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. 66-67
## Section 1.4.4.4. Eigensymmetry groups and non-characteristic orbits B. Souvignier
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A crystallographic orbit has been defined as the set of points obtained by applying all operations of some space group to a point . From that it is clear that the set is invariant as a whole under the action of operations in , since for some point in the orbit and one has , which is again contained in because belongs to . However, it is possible that the orbit is also invariant under some isometries of that are not contained in . Since the composition of two such isometries still keeps the orbit invariant, the set of all isometries leaving invariant forms a group which contains as a subgroup.

*Definition*

Let be the orbit of a point under a space group . Then the group of isometries of which leave invariant as a whole is called the *eigensymmetry group* of .

Since the orbit is a discrete set, the eigensymmetry group has to be a space group itself. One distinguishes the following cases:

Non-characteristic orbits are closely related to the concept of *lattice complexes*, which are discussed in Chapter 3.4
. An extensive listing of non-characteristic orbits of space groups can be found in Engel *et al.* (1984).

The fact that an orbit of a space group has a larger eigensymmetry group is an important example of a pair of groups that are in a group–subgroup relation. Knowledge of subgroups and supergroups of a given space group play a crucial role in the analysis of phase transitions, for example, and are discussed in detail in Chapter 1.7 .

The occurrence of non-characteristic orbits does not require the point *X* to be chosen at a special position. Even the general position of a space group may give rise to a non-characteristic orbit. Moreover, special values of the coordinates of the general position may give rise to additional eigensymmetries without the position becoming a special position. Conversely, the orbit of a point at a special position need not be non-characteristic.

*Example*

We compare space groups of types (76) and (77). For a space group of type , the general position with generic coordinates gives rise to a characteristic orbit, whereas the general-position orbit for a space group of type consists of the points , , and . An inversion in 0, 0, *z* interchanges and , and maps to , which is clearly equivalent to under a translation. This shows that the general-position orbit for a space group of type is a non-characteristic orbit, and the eigensymmetry group of this orbit is of type (84), where the origin has to be shifted to the inversion point 0, 0, *z* to obtain the conventional setting. Since the unit cell and the orbit are unchanged, but the point group of is a subgroup of index 2 in the point group of , the orbit points must belong to a special position for , namely the position labelled 4*j*. In the conventional setting of , a point belonging to this Wyckoff position is given by *x*, *y*, 0 and one finds that the orbit of this point in special position is characteristic, *i.e.* its eigensymmetry group is just .

If we assume that the metric of the space group is not special, the eigensymmetry group is restricted to the same crystal family (for the definition of `specialized' metrics, *cf.* Section 1.3.4.3
and Chapter 3.5
). Therefore, a space group for which the point group is a holohedry can only have non-characteristic orbits by additional translations, *i.e.* extraordinary orbits. However, if we allow specialized metrics, the eigensymmetry group may belong to a higher crystal family. For example, if a space group belongs to the orthorhombic family, but the unit cell has equal parameters *a* = *b*, then the eigensymmetry group of an orbit can belong to the tetragonal family.

*Note*: A space group is equal to the intersection of the eigensymmetry groups of the orbits of all its positions. If none of the positions of a space group gives rise to a characteristic orbit, this means that each single orbit under does not have as its symmetry group, but a larger group that contains as a proper subgroup. It may thus be necessary to have the union of at least two orbits under to obtain a structure that has precisely as its group of symmetry operations.

*Examples*

Knowledge of the eigensymmetry groups of the different positions for a group is of utmost importance for the analysis of diffraction patterns. Atoms in positions that give rise to non-characteristic orbits, in particular extraordinary orbits, may cause systematic absences that are not explained by the space-group operations. These absences are specified as *special reflection conditions* in the space-group tables of this volume, but only as long as no specialization of the coordinates is involved. For the latter case, the possible existence of systematic absences has to be deduced from the tables of noncharacteristic orbits. Reflection conditions are discussed in detail in Chapter 1.6
.

*Example*

For the group of type *Pccm* (49) the special position (Wyckoff position 4*p*) gives rise to an extraordinary orbit, since it allows the additional translation . The special reflection condition corresponding to this additional translation is the integral reflection condition *hkl*: *l* = 2*n*. However, if the *z* coordinate in position 4*p* is set to , the eigensymmetry group also contains the translation . In this case, the special reflection condition becomes *hkl*: *l* = 4*n*.

### References

Engel, P., Matsumoto, T., Steinmann, G. & Wondratschek, H. (1984). The Non-characteristic Orbits of the Space Groups.*Z. Kristallogr.*Supplement issue No. 1.