## Brillouin zones of two-dimensional Bravais lattices

A two-dimensional Bravais lattice can be specified by giving the lattice parameters $a$, $b$, and $\gamma$ or by specifying the primitive lattice vectors in real space $\vec{a}_1$ and $\vec{a}_2$. If we align $\vec{a}_1$ with the $x$-axis, the primitive lattice vectors in real space are,

$$\vec{a}_1 = a\,\hat{x},\qquad\vec{a}_2 = b\cos\gamma\,\hat{x} + b\sin\gamma\,\hat{y}.$$

The primitive lattice vectors in reciprocal space can then be calculated using the conditions,

$$\vec{a}_i\cdot\vec{b}_j=2\pi\delta_{ij}.$$

The primitive lattice vectors in reciprocal space are,

$$\vec{b}_1 = \frac{2\pi}{a}\,\hat{k_x}-\frac{2\pi\cos\gamma}{a\sin\gamma}\,\hat{k_y},\qquad\vec{b}_2 = \frac{2\pi}{b\sin\gamma}\,\hat{k_y}.$$

The reciprocal lattice vectors are $\vec{G}_{hk}=h\vec{b}_1+k\vec{b}_2$, where $h$ and $k$ are integers. To construct the Brillouin zones we draw lines normal to each reciprocal lattice vector that passes through $\vec{G}_{hk}/2$. These lines have the form,

$$G_{hk,x}k_x+G_{hk,y}k_y = \frac{G_{hk,x}^2}{2}+\frac{G_{hk,y}^2}{2},$$

where $G_{hk,x}=hb_{1,x}+kb_{2,x}$ and $G_{hk,y}=hb_{1,y}+kb_{2,y}$.

The form below can be used to adjust the ratio $b/a$ and the angle $\gamma$. Using these values the primitive lattice vectors in real space and primitive lattice vectors in reciprocal space are calculated. The reciprocal lattice points are plotted in red. The lines that form the Brillouin zone boundaries are then drawn. For the drawings the normalization $a=1$ is used.

$b/a=$1
$\gamma=$90

$\vec{a}_1=1\,\hat{x}$   $\vec{a}_2=$ $\hat{x} + ($ $) \hat{y}$

$\vec{b}_1=$ $\hat{k}_x + ($ $)\hat{k}_y$   $\vec{b}_2=$ $\hat{k}_y$

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