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 case

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 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

In crystallography, the lengths of and angles between the edge 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.
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
Once the volume is calculated as above, one has
Now let us summarize the expression of the edge vectors

Conversion from Cartesian coordinates

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

Conversion to Cartesian coordinates

To return the orthogonal coordinates in ångströms from fractional coordinates, one can employ the first equation on top and the expression of the edge vectors
For the special case of a monoclinic cell where and, this gives:

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