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PHY.K02UF Molecular and Solid State Physics

## Drawing Wigner-Seitz Cells

To draw a Wigner-Seitz cell, first the Bravais lattice vectors must be constructed from the primitive lattice vectors. The Bravais lattice vectors are, $\vec{R}_{hkl}=h\vec{a}_1+k\vec{a}_2+l\vec{a}_3$, where $\vec{a}_1,\,\vec{a}_2,\,\vec{a}_3$ are the primitive lattice vectors and $h,\,k$ and $l$ are integers.

The Bravais lattice vectors are,

$$\vec{R}_{hkl}=h\vec{a}_1+k\vec{a}_2+l\vec{a}_3=\left(ha_{1x}+ka_{2x}+la_{3x}\right)\hat{x}+\left(ha_{1y}+ka_{2y}+la_{3y}\right)\hat{y}+\left(ha_{1z}+ka_{2z}+la_{3z}\right)\hat{z}.$$

A plane normal to each Bravais lattice vector is drawn that passes through $\vec{R}_{hkl}/2$. All of the points that can be reached from the origin without crossing any of these planes is in the Wigner-Seitz cell.

A brief review of vectors and planes
The set of planes perpendicular to a vector $A_x\hat{x}+A_y\hat{y}+A_z\hat{z}$ is,

$$A_xx+A_yy+A_zz = C,$$

where $C$ is any constant. If a point $(x_0,y_0,z_0)$ on the plane is known, $C$ can be calculated,

$$C=A_xx_0+A_yy_0+A_zz_0.$$

To find a point where three planes intersect, write down the equations that describe these three planes. These will be three linear equations with three unknowns. The app below will solve for the three unknowns and determine the point of intersection of the planes.

$x+$  $y+$  $z=$
$x+$  $y+$  $z=$
$x+$  $y+$  $z=$

The form below takes the primitive lattice vectors in real space as input and calculates the Bravais lattice vectors $\vec{R}_{hkl}$, the planes $(hkl)$ that form the Wigner-Seitz cell boundaries, and the corners of the Wigner-Seitz cell.

Primitive lattice vectors:

 $\vec{a}_1=$ $\hat{x}+$ $\hat{y}+$ $\hat{z}$ [Å] $\vec{a}_2=$ $\hat{x}+$ $\hat{y}+$ $\hat{z}$ [Å] $\vec{a}_3=$ $\hat{x}+$ $\hat{y}+$ $\hat{z}$ [Å]

A boundary of the Wigner-Seitz cell is a plane normal to $\vec{R}_{hkl}$, that passes through the point $\frac{\vec{R}_{hkl}}{2}$. For the planes that make up the Wigner-Seitz cell boundary, the distance from $\frac{\vec{R}_{hkl}}{2}$ to the origin is smaller than the distance from $\frac{\vec{R}_{hkl}}{2}$ to any of the other Bravais lattice vectors. By computing these distances, the planes that make up the Wigner-Seitz cell can be determined.

Once the planes are known, the points at the corners of the Wigner-Seitz cell can be determined by considering the intersections of the planes. The formula for the $(hkl)$ plane is,

$$R_{hkl,x}k_x+R_{hkl,y}k_y+R_{hkl,z}k_z=\frac{R_{hkl,x}^2}{2}+\frac{R_{hkl,y}^2}{2}+\frac{R_{hkl,z}^2}{2}.$$

By solving the sets of linear equations, the corners can be determined.

The edges of the Wigner-Seitz cell can be determined by considering all pairs of corners. If a pair of corners share two planes, there is an edge between these two corners.

Rotatable model of the Wigner-Seitz cell

Right-click and select 'view source' to see the code that generates this page. For some primitive lattice vectors, numerical errors occur and not all corners and edges are found.