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PHY.K02UF Molecular and Solid State Physics

The first Brillouin zone of a body centered 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},\frac{\pi}{a},0]
Z:\, (\frac{1}{2},\frac{1}{2},-\frac{1}{2})[\frac{2\pi}{a},0,0]
N:\, (0,\frac{1}{2},0)[\frac{\pi}{a},0,\frac{\pi}{c}]
P:\, (\frac{1}{4},\frac{1}{4},\frac{1}{4})[\frac{\pi}{a},\frac{\pi}{a},\frac{\pi}{c}]
 

\overline{\Gamma X} = \frac{\sqrt{2}\pi}{a}

\overline{\Gamma Z} = \frac{2\pi}{a}

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

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

 
 Symmetry lines 
\Delta :\,(v,0,0)0\lt v\lt\frac{1}{2}
\Sigma :\,(v,v,-v)0\lt v\lt\frac{1}{2}
Y :\,(\frac{1}{2},v,-v)0\lt v\lt\frac{1}{2}
\Lambda :\,(-v,v,v)0\lt v\lt\frac{1}{4}+\frac{c^2}{a^2}
W :\,(\frac{1}{2}-v,v,v)0\lt v\lt\frac{1}{4}
Q :\,(v,\frac{1}{2}-v,v)0\lt v\lt\frac{1}{4}

The real space and reciprocal space primitive translation vectors are:

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

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