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PHY.K02UF Molecular and Solid State Physics

## Drawing Wigner-Seitz cells in two-dimensions

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 form below can be used to adjust the ratio $b/a$ and the angle $\gamma$. The Bravais lattice points are plotted in blue. Light gray lines are drawn from the central Bravais lattice point to the neighboring Bravais lattice points. The black lines bisect the gray lines. The Wigner-Seitz cell consists of the space that can be reached from the central Bravais lattice point without crossing any of the black lines. For the drawings the normalization $a=1$ is used.

The WIgner-Seitz cell is the primitive unit cell with the maximum symmetry.

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

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

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