PHY.K02UF Molecular and Solid State Physics

## The first Brillouin zone of a (simple) hexagonal lattice

$\vec{k}=u\vec{b}_1+v\vec{b}_2+w\vec{b}_3$

 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.