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2010-07-14T01:44:20Z

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{{Expert-subject|Chemistry|reason=An editor has questioned the accuracy of the transformation matrix shown in the "Conversion to cartesian coordinates" section (see article talk page)|date=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 [[Vector (mathematics and physics)|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.

[[Image:UnitCell.png|center|thumb|300px|Unit cell definition using parallelepiped with lengths ''a'', ''b'', ''c'' and angles between the sides given by α,β,γ<ref>[http://www.ccdc.cam.ac.uk/support/documentation/mercury_csd/portable/mercury_portable-4-70.html Unit cell definition using parallelepiped with lengths ''a'', ''b'', ''c'' and angles between the edges given by α,β,γ]</ref>]]

==Conversion to cartesian coordinates==
If the fractional coordinate system has the same origin as the [[cartesian coordinate system]], the a-axis is collinear with the x-axis, and the b-axis lies in the xy-plane, fractional coordinates can be converted to cartesian coordinates through the following transformation matrix:<ref>[http://graphics.med.yale.edu:5080/TriposBookshelf/sybyl/crystal/crystal_appendix2.html http://graphics.med.yale.edu:5080/TriposBookshelf/sybyl/crystal/crystal_appendix2.html] Probably a slightly unstable reference for the transformation matrix</ref><ref>[[OpenBabel]] source code</ref><ref>[http://www.angelfire.com/linux/myp/FracCor/fraccor.html http://www.angelfire.com/linux/myp/FracCor/fraccor.html] Another transformation matrix that is defined differently</ref>

:<math>\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} \hat{a} \\ \hat{b} \\ \hat{c} \\ \end{bmatrix}
</math>

where <math>v</math> is the volume of a unit parallelepiped defined as

:<math>
v =\sqrt{1-\cos^2(\alpha)-\cos^2(\beta)-\cos^2(\gamma)+2\cos(\alpha)\cos(\beta)\cos(\gamma)}

</math>

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

:<math>
x=a\,x_{frac} + c\,z_{frac}\,\cos(\beta)
</math>
:<math>
y=b\,y_{frac}
</math>
:<math>
z=c\,z_{frac}\,\sin(\beta)
</math>

==Conversion from cartesian coordinates==

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

:<math>\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}
</math>

==Supporting file formats==
*[[CPMD]] input
*[[Crystallographic Information File|CIF]]

==References==
{{reflist}}
http://www.ruppweb.org/Xray/tutorial/Coordinate%20system%20transformation.htm
{{DEFAULTSORT:Fractional Coordinates}}
[[Category:Molecular modelling]]
[[Category:Computational chemistry]]
[[Category:Crystallography]]

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en>David Eppstein: cat