# 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 ${\displaystyle a,b,c}$ and angles between them ${\displaystyle \alpha ,\beta ,\gamma }$. In terms of the lattice vectors ${\displaystyle {\mathbf {a} }}$ , ${\displaystyle \mathbf {b} }$, and ${\displaystyle \mathbf {c} }$, the fractional coordinates ${\displaystyle (u,v,w)}$ of a point in space are defined as[citation needed]

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

## Conversion

The fractional coordinates may be converted from Cartesian coordinates using the following matrix:[1]

${\displaystyle \left[{\begin{matrix}u\\v\\w\end{matrix}}\right]=\left[{\begin{matrix}{\frac {1}{a}}&-{\frac {\cos \gamma }{a\sin \gamma }}&bc{\frac {\cos \alpha \cos \gamma -\cos \beta }{\Omega \sin \gamma }}\\0&{\frac {1}{b\sin \gamma }}&ac{\frac {\cos \beta \cos \gamma -\cos \alpha }{\Omega \sin \gamma }}\\0&0&{\frac {ab\sin \gamma }{\Omega }}\end{matrix}}\right]\left[{\begin{matrix}x\\y\\z\end{matrix}}\right]}$

where ${\displaystyle (x}$, ${\displaystyle y}$, ${\displaystyle z)}$ are the components of the arbitrary vector ${\displaystyle \mathbf {r} }$ in Cartesian coordinates, and

${\displaystyle \Omega ={\mathbf {a} }\cdot ({\mathbf {b} }\times {\mathbf {c} })=abc{\sqrt {1-\cos ^{2}\alpha -\cos ^{2}\beta -\cos ^{2}\gamma +2\cos \alpha \cos \beta \cos \gamma }}}$

is the volume of the unit cell.[2]

Similarly, they may be converted to Cartesian coordinates using:[3][4]

${\displaystyle \left[{\begin{matrix}x\\y\\z\end{matrix}}\right]=\left[{\begin{matrix}a&b\cos \gamma &c\cos \beta \\0&b\sin \gamma &c{\frac {\cos \alpha -\cos \beta \cos \gamma }{\sin \gamma }}\\0&0&{\frac {\Omega }{ab\sin \gamma }}\end{matrix}}\right]\left[{\begin{matrix}u\\v\\w\end{matrix}}\right].}$

For the special case of a monoclinic cell (a common case) where ${\displaystyle \alpha =\gamma =90^{\circ }}$ and ${\displaystyle \beta >90^{\circ }}$, this gives:

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

## References

1. ^ "Coordinate system transformation". Ruppweb.org. Retrieved 2016-10-19.
2. ^ "Coordinate system transformation". www.ruppweb.org. Retrieved 2016-10-19.
3. ^ Sussman, J.; Holbrook, S.; Church, G.; Kim, S (1977). "A Structure-Factor Least-Squares Refinement Procedure For Macromolecular Structures Using Constrained And Restrained Parameters". Acta Crystallogr. A. 33 (5): 800–804. Bibcode:1977AcCrA..33..800S. CiteSeerX 10.1.1.70.8631. doi:10.1107/S0567739477001958.
4. ^ Rossmann, M.; Blow, D. (1962). "The Detection Of Sub-Units Within The Crystallographic Asymmetric Unit". Acta Crystallogr. 15: 24–31. CiteSeerX 10.1.1.319.3019. doi:10.1107/S0365110X62000067.