. [5 marks] (c) The density of states of a 3D free electron gas is given below. Derive an expression for the Fermi vector, KF, of a free electron gas in 3D in terms of N and V, where N is the total number of electrons and V is the volume of the sample. [10 marks] V 3/2 g(e) - 2 / 2 (m )