Wyckoff Positions 2
Consider the space group \(P\frac{2}{b}\frac{2_{1}}{c}\frac{2_{1}}{m}\). Glide planes cannot contribute to Wyckoff positions due to the translational associated with this symmetry element (an atom on these planes still moves!). Screw axes also do not typically contribute for the same reason, except in the other inherent symmetry they produce. For example, a \(4_{2}\) screw axis also mandates a two-fold rotation in the same position, which is used to define the Wyckoff position.
In the space group \(P\frac{2}{b}\frac{2_{1}}{c}\frac{2_{1}}{m}\), atoms can lie on the mirror plane, on the two-fold axis or on one of the inversion centres. The Wyckoff positions are given in the table below.
| Multiplicity | Wyckoff Letter | Site Symmetry | Positions |
| 8 | e | 1 | \((x, y, z)\) \((\bar{x}, \bar{y}, z+\frac{1}{2})\) \((\bar{x}, y+\frac{1}{2}, \bar{z}+\frac{1}{2})\) \((x, \bar{y}+\frac{1}{2}, \bar{z})\) \((\bar{x}, \bar{y}, \bar{z})\) \((x, y, \bar{z}+\frac{1}{2})\) \((x, \bar{y}+\frac{1}{2}, z+\frac{1}{2})\) \((\bar{x}, y+\frac{1}{2}, z)\) |
| 4 | d | . . m | \((x, y, \frac{1}{4})\) \((\bar{x}, \bar{y}, \frac{3}{4})\) \((\bar{x}, y+\frac{1}{2}, \frac{1}{4})\) \((x, \bar{y}+\frac{1}{2}, \frac{3}{4})\) |
| 4 | c | 2 . . | \((x, \frac{1}{4}, 0)\) \((\bar{x}, \frac{3}{4}, \frac{1}{2})\) \((\bar{x}, \frac{3}{4}, 0)\) \((x, \frac{1}{4}, \frac{1}{2})\) |
| 4 | b | \(\bar{1}\) | \((\frac{1}{2}, 0, 0)\) \((\frac{1}{2}, 0, \frac{1}{2})\) \((\frac{1}{2}, \frac{1}{2}, \frac{1}{2})\) \((\frac{1}{2}, \frac{1}{2}, 0)\) |
| 4 | a | \(\bar{1}\) | \((0, 0, 0)\) \((0, 0, \frac{1}{2})\) \((0, \frac{1}{2}, \frac{1}{2})\) \((0, \frac{1}{2}, 0)\) |
The site symmetry is the symmetry elements the position lies on. Dots are used to inform about which direction the symmetry element corresponds to. For example, ‘. . m‘ means the m corresponds to the third space group symbol, which in orthorhombic is the \(c\) axis. Therefore the position lies on the mirror plane perpendicular to the \(c\) axis.
Final Notes
In the positions associated with each Wyckoff position, remember \(\bar{x}\) is equivalent to \(1-x\).
‘General’ positions refer to the Wyckoff position with highest multiplicity while ‘special positions’ refer to all other Wyckoff positions in the space group. It is common for crystallographers to quote Wyckoff positions with just the multiplicity and letter e.g. 4c site.
In some space groups, a Wyckoff position may have coordinates such as \((x, 2x, z)\) e.g. \(P\frac{6_{3}}{m}mc\). This is because the mirror plane is aligned perpendicular to the vector \(\mathbf{a}\), and the unit cell vectors are not perpendicular, so coordinates such as \((0.1, 0.2, z)\) or \((0.07, 0.14, z)\) lie on this mirror plane. It might be worth looking at a space group diagram of \(P\frac{6_{3}}{m}mc\) to confirm this for yourself.