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.