## Empty lattice approximation for a body centered tetragonal crystal

 $\Large \frac{E}{\frac{\hbar^2}{2ma^2}}$ Γ X Z N P Γ X Z N P Γ X Z N P Γ X Z N P Γ X Z N P Γ X Z N P a/c:

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

 Symmetry points $(u,v,w)$ $[k_x,k_y,k_z]$ $\Gamma:\,(0,0,0)$ $[0,0,0]$ $X:\,(\frac{1}{2},0,0)$ $[\frac{\pi}{a},\frac{\pi}{a},0]$ $Z:\, (\frac{1}{2},\frac{1}{2},-\frac{1}{2})$ $[\frac{2\pi}{a},0,0]$ $N:\, (0,\frac{1}{2},0)$ $[\frac{\pi}{a},0,\frac{\pi}{c}]$ $P:\, (\frac{1}{4},\frac{1}{4},\frac{1}{4})$ $[\frac{\pi}{a},\frac{\pi}{a},\frac{\pi}{c}]$ $\overline{\Gamma X} = \frac{\sqrt{2}\pi}{a}$ $\overline{\Gamma Z} = \frac{2\pi}{a}$ $\overline{\Gamma N} = \frac{\pi}{ac}\sqrt{a^2+c^2}$ $\overline{\Gamma P} = \frac{\pi}{ac}\sqrt{a^2+2c^2}$

The real space and reciprocal space primitive translation vectors are:

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

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