Relationship Between Symmetry Elements and Space Groups
Hermann-Mauguin notation is typically the method used to describe space groups. There are short Hermann-Mauguin symbols and long Hermann-Mauguin symbols. The short Hermann-Mauguin symbols detail the minimum symmetry needed to determine the space group, while the long Hermann-Mauguin symbols detail more symmetry elements. Depending on the exact symmetry, the long Hermann-Mauguin symbol may give all symmetry elements present in the lattice. This is easiest to begin learning using orthorhombic space groups.
Consider the space group \(Pma2\). The first symbol denotes the lattice centring present (primitive). The second symbol gives the symmetry elements defined by the \(a\) axis. The symbol is \(m\), therefore this space group has a mirror plane perpendicular to the \(a\) axis. The third symbol is \(a\). This means there is an \(a\) glide perpendicular to the \(b\) axis. The fourth symbol is \(2\). This means there is a two-fold rotation axis parallel to the \(c\) axis.
The space group symbols for orthorhombic space groups represent the centring, symmetry elements with unique axis \(a\), symmetry elements with unique axis \(b\), and symmetry elements with unique axis \(c\), in that order.
Full and Short Space Group Symbols
In the example above, it just so happens the long and short space group symbols are identical. For orthorhombic space groups, the full symbol gives extra information about the symmetry elements. Consider the space group \(Cmcm\). This is a \(C\)-centred lattice with mirror planes perpendicular to the \(a\) and \(c\) axes, and a \(c\) glide perpendicular to the \(b\) axis. However, the full space group symbol is \(C\frac{2}{m} \frac{2}{c} \frac{2_{1}}{m}\), to show the rotation and screw axes present also. In the short space group symbol, mirror and glide planes take priority. Inversion centres are not shown in orthorhombic space groups. This is because they can be derived from the other symmetry present in the unit cell.
Symmetry Elements to Space Groups
A lattice has the following symmetry elements:
- Primitive centring
- A \(2_{1}\) screw axis parallel to \(a\)
- A \(2_{1}\) screw axis parallel to \(b\)
- A \(2\) rotation axis parallel to \(c\)
- An \(n\) glide plane perpendicular to \(a\)
- An \(m\) mirror plane perpendicular to \(b\)
- An \(a\) glide plane perpendicular to \(c\)
- An inversion centre (\(\bar{1}\))
What is the long and short space group symbol? The answers will be on the next page.