The first Brillouin zone of a simple cubic lattice

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

m3m

  R: (1/2,1/2,1/2)    [π/a,π/a,π/a]

m3m

  X: (0,1/2,0)    [0,π/a,0]

4/mmm

  M: (1/2,1/2,0)    [π/a,π/a,0]

4/mmm

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

4mm

  T: (1/2,1/2,w)  0 < w < 1/2  

4mm

  Λ: (w,w,w)  0 < w < 1/2  

3m

  Σ: (u,u,0)  0 < u < 1/2  

mm2

  S: (u,1/2,u)  0 < u < 1/2  

mm2

  Z: (u,1/2,0)  0 < u < 1/2  

mm2

The real space and reciprocal space primitive translation vectors are:

Cut-out pattern to make a paper model of the simple cubic Brillouin zone.