Student Project Solid State Physics WS 2014/15 Thomas Knoll
Example of Chapter 3: Wigner-Seitz Cell
1. Problem
How many corners does the Wigner-Seitz cell of a Base-centred Orthorhombic C lattice have?
2. Basics
Orthorhombic lattices result from stretching a cubic lattice along two of its orthogonal pairs by two different factors, resulting in a rectangular prism with a rectangular base (a by b) and height (c), such that a, b, and c are distinct. All three bases intersect at 90° angles. For the Base- centred orthorhombic lattice there is also one lattice point in the centre of each rectangular base.
A Wigner-Seitz cell is a primitive unit cell built around one lattice point, so that all points inside the cell are closer to this lattice point than to any other. The primitive cell of the reciprocal lattice in momentum space is called the Brillouin zone. The Wigner–Seitz cell in the reciprocal space is known as the first Brillouin zone. Figure 1: Base centred Orthorhombic C lattice
The cell may be chosen by first picking a lattice point. Then, lines are drawn to all closest lattice points. At the midpoint of each line, a perpendicular plane is drawn. These planes cut each other and form an enclosed volume, which is called the Wigner-Seitz cell. All space within the lattice will be filled by this type of primitive cells and will leave no gaps.
Figure 2: Construction sample of Wigner-Seitz cell
Student Project – Wigner-Seitz cell Thomas Knoll
1 3. Calculation
At first I write the vectors from the centred lattice point O to the closest neighbours. As examples I have chosen P,Q,R and S. All the other points are calculated similar to them. Without loss of generality I set and therefore can ignore another lattice point neighbour Y, because it wouldn’t change the shape of the Wigner-Seitz cell. I divide all these vectors by 2, so I get immediately also one point of each perpendicular plane.
Then I calculate the perpendicular planes.
Student Project – Wigner-Seitz cell Thomas Knoll
2 Now I cut with and calculate the line of intersection.
Take
Then I cut with and get the first corner point A of the Wigner-Seitz cell.
Another calculation is necessary for cutting with . I equal the left sides of the equations and get.
Insert x in .
The line of intersection is now
The corner point B follows
All the other corner points can be calculated similarly only with change in sign.
And all the corner points in the negative z-direction are the same with – instead of .
Student Project – Wigner-Seitz cell Thomas Knoll
3 4. Summary
The Wigner-Seitz cell of a Base-centred Orthorhombic C lattice has 12 corners, 18 edges and 8 planes (6 rectangular and 2 hexagonal). It is the form of a hexagonal prism.
For the special case that the Wigner-Seitz cell turns out to have 8 corners, 12 edges and 6 rectangular planes. It is the form of a quadratic prism.
5. Sketch
This orthorhombic example is drawn for some arbitrary values of a, b and c. The scale is 1:1 and the contraction factor in x-direction is 0.5.
Student Project – Wigner-Seitz cell Thomas Knoll
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