Unit Cells

A unit cell is the smallest repeating unit of a crystal structure that can be translated in three dimensions to reproduce the entire crystal lattice. It must capture both particle positions and the spacing between them. The unit cell is fundamental because its geometry and composition determine all macroscopic properties of the solid.

Unit cell dimensions are defined by three edge lengths denoted a, b, and c, and three angles between them denoted α, β, and γ. These parameters fully describe the size and shape of the parallelepiped. The chosen unit cell should have maximum symmetry and the smallest volume consistent with that symmetry.

Primitive unit cells contain one lattice point per cell, with eight corner points each shared by eight neighbors. Non-primitive unit cells contain additional lattice points at centers or on faces, making them larger but sometimes more convenient for describing crystal symmetry. Both types can represent the same crystal structure.

Each corner atom of a unit cell is shared by eight adjacent unit cells, contributing only 1/8 of an atom to any single cell. When eight corners are counted, they collectively yield one complete atom per unit cell. This fractional counting ensures accurate stoichiometry when the unit cell is repeated throughout the crystal.

A unit cell must capture both atomic positions and interatomic distances because these features determine the crystal's physical and chemical properties. Reproducing only positions without spacing would create an incomplete structural description. The complete spatial arrangement ensures that translating the unit cell generates an accurate three-dimensional crystal structure.

A space lattice is an infinite three-dimensional array of points where each point is surrounded by its neighbors in an identical way. It represents the geometric framework of a crystal, with each lattice point marking the position of an atom, ion, or molecule. The space lattice defines the periodicity and symmetry of the entire crystal structure.

In cesium chloride, the unit cell contains one cesium ion at the center and eight chloride ions at corners, each contributing 1/8 to the cell. This arrangement yields a 1:1 ratio matching the formula CsCl. The example illustrates how fractional corner contributions combine to produce the correct stoichiometry and point line and plane defects can disrupt this ideal arrangement.