Fractional coordinates

In crystallography, fractional coordinates [ [http://www.ccdc.cam.ac.uk/support/documentation/mercury_csd/portable/mercury_portable-4-28.html Cambridge Crystallographic Data Center Glossary of terms for Mercury program] ] are often used to represent the positions of the atomic nuclei in the coordinate space of the unit cell given by sides a, b, c and angles between the sides of α,β,γ as shown in the figure below.

Conversion to Cartesian Coordinates

Fractional coordinates can be converted to cartesian coordinates through the following transformation matrix [ [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] [OpenBabel source code] [ [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]

: $mathbf{T_{frac
ightarrow cartesian} =egin{bmatrix} a & 0 & 0 \ b cos(gamma) & b sin(gamma) & 0 \ccos(eta) & -csin(eta)cos(alpha_a) & left ( c,v sin(gamma)
ight ) ^{-1} \end{bmatrix$

where

: $mathbf{alpha_a =arccos left ( frac {cos(eta )cos(gamma)-cos(alpha)} {sin(eta)sin(gamma)}
ight ) }$

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

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

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

upporting file formats

:CPMD input:cif- crystallographic information file

External Links

* [http://lec.ugr.es/trans/ Practical transformation matrices in crystallography] - includes an online coordinate converter

References

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