KAHIBARO
Table of Contents
The repeating pattern behind crystals
A crystal is not just a solid with atoms packed closely together. Its key feature is order that repeats through space. Bravais lattices are the mathematical way to describe that repeating order.
A Bravais lattice is an infinite array of points arranged so that every point has exactly the same environment as every other point. If you stand on one lattice point and look around, the pattern of neighboring points is identical to what you would see from any other lattice point.
This idea is more abstract than a real crystal. The lattice is made of points, not atoms. Real crystals are built by attaching one or more atoms to each lattice point. That attached set of atoms is called a basis, but the detailed treatment of bases belongs more naturally with unit cells and crystal lattices. Here, the main goal is to understand the possible kinds of repeating point arrangements.
A Bravais lattice is defined by translational symmetry. If a set of points can be shifted by certain vectors and remain unchanged, it is a lattice.
Translational symmetry
The defining property of a Bravais lattice is that the whole pattern can be generated from one point by translation. In three dimensions, any lattice point can be reached from an origin by
$$
\vec{R} = n_1 \vec{a}_1 + n_2 \vec{a}_2 + n_3 \vec{a}_3
$$
where $n_1$, $n_2$, and $n_3$ are integers, and $\vec{a}_1$, $\vec{a}_2$, and $\vec{a}_3$ are primitive translation vectors.
These three vectors define the repeating structure. Different choices of vectors may describe the same lattice, but the lattice itself is the full set of all points generated by integer combinations of them.
The lattice points are located at
$$
\vec{R} = n_1 \vec{a}_1 + n_2 \vec{a}_2 + n_3 \vec{a}_3, \qquad n_i \in \mathbb{Z}
$$
This is the basic mathematical definition of a Bravais lattice.
A simple two dimensional picture helps. If two vectors define the repeat directions, then every lattice point is obtained by moving whole numbers of steps along those two directions.
Why there are only certain kinds
At first, it may seem that infinitely many different lattice types should exist. In one sense they do, because lengths and angles can vary continuously. But when physicists classify Bravais lattices, they group lattices by symmetry and centering type. In three dimensions, this classification gives exactly 14 distinct Bravais lattices.
These 14 lattices are distributed among the 7 crystal systems. The crystal systems describe the general shape constraints of the unit cell, such as whether its edges are equal or whether the angles are right angles.
The 7 crystal systems and the 14 Bravais lattices
The following table shows the standard classification.
| Crystal system | Cell constraints | Bravais lattices |
|---|---|---|
| Triclinic | $a \neq b \neq c$, $\alpha \neq \beta \neq \gamma \neq 90^\circ$ | Simple |
| Monoclinic | $a \neq b \neq c$, $\alpha = \gamma = 90^\circ$, $\beta \neq 90^\circ$ | Simple, Base-centered |
| Orthorhombic | $a \neq b \neq c$, $\alpha = \beta = \gamma = 90^\circ$ | Simple, Base-centered, Body-centered, Face-centered |
| Tetragonal | $a = b \neq c$, $\alpha = \beta = \gamma = 90^\circ$ | Simple, Body-centered |
| Trigonal | $a = b = c$, $\alpha = \beta = \gamma \neq 90^\circ$ | Simple |
| Hexagonal | $a = b \neq c$, $\alpha = \beta = 90^\circ$, $\gamma = 120^\circ$ | Simple |
| Cubic | $a = b = c$, $\alpha = \beta = \gamma = 90^\circ$ | Simple, Body-centered, Face-centered |
The word simple is also called primitive, often written as $P$. Other common centering labels are base-centered $C$, body-centered $I$, and face-centered $F$.
Primitive and centered lattices
A lattice can be described using different unit cells. Some unit cells contain lattice points only at the corners. These are primitive cells in the conventional sense of centering type. Others include extra lattice points at special positions.
The main centering types are shown below.
| Symbol | Name | Extra lattice points |
|---|---|---|
| $P$ | Primitive | None besides corners |
| $C$ | Base-centered | Centers of one pair of opposite faces |
| $I$ | Body-centered | One at the center of the cell |
| $F$ | Face-centered | One at the center of each face |
For example, in a cubic system, the three possible Bravais lattices are:
| Lattice | Lattice points per conventional cell |
|---|---|
| Simple cubic, $P$ | $1$ |
| Body-centered cubic, $I$ | $2$ |
| Face-centered cubic, $F$ | $4$ |
These counts come from sharing of corner and face points between neighboring cells. The detailed counting method is often discussed with unit cells, but the numbers are useful here because they distinguish the lattice types.
Examples of important Bravais lattices
Some Bravais lattices appear often in physics and materials science.
The simple cubic lattice is the most straightforward. Points lie only at the cube corners. It is easy to picture, but relatively few elements crystallize in this form.
The body-centered cubic lattice, abbreviated BCC, has one lattice point at the center in addition to the corners. Metals such as iron at room temperature can have this structure.
The face-centered cubic lattice, abbreviated FCC, has lattice points at the center of each face as well as at the corners. Many metals, such as copper and aluminum, form this lattice.
The hexagonal Bravais lattice is also very important. Its symmetry is different from the cubic lattices, and it is especially useful for describing crystals with sixfold rotational features.
Symmetry and equivalence of lattice points
What makes a Bravais lattice special is not just repetition, but equivalence. Every lattice point must be indistinguishable from every other by translation alone. This condition rules out many patterns that may look regular but are not Bravais lattices.
For instance, if a pattern has alternating types of points, then not all points have the same surroundings. Such a structure may still describe a crystal, but it is not itself a Bravais lattice unless the different objects are treated as part of the basis rather than as different lattice points.
In a Bravais lattice, all lattice points are equivalent.
If two points have different local surroundings, the point pattern is not a Bravais lattice.
Conventional cell and primitive cell
A primitive cell is the smallest volume that can reproduce the entire lattice by translation. A conventional cell is often chosen to make the symmetry easier to see, even if it is larger than necessary.
For example, the face-centered cubic lattice is commonly drawn as a cube because its cubic symmetry is obvious in that form. However, that cubic cell is not the smallest possible primitive cell.
This distinction matters because one lattice can have several valid descriptions, but it remains the same Bravais lattice.
Why Bravais lattices matter in physics
Bravais lattices are the starting point for understanding crystalline solids. Once the lattice is known, physicists can study how atoms are arranged, how electrons move through the solid, and how waves travel through the structure.
Many properties of materials, such as electrical conduction, elastic behavior, and diffraction patterns, depend strongly on the symmetry of the lattice. The Bravais lattice captures the translational skeleton of the crystal.
Final picture
A Bravais lattice is the most basic geometric description of crystal order. It is an infinite repeating set of equivalent points generated by integer combinations of translation vectors. In three dimensions, symmetry allows exactly 14 distinct types, grouped into 7 crystal systems.
Key facts to remember:
$$
\vec{R} = n_1 \vec{a}_1 + n_2 \vec{a}_2 + n_3 \vec{a}_3
$$
There are exactly 14 Bravais lattices in three dimensions.
All lattice points in a Bravais lattice are equivalent under translation.