PHY.K02UF 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:

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

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.