11.1: Unit Cells

A crystal's internal structure is an orderly array of atoms, ions, or molecules, and the details of this array significantly influence the solid's properties. In a crystal, periodically repeating 'structural motifs' - which could be atoms, molecules, or groups thereof - create a 'space lattice.' This is essentially a three-dimensional, infinite array of points, each surrounded by its neighbors in an identical way, forming the basic structure of the crystal.

A 'unit cell' is a theoretical parallel-sided figure that through repetitive translations can be used to construct the entire space lattice. These cells are often formed by connecting neighboring lattice points with straight lines, creating what is called 'primitive' unit cells. In three dimensions, each of the eight points of a primitive unit cell is shared by eight neighbors, resulting in one lattice point per unit cell. However, larger non-primitive unit cells with additional lattice points at their centers or on opposite faces can also be used for convenience.

There are infinite ways to describe the same lattice using different unit cells, but usually, the unit cell with the shortest sides and angles most nearly perpendicular to each other is chosen. The sides of a unit cell are denoted as a, b, and c, and the angles between them are α, β, and γ.

The unit cell must be able to reproduce not just the positions of the particles but also the space between them. Conventionally, the unit cell should be the smallest part of the crystal that can reproduce the entire crystal structure.

In a unit cell, the same species is usually found at the corners and may also be in other positions. When the unit cell is propagated in three dimensions, the partial atoms at the corners combine to form complete atoms in the macroscopic crystal.