||This article needs attention from an expert in Chemistry. The specific problem is: An editor has questioned the accuracy of the transformation matrix shown in the "Conversion to cartesian coordinates" section (see article talk page). (June 2012)|
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.
Conversion to cartesian coordinates
where a, b, c, α, β, and γ are the unit-cell parameters, and ω is given by:
Also, v is the volume of a unit parallelepiped defined as:
For the special case of a monoclinic cell (a common case) where α=γ=90° and β>90°, this gives:
Conversion from cartesian coordinates
The above fractional-to-cartesian transformation can be inverted as follows
Supporting file formats
- Unit cell definition using parallelepiped with lengths a, b, c and angles between the edges given by α,β,γ
-  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
-  Rossmann, M.; Blow, D. The Detection Of Sub-Units Within The Crystallographic Asymmetric Unit. Acta Cryst 1962, 15, 24-31.
- 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)