PHY.K02UF Molecular and Solid State Physics

The first Brillouin zone of a simple 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

 $X:\, (\frac{1}{2},0,0)$   $[\frac{\pi}{a},0,0]$ 

mmm

 $Y:\, (0,\frac{1}{2},0)$   $[0,\frac{\pi}{b},0]$ 

mmm

 $Z:\, (0,0,\frac{1}{2})$   $[0,0,\frac{\pi}{c}]$ 

mmm

 $T:\, (0,\frac{1}{2},\frac{1}{2})$   $[0,\frac{\pi}{b},\frac{\pi}{c}]$ 

mmm

 $U:\, (\frac{1}{2},0,\frac{1}{2})$   $[\frac{\pi}{a},0,\frac{\pi}{c}]$ 

mmm

 $S:\, (\frac{1}{2},\frac{1}{2},0)$   $[\frac{\pi}{a},\frac{\pi}{b},0]$

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 $R:\, (\frac{1}{2},\frac{1}{2},\frac{1}{2})$   $[\frac{\pi}{a},\frac{\pi}{b},\frac{\pi}{c}]$

mmm

 

$\overline{\Gamma Y} = \overline{ZT}= \overline{XS}= \overline{UR} = \frac{\pi}{b}$

$\overline{\Gamma X} = \overline{YS}= \overline{ZU}= \overline{TR} = \frac{\pi}{a}$

$\overline{\Gamma Z} = \overline{YT}= \overline{SR}= \overline{XU}= \frac{\pi}{s}$

$\overline{\Gamma T} = \frac{\pi}{bc}\sqrt{b^2+c^2}$

$\overline{\Gamma U} = \frac{\pi}{ac}\sqrt{a^2+c^2}$

$\overline{\Gamma R} = \pi\sqrt{\frac{1}{a^2}+\frac{1}{b^2}+\frac{1}{c^2}}$

 
 Symmetry lines   Point group  
 $\Lambda :\,(0,0,w)$ $0\lt w\lt\frac{1}{2}$ 

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 $H :\,(0,\frac{1}{2},w)$ $0\lt w\lt\frac{1}{2}$ 

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 $G :\,(\frac{1}{2},0,w)$ $0\lt w\lt\frac{1}{2}$ 

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 $Q :\,(\frac{1}{2},\frac{1}{2},w)$ $0\lt w\lt\frac{1}{2}$ 

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 $\Delta :\,(0,v,0)$ $0\lt v\lt\frac{1}{2}$ 

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 $B :\,(0,v,\frac{1}{2})$ $0\lt v\lt\frac{1}{2}$ 

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 $D :\,(\frac{1}{2},v,0)$ $0\lt v\lt\frac{1}{2}$ 

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 $P :\,(\frac{1}{2},v,\frac{1}{2})$ $0\lt v\lt\frac{1}{2}$ 

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 $\Sigma :\,(u,0,0)$ $0\lt u\lt\frac{1}{2}$ 

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 $A :\,(u,0,\frac{1}{2})$ $0\lt u\lt\frac{1}{2}$ 

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 $C :\,(u,\frac{1}{2},0)$ $0\lt u\lt\frac{1}{2}$ 

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 $E :\,(u,\frac{1}{2},\frac{1}{2})$ $0\lt u\lt\frac{1}{2}$ 

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The real space and reciprocal space primitive translation vectors are:

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

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

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