Ladder graph

Template:Expert-subject 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

Conversion to cartesian coordinates

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:

$\left[\begin{array}{c}\mathbf{x}\\ \mathbf{y}\\ \mathbf{z}\\ \end{array}\right]=\left[\begin{array}{ccc}\mathbf{a}& \mathbf{b}\cos \mathbf{(}\mathit{\gamma }\mathbf{)}& \mathbf{c}\cos \mathbf{(}\mathit{\beta }\mathbf{)}\\ \mathbf{0}& \mathbf{b}\sin \mathbf{(}\mathit{\gamma }\mathbf{)}& \mathbf{c}\frac{\cos (\alpha )-\cos (\beta )\cos (\gamma )}{\sin (\gamma )}\\ \mathbf{0}& \mathbf{0}& \mathbf{c}\frac{v}{\sin (\gamma )}\\ \end{array}\right]\left[\begin{array}{c}\hat{a}\\ \hat{b}\\ \hat{c}\\ \end{array}\right]$

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:

$x=a{x}_{frac}+c{z}_{frac}\cos (\beta )$

$y=b{y}_{frac}$

$z=c{z}_{frac}\sin (\beta )$

Conversion from cartesian coordinates

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

$\left[\begin{array}{c}\hat{\mathbf{a}}\\ \hat{\mathbf{b}}\\ \hat{\mathbf{c}}\\ \end{array}\right]=\left[\begin{array}{ccc}\frac{1}{a}& \mathbf{-}\frac{\cos (\gamma )}{a\sin (\gamma )}& \frac{\cos (\alpha )\cos (\gamma )-\cos (\beta )}{av\sin (\gamma )}\\ \mathbf{0}& \frac{1}{b\sin (\gamma )}& \frac{\cos (\beta )\cos (\gamma )-\cos (\alpha )}{bv\sin (\gamma )}\\ \mathbf{0}& \mathbf{0}& \frac{\sin (\gamma )}{cv}\\ \end{array}\right]\left[\begin{array}{c}x\\ y\\ z\\ \end{array}\right]$

Supporting file formats

• CPMD input
• CIF

References

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