1.4: Unit Cells with Rotational Symmetry

Unit cells are non-overlapping regions of crystalline structures that completely fill real space by repeating periodically. For any crystalline lattice, the smallest fundamental region is a primitive unit cell with edges that connect adjacent lattice points in the independent directions. For 2-d lattices, these regions are parallelograms with 4 edges and 4 corners with area \(A_{1}\); for 3-d lattices, they are parallelohedra with 6 faces, 12 edges, and 8 corners with volume \(V_{1}\). Therefore, the primitive unit cell of any lattice contains one and only one lattice point. Using the standard nomenclature for primitive cell edge lengths and angles, \(A_{1}\) and \(V_{1}\) are:

2-d: \(A_{1} = a_{1}a_{2}\sin\alpha_{3}\);

3-d: \(V_{1} = a_{1}a_{2}a_{3}\left\lbrack 1 - \cos^{2}\alpha_{1} - \cos^{2}\alpha_{2} - \cos^{2}\alpha_{3} + 2\cos\alpha_{1}\cos\alpha_{2}\cos\alpha_{3} \right\rbrack^{\frac{1}{2}}\).

Although any primitive unit cell of a crystalline structure contains all atoms needed to generate the entire structure by translational periodicity, the cell’s shape may not display the full rotational symmetry of the lattice. Since rotationally symmetric unit cells are useful for many applications, there are two important types of unit cells that account for both translational and rotational symmetry (1) Crystallographic unit cells and (2) Wigner-Seitz cells.

Crystallographic unit cells

Crystallographic unit cells are rotationally symmetric primitive or non-primitive cells. If the primitive unit cell does not display the complete rotational symmetry of its lattice, then a non-primitive cell can be identified that does. Such cells are called centered because of the locations of additional lattice points. For the 2-d lattice shown here, the primitive cell is diamond-shaped, but a larger rectangular-shaped cell is also possible. The larger cell contains 2 lattice points, one at the corners and one at the center, so that this lattice is described as centered-rectangular. Although the primitive, diamond-shaped cell also satisfies the rotational constraints, the centered rectangular cell directly mimics the rectangular symmetry of this lattice.

Figure 1.26: 2-d lattice showing primitive (in red) and centered rectangular (in green) unit cells.  The area of the centered rectangular cell is twice the area of the primitive cell.

Crystallographic unit cells, whether primitive or centered, are used to describe the structures of crystalline solids. Their unit cell vectors are designated as a, b, and c, so that the unit cell sides have lengths a, b, c and interior angles α (between b and c), β (between a and c), γ (between a and b). If a non-primitive crystallographic cell contains n lattice points, then its size is n times the size of the primitive cell. For 2-d and 3-d lattices, the corresponding area and volume of the crystallographic cells are:

2-d: \(A_{n} = ab\sin\gamma = nA_{1}\);

3-d: \(V_{n} = abc\left\lbrack 1 - \cos^{2}\alpha - \cos^{2}\beta - \cos^{2}\gamma + 2\cos\alpha\cos\beta\cos\gamma \right\rbrack^{\frac{1}{2}} = nV_{1}\).

These unit cells are used by crystallographers, but the primitive unit cells are sufficient for computational scientists to use as structural input for most electronic structure calculations.

Wigner-Seitz Cells

Wigner-Seitz cells are rotationally symmetric primitive unit cells that enclose the regions of space closest to one lattice point. To construct a Wigner-Seitz cell, choose one lattice point and identify all surrounding lattice points, which extend to nearest and possibly next-nearest neighbor points. Then, construct the perpendicular bisectors between the selected origin point and every neighbor. The region formed by these bisectors completely encloses the origin lattice point and is the Wigner-Seitz cell for the lattice. Like crystallographic cells, Wigner-Seitz cells completely fill the space of the lattice without overlapping or gaps. Being a primitive cell, its size is \(A_{1}\) in 2-d or \(V_{1}\) in 3-d. These cells are important for electronic structure calculations, but their generally complex shapes have limited their application in crystallography. In real space, these regions are called Voronoi polyhedra (polygons) and Dirichlet domains and can be useful to evaluate coordination numbers at atoms.

Figure 1.27: Same lattice as in Fig. 1.26 with honeycomb-shaped Wigner-Seitz unit cells (in red) surrounding every lattice point.