PHY.K02UF Molecular and Solid State Physics

## The first Brilluoin zone of a body 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 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 $\large \overline{\Gamma N}=\frac{\sqrt{2}\pi}{a},\,\overline{\Gamma P}=\frac{\sqrt{3}\pi}{a},\,\overline{\Gamma H}=\frac{2\pi}{a}$ 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:

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

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.