Brillouin zone, rectangular lattice a two-dimensional metal has one atom of valency one in a simple rectangular primitive cell a = 2 A; b = 4 A. 

(a) Draw the first Brillouin zone give its dimensions, in cm^-1. 

(b) Calculate the radius of the free electron Fermi sphere, in cm^–1. 

(c) Draw this sphere to scale on a drawing of the first Brillouin zone, make another sketch to show the first few periods of the free electron band in the periodic zone scheme, for both the first and second energy bands. Assume there is a small energy gap at the zone boundary.