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 ångströms from fractional coordinates, one can multiply the fractional coordinates by the operation matrix below:^[2]^[3]

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

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=abc{\sqrt {1-\cos ^{2}(\alpha )-\cos ^{2}(\beta )-\cos ^{2}(\gamma )+2\cos(\alpha )\cos(\beta )\cos(\gamma )}}

^[4]

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

{\begin{aligned}x&=ax_{\mathrm {frac} }+cz_{\mathrm {frac} }\cos(\beta )\\y&=by_{\mathrm {frac} }\\z&=cvz_{\mathrm {frac} }=cz_{\mathrm {frac} }\sin(\beta )\end{aligned}}

Conversion from cartesian coordinates

The above fractional-to-cartesian transformation can be inverted as follows^[5]

\mathbf {{\begin{bmatrix}x_{\mathrm {frac} }\\y_{\mathrm {frac} }\\z_{\mathrm {frac} }\\\end{bmatrix}}={\begin{bmatrix}{\dfrac {1}{a}}&-{\dfrac {\cos(\gamma )}{a\sin(\gamma )}}&bc{\dfrac {\cos(\alpha )\cos(\gamma )-\cos(\beta )}{v\sin(\gamma )}}\\0&{\dfrac {1}{b\sin(\gamma )}}&ac{\dfrac {\cos(\beta )\cos(\gamma )-\cos(\alpha )}{v\sin(\gamma )}}\\0&0&ab{\dfrac {\sin(\gamma )}{v}}\\\end{bmatrix}}} {\begin{bmatrix}x\\y\\z\\\end{bmatrix}}

Supporting file formats

CPMD input
CIF

References

↑ "Unit cell definition using parallelepiped with lengths a, b, c and angles between the edges given by α, β, γ". Ccdc.cam.ac.uk. Retrieved 2016-08-17.
↑ Sussman, J.; Holbrook, S.; Church, G.; Kim, S (1977). "A Structure-Factor Least-Squares Refinement Procedure For Macromolecular Structures Using Constrained And Restrained Parameters". Acta Crystallogr. A. 33: 800–804. doi:10.1107/S0567739477001958.
↑ Rossmann, M.; Blow, D. (1962). "The Detection Of Sub-Units Within The Crystallographic Asymmetric Unit". Acta Crystallogr. 15: 24–31. doi:10.1107/S0365110X62000067.
↑ "Coordinate system transformation". www.ruppweb.org. Retrieved 2016-10-19.
↑ "Coordinate system transformation". Ruppweb.org. Retrieved 2016-10-19.

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