Coordinates on a Crystal Lattice

Atoms in a crystal sit on a slanted grid built from two basis vectors. Any position can be written as a combination of them, and those two numbers are its coordinates in the crystal's own basis. Change the basis and the same point gets new coordinates.

How it works

Bases and coordinate systems

In one sentence

If $\mathcal{B} = \{\mathbf{b}_1, \ldots, \mathbf{b}_n\}$ is a basis, every $\mathbf{x}$ has unique coordinates $[\mathbf{x}]_{\mathcal{B}}$ with $\mathbf{x} = P_{\mathcal{B}}[\mathbf{x}]_{\mathcal{B}}$ , where $P_{\mathcal{B}} = [\mathbf{b}_1 \cdots \mathbf{b}_n]$ is invertible.

General formula

{\htmlData{sym=0}{\mathbf{x}}} = {\htmlData{sym=1}{c_1}}{\htmlData{sym=3}{\mathbf{b}_1}} + {\htmlData{sym=2}{c_2}}{\htmlData{sym=4}{\mathbf{b}_2}}, \qquad \begin{bmatrix} {\htmlData{sym=1}{c_1}} \\ {\htmlData{sym=2}{c_2}} \end{bmatrix} = \begin{bmatrix} {\htmlData{sym=3}{\mathbf{b}_1}} & {\htmlData{sym=4}{\mathbf{b}_2}} \end{bmatrix}^{-1}{\htmlData{sym=0}{\mathbf{x}}}

A basis is a set of measuring sticks. Coordinates count how many of each stick it takes to reach a point.

What the symbols mean

$\mathbf{x}$
Position
Where the point is, in ordinary coordinates.
$c_1$
First coordinate
How many b₁ steps to the point.
$c_2$
Second coordinate
How many b₂ steps to the point.
$\mathbf{b}_1$
First lattice vector
One edge of the crystal's unit cell.
$\mathbf{b}_2$
Second lattice vector
The other edge, at an angle.

The lattice vectors $\mathbf{b}_1, \mathbf{b}_2$ are independent and span the plane, so they form a basis.
Every point is exactly one combination $c_1\mathbf{b}_1 + c_2\mathbf{b}_2$ : the coordinates $(c_1, c_2)$ relative to that basis.
To find them, solve with the matrix whose columns are the basis vectors: $[\mathbf{x}]_B = P_B^{-1}\mathbf{x}$ .
Atoms sit where both coordinates are whole numbers; a point between atoms has fractional coordinates.

Learn the math

Vector Spaces and SubspacesLin Alg · Lesson 19, Vector Spaces Linearly Independent Sets and BasesLin Alg · Lesson 21, Vector Spaces Coordinate SystemsLin Alg · Lesson 22, Vector Spaces