# 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 α, β, γ as shown in the figure below.

Unit cell definition using parallelepiped with lengths a, b, c and angles between the sides given by α,β,γ[1]

## Conversion to cartesian coordinates

To return the orthogonal coordinates in Å from fractional coordinates, one can multiply the fractional coordinates by the operation matrix below:[2][3]

$\begin{bmatrix} a \sin \gamma \sin \omega & 0 & 0 \\ a \cos \gamma & b & c \cos \alpha \\ a \sin \gamma \cos \omega & 0 & c \sin \alpha \\ \end{bmatrix}$

where a, b, c, α, β, and γ are the unit-cell parameters, and ω is given by:

$\cos \omega = \frac{(\cos \beta - \cos \alpha\ cos \gamma)}{\sin \alpha\ sin \gamma}$

Also, v is the volume of a unit parallelepiped defined as:

$v =\sqrt{1-\cos^2(\alpha)-\cos^2(\beta)-\cos^2(\gamma)+2\cos(\alpha)\cos(\beta)\cos(\gamma)}$

For the special case of a monoclinic cell (a common case) where α=γ=90° and β>90°, this gives:

$x=a\,x_{frac} + c\,z_{frac}\,\cos(\beta)$
$y=b\,y_{frac}$
$z=c\,v\,z_{frac} = c\, z_{frac}\,\sin(\beta)$

## Conversion from cartesian coordinates

The above fractional-to-cartesian transformation can be inverted as follows[4]

$\mathbf{\begin{bmatrix} \hat{a} \\ \hat{b} \\ \hat{c} \\ \end{bmatrix} = \begin{bmatrix} \frac{1}{a} & -\frac{\cos(\gamma)} {a\sin(\gamma)} & \frac{\cos(\alpha)\cos(\gamma)-\cos(\beta)}{av\sin(\gamma)} \\ 0 & \frac{1}{b\sin(\gamma)} & \frac{\cos(\beta)\cos(\gamma)-\cos(\alpha)}{bv\sin(\gamma)} \\ 0 & 0 & \frac {\sin(\gamma)} {cv} \\ \end{bmatrix}} \begin{bmatrix} x \\ y \\ z \\ \end{bmatrix}$

## References

1. ^ Unit cell definition using parallelepiped with lengths a, b, c and angles between the edges given by α,β,γ
2. ^ [1] Sussman, J.; Holbrook, S.; Church, G.; Kim, S. A Structure-Factor Least-Squares Refinement Procedure For Macromolecular Structures Using Constrained And Restrained Parameters. Acta Cryst Sect A 1977, 33, 800-804. 10.1107/s0567739477001958
3. ^ [2] Rossmann, M.; Blow, D. The Detection Of Sub-Units Within The Crystallographic Asymmetric Unit. Acta Cryst 1962, 15, 24-31.
4. ^ http://www.ruppweb.org/Xray/tutorial/Coordinate%20system%20transformation.htm (note that the V defined there differs from the v used here by a factor abc)

http://www.ruppweb.org/Xray/tutorial/Coordinate%20system%20transformation.htm