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]

$\mathbf{\begin{bmatrix} x \\ y \\ z \\ \end{bmatrix} = \begin{bmatrix} a & b\cos(\gamma) & c\cos(\beta) \\ 0 & b\sin(\gamma) & c\frac {\cos(\alpha)-\cos(\beta)\cos(\gamma)} {\sin(\gamma)} \\ 0 & 0 & c\frac {v} {\sin(\gamma)} \\ \end{bmatrix}} \begin{bmatrix} x_{frac} \\ y_{frac} \\ z_{frac} \\ \end{bmatrix}$

where a, b, c, α, β, and γ are the unit-cell parameters. 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]

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

Supporting file formats

• CPMD input
• CIF

References

1. Unit cell definition using parallelepiped with lengths a, b, c and angles between the edges given by α,β,γ
2. Sussman, J.; Holbrook, S.; Church, G.; Kim, S. A Structure-Factor Least-Squares Refinement Procedure For Macromolecular Structures Using Constrained And Restrained Parameters. Acta Crystallogr A 1977, 33, 800-804. 10.1107/s0567739477001958
3. Rossmann, M.; Blow, D. The Detection Of Sub-Units Within The Crystallographic Asymmetric Unit. Acta Crystallogr 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