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:

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

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:

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

$y=b\,y_{frac}$

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

Conversion from cartesian coordinates

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

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

Supporting file formats

• CPMD input
• CIF

References

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