PHY.K02UF Molecular and Solid State Physics

## The first Brillouin zone of a tetragonal lattice

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

 Symmetry points $(u,v,w)$ $[k_x,k_y,k_z]$ $\Gamma:\,(0,0,0)$ $[0,0,0]$ $X:\,(\frac{1}{2},0,0)$ $[\frac{\pi}{a},0,0]$ $M:\, (\frac{1}{2},\frac{1}{2},0)$ $[\frac{\pi}{a},\frac{\pi}{a},0]$ $Z:\, (0,0,\frac{1}{2})$ $[0,0,\frac{\pi}{c}]$ $R:\, (\frac{1}{2},0,\frac{1}{2})$ $[\frac{\pi}{a},0,\frac{\pi}{c}]$ $A:\, (\frac{1}{2},\frac{1}{2},\frac{1}{2})$ $[\frac{\pi}{a},\frac{\pi}{a},\frac{\pi}{a}]$ $\overline{\Gamma X} = \overline{ZR}= \overline{MX}= \overline{AR} = \frac{\pi}{a}$ $\overline{\Gamma Z} = \overline{MA}= \overline{XR}= \frac{\pi}{c}$ $\overline{\Gamma M} = \overline{ZA}= \frac{\sqrt{2}\pi}{a}$ $\overline{\Gamma A} = \frac{\pi}{ac}\sqrt{2c^2+a^2}$ $\overline{\Gamma R} = \frac{\pi}{ac}\sqrt{c^2+a^2}$ Symmetry lines $\Delta :\,(v,0,0)$ $0\lt v\lt\frac{1}{2}$ $\Sigma :\,(v,v,0)$ $0\lt v\lt\frac{1}{2}$ $Y :\,(\frac{1}{2},v,0)$ $0\lt v\lt\frac{1}{2}$ $\Lambda :\,(0,0,v)$ $0\lt v\lt\frac{1}{2}$ $U :\,(v,0,\frac{1}{2})$ $0\lt v\lt\frac{1}{2}$ $S :\,(v,v,\frac{1}{2})$ $0\lt v\lt\frac{1}{2}$ $T :\,(\frac{1}{2},v,\frac{1}{2})$ $0\lt v\lt\frac{1}{2}$ $V :\,(\frac{1}{2},\frac{1}{2},v)$ $0\lt v\lt\frac{1}{2}$ $W :\,(\frac{1}{2},0,v)$ $0\lt v\lt\frac{1}{2}$

The real space and reciprocal space primitive translation vectors are:

$\large \vec{a}_1 = a\hat{x}$  $\large \vec{a}_2 = a\hat{y}$  $\large \vec{a}_3 = c\hat{z}$,

$\large \vec{b}_1 = \frac{2\pi}{a}\hat{k_x}$  $\large \vec{b}_2 =\frac{2\pi}{a}\hat{k_y}$  $\large \vec{b}_3 = \frac{2\pi}{c}\hat{k_z}$.