As noted before, some space groups have locations where if an atom is placed, some symmetry operations will generate the same position again rather than a different location. These are called special positions, which may be a point, line or even plane. For every space group, the International Tables has a tabulation of special positions known as the Wyckoff sites. These consist of a number followed by a letter. The number indicates the number of these type of positions that occur in the cell, which is known as the multiplicity. The letter is assigned sequentially. The first item, will have the largest multiplicity, and the last assigned letter will be the the lowest symmetry locations in the cell and with it will be listed the equivalent locations. The subsequent Wyckoff sites will have higher symmetry and lower multiplicity and the locations where that occurs. If a line, it will be expressed as a formula such as (x,0,0) or (x,x,0) and if a plane (x,y,0). As an example, the space group \(P 2_1/c\) has the following Wyckoff sites listed:
The 4e Wyckoff site is the general position and the other four Wyckoff sites are pairs of discrete points in the unit cell. Alas, the listing order of Wyckoff sites, other than in decreasing multiplicity, is not in any derivable sequence and thus is only a reference to the International Tables tabulations. I tend to avoid use of Wyckoff labels, as I consider the letter assignment to be arbitrary, preferring to describe sites by their multiplicity and the site symmetry, but sometimes use of Wyckoff labels is unavoidable.
Related to site multiplicities is the concept of the asymmetric unit. This can be expressed in two different ways. The International Tables lists an asymmetric unit for each space group as a unique volume within the unit cell and if that unique volume is duplicated using the symmetry operators, the entire contents of the unit cell can be constructed. The other way that the asymmetric unit is described is in terms of the unique atoms that can be used to build up the unit cell contents. Note that each atom in the asymmetric unit will then have a multiplicity. The previous expression for the structure factor can then have the number of terms reduced
\[F_{hkl}=\sum _j^{N^\prime } f_j m_j\exp [-2\pi i(hx_j + ky_j + lz_j)] \exp [-2\pi i(\vec {Q}\cdot \vec {U_j})]\]
where the summation over \(N^\prime \) reduced to a summation over the number of atoms in the asymmetric unit rather than the entire unit cell contents and an extra term, \(m_j\), is included for the multiplicity of each atom site.