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 and angles between them . In terms of the lattice vectors , , and , the fractional coordinates of a point in space are defined as[citation needed]


The fractional coordinates may be converted from Cartesian coordinates using the following matrix:[1]

where , , are the components of the arbitrary vector in Cartesian coordinates, and

is the volume of the unit cell.[2]

Similarly, they may be converted to Cartesian coordinates using:[3][4]

For the special case of a monoclinic cell (a common case) where and , this gives:

Supporting file formats[edit]


  1. ^ "Coordinate system transformation". Retrieved 2016-10-19.
  2. ^ "Coordinate system transformation". Retrieved 2016-10-19.
  3. ^ 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 (5): 800–804. Bibcode:1977AcCrA..33..800S. CiteSeerX doi:10.1107/S0567739477001958.
  4. ^ Rossmann, M.; Blow, D. (1962). "The Detection Of Sub-Units Within The Crystallographic Asymmetric Unit". Acta Crystallogr. 15: 24–31. CiteSeerX doi:10.1107/S0365110X62000067.