Derivation of ‘Unknown’ Symmetry Elements
For the symmetry elements on the previous page, the long space group symbol is \(P\frac{2_{1}}{n} \frac{2_{1}}{m} \frac{2_{1}}{a}\). The short space group symbol, in which mirror and glide planes take precedence, is \(Pnma\).
To derive the presence of other symmetry elements, consider how these symmetry elements interact with one another. Even if we do not know the position of these symmetry elements, we can still do this (the position of the origin is convention, rather than being fixed!). The mirror plane converts \((x, y, z)\) to \((x, -y, z)\). If we then apply the \(a\) glide to this new position, we arrive at \((x+\frac{1}{2}, -y, -z)\). This is equivalent to a \(2_{1}\) screw axis parallel to \(a\) acting on our original position, \((x, y, z)\).
Which symmetry element is produced by the combination of the \(n\) glide and the \(a\) glide? Which symmetry element is produced by the combination of the \(n\) glide and the mirror plane. The answers can be found on the next page.