PHY.K02UF Molecular and Solid State Physics

## The first Brillouin zone of a simple cubic lattice

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

 Symmetry points  (u,v,w) [kx,ky,kz] Point group Γ: (0,0,0) [0,0,0] m3m 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 $\large \overline{\Gamma X}=\frac{\pi}{a},$ $\large \overline{\Gamma M}=\frac{\sqrt{2}\pi}{a},$ $\large \overline{\Gamma R}=\frac{\sqrt{3}\pi}{a}$ 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:

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

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