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PHY.K02UF Molecular and Solid State Physics | ||||
$\vec{k}=u\vec{b}_1+v\vec{b}_2+w\vec{b}_3$
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The real space and reciprocal space primitive translation vectors are:
$$\vec{a}_1=a\hat{x},\qquad\vec{a}_2=\frac{a}{2}\hat{x}+\frac{\sqrt{3}a}{2}\hat{y},\qquad\vec{a}_3=c\hat{z}$$ $$\vec{b}_1=\frac{2\pi}{\sqrt{3}a}\left(\sqrt{3}\hat{k}_x-\hat{k}_y\right),\qquad\vec{b}_2=\frac{4\pi}{\sqrt{3}a}\hat{k}_y,\qquad\vec{b}_3=\frac{2\pi}{c}\hat{k}_z$$The first Brillouin zone of an hexagonal lattice is hexagonal again. Some crystals with an (simple) hexagonal Bravais lattice are Mg, Nd, Sc, Ti, Zn, Be, Cd, Ce, Y.