# Fractional coordinates

In crystallography, a fractional coordinate system is a coordinate system in which the edges of the unit cell are used as the basic vectors to describe the positions of atomic nuclei. The unit cell is a parallelepiped defined by the lengths of its edges $a,b,c$ and angles between them $\alpha ,\beta ,\gamma$ . In terms of the lattice vectors ${\mathbf {a} }$ , $\mathbf {b}$ , and $\mathbf {c}$ , the fractional coordinates $(u,v,w)$ of a point in space are defined as[citation needed]

${\mathbf {r} }=u{\mathbf {a} }+v{\mathbf {b} }+w{\mathbf {c} }.$ ## Conversion

The fractional coordinates may be converted from Cartesian coordinates using the following matrix:

$\left[{\begin{matrix}u\\v\\w\end{matrix}}\right]=\left[{\begin{matrix}{\frac {1}{a}}&-{\frac {\cos \gamma }{a\sin \gamma }}&bc{\frac {\cos \alpha \cos \gamma -\cos \beta }{\Omega \sin \gamma }}\\0&{\frac {1}{b\sin \gamma }}&ac{\frac {\cos \beta \cos \gamma -\cos \alpha }{\Omega \sin \gamma }}\\0&0&{\frac {ab\sin \gamma }{\Omega }}\end{matrix}}\right]\left[{\begin{matrix}x\\y\\z\end{matrix}}\right]$ where $(x$ , $y$ , $z)$ are the components of the arbitrary vector $\mathbf {r}$ in Cartesian coordinates, and

$\Omega ={\mathbf {a} }\cdot ({\mathbf {b} }\times {\mathbf {c} })=abc{\sqrt {1-\cos ^{2}\alpha -\cos ^{2}\beta -\cos ^{2}\gamma +2\cos \alpha \cos \beta \cos \gamma }}$ is the volume of the unit cell.

Similarly, they may be converted to Cartesian coordinates using:

$\left[{\begin{matrix}x\\y\\z\end{matrix}}\right]=\left[{\begin{matrix}a&b\cos \gamma &c\cos \beta \\0&b\sin \gamma &c{\frac {\cos \alpha -\cos \beta \cos \gamma }{\sin \gamma }}\\0&0&{\frac {\Omega }{ab\sin \gamma }}\end{matrix}}\right]\left[{\begin{matrix}u\\v\\w\end{matrix}}\right].$ For the special case of a monoclinic cell (a common case) where $\alpha =\gamma =90^{\circ }$ and $\beta >90^{\circ }$ , this gives:

{\begin{aligned}x&=au+cw\cos \beta ,\\y&=bv,\\z&={\frac {\Omega }{ab}}w=cw\sin \beta .\end{aligned}} 