Unit cell

In fields like geometry, biology, mineralogy, and solid state physics, scientists study repeating patterns. These patterns are called a lattice. To understand these complex structures, researchers use a concept known as a unit cell. A unit cell is a repeating unit formed by vectors that span the points of a lattice.

You can think of the unit cell as the basic building block of a larger tiling. By using only translations, which means sliding the shape without rotating it, a single cell can generate an entire lattice. This concept helps describe crystal structures in two and three dimensions, though it also works in higher dimensions.

There are two specific types of unit cells: the primitive cell and the conventional cell. The primitive cell is the smallest possible unit cell for a given lattice. It is defined as a cell that contains exactly one lattice point.

Because it is so small, the full symmetry of a crystal structure might not be obvious when looking only at a primitive cell. To solve this, scientists use a conventional cell. A conventional cell is a unit cell that shows the full symmetry of the lattice. While a primitive cell is always the smallest, a conventional cell may include more than one lattice point.

To build a primitive cell, scientists use primitive translation vectors. These vectors, often labeled as a, b, and c, span a lattice cell of the smallest volume. By using these vectors with integers, one can define a crystal translation vector. This ensures that the arrangement of points looks exactly the same from one point as it does from another.

In two and three dimensions, primitive cells are often shaped like parallelograms or parallelepipeds. These shapes usually have an atom located at each corner. While the choice of a primitive cell is not unique, the volume of these cells is always determined by the axes of the primitive translation vectors.

Another unique way to define a primitive cell is through the Wigner–Seitz cell. This is a specific type of Voronoi cell used for every Bravais lattice. Unlike the standard parallelepiped shape, the lattice point sits at the very center of a Wigner–Seitz cell. For most Bravais lattices, the resulting shape is not a simple parallelogram or parallelepiped.

Interestingly, when this concept is applied to a reciprocal lattice in momentum space, the resulting cell is called a Brillouin zone.

In two dimensions, the unit cells are always parallelograms. These can take special forms, such as rectangles or squares, if they have equal lengths or orthogonal angles. There are five different two-dimensional Bravais lattices. These include the oblique, rectangular, square, hexagonal, and centered rectangular lattices.

For the centered rectangular lattice, a rhombus is often used as a primitive cell. However, to make the symmetry easier to see, crystallographers often use a conventional cell containing two lattice points.

Three-dimensional lattices are even more complex, featuring fourteen different Bravais lattices. The conventional unit cells in 3D are parallelepipeds. These can be cubes, rectangular cuboids, or various types of prisms.

Seven of these lattices are represented using conventional primitive cells, such as the cubic or hexagonal types.

The other seven are known as centered lattices. For these, the conventional cell is used to show symmetry and contains more than one lattice point.

The volume of these conventional cells is always an integer multiple of the primitive cell volume, such as 1, 2, 3, or 4 times larger.

Crystallographers choose these conventional cells on a case-by-case basis. Their main goal is to make calculations more convenient. By choosing a cell that highlights the symmetry of the lattice, they can better understand the material. This mathematical approach allows scientists to map out the tiny, repeating structures that make up the physical world. Whether studying a mineral or a complex biological structure, the unit cell provides the essential framework for understanding order in nature.