Isomorphic Space Groups
Centring relationships will also contribute to the derivation of unknown symmetry elements. For example, consider \(Ccca\). Applying the \(C\)-centring and \(a\) glide leads from \((x, y, z)\) to \( (x + \frac{1}{2}, y + \frac{1}{2}, z)\) to \((x, y + \frac{1}{2}, -z)\). This is equivalent to a \(b\) glide plane perpendicular to \(c\). Indeed this space group actually has both \(a\) and \(b\) glides perpendicular to the \(c\) axis (it does not have an \(e\) or \(n\) glide, this is different).
As such, the space group \(Ccca\) can also be written \(Cccb\). This is the case for many orthorhombic space groups. For an easy example, as \(a\), \(b\) and \(c\) axes are all independent, these can be exchanged. \(Pnnm\) is identical to the \(Pnmn\) and \(Pmnn\), and these are not listed as new space groups in The International Tables.
For a more complicated example, other glide planes can have the translation vector changed when exchanging axes. For example, \(Pnma\) is identical to \(Pnam\), \(Pmnb\), \(Pcmn\) etc. These are determined by exchanging the axes.
These space groups are called isomorphic. They are identical space groups, but are written in a different form. Have a go at identifying as many isomorphic space groups of \(Pma2\) as you can. Scroll down for the answers.
The isomorphic space groups of \(Pma2\) are \(Pm2a\), \(Pbm2\), \(P2mb\), \(Pc2m\), \(P2cm\). In each case, the glide direction is along the axis with the mirror plane. This keeps the symmetry consistent even with exchanging of axes. This is not a skill you will need too often, but it is useful to be aware of isomorphic space groups and consider how symmetry elements fit together.
If you would like to, have another go with \(Pmna\). The answers are below.
Isomorphic space groups of \(Pmna\) are: \(Pman\), \(Pnmb\), \(Pncm\), \(Pbmn\) and \(Pcnm\).