Hexagonal space lattice the primitive translation vectors of the hexagonal space lattice may be taken as

a₁ = (3^1/2 a/2)x + (a/2)y       ;                        a₂ = – (3^1/2 a/2)x + (a/2)y ;      a₃ = cz

(a) Show that the volume of the primitive cell is (3^1/2 /2) a²c.

(b) Show that the primitive translations of the reciprocal lattice are 

b₁ = (2π/3^1/2 a)x + (2π/a)y   ;                       b2 = – (2π/3^1/2 a)x + (2π/a) y ; b3 = (2π/c)z,

so that the lattice is its own reciprocal, but with a rotation of axes.

(c) Describe and sketch the first Brillouin zone of the hexagonal space lattice.