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PHY.K02UF Molecular and Solid State Physics

## The first Brillouin zone of a base centered orthorhombic lattice

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

 Symmetry points $(u,v,w)$ $[k_x,k_y,k_z]$ Point group $\Gamma:\,(0,0,0)$ $[0,0,0]$ mmm $Y:\, (\frac{1}{2},\frac{1}{2},0)$ $[\frac{\pi}{a},0,0]$ mmm $Y':\, (-\frac{1}{2},\frac{1}{2},0)$ $[0,\frac{\pi}{b},0]$ mmm $Z:\, (0,0,\frac{1}{2})$ $[0,0,\frac{\pi}{c}]$ mmm $T:\, (\frac{1}{2},\frac{1}{2},\frac{1}{2})$ $[\frac{\pi}{a},0,\frac{\pi}{c}]$ mmm $T':\, (-\frac{1}{2},\frac{1}{2},\frac{1}{2})$ $[0,\frac{\pi}{b},\frac{\pi}{c}]$ mmm $S:\, (0,\frac{1}{2},0)$ $[\frac{\pi}{2a},\frac{\pi}{2b},0]$ 2/m $R:\, (0,\frac{1}{2},\frac{1}{2})$ $[\frac{\pi}{2a},\frac{\pi}{2b},\frac{\pi}{c}]$ 2/m $\overline{\Gamma Y} = \frac{\pi}{a}$ $\overline{\Gamma Z} = \overline{YT}= \overline{SR}= \frac{\pi}{c}$ $\overline{\Gamma T} = \frac{\pi}{ac}\sqrt{a^2+c^2}$ Symmetry lines Point group $\Lambda :\,(0,0,w)$ $0\lt w\lt\frac{1}{2}$ mm2 $H :\,(\frac{1}{2},\frac{1}{2},w)$ $0\lt w\lt\frac{1}{2}$ mm2 $\Sigma :\,(u,u,0)$ $0\lt u\lt\frac{1}{2}$ mm2 $A :\,(u,u,\frac{1}{2})$ $0\lt u\lt\frac{1}{2}$ mm2 $\Delta :\,(-v,v,0)$ $0\lt v\lt\frac{(a^2+b^2)}{4a^2}$ mm2 $B :\,(-v,v,\frac{1}{2})$ $0\lt v\lt\frac{(a^2+b^2)}{4a^2}$ mm2 $F :\,(\frac{1}{2}-v,\frac{1}{2}+v,0)$ $0\lt v\lt\frac{1}{2}-\frac{(a^2+b^2)}{4a^2}$ mm2 $G :\,(\frac{1}{2}-v,\frac{1}{2}+v,\frac{1}{2})$ $0\lt v\lt\frac{1}{2}-\frac{(a^2+b^2)}{4a^2}$ mm2 $D :\,(0,\frac{1}{2},w)$ $0\lt w\lt\frac{1}{2}$ 2

The real space and reciprocal space primitive translation vectors are:

$\large \vec{a}_1 = \frac{a}{2} \hat{x}-\frac{b}{2}\hat{y}$  $\large \vec{a}_2 = \frac{a}{2} \hat{x}+\frac{b}{2}\hat{y}$  $\large \vec{a}_3 = c\hat{z}$,

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

Cut-out pattern to make a paper model of the body centered orthorhombic Brillouin zone.