The

**Wigner-Seitz cell** is a

primitive unit cell about a

Bravais lattice point with the the property that the space it encloses is closer to that point than to any other lattice point (see

Bravais lattice for background on this topic). A

primitive unit cell is a region of space that, when translated through all

Bravais lattice vectors, fills space completely with no two cells overlapping. Consider the following 2-d example of four adjoining Wigner-Seitz cells of a simple square

lattice.

X X X X X X X
-------------
| | |
X X X | X | X | X X
| | |
|-----|-----|
| | |
X X X | X | X | X X
| | |
----- -----
X X X X X X X
X X X X X X X

Notice that the Wigner-Seitz cell has the same 90-degree rotational symmetry as the square lattice. *All Wigner-Seitz cells have the same symmetries as their corresponding Bravais lattices.* The Wigner-Seitz cells of 3-d Bravais lattices can have quite complex structure (i.e. there's no way i can draw them with ASCII art).

The Wigner-Seitz cell of a reciprocal lattice is extremely important in solid state physics. This cell has a special name--the first Brillouin zone. Bragg diffraction of electron wavefunctions occurs when their Bloch wavevectors (see Bloch's theorem) lie on the boundary of the first Brillouin zone. This leads to the formation of energy bandgaps in solids.