10 | Crystal Systems and Lattices

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The unit cell describes the fundamental repeat unit and is useful as starting point to think about the structure of crystals from a local perspective. To make the work of describing an ideally infinite periodic structure manageable, it is useful to conceptually project a lattice onto the real crystal structure to serve as a guide to discuss the dimensions and symmetry independent of the particular locations of individual atoms.

In a infinitely periodic system we can always find a set of points that are structurally indistinguishable from each other both locally as well as the relative locations of every other object in the crystal. This set of points is called a lattice. For example, in the 1D periodic waveform shown in Fig. 15, in orange are shown a set of lattice points. It doesn’t matter exactly where the lattice is superimposed the points remain mutually indistinguishable regardless of relative position of the periodic structure. In 1D, the distance between these points is the only lattice parameter and the content between points describe a unit cell.

Fig. 15 (Click image) A Lattice is a set of lattice points (orange) that can be superimposed on a periodic structure. In an infinite periodic structure all lattice points are perfectly indistinguishable from one another.

2D Crystals

The four 2D crystal systems

In 2D there are 4 crystal systems. Recall a crystal system describe a coordinate system that be can used to define positions in a crystal (or a periodic lattice). The square system is formed when the two basis vectors are orthogonal and the same length. The rectangular system is also composed of two orthogonal basis vectors but they are not the same length. And in an oblique crystal system basis vectors are not orthogonal. In the hexagonal crystal system the angle between the basis vectors, \(\gamma\), is 120° and \(|\vec{a}|=|\vec{b}|\). The four crystal systems are shown in Fig. 16

Fig. 16 The five possible lattices in two dimensions. Regions shaded in orange show alternative tilings. Note, rectangular-C is a special case of an oblique-P.

The five 2D crystal lattices

A lattice type is described by a combination of (1) a crystal system and (2) a centering. The centering of a lattice describes the number and location of lattice points per unit cell. All lattices have a set of lattice points at the vertices of the unit cell. If these are the only lattice points in the unit cell then the lattice is primitive and symbolized as the letter \(P\). In 2D all lattices that can be described in an oblique, square, or hexagonal crystal system are primitive. However, there is a special case of an oblique-\(P\) lattice, Fig. 16, where it is possible to draw a unit cell with twice the area where \(\gamma = 90°\) and calling this new lattice rectangular-\(C\). Doing so allows use the convenience of only describing two lattice parameters \(a,\, b\) instead of three and in this case also provides use with a more intuitive orthogonal coordinate system.

Note

It is always possible to describe a centered lattice as a lower symmetry primitive lattice with a smaller unit cell. Doing so is advantageous in electronic structure calculations to reduce the number of atoms that need to be include in the calculation.

The lattices points shown in Fig. 16 can each be assigned a point symmetry within the infinite lattice. All 2D lattices contain a least a 2-fold rotation. The symmetry and contraints for all five 2D lattices are shown in Table 5

Table 5 Summary of the four crystal systems, their allowed centerings, and lattice parameters.

Crystal System	Lattice parameters	Constraints	Allowed cell centerings	Point symmetry at lattice points
Oblique	\(a, \,b,\, \gamma\)	\(a \neq b, \gamma \neq 90°, 120°\)	\(P\)	\(2\)
Rectangular	\(a,\, b\)	\(a \neq b, \gamma = 90°\)	\(P,\, C\)	\(2mm\)
Square	\(a\)	\(a = b, \gamma = 90°\)	\(P\)	\(4mm\)
Hexagonal	\(a\)	\(a = b, \gamma = 120°\)	\(P\)	\(6mm\)

3D crystals

The lattices of three dimensional crystals can be constructed by stacking the 2D lattices. Where there were 4 crystal systems in 2D there are 7 in 3D. And where there were only 5 lattice types in 2D there are 14 in 3D.

The seven crystal systems

Fig. 17 (Click image) Relationships between five of the seven crystal systems: cubic, tetragonal, orthogonal, monoclinic, and triclinic animated by descents in symmetry.

Fig. 18 (Click image) Relationships between the cubic and trigonal crystal system by elongation of the cubic unit cell along the diagonal (dashed line). The resulting trigonal lattice is shown in the rhombohedral setting.

Fig. 19 In the trigonal crystal system a rhombohedral cell can either be describe as either a primitive rhombohedral lattice (grey circles) with the unit cell parameters \(a,\, \alpha\) or as a centered cell in a hexagonal setting (orange circles) with the lattice parameters \(a,\, c,\, \gamma = 120°\).

The fourteen Bravias lattices

Fig. 20 The 14 Bravais lattices.

Table 6 Summary of the seven crystal systems and their lattice parameters

Crystal System	lattice parameters	Constraints	Allowed cell centerings	Point symmetry at lattice points
Triclinic	\(a, \, b, \,c,\, \alpha, \, \beta, \, \gamma\)	none	\(P\)	\(1, \bar{1}\)
Monoclinic	\(a, \, b, \,c,\, \beta\)	\(\alpha = \gamma = 90°\)	\(P, \, C\)	\(\frac{2}{m}\)
Orthorhombic	\(a, \, b, \,c\)	\(\alpha = \beta = \gamma = 90°\)	\(P, \, A, \, B, \, C, \, I, \, F\)	\(mmm\)
Tetragonal	\(a, \,c\)	\(a=b;\) \(\,\alpha = \beta = \gamma = 90°\)	\(P, \, I\)	\(\frac{4}{m}mm\)
Cubic	\(a\)	\(a=b=c;\) \(\,\alpha = \beta = \gamma = 90°\)	\(P, \, I, \, F\)	\(m\bar{3}m\)
Trigonal	\(a, \alpha\) or \(a,c\) (on hexagonal axes)	\(a=b=c;\) \(\,\alpha = \beta = \gamma \neq 90°\) or \(a=b\neq c;\) \(\,\alpha = \beta=90°,\, \gamma = 120°\)	\(P, \, R\)	\(\bar{3}m\)
Hexagonal	\(a,c\)	\(a=b\neq c;\) \(\,\alpha = \beta=90°,\, \gamma = 120°\)	\(P\)	\(\frac{6}{m}mm\)