InternationalSpace-group symmetryTables for Crystallography Volume A Edited by M. I. Aroyo © International Union of Crystallography 2015 |
International Tables for Crystallography (2015). Vol. A, ch. 1.4, pp. 62-64
## Section 1.4.4.2. Wyckoff positions B. Souvignier
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As already mentioned, one of the first issues in the analysis of crystal structures is the determination of the actual atom positions. Energetically favourable configurations in inorganic compounds are often achieved when the atoms occupy positions that have a nontrivial site-symmetry group. This suggests that one should classify the points in into equivalence classes according to their site-symmetry groups.

*Definition*

A point is called a point in a *general position* for the space group if its site-symmetry group contains only the identity element of . Otherwise, *X* is called a point in a *special position*.

The distinctive feature of a point in a general position is that the points in its orbit are in one-to-one correspondence with the symmetry operations of the group by associating the orbit point with the group operation . For different group elements and , the orbit points and must be different, since otherwise would be a non-trivial operation in the site-symmetry group of *X*. Therefore, the entries listed in the space-group tables for the general positions can not only be interpreted as a shorthand notation for the symmetry operations in (as seen in Section 1.4.2.3), but also as coordinates of the points in the orbit of a point *X* in a general position with coordinates *x*, *y*, *z* (up to translations).

Whereas points in general positions exist for every space group, not every space group has points in a special position. Such groups are called *fixed-point-free space groups* or *Bieberbach groups* and are precisely those groups that may contain glide reflections or screw rotations, but no proper reflections, rotations, inversions and rotoinversions.

*Example*

The group of type (33) has a point group of order 4 and representatives for the non-trivial cosets relative to the translation subgroup are the twofold screw rotation , the *a* glide and the *n* glide . No operation in the coset of the twofold screw rotation can have a fixed point, since such an operation maps the *z* component to for an integer , and this is never equal to *z*. The same argument applies to the *x* component of the *a* glide and to the *y* component of the *n* glide, hence this group contains no operation with a fixed point (apart from the identity element) and is thus a fixed-point-free space group.

The distinction into general and special positions is of course very coarse. In a finer classification, it is certainly desirable that two points in the same orbit under the space group belong to the same class, since they are symmetry equivalent. Such points have *conjugate* site-symmetry groups (*cf.* the orbit–stabilizer theorem in Section 1.1.7
).

*Lemma*

Let *X* and *Y* be points in the same orbit of a space group and let such that . Then the site-symmetry groups of *X* and *Y* are conjugate by the operation mapping *X* to *Y*, *i.e.* one has .

The classification motivated by the conjugacy relation between the site-symmetry groups of points in the same orbit is the classification into *Wyckoff positions*.

*Definition*

Two points *X* and *Y* in belong to the same *Wyckoff position* with respect to if their site-symmetry groups and are conjugate subgroups of .

In particular, the Wyckoff position containing a point *X* also contains the full orbit of *X* under .

*Remark*: It is built into the definition of Wyckoff positions that points that are related by a symmetry operation of belong to the same Wyckoff position. However, a single site-symmetry group may have more than one fixed point, *e.g.* points on the same rotation axis or in the same reflection plane. These points are in general not symmetry related but, having identical site-symmetry groups, clearly belong to the same Wyckoff position. This situation can be analyzed more explicitly:

Let be the site-symmetry group of the point *X* and assume that *Y* is another point with the same site-symmetry group . Choosing a coordinate system with origin *X*, the operations in all have translational part equal to zero and are thus matrix–column pairs of the form . In particular, these operations are *linear* operations, and since both points *X* and *Y* are fixed by all operations in , the vector is also fixed by the linear operations in . But with the vector **v** each scaling of **v** is fixed as well, and therefore all the points on the line through *X* and *Y* are fixed by the operations in . This shows that the Wyckoff position of *X* is a union of infinitely many orbits if has more than one fixed point.

*Lemma*

Let be the site-symmetry group of *X* in :

The space-group tables of Chapter 2.3 contain the following information about the Wyckoff positions of a space group :

The entries in the last column, the *reflection conditions*, are discussed in detail in Chapter 1.6
. This column lists the conditions for the reflection indices *hkl* for which the corresponding structure factor is not systematically zero.

*Examples*