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 and angles between them .

General caseEdit

Consider a system of periodic structure in space and use   ,  , and   as the three independent period vectors, forming a right-handed triad, which are also the edge vectors of a cell of the system. Then any vector   in Cartesian coordinates can be written as a linear combination of the period vectors

 

Our task is to calculate the scalar coefficients known as fractional coordinates  ,  , and  , assuming  ,  ,  , and   are known.

For this purpose, let us calculate the following cell surface area vector

 

then

 

and the volume of the cell is

 

If we do a vector inner (dot) product as follows

 

then we get

 

Similarly,

 
 

we arrive at

 

and

 
 
 

If there are many  s to be converted with respect to the same period vectors, to speed up, we can have

 

where

 

In crystallographyEdit

In crystallography, the lengths ( ,  ,  ) of and angles ( ,  ,  ) between the edge (period) vectors ( ,  ,  ) of the parallelepiped unit cell are known. For simplicity, it is chosen so that edge vector   in the positive  -axis direction, edge vector   in the   plane with positive  -axis component, edge vector   with positive  -axis component in the Cartesian-system, as shown in the figure below.

 
Unit cell definition using parallelepiped with lengths  ,  ,   and angles between the sides given by  ,  , and   [1]

Then the edge vectors can be written as

 

where all  ,  ,  ,  ,   are positive. Next, let us express all   components with known variables. This can be done with

 

Then

 

The last one continues

 

where

 

Remembering  ,  , and   being positive, one gets

 

Since the absolute value of the bottom surface area of the cell is

 

the volume of the parallelepiped cell can also be expressed as

 .[2]

Once the volume is calculated as above, one has

 

Now let us summarize the expression of the edge (period) vectors

 

Conversion from Cartesian coordinatesEdit

Let us calculate the following surface area vector of the cell first

 

where

 

Another surface area vector of the cell

 

where

 

The last surface area vector of the cell

 

where

 

Summarize

 

As a result[3]

 

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

Conversion to Cartesian coordinatesEdit

To return the orthogonal coordinates in ångströms from fractional coordinates, one can employ the first equation on top and the expression of the edge (period) vectors[4][5]

 

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

 

Supporting file formatsEdit

ReferencesEdit

  1. ^ "Unit cell definition using parallelepiped with lengths a, b, c and angles between the edges given by α, β, γ". Ccdc.cam.ac.uk. Archived from the original on 2008-10-04. Retrieved 2016-08-17.
  2. ^ "Coordinate system transformation". www.ruppweb.org. Retrieved 2016-10-19.
  3. ^ "Coordinate system transformation". Ruppweb.org. Retrieved 2016-10-19.
  4. ^ 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 10.1.1.70.8631. doi:10.1107/S0567739477001958.
  5. ^ Rossmann, M.; Blow, D. (1962). "The Detection Of Sub-Units Within The Crystallographic Asymmetric Unit". Acta Crystallogr. 15: 24–31. CiteSeerX 10.1.1.319.3019. doi:10.1107/S0365110X62000067.