InternationalSpace-group symmetryTables for Crystallography Volume A Edited by Th. Hahn © International Union of Crystallography 2006 |
International Tables for Crystallography (2006). Vol. A, ch. 14.3, pp. 873-876
doi: 10.1107/97809553602060000532 ## Chapter 14.3. Applications of the lattice-complex concept
Chapter 14.3 gives a short introduction to some applications of lattice complexes: (i) The knowledge of the assignment of the Wyckoff positions to lattice complexes considerably facilitates the study of geometrical properties of point configurations. (ii) Relations between crystal structures with different symmetries are often discernible because the corresponding Wyckoff positions either belong to the same lattice complex or because a limiting-complex relationship exists. (iii) Wyckoff positions belonging to the same lattice complex show analogous reflection conditions. (iv) If a phase transition of a crystal is connected with a group–subgroup degradation, comparison of the lattice complexes corresponding to the Wyckoff positions of the original space group on the one hand and of its various subgroups on the other hand very often shows which of these subgroups are suitable for the low-symmetry modification. (v) Many incorrect space-group assignments to crystal structures could be avoided by simply looking at the lattice complexes (and their descriptive symbols) that correspond to the Wyckoff positions occupied by the different kinds of atoms. |

To study the geometrical properties of all point configurations in three-dimensional space, it is not necessary to consider all Wyckoff positions of the space groups or all 1128 types of Wyckoff set. Instead, one may restrict the investigations to the characteristic Wyckoff positions of the 402 lattice complexes. The results can then be transferred to all noncharacteristic Wyckoff positions of the lattice complexes, as listed in Tables 14.2.3.1 and 14.2.3.2 .

The determination of all types of sphere packings with cubic or tetragonal symmetry forms an example for this kind of procedure (Fischer, 1973, 1974, 1991*a*,*b*, 1993). The cubic lattice complex *I*4*xxx*, for example, allows two types of sphere packings within its characteristic Wyckoff position 8*c* *xxx* .3*m*, namely for and for (*cf.* Fischer, 1973). Ag_{3}PO_{4} crystallizes with symmetry (Deschizeaux-Cheruy *et al.*, 1982) and the oxygen atoms occupy Wyckoff position 8*e xxx* .3., which also belongs to *I*4*xxx*. Comparison of the coordinate parameter for the oxygen atoms with the sphere-packing parameters listed for *m c* shows directly that the oxygen arrangement in this crystal structure does not form a sphere packing.

Other examples for this approach are the derivation of crystal potentials (Naor, 1958), of coordinate restrictions in crystal structures (Smirnova, 1962), of Patterson diagrams (Koch & Hellner, 1971), of Dirichlet domains (Koch, 1973, 1984) and of sphere packings for subperiodic groups (Koch & Fischer, 1978).

The 30 lattice complexes in two-dimensional space correspond uniquely to the `henomeric types of dot pattern' introduced by Grünbaum and Shephard (*cf. e.g.* Grünbaum & Shephard, 1981; Grünbaum, 1983).

Frequently, different crystal structures show the same geometrical arrangement for some of their atoms, even though their space groups do not belong to the same type. In these cases, the corresponding Wyckoff positions either belong to the same lattice complex or there exist close relationships between them, *e.g.* limiting-complex relations.

*Examples*

Publications by Hellner (1965, 1976*a*,*b*,*c*, 1977, 1979), Loeb (1970), Smirnova & Vasserman (1971), Sakamoto & Takahasi (1971), Niggli (1971), Fischer & Koch (1974), Hellner *et al.* (1981) and Hellner & Sowa (1985) refer to this aspect.

Wyckoff positions belonging to the same lattice complex show analogous reflection conditions. Therefore, lattice complexes have also been used to check the reflection conditions for all Wyckoff positions in the space-group tables of this volume.

The descriptive symbols may supply information on the reflection conditions. If the symbol does not contain any distribution-symmetry part, the reflection conditions of the Wyckoff position are indicated by the symbol of the invariant lattice complex in the central part (*e.g.* : *C*4*xx* shows that the reflection condition is that of a *C* lattice, ). In the case that the site set consists of only one point, *i.e.* the Wyckoff position belongs to a Weissenberg complex, all conditions for general reflections *hkl* that may arise from special choices of the coordinates can be read from the central part of the symbol (*e.g.* indicates that, by special choice of *z*, either or may be produced).

If a crystal undergoes a phase transition from a high- to a low-symmetry modification, the transition may be connected with a group–subgroup degradation. In such a case, the comparison of the lattice complexes corresponding to the Wyckoff positions of the original space group on the one hand and of its various subgroups on the other hand very often shows which of these subgroups are suitable for the low-symmetry modification.

This kind of procedure will be demonstrated with the aid of a space group and its three translation-equivalent subgroups with index 2, namely *R*32, and *R*3*m*. In the course of the subgroup degradation, the Wyckoff positions of behave differently:

The descriptive symbols *R* and refer to Wyckoff positions and 3*b* as well as to Wyckoff positions *R*32 3*a* and 3*b* and and 3*b*. Therefore, all corresponding point configurations and atomic arrangements remain unchanged in these subgroups. In subgroup *R*3*m*, however, the respective Wyckoff position is 3*a* with descriptive symbol *R*[*z*], *i.e.* a shift parallel to [001] of the entire point configuration is allowed.

The descriptive symbol *R*2*z* for occurs also for *R*32 6*c* and . Again both subgroups do not allow any deformations of the corresponding point configurations or atomic arrangements. Symmetry reduction to *R*3*m*, however, yields a splitting of each *R*2*z* configuration into two *R*[*z*] configurations. The two *z* parameters may be chosen independently.

As *M* and are the descriptive symbols not only of and 9*d* but also of and 9*d*, does not enable any deformation of the corresponding atomic arrangements. In *R*32 and in *R*3*m*, however, the respective point configurations may be deformed differently, as the descriptive symbols show: *R*3*x* and (*R*32 9*d* and 9*e*), (*R*3*m* 9*b*).

Wyckoff positions and 18*g* (*R*6*x* and ) correspond to *R*32 9*d* and 9*e* (*R*3*x* and ), to , and to *R*3*m* 18*c* . In *R*32, the hexagons 6*x* around the points of the *R* lattice are split into two oppositely oriented triangles 3*x*, which may have different size. In and in *R*3*m*, the hexagons may be deformed differently.

Wyckoff position corresponds to sets of trigonal antiprisms around the points of an *R* lattice. These antiprisms may be distorted in *R*32 18*f* (*R*3*x*2*yz*) or rotated in 18*f* (*R*6*xyz*). In *R*3*m* 9*b* , each antiprism is split into two parallel triangles that may differ in size.

In each of the three subgroups, any point configuration belonging to the general position 36*i* splits into two parts. Each of these parts may be deformed differently.

In the literature, some crystal structures are still described within space groups that are only subgroups of the correct symmetry groups. Many such mistakes (but not all of them) could be avoided by simply looking at the lattice complexes (and their descriptive symbols) that correspond to the Wyckoff positions of the different kinds of atoms. Whenever the same (or an analogous) lattice-complex description of a crystal structure is also possible within a supergroup, then the crystal structure has at least that symmetry.

*Examples*

Descriptive symbols of lattice complexes – at least those of the invariant lattice complexes – have been used for the description of crystal structures (*cf.* Section 14.3.2 and the literature cited there), for the nomenclature of three-periodic surfaces (von Schnering & Nesper, 1987) and in connection with orbifolds of space groups (Johnson *et al.*, 2001).

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