# 10.3: Crystal Systems

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- 49658

Unit cells need not be cubes, but they *must* be parallel-sided, three-dimensional figures. A general example is shown in Figure \(\PageIndex{1}\). Such a cell can be described in terms of the lengths of three adjacent edges, *a, b*, and *c*, and the angles between them, α, β, and γ.

**Figure** \(\PageIndex{1}\) * A generalized unit cell with sides a, b, and c, and angles α, β, and γ.*

Crystals are usually classified as belonging to one of seven crystal systems, depending on the shape of the unit cell. These seven systems are shown in the image below.

**Figure ** \(\PageIndex{1}\) *The Seven Crystal Systems.*

The simplest is the cubic system, in which all edges of the unit cell are equal and all angles are 90°. The tetragonal and orthorhombic classes also feature rectangular cells, but the edges are not all equal. In the remaining classes some or all of the angles are not 90°. The least symmetrical is the triclinic, in which no edges are equal and no angles are equal to each other or to 90°. Special note should be made of the hexagonal system whose unit cell is shown in Figure \(\PageIndex{2}\) . It is related to the two-dimensional cell encountered previously as the second example of a 2D crystal lattice structure, in that two edges of the cell equal and subtend an angle of 120°. Hexagonal crystals are quite common among simple compounds, like quartz, seen here below.

**Figure ** \(\PageIndex{2}\) * The hexagonal unit cell a = b ≠ c, α = β = 90°, γ = 120° and an example of a material, quartz, with a hexagonal unit cell.*

### Contributors

Ed Vitz (Kutztown University), John W. Moore (UW-Madison), Justin Shorb (Hope College), Xavier Prat-Resina (University of Minnesota Rochester), Tim Wendorff, and Adam Hahn.