Drawing Wigner-Seitz Cells

In three dimensions, there are 14 Bravais lattices. To draw a Wigner-Seitz cells of these Bravais lattices, first the Bravais lattice vectors must be constructed from the primitive lattice vectors. The Bravais lattice vectors are, $\vec{T}_{uvw}=u\vec{a}_1+v\vec{a}_2+w\vec{a}_3$, where $\vec{a}_1,\,\vec{a}_2,\,\vec{a}_3$ are the primitive lattice vectors and $u,\,v$ and $w$ are integers. If $u$, $v$, and $w$ are restricted to the values 1, 0, and -1; there are 26 Bravais lattice points around the point at 0,0,0 to consider.

The Bravais lattice vectors are,

$$\vec{T}_{uvw}=u\vec{a}_1+v\vec{a}_2+w\vec{a}_3=\left(ua_{1x}+va_{2x}+wa_{3x}\right)\hat{x}+\left(ua_{1y}+va_{2y}+wa_{3y}\right)\hat{y}+\left(ua_{1z}+va_{2z}+wa_{3z}\right)\hat{z}.$$

A plane normal to each Bravais lattice vector is drawn that passes through $\vec{T}_{uvw}/2$. All of the points that can be reached from the origin without crossing any of these planes is in the Wigner-Seitz cell. Before we continue, there is a brief review of how you find the intersection point of three planes.

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. This is a standard linear algebra problem. The app below will solve for the three unknowns and determine the point of intersection of the planes.

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

A boundary of the Wigner-Seitz cell is a plane normal to $\vec{T}_{uvw}$, that passes through the point $\frac{\vec{T}_{uvw}}{2}$. For the planes that make up the Wigner-Seitz cell boundary, the distance from $\frac{\vec{T}_{uvw}}{2}$ to the origin is smaller than the distance from $\frac{\vec{T}_{uvw}}{2}$ 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 plane perpendicular to $T_{uvw}$ is,

$$T_{uvw,x}x+T_{uvw,y}y+T_{uvw,z}z=\frac{T_{uvw,x}^2}{2}+\frac{T_{uvw,y}^2}{2}+\frac{T_{uvw,z}^2}{2}.$$

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

//set up the three.js variables

var canvas1 = document.getElementById("canvas_bz");
var width = canvas1.clientWidth;
var height = canvas1.clientHeight;

var scene = new THREE.Scene();
scene.background = new THREE.Color(0xffffff);
var camera = new THREE.PerspectiveCamera(3, width/height, 1, 1000);

var renderer = new THREE.WebGLRenderer();
renderer.setSize(width, height);
canvas1.appendChild(renderer.domElement);

var controls = new THREE.OrbitControls(camera);

var material = new THREE.MeshBasicMaterial({
    color: 0x000000
});

var geometry;

//this while loop erases the previous BZ
  //while(scene.children.length > 0){ scene.remove(scene.children[0]); }

for (i = 0; i < Ne; i++) {
    geometry = new THREE.Geometry();
    geometry.vertices.push(new THREE.Vector3(edges[i][0], edges[i][1], edges[i][2]));
    geometry.vertices.push(new THREE.Vector3(edges[i][3], edges[i][4], edges[i][5]));

var line = new THREE.Line(geometry, material);

scene.add(line);

}

camera.position.z = 50;
  var animate = function() {
    
    requestAnimationFrame(animate);

//controls.update() must be called after any manual changes to the camera's transform
    controls.update();
    renderer.render(scene, camera);

}

animate();

</script>

Right-click and select 'view source' to see the code that generates this page.