Past exam of the materials science course of the University of Cambridge / 2015 / Paper 1 / 1 / a / i / Solution

The vertical mirrors of the diamond cubic structure have the required orientation: their normals lie in the $(001)$ plane. In the chosen atomic-origin convention, their traces satisfywith $x,y$ expressed as fractions of $a$. These are the two orientations of vertical $\{110\}$ mirrors and their translated copies. For example, the plane $x=y$ interchanges the first two coordinates and leaves every diamond site in an equivalent site. Reflection in $x+y=1/2$ sends $(x,y,z)$ to $(1/2-y,1/2-x,z)$, which also preserves the full three-dimensional motif.

The projected traces alone do not test a mirror: the atom heights must also agree under reflection. In particular, a vertical $\{100\}$ trace can appear plausible in an unlabelled projection but does not preserve all heights as a pure mirror; glide operations are different symmetry elements.