(Solved): 2. Hexagonal space lattice. The primitive translation vectors of the hexagonal space lattice ma ...

2. Hexagonal space lattice. The primitive translation vectors of the hexagonal space lattice may be taken as ${\mathrm{a}}_1=\left(3^{1/2}a/2\right)\hat{\mathbf{x}}+(a/2)\hat{\mathbf{y}};{\mathbf{a}}_2=?\left(3^{1/2}a/2\right)\hat{\mathbf{x}}+(a/2)\hat{\mathbf{y}};{\mathrm{a}}_3=c\hat{\mathbf{z}}$ (a) Show that the volume of the primitive cell is $\left(3^{1/2}/2\right)a^{2}c$. (b) Show that the primitive translations of the reciprocal lattice are ${\mathbf{b}}_1=\left(2?/3^{1/2}a\right)\hat{\mathbf{x}}+(2?/a)\hat{\mathbf{y}};{\mathbf{b}}_2=?\left(2?/3^{1/2}a\right)\hat{\mathbf{x}}+(2?/a)\hat{\mathbf{y}};{\mathbf{b}}_3=(2?/c)\hat{\mathbf{z}}$ 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.