Learn Space Groups

Test Your Understanding – Answers

1. A two-fold rotation axis parallel to \(a\) is located at \((a, 0, \frac{1}{4})\). What effect does this have on the coordinates \((x, y, z)\)?

The coordinates are transformed to the position \((x, -y, -z+\frac{1}{2})\).

2. A mirror plane perpendicular to \(b\) is located at \(b = 0\). What effect does this have on the coordinates \((x, y, z)\)?

The coordinates are transformed to the position \((x, -y, z)\).

3. An inversion centre is located at \((\frac{1}{2}, 0, \frac{1}{4})\). What effect does this have on the coordinates \((x, y, z)\)?

The coordinates are transformed to the position \((1-x, -y, -z+\frac{1}{2})\). However, this is identical to \((-x, -y, -z+\frac{1}{2})\) when the periodicity of the lattice is accounted for.

4. An \(a\) glide plane perpendicular to \(b\) is located at \(b = \frac{1}{4}\). What effect does this have on the coordinates \((x, y, z)\)?

The coordinates are transformed to the position \((x+\frac{1}{2}, -y+\frac{1}{2}, z)\).

5. An \(n\) glide plane perpendicular to \(c\) is located at \(c = 0\). What effect does this have on the coordinates \((x, y, z)\)?

The coordinates are transformed to the position \((x+\frac{1}{2}, y+\frac{1}{2}, -z)\).

6. A \(2_{1}\) screw axis parallel to \(c\) is located at \((\frac{1}{4}, 0, c)\). What effect does this have on the coordinates \((x, y, z)\)?

The coordinates are transformed to the position \((-x+\frac{1}{2}, -y, z+\frac{1}{2})\).

7. What effect does \(I\)-centring have on the coordinates \((x, y, z)\)?

The coordinates are transformed to the position \((x+\frac{1}{2}, y+\frac{1}{2}, z+\frac{1}{2})\).

8. What effect does \(B\)-centring have on the coordinates \((x, y, z)\)?

The coordinates are transformed to the position \((x+\frac{1}{2}, y, z+\frac{1}{2})\).

9. What symmetry elements are present in the space group \(Ccc2\)? Give unique axes of symmetry elements but not positions.

1. C-centring
2. A \(c\) glide perpendicular to \(a\)
3. A \(c\) glide perpendicular to \(b\)
4. A two-fold rotation axis parallel to \(c\)

10. What symmetry elements are present in the space group \(Pca2_{1}\)? Give unique axes of symmetry elements but not positions.

1. A \(c\) glide perpendicular to \(a\)
2. An \(a\) glide perpendicular to \(b\)
3. A \(2_{1}\) screw axis parallel to \(c\)

11. What symmetry elements are present in the short space group symbol \(Pbca\)?

1. A \(b\) glide perpendicular to \(a\)
2. A \(c\) glide perpendicular to \(b\)
3. An \(a\) glide perpendicular to \(c\)

12. Derive the other symmetry elements present in \(Pbca\) that are not listed in the short space group symbol. What is the full space group symbol of \(Pbca\)?

1. Combining the \(a\) and \(b\) glide and applying them to \((x, y, z)\) produces the position \((-x+\frac{1}{2}, y + \frac{1}{2}, -z)\), equivalent to the application of a \(2_{1}\) screw axis parallel to \(b\).
2. Combining the \(a\) and \(c\) glide and applying them to \((x, y, z)\) produces the position \((x+\frac{1}{2}, -y, -z + \frac{1}{2})\), equivalent to the application of a \(2_{1}\) screw axis parallel to \(a\).
3. Combining the \(b\) and \(c\) glide and applying them to \((x, y, z)\) produces the position \((-x, -y + \frac{1}{2}, z+\frac{1}{2})\), equivalent to the application of a \(2_{1}\) screw axis parallel to \(c\).
4. The full space group symbol is thus \(P\frac{2_{1}}{b}\frac{2_{1}}{c}\frac{2_{1}}{a}\).

13. What symmetry element is produced by the combination of \(A\)-centring and a \(2_{1}\) screw axis parallel to \(c\)?

Combining these symmetry elements produces the position \((-x, -y + \frac{1}{2}, z)\). This is equivalent to a two-fold rotation axis parallel to \(c\). This two-fold rotation axis is located at a different position to the \(2_{1}\) screw axis.