International
Tables for
Crystallography
Volume A
Space-group symmetry
Edited by M. I. Aroyo

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. Souvignierd

1.4.4.4. Eigensymmetry groups and non-characteristic orbits

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A crystallographic orbit [{\cal O}] has been defined as the set of points [\ispecialfonts{\sfi g}(X)] obtained by applying all operations of some space group [{\cal G}] to a point [X \in {\bb E}^3]. From that it is clear that the set [{\cal O}] is invariant as a whole under the action of operations in [{\cal G}], since for some point [\ispecialfonts Y = {\sfi g}(X)] in the orbit and [\ispecialfonts{\sfi h} \in {\cal G}] one has [\ispecialfonts{\sfi h}(Y) = ({\sfi h} {\sfi g})(X)], which is again contained in [{\cal O}] because [\ispecialfonts{\sfi h} {\sfi g}] belongs to [{\cal G}]. However, it is possible that the orbit [{\cal O}] is also invariant under some isometries of [{\bb E}^3] that are not contained in [{\cal G}]. Since the composition of two such isometries still keeps the orbit invariant, the set of all isometries leaving [{\cal O}] invariant forms a group which contains [{\cal G}] as a subgroup.

Definition

Let [\ispecialfonts{\cal O} = \{ {\sfi g}(X) | {\sfi g} \in {\cal G} \}] be the orbit of a point [X \in {\bb E}^3] under a space group [{\cal G}]. Then the group [{\scr E}] of isometries of [{\bb E}^3] which leave [{\cal O}] invariant as a whole is called the eigensymmetry group of [{\cal O}].

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

  • (i) The eigensymmetry group [{\scr E}] equals the group [{\cal G}] by which the orbit was generated. In this case the orbit is called a characteristic orbit of [{\cal G}].

  • (ii) The eigensymmetry group [{\scr E}] contains [{\cal G}] as a proper sub­group. Then the orbit is called a non-characteristic orbit.

  • (iii) If the eigensymmetry group [{\scr E}] contains translations that are not contained in [{\cal G}], i.e. if [{\cal T_G}] is a proper subgroup of [{\cal T}_{\!\!\!\scr E}], the orbit is called an extraordinary orbit. Of course, extraordinary orbits are a special kind of non-characteristic orbits.

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

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[link] .

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 [{\cal G}] 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 [P4_1] (76) and [P4_2] (77). For a space group of type [P4_1], the general position with generic coordinates [x, y, z] gives rise to a characteristic orbit, whereas the general-position orbit for a space group of type [P4_2] consists of the points [X_1 = \pmatrix{x\cr y\cr z}], [X_2 = \pmatrix{\overline{x}\cr \overline{y}\cr z}], [X_3 = \pmatrix{\overline{y}\cr x\cr z+\textstyle{{1 \over 2}}}] and [X_4 = \pmatrix{y\cr \overline{x}\cr z+\textstyle{{1 \over 2}}}]. An inversion [\{\overline{1}| 0,0,2z\}] in 0, 0, z interchanges [X_1] and [X_2], and maps [X_3] to [y,\overline{x},z-\textstyle{{1 \over 2}}], which is clearly equivalent to [X_4] under a translation. This shows that the general-position orbit for a space group of type [P4_2] is a non-characteristic orbit, and the eigensymmetry group of this orbit is of type [P4_2/m] (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 [P4_2] is a subgroup of index 2 in the point group of [P4_2/m], the orbit points must belong to a special position for [P4_2/m], namely the position labelled 4j. In the conventional setting of [P4_2/m], 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 [P4_2/m].

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[link] and Chapter 3.5[link] ). Therefore, a space group [{\cal G}] 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 [{\cal G}] 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 [{\cal G}] gives rise to a characteristic orbit, this means that each single orbit under [{\cal G}] does not have [{\cal G}] as its symmetry group, but a larger group that contains [{\cal G}] as a proper subgroup. It may thus be necessary to have the union of at least two orbits under [{\cal G}] to obtain a structure that has precisely [{\cal G}] as its group of symmetry operations.

Examples

  • (1) For the group [{\cal G}] of type Pmmm (47) all Wyckoff positions with no further special values of the coordinates give rise to characteristic orbits, because the point group of [{\cal G}] is a holohedry and the general coordinates allow no further translations. However, there are various `specializations' of the positions that give rise to extraordinary orbits. For example, setting x to the special value [\textstyle{{1 \over 4}}] for the general position introduces the additional translation [t(\textstyle{{1 \over 2}},0,0)]. In fact, for all positions in which the first coordinate has no specified value (positions 2i–2l, 4w–4z, 8α), setting [x = \textstyle{{1 \over 4}}] introduces the translation [t(\textstyle{{1 \over 2}},0,0)] and thus gives rise to an extraordinary orbit. In all these cases, the resulting eigensymmetry group is of type Pmmm with primitive lattice basis [\textstyle{{1 \over 2}} {\bf a}, {\bf b}, {\bf c}].

  • (2) For the group [{\cal G}] of type Pmm2 (25) no Wyckoff position gives rise to a characteristic orbit, because this is a polar group (with respect to the c axis). Any orbit of a point with third coordinate z allows an additional mirror plane normal to the c axis and located at 0, 0, z. For example, the general position gives rise to a non-characteristic orbit with eigensymmetry group Pmmm (47). Since the general coordinates allow no additional translation, this is not an extraordinary orbit. However, setting [x = \textstyle{{1 \over 4}}] for the general position introduces the translation [t(\textstyle{{1 \over 2}},0,0)] (as in the above example) and thus gives rise to an extraordinary orbit. The eigensymmetry group is Pmmm with primitive lattice basis [\textstyle{{1 \over 2}} {\bf a}, {\bf b}, {\bf c}].

    On the other hand, the special positions x, 0, z (Wyckoff position 2e) and [x, \textstyle{{1 \over 2}}, z] (Wyckoff position 2f) both have the same eigensymmetry group as the general position and setting [x= \textstyle{{1 \over 4}}] for each, giving [\textstyle{{1 \over 4}}, 0, z] and [\textstyle{{1 \over 4}}, \textstyle{{1 \over 2}}, z], results in these positions having the same eigensymmetry group as the [\textstyle{{1 \over 4}}, y, z] case of the general position.

  • (3) For a group [{\cal G}] of type [P\bar{4}c2] (116) the general-position coordinates are[\quad\matrix{(1) \ x,y,z\hfill & (2) \ \overline{x}, \overline{y}, z\hfill& (3) \ y, \overline{x}, \overline{z}\hfill& (4) \ \overline{y}, x, \overline{z}\hfill \cr (5)\ x, \overline{y}, z + \textstyle{{1 \over 2}}\hfill&(6)\ \overline{x}, y, z + \textstyle{{1 \over 2}}\hfill&(7)\ y, x, \overline{z} + \textstyle{{1 \over 2}}\hfill&(8)\ \overline{y}, \overline{x}, \overline{z} + \textstyle{{1 \over 2}}\hfill}]

    A point x, y, z in a general position does not give rise to an extraordinary orbit because, owing to the general coordinates, there can not be any additional translation. Furthermore, the point group [\bar{4}m2] of [{\cal G}] has index 2 in the holohedry 4/mmm. Thus, in order to have a non-characteristic orbit one would require an inversion in some point as an additional operation. But an inversion in [p_1, p_2, p_3] would map [x, y, z] to [\bar{x}+2p_1, \bar{y}+2p_2,] [ \bar{z}+2p_3] and no such point is contained in the orbit for generic [x, y, z]. The point [x, y, z] therefore gives rise to a characteristic orbit.

    However, if the point in a general position is chosen with x = y, one indeed obtains an additional inversion at [0,0,\textstyle{{1 \over 4}}] which maps [x, x, z] to the orbit point [\bar{x},\bar{x},\bar{z}+\textstyle{{1 \over 2}}] (general position point No. 8). This orbit thus is non-characteristic, but it is not extraordinary, since no additional translation is introduced. The eigensymmetry group obtained is [P4_2/mcm] (132).

    On the other hand, if the general position is chosen with y = 0, no additional inversion is obtained, but the translation by [\textstyle{{1 \over 2}} {\bf c}] maps [x, 0, z] to [x,0,z+\textstyle{{1 \over 2}}] (general-position point No. 5). The position [x, 0, z] therefore gives rise to an extraordinary orbit with eigensymmetry group [P\bar{4}m2] (115).

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[link] .

Example

For the group [{\cal G}] of type Pccm (49) the special position [\textstyle{{1 \over 2}}, 0, z] (Wyckoff position 4p) gives rise to an extraordinary orbit, since it allows the additional translation [\textstyle{{1 \over 2}} {\bf c}]. 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 [z = \textstyle{{1}\over{8}}], the eigensymmetry group also contains the translation [\textstyle{{1 \over 4}} {\bf c}]. In this case, the special reflection condition becomes hkl: l = 4n.

References

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








































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