Unconventional Unit Cells

The unit cells underlying the 14 Bravais lattices have the characteristics conventionally required for a unit cell: the smallest possible volume, consistent with the symmetry of the crystal system.

However, it has been shown that primitive unit cells P can be identified for all centered cells C, F, or I. Unlike the conventional non-primitive unit cells, these primitive cells do not adhere to the symmetry constraints of the system in question.

For example, the primitive cell for a cubic F lattice is a rhombohedral cell with all equal axes and all angles equal to 60°, while the primitive cell for a cubic I lattice is a rhombohedral cell with α = 109°28'.

Naturally, these primitive cells, though unconventional, correctly generate the entire lattice and are sometimes used because they have the advantage (especially from the perspective of solid-state physicists) of containing only one lattice node, compared to the 2 nodes in C and I cells and the 4 nodes in F cells.

Wigner-Seitz Cells

In general, the conventional unit cell is a parallelepiped and, as such, can be considered a particular polyhedron.

Many families of polyhedra can fill space by translation.

One important type of polyhedron is obtained through the Dirichlet construction: Each lattice point is connected by line segments to its nearest neighbors;  At the midpoints of these segments, planes perpendicular to the segments are constructed; The intersection of these planes defines a region of space known as the Dirichlet region, or more commonly, the Wigner-Seitz cell.

Example of a Wigner-Seitz cell in two-dimensional space.

Esempio tridimensionale di cella Wigner-Seitz

The Wigner-Seitz cell is always a primitive cell.

It coincides with a Bravais cell, only when the latter is rectangular and primitive.

When applied in reciprocal space, an identical construction to that of the Wigner-Seitz cell defines a primitive cell conventionally called the first Brillouin zone.

There are 14 first Brillouin zones corresponding to the 14 Bravais lattices.

Brillouin zones are of great importance in lattice dynamics and band theory.

 

Last modified: Monday, 7 April 2025, 11:09 AM