Effects of Higher Rotation Axes
Higher rotation axes produce more complicated coordinate transformations in Cartesian coordinates, though these often appear far simpler when presented in the coordinate system used for the conventional unit cell. The derivations of these can be seen in the maths-focused course.
Three-fold rotation axes
In the space group \(P3\), the positions produced by the three-fold rotation, in the hexagonal coordinate system are:
$$(x, y, z), (-y, x – y, z), (y – x, -x, z)$$
For example, if an atom in a \(P3\) unit cell is located at \((0.1, 0.4, 0.3)\), there is an equivalent atom at \((-0.4, -0.3, 0.3)\) and \((0.3, -0.1, 0.3)\). Remember that combined with the periodicity of the unit cell, these atoms are located within the unit cell at \((1-0.4, 1-0.3, 0.3)\) and \((0.3, 1-0.1, 0.3)\).
Four-fold rotation axes
The positions associated with a four-fold rotation axis in right-angled coordinate systems, such as those used for tetragonal and cubic space groups are:
$$(x, y, z), (-x, -y, z), (-y, x, z), (y, -x, z)$$
Six-fold rotation axes
In hexagonal coordinates, the positions associated with a six-fold rotation are:
$$(x, y, z), (-y, x-y, z), (y-x, -x, z), (-x, -y, z), (y, y-x, z), (x-y, x, z)$$
There is no need to remember these coordinates. They can be found in The International Tables. Remember to adjust the \(z\) coordinate when screw axes are present.