\(\Large \frac{E}{\frac{\hbar^2}{2ma^2}}\) | |
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$\large
\vec{k}=u\vec{b}_1+v\vec{b}_2+w\vec{b}_3\,:\,(u,v,w)$
Symmetry points $(u,v,w)$ | $[k_x,k_y,k_z]$ |
$\Gamma:\,(0,0,0)$ | $[0,0,0]$ |
$X:\,(\frac{1}{2},0,0)$ | $[\frac{\pi}{a},\frac{\pi}{a},0]$ |
$Z:\, (\frac{1}{2},\frac{1}{2},-\frac{1}{2})$ | $[\frac{2\pi}{a},0,0]$ |
$N:\, (0,\frac{1}{2},0)$ | $[\frac{\pi}{a},0,\frac{\pi}{c}]$ |
$P:\, (\frac{1}{4},\frac{1}{4},\frac{1}{4})$ | $[\frac{\pi}{a},\frac{\pi}{a},\frac{\pi}{c}]$ |
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$\overline{\Gamma X} =
\frac{\sqrt{2}\pi}{a}$ |
$\overline{\Gamma Z} =
\frac{2\pi}{a}$ |
$\overline{\Gamma N} =
\frac{\pi}{ac}\sqrt{a^2+c^2}$ |
$\overline{\Gamma P} =
\frac{\pi}{ac}\sqrt{a^2+2c^2}$ |
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The real space and reciprocal space primitive translation vectors
are:
$\large \vec{a}_1 =
\frac{a}{2}(\hat{x}+\hat{y})-\frac{c}{2}\hat{z}$ $\large \vec{a}_2 = \frac{a}{2}(\hat{x}-\hat{y})+\frac{c}{2}\hat{z}$ $\large
\vec{a}_3 = \frac{a}{2}(-\hat{x}+\hat{y})+\frac{c}{2}\hat{z}$,
$\large \vec{b}_1 =
\frac{2\pi}{a}(\hat{k_x}+\hat{k_y})$ $\large \vec{b}_2
=\frac{2\pi}{a}\hat{k_x}+\frac{2\pi}{c}\hat{k_z}$ $\large \vec{b}_3 =
\frac{2\pi}{a}\hat{k_y}+\frac{2\pi}{c}\hat{k_z}$.
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