PHY.K02UF Molecular and Solid State Physics

The first Brilluoin zone of a body centered cubic lattice

    

k=ub1+vb2+wb3k=ub1+vb2+wb3 : (u,v,w)(u,v,w)

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

m3m

  H: (-1/2,1/2,1/2)    [0,0,2π/a]  

m3m

  P: (1/4,1/4,1/4)    [π/a,π/a,π/a]  

43m

  N: (0,1/2,0)    [0,π/a,π/a]  

mmm

¯ΓN=2πa,¯ΓP=3πa,¯ΓH=2πa¯¯¯¯¯¯¯¯ΓN=2πa,¯¯¯¯¯¯¯¯ΓP=3πa,¯¯¯¯¯¯¯¯ΓH=2πa

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

4mm

  Λ: (w,w,w)  0 < w < 1/4  

3m

  Σ: (0,v,0)  0 < v < 1/2  

mm2

  F: (-1/2 +3w,1/2-w,1/2-w)  0 < w < 1/4  

3m

  D: (u,1/2-u,u)  0 < u < 1/4  

mm2

  G: (-u,1/2,u)  0 < u < 1/2  

mm2

The real space and reciprocal space primitive translation vectors are:

a1=a2(ˆx+ˆyˆz),a2=a2(ˆx+ˆy+ˆz),a3=a2(ˆxˆy+ˆz),b1=2πa(ˆkx+ˆky),b2=2πa(ˆky+ˆkz),b3=2πa(ˆkx+ˆkz).

The first Brillouin zone of an bcc lattice has the same shape (a rhombic dodecahedron) as the Wigner-Seitz cell of a fcc lattice. Some crystals with an bcc Bravais lattice are Li, Na, K, Cs, V, Cr, Fe, Nb, Mo, Rb, Ba, Ta.