PHY.F20 Molecular and Solid State Physics

The first Brillouin zone of a face centered cubic lattice

    

$\large \vec{k}=u\vec{b}_1+v\vec{b}_2+w\vec{b}_3$ : $(u,v,w)$

  Symmetry points  (u,v,w)  [kx,ky,kz]  Point group    
  Γ: (0,0,0)    [0,0,0]

m3m

  X: (0,1/2,1/2)    [0,2π/a,0]  

4/mmm

  L: (1/2,1/2,1/2)    [π/a,π/a,π/a]  

3m

  W: (1/4,3/4,1/2)    [π/a,2π/a,0]  

42m

  U: (1/4,5/8,5/8)    [π/2a,2π/a,π/2a]  

mm2

  K: (3/8,3/4,3/8)    [3π/2a,3π/2a,0]  

mm2

  Symmetry lines    Point group  
  Δ: (0,v,v)  0 < v < 1/2  

4mm

  Λ: (w,w,w)  0 < w < 1/2  

3m

  Σ: (u,2u,u)  0 < u < 3/8  

mm2

  S: (2u,1/2+u,1/2+u)  0 < u < 1/8  

mm2

  Z: (u,1/2+u,1/2)  0 < u < 1/4  

mm2

  Q: (1/2-u,1/2+u,1/2)  0 < u < 1/4  

2

The real space and reciprocal space primitive translation vectors are:

\begin{equation} \large \vec{a}_1=\frac{a}{2}(\hat{x}+\hat{z}),\quad \vec{a}_2=\frac{a}{2}(\hat{x}+\hat{y}),\quad\vec{a}_3=\frac{a}{2}(\hat{y}+\hat{z}),\\ \large \vec{b}_1=\frac{2\pi}{a}(\hat{k}_x-\hat{k}_y+\hat{k}_z),\quad \vec{b}_2=\frac{2\pi}{a}(\hat{k}_x+\hat{k}_y-\hat{k}_z),\quad\vec{b}_3=\frac{2\pi}{a}(-\hat{k}_x+\hat{k}_y+\hat{k}_z). \end{equation}

The first Brillouin zone of an fcc lattice has the same shape (a truncated octahedron) as the Wigner-Seitz cell of a bcc lattice. Some crystals with an fcc Bravais lattice are Al, Cu, C (diamond), Si, Ge, Ni, Ag, Pt, Au, Pb, NaCl.