Fractional coordinates

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General case[edit]

Let us 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 , , 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 Crystallography[edit]

In crystallography, the lengths (, , ) of and angles (, , ) between the edge (period) vectors (, , ) of the parallelepiped unit cell are known. For simplicty, 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 coordinates[edit]

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 coordinates[edit]

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 formats[edit]

References[edit]

  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: 800–804. 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. doi:10.1107/S0365110X62000067.