International
Tables for Crystallography Volume A Spacegroup symmetry Edited by M. I. Aroyo © International Union of Crystallography 2015 
International Tables for Crystallography (2015). Vol. A, ch. 1.4, pp. 6167
Section 1.4.4. General and special Wyckoff positions
B. Souvignier^{d}

One of the first tasks in the analysis of crystal patterns is to determine the actual positions of the atoms. Since the full crystal pattern can be reconstructed from a single unit cell or even an asymmetric unit, it is clearly sufficient to focus on the atoms inside such a restricted volume. What one observes is that the atoms typically do not occupy arbitrary positions in the unit cell, but that they often lie on geometric elements, e.g. reflection planes or lines along rotation axes. It is therefore very useful to analyse the symmetry properties of the points in a unit cell in order to predict likely positions of atoms.
We note that in this chapter all statements and definitions refer to the usual threedimensional space , but also can be formulated, mutatis mutandis, for plane groups acting on and for higherdimensional groups acting on ndimensional space .
Since the operations of a space group provide symmetries of a crystal pattern, two points X and Y that are mapped onto each other by a spacegroup operation are regarded as being geometrically equivalent. Starting from a point , infinitely many points Y equivalent to X are obtained by applying all spacegroup operations to X: = .
Definition
For a space group acting on the threedimensional space , the (infinite) setis called the orbit of X under .
The orbit of X is the smallest subset of that contains X and is closed under the action of . It is also called a crystallographic orbit.
Every point in direct space belongs to precisely one orbit under and thus the orbits of partition the direct space into disjoint subsets. It is clear that an orbit is completely determined by its points in the unit cell, since translating the unit cell by the translation subgroup of entirely covers .
It may happen that two different symmetry operations and in map X to the same point. Since implies that , the point X is fixed by the nontrivial operation in .
Definition
The subgroup of symmetry operations from that fix X is called the sitesymmetry group of X in .
Since translations, glide reflections and screw rotations fix no point in , a sitesymmetry group never contains operations of these types and thus consists only of reflections, rotations, inversions and rotoinversions. Because of the absence of translations, contains at most one operation from a coset relative to the translation subgroup of , since otherwise the quotient of two such operations and would be the nontrivial translation (see Chapter 1.3 for a discussion of coset decompositions). In particular, the operations in all have different linear parts and because these linear parts form a subgroup of the point group of , the order of the sitesymmetry group is a divisor of the order of the point group of .
The sitesymmetry group of a point X is thus a finite subgroup of the space group , a subgroup which is isomorphic to a subgroup of the point group of .
Example
For a space group of type , the sitesymmetry group of the origin is clearly generated by the inversion in the origin: . On the other hand, the point is fixed by the inversion in Y, i.e.The symmetry operation also belongs to and generates the sitesymmetry group of Y. The sitesymmetry groups of X and of Y are thus different subgroups of order 2 of which are isomorphic to the point group of (which is generated by ).
The order of the sitesymmetry group is closely related to the number of points in the orbit of X that lie in the unit cell. An application of the orbit–stabilizer theorem (see Section 1.1.7 ) yields the crucial observation that each point in the orbit of X under is obtained precisely times as an orbit point: for each one has = and conversely implies that and thus for an operation in .
Assuming first that we are dealing with a space group described by a primitive lattice, each coset of relative to the translation subgroup contains precisely one operation such that lies in the primitive unit cell. Since the number of cosets equals the order of the point group of and since each orbit point is obtained times, it follows that the number of orbit points in the unit cell is .
If we deal with a space group with a centred unit cell, the above result has to be modified slightly. If there are k − 1 centring vectors, the lattice spanned by the conventional basis is a sublattice of index k in the full translation lattice. The conventional cell therefore is built up from k primitive unit cells (spanned by a primitive lattice basis) and thus in particular contains k times as many points as the primitive cell (see Chapter 1.3 for a detailed discussion of conventional and primitive bases and cells).
Proposition
Let be a space group with point group and let be the sitesymmetry group of a point X in . Then the number of orbit points of the orbit of X which lie in a conventional cell for is equal to the product , where k is the volume of the conventional cell divided by the volume of a primitive unit cell.
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 sitesymmetry group. This suggests that one should classify the points in into equivalence classes according to their sitesymmetry groups.
Definition
A point is called a point in a general position for the space group if its sitesymmetry 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 onetoone 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 nontrivial operation in the sitesymmetry group of X. Therefore, the entries listed in the spacegroup 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 fixedpointfree 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 nontrivial 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 fixedpointfree 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 sitesymmetry 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 sitesymmetry 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 sitesymmetry 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 sitesymmetry 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 sitesymmetry 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 sitesymmetry groups, clearly belong to the same Wyckoff position. This situation can be analyzed more explicitly:
Let be the sitesymmetry group of the point X and assume that Y is another point with the same sitesymmetry 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 sitesymmetry group of X in :
The spacegroup 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
Points belonging to the same Wyckoff position have conjugate sitesymmetry groups and thus in particular all those points are collected together that lie in one orbit under the space group . However, in addition, points that are not symmetryrelated by a symmetry operation in may still play geometrically equivalent roles, e.g. as intersections of rotation axes with certain reflection planes.
Example
In the conventional setting, the fourfold axes of a space group of type P4 (75) intersect the ab plane in the points and for integers , as can be seen from the spacegroup diagram in Fig. 1.4.4.3.
The points lie in one orbit under the translation subgroup of , and thus belong to the same Wyckoff position, labelled 1a. For the same reason, the points belong to a single Wyckoff position, namely to position 1b. The points and do not belong to the same Wyckoff position, because the sitesymmetry group is generated by the fourfold rotation 4_{001} and conjugating this by an operation results in a fourfold rotation with axis parallel to the c axis and running through w. But since the translation parts of all operations in are integral, such an axis can not contain and thus and are not conjugate in .
However, the translation by conjugates to , while fixing the group as a whole. This shows that there is an ambiguity in choosing the origin either at or , since these points are geometrically indistinguishable (both being intersections of a fourfold axis with the ab plane).
The ambiguity in the origin choice in the above example can be explained by the affine normalizer of the space group (see Section 1.1.8 for a general introduction to normalizers). The full group of affine mappings acts via conjugation on the set of space groups and the space groups of the same affine type are obtained as the orbit of a single group of that type under .
Definition
The group of affine mappings that fix a space group under conjugation is called the affine normalizer of , i.e.The affine normalizer is the largest subgroup of such that is a normal subgroup of .
Conjugation by operations of the affine normalizer results in a permutation of the operations of , i.e. in a relabelling without changing their geometric properties. The additional translations contained in the affine normalizer can in fact be derived from the spacegroup diagrams, because shifting the origin by such a translation results in precisely the same diagram. More generally, an element of the affine normalizer can be interpreted as a change of the coordinate system that does not alter the spacegroup diagrams.
A more thorough description of the affine normalizers of space groups is given in Chapter 3.5 , where tables with the affine normalizers are also provided.
Since the affine normalizer of a space group is in general a group containing as a proper subgroup, it is possible that subgroups of that are not conjugate by any operation of may be conjugate by an operation in the affine normalizer. As a consequence, the sitesymmetry groups and of two points X and Y belonging to different Wyckoff positions of may be conjugate under the affine normalizer of . This reveals that the points X and Y are in fact geometrically equivalent, since they fall into the same orbit under the affine normalizer of . Joining the equivalence classes of these points into a single class results in a coarser classification with larger classes, which are called Wyckoff sets.
Definition
Two points X and Y belong to the same Wyckoff set if their sitesymmetry groups and are conjugate subgroups of the affine normalizer of .
In particular, the Wyckoff set containing a point X also contains the full orbit of X under the affine normalizer of .
Example
Let be the space group of type (17) generated by the translations of an orthorhombic lattice, the twofold rotation and the twofold screw rotation . Note that the composition of these two elements is the twofold rotation with the line as its geometric element. The group has four different Wyckoff positions with a sitesymmetry group generated by a twofold rotation; representatives of these Wyckoff positions are the points (Wyckoff position 2a, sitesymmetry symbol 2..), (position 2b, symbol 2..), (position 2c, symbol .2.) and (position 2d, symbol .2.).
From the tables of affine normalizers in Chapter 3.5 , but also by a careful analysis of the spacegroup diagrams in Fig. 1.4.4.4, one deduces that the affine normalizer of contains the additional translations , and , since all the diagrams are invariant by a shift of along any of the coordinate axes. Moreover, the symmetry operation which interchanges the a and b axes and shifts the origin by along the c axis belongs to the affine normalizer, because it precisely interchanges the twofold rotations around axes parallel to the a and to the b axes. The translation maps to , and hence and have sitesymmetry groups which are conjugate under the affine normalizer of and thus belong to the same Wyckoff set. Analogously, and belong to the same Wyckoff set, because maps to . Finally, the operation found in the affine normalizer maps to . This shows that the points of all four Wyckoff positions actually belong to the same Wyckoff set.

Symmetryelement diagrams for the space group (17) for orthogonal projections along [001], [010], [100] (left to right). 
Geometrically, the positions in this Wyckoff set can be described as those points that lie on a twofold rotation axis.
The assignments of Wyckoff positions of plane and space groups to Wyckoff sets are discussed and tabulated in Chapter 3.4 .
Remark: The previous example deserves some further discussion. The group of type belongs to the orthorhombic crystal family, and the conventional unit cell is spanned by three basis vectors a, b, c with lengths a, b, c and right angles between each pair of basis vectors. Unless the parameters a and b are equal because of some metric specialization, the operation of the affine normalizer is not an isometry but changes lengths. If it is desired that the metric properties are preserved, the full affine normalizer cannot be taken into account, but only the subgroup that consists of isometries. This subgroup is called the Euclidean normalizer of . (A detailed discussion of Euclidean normalizers of space groups and their tabulation are given in Chapter 3.5 .)
Taking conjugacy of the sitesymmetry groups under the Euclidean normalizer as a condition results in a notion of equivalence which lies between that of Wyckoff positions and Wyckoff sets. In the above example, the four Wyckoff positions would be merged into two classes represented by and , but and would not be regarded as equivalent, since they are not related by an operation of the Euclidean normalizer.
It turns out, however, that in many cases this intermediate classification coincides with the Wyckoff sets, because points belonging to different Wyckoff positions are often related to each other by a translation contained in the affine normalizer. Since translations are always isometries, the translations contained in the affine normalizer always belong to the Euclidean normalizer as well.
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:

Noncharacteristic orbits are closely related to the concept of lattice complexes, which are discussed in Chapter 3.4 . An extensive listing of noncharacteristic 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 noncharacteristic 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 noncharacteristic 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 noncharacteristic.
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 generalposition 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 generalposition orbit for a space group of type is a noncharacteristic 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 4j. 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 noncharacteristic 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 noncharacteristic orbits, in particular extraordinary orbits, may cause systematic absences that are not explained by the spacegroup operations. These absences are specified as special reflection conditions in the spacegroup 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 4p) 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 = 2n. However, if the z coordinate in position 4p is set to , the eigensymmetry group also contains the translation . In this case, the special reflection condition becomes hkl: l = 4n.
References
Engel, P., Matsumoto, T., Steinmann, G. & Wondratschek, H. (1984). The Noncharacteristic Orbits of the Space Groups. Z. Kristallogr. Supplement issue No. 1.