$\vec{k}=u\vec{b}_1+v\vec{b}_2+w\vec{b}_3$   Symmetry points  (u,v,w)   [kx,ky,kz]   Point group       Γ: (0,0,0)     [0,0,0] 6/mmm   A: (0,0,1/2)     [0,0,π/c]   6/mmm   K: (2/3,1/3,0)     [4π/3a,0,0]   62m   H: (2/3,1/3,1/2)     [4π/3a,0,π/c]   62m   M: (1/2,0,0)     [π/a,-π/√3a,0]   mmm   L: (1/2,0,1/2)     [π/a,-π/√3a,π/c]   mmm $\overline{\Gamma A}=\frac{\pi}{c},\quad\overline{\Gamma K}=\frac{4\pi}{3a},\quad\overline{\Gamma M}=\frac{2\pi}{\sqrt{3}a},\quad\overline{MK}=\frac{2\pi}{3a}$   Symmetry lines     Point group     Δ: (0,0,w)  0 < w < 1/2   6mm   P: (2/3,1/3,w)  0 < w < 1/2   3m   U: (1/2,0,w)  0 < w < 1/2   mm2   Λ: (2v,v,0)  0 < v < 1/3   mm2   Q: (2v,v,1/2)  0 < v < 1/3   mm2   Σ: (u,0,0)  0 < u < 1/2   mm2   R: (u,0,1/2)  0 < u < 1/2   mm2   T: (1/2+v/2,v,0)  0 < v < 1/3   mm2   S: (1/2+v/2,v,1/2)  0 < v < 1/3   mm2

The real space and reciprocal space primitive translation vectors are:

$$\vec{a}_1=a\hat{x},\qquad\vec{a}_2=\frac{a}{2}\hat{x}+\frac{\sqrt{3}a}{2}\hat{y},\qquad\vec{a}_3=c\hat{z}$$ $$\vec{b}_1=\frac{2\pi}{\sqrt{3}a}\left(\sqrt{3}\hat{k}_x-\hat{k}_y\right),\qquad\vec{b}_2=\frac{4\pi}{\sqrt{3}a}\hat{k}_y,\qquad\vec{b}_3=\frac{2\pi}{c}\hat{k}_z$$

The first Brillouin zone of an hexagonal lattice is hexagonal again. Some crystals with an (simple) hexagonal Bravais lattice are Mg, Nd, Sc, Ti, Zn, Be, Cd, Ce, Y.

• Cut-out pattern to make a paper model of the hexagonal Brillouin zone.