## Solid State 2D Wigner-Seitz Cell and Brillouin Zone

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Calculate the direct and reciprocal lattice, and the points of symmetry of the Wigner-Seitz Cell and the Brillouin Zone
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Purpose
This is a matlab solution to a computer project given in Solid State class at University of Massachusetts, Lowell,
around 2007.

Problem
Consider the generation fo the direct and reciprocal two dimensional rectangular Bravais lattice with spacing 'a' parallel to the x-axis and '2a' parallel to the y-axis (a primitive unit cell of sides a and 2a).
Write a program to:
1) Calculate the coordinates of each lattice point in both direct and reciprocal space out to a given (input) distance from the origin.
2) Calculate the coordinates of the corners of the Wigner-Seitz cell and the first Brillouin zone.
3) Determine the coordinates of the corner and the equations of the lines from the origin to the corner, from the orgiin to the center of the edge, and from the center of the edge to the corner of the first quadrant (high symmetry directions) for both the Wigner-Seitz cell and the first Brillouin zone.

Run-Time Cases
The assignment called for setting the input distance equal to three times the nearest neighbor distance. Then calculating:
Table 1: The Coordinates of the Direct Lattice
Table 2: The Coordinates of the Reciprocal Lattice
Table 3: The lines associated with pairs of latitce points or origin and edge points for the Wigner-Seitz Cell
Table 4: The lines associated with pairs of latitce points or origin and edge points for the first Brillouin Zone
Plots of the Wigner-Seitz cell and Brillouin Zone including high symmetry directions labeled.

Algortihm
The basic problem for determining the Wigner-Seitz Cell (and the Brillouin Zone) is to:
1) Find the lattice points and reciprocal lattice points.
2) Compute the midpoints of the 8 vectors from the origin to the outer points.
3) Compute two end points of a line segment that intersects the midpoint and is normal to the vector.
4) Along each vector, determine which midpoint line segment intersects that vector at a point closest to the origin. Is it the line associated with the vector or a line associated with one of the neighboring vectors?
5) Find the intersection points of all the midpoint lines that intersect at least one vector at the minimum. This is the minimum bounded polygon (or volume in 3D space). These ar the corner points of the Wigner-Seitz cell and for the Brillouin Zone.

### 引用

Meg Noah (2024). Solid State 2D Wigner-Seitz Cell and Brillouin Zone (https://www.mathworks.com/matlabcentral/fileexchange/70593-solid-state-2d-wigner-seitz-cell-and-brillouin-zone), MATLAB Central File Exchange. 取得済み .

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