Fractional coordinates

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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[edit]

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

 a \sin \gamma \sin \omega   & 0   &  0      \\
 a \cos \gamma   & b   & c \cos \alpha  \\
 a \sin \gamma \cos \omega & 0 & c \sin \alpha \\

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)


z=c\,v\,z_{frac} = c\, z_{frac}\,\sin(\beta)

Conversion from cartesian coordinates[edit]

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

\mathbf{\begin{bmatrix} \hat{a} \\ \hat{b} \\ \hat{c} \\ \end{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} \\
\begin{bmatrix} x \\ y \\ z \\ \end{bmatrix}

Supporting file formats[edit]


  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. ^ (note that the V defined there differs from the v used here by a factor abc)