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 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[edit]

If the fractional coordinate system has the same origin as the cartesian coordinate system, the a-axis is collinear with the x-axis, and the b-axis lies in the xy-plane, fractional coordinates can be converted to cartesian coordinates through the following transformation matrix:[2][3][4]

\begin{bmatrix} x \\ y \\ z \\ \end{bmatrix} =
 a     & b\cos(\gamma)     & c\cos(\beta)     \\
 0     & b\sin(\gamma)     & c\frac {\cos(\alpha)-\cos(\beta)\cos(\gamma)} {\sin(\gamma)}  \\
 0 & 0 & c\frac {v} {\sin(\gamma)} \\
\begin{bmatrix} \hat{a} \\ \hat{b} \\ \hat{c} \\ \end{bmatrix}

where v is the volume of a unit parallelepiped defined as

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

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

x=a\,x_{frac} + c\,z_{frac}\,\cos(\beta)


z=c\,v\,z_{frac} = c\, z_{frac}\,\sin(\beta)

Conversion from cartesian coordinates[edit]

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

\mathbf{\begin{bmatrix} \hat{a} \\ \hat{b} \\ \hat{c} \\ \end{bmatrix} =
 \frac{1}{a}     & -\frac{\cos(\gamma)} {a\sin(\gamma)}     & \frac{\cos(\alpha)\cos(\gamma)-\cos(\beta)}{av\sin(\gamma)}     \\
 0     & \frac{1}{b\sin(\gamma)}     & \frac{\cos(\beta)\cos(\gamma)-\cos(\alpha)}{bv\sin(\gamma)}  \\
 0 & 0 & \frac {\sin(\gamma)} {cv} \\
\begin{bmatrix} x \\ y \\ z \\ \end{bmatrix}

Supporting file formats[edit]