International
Tables for Crystallography Volume A Spacegroup symmetry Edited by M. I. Aroyo © International Union of Crystallography 2016 
International Tables for Crystallography (2016). Vol. A, ch. 3.4, pp. 796798
Section 3.4.2. The concept of characteristic and noncharacteristic orbits, comparison with the latticecomplex concept^{a}Institut für Mineralogie, Petrologie und Kristallographie, PhilippsUniversität, D35032 Marburg, Germany 
3.4.2. The concept of characteristic and noncharacteristic orbits, comparison with the latticecomplex concept
The generating space group of any crystallographic orbit may be compared with the eigensymmetry of its point configuration. If both groups coincide, the orbit is called a characteristic crystallographic orbit, otherwise it is named a noncharacteristic crystallographic orbit (Wondratschek, 1976; Engel et al., 1984; see also Section 1.1.7 ). If the eigensymmetry group contains additional translations in comparison with those of the generating space group, the term extraordinary orbit is used (cf. also Matsumoto & Wondratschek, 1979). Each class of configurationequivalent orbits contains exactly one characteristic crystallographic orbit.
The set of all point configurations in can be divided into 402 equivalence classes by means of their eigensymmetry: two point configurations belong to the same symmetry type of point configuration if and only if their characteristic crystallographic orbits belong to the same type of Wyckoff set.
As each crystallographic orbit is uniquely related to a certain point configuration, each equivalence relationship on the set of all point configurations also implies an equivalence relationship on the set of all crystallographic orbits: two crystallographic orbits are assigned to the same orbit type (cf. also Engel et al., 1984) if and only if the corresponding point configurations belong to the same symmetry type.
In contrast to lattice complexes, neither symmetry types of point configuration nor orbit types can be used to define equivalence relations on Wyckoff positions, Wyckoff sets or types of Wyckoff set. Two crystallographic orbits coming from the same Wyckoff position belong to different orbit types, if – owing to special coordinate values – they differ in the eigensymmetry of their point configurations. Furthermore, two crystallographic orbits with the same coordinate description, but stemming from different space groups of the same type, may belong to different orbit types because of a specialization of the metrical parameters.
Example
The eigensymmetry of orbits from Wyckoff position with or is enhanced to and hence they belong to a different orbit type to those with .
Example
In general, an orbit belonging to the type of Wyckoff set I4/m 2a, b corresponds to a point configuration with eigensymmetry I4/mmm 2a, b. If, however, the space group I4/m has specialized metrical parameters, e.g. c/a = 1 or c/a = 2^{1/2}, then the eigensymmetry of the point configuration is enhanced to or , respectively.
It is the common intention of the latticecomplex and the orbittype concepts to subdivide the point configurations and crystallographic orbits in into subsets with certain common properties. With only a few exceptions, the two concepts result in different subsets. As similar but not identical symmetry considerations are used, each lattice complex is uniquely related to a certain symmetry type of point configuration and to a certain orbit type, and vice versa. Therefore, the two concepts result in the same number of subsets: there exist 402 lattice complexes and 402 symmetry types of point configuration and orbit types. The differences between the subsets are caused by the different properties of the point configurations and crystallographic orbits used for the classifications (cf. also Koch & Fischer, 1985).
The concept of orbit types is entirely based on the eigensymmetry of the particular point configurations: a crystallographic orbit is regarded as an isolated entity, i.e. detached from its Wyckoff position and its type of Wyckoff set. On the contrary, lattice complexes result from a hierarchy of classifications of crystallographic orbits into Wyckoff positions, Wyckoff sets, types of Wyckoff set and classes of configurationequivalent types of Wyckoff set, i.e. a crystallographic orbit is always considered as being embedded in its type of Wyckoff set, and the eigensymmetry of a particular point configuration is disregarded. The differences between the two concepts become clear if limiting complexes are considered.
Fortynine lattice complexes without any limiting complex exist (cf. Table 3.4.2.1). They coincide completely with the corresponding symmetry types of point configurations. As can be extracted from the tables by Engel et al. (1984) there exist 15 additional lattice complexes without limiting complexes due to specialized coordinates. For fundamental reasons, no cubic or hexagonal complexes allow any metrical specialization.
Example
The lattice complex of all triclinic point lattices includes as limiting complexes the 13 other lattice complexes that refer to Bravais lattices. Hence the crystallographic orbits of belong to 14 different orbit types.

Example
The lattice complex Fddd a of all orthorhombic diamond patterns includes as limiting complexes those of the tetragonal and the cubic diamond patterns I4_{1}/amd a and , respectively. The orbits of Fddd a with specialized metric, therefore, belong to the orbit types I4_{1}/amd a or .
353 lattice complexes comprise at least one limiting complex. Each of them includes additional point configurations in comparison to the corresponding symmetry type of point configuration (and orbit type), namely those belonging to the limiting complex.
Example
Lattice complex 24g comprises for y = z the limiting complex 24h, and for the limiting complex 3c. The corresponding orbits with y = z and do not belong to orbit type 24g.
Example
P4/mmm 8r comprises for the limiting complex P4/mmm 4j, for the limiting complex P4/mmm 2g, for the limiting complex P4/mmm 1a, for a = c and x = z the limiting complex 8g, and for a = c and the limiting complex la. Again, none of the corresponding orbits belong to orbit type P4/mmm 8r.
The comparison of an orbit type with its corresponding lattice complex is more intricate. Again, the concept of limiting complexes and comprehensive complexes elucidates the interrelation.
Let A be a lattice complex with a limiting complex B and a comprehensive complex C. The respective orbit types will also be designated A, B and C (e.g. A = 24h ; B = 3c, d , ; C = 24g ). Then a crystallographic orbit from a Wyckoff position of lattice complex A belongs to orbit type A only if it does not correspond to a point configuration of the limiting complex B (i.e. only the crystallographic orbits of 24h with belong to orbit type 24h). The crystallographic orbits of lattice complex A, however, that do correspond to the limiting complex B belong to orbit type B (i.e. all crystallographic orbits from 24h with belong to orbit type 3c, d). On the contrary, those orbits that refer to lattice complex C and that happen to correspond to the limiting complex A of C belong to orbit type A instead of orbit type C. All crystallographic orbits of 24g with y = z or create point configurations of lattice complex 24g but belong to orbit type 24h or 3c, d, respectively.
For the comparison of lattice complexes and orbit types the concept of noncharacteristic orbits is less helpful than the concept of limiting complexes. In terms of lattice complexes, there exist two basically different reasons for a crystallographic orbit to be noncharacteristic:
As a consequence, three kinds of noncharacteristic orbits may be distinguished:
As these considerations illustrate, limiting complexes and noncharacteristic orbits do not coincide and a statement by Engel (1983) proposing this correspondence, therefore, is not correct.
The concept of lattice complexes and limiting complexes on the one hand and of orbit types and noncharacteristic orbits on the other hand are complementary in a certain sense: it is possible to derive all orbit types and all noncharacteristic orbits from the complete knowledge of lattice complexes and limiting complexes and vice versa.
Engel et al. (1984) enumerated for all spacegroup types those noncharacteristic orbits that refer to special coordinates, but they excluded all further ones that are based on specialized metrical parameters of the generating space groups or on the simultaneous specialization of metrical and coordinate parameters. A computer program which enables the determination of noncharacteristic orbits is now available (NONCHAR on the Bilbao Crystallographic Server at http://www.cryst.ehu.es ). Lawrenson & Wondratschek (1976) listed the extraordinary orbits of the plane groups, and Matsumoto & Wondratschek (1987) listed the noncharacteristic orbits of the plane groups.
The special, but not exceptional, case in which a noncharacteristic orbit is produced only if both the coordinates and metric are specialized deserves extra concern. The crystallographic orbits from 6f with or and with the rhombohedral angle α = 90° may be used as an example. The eigensymmetry of the corresponding point configurations is 6c, d (corresponding to the position of the Cr atoms in the crystal structure of Cr_{3}Si). Accordingly, the lattice complex f comprises as limiting complex. shows special integral reflection conditions (hkl: h + k + l = 2n or h = 2n + 1, k = 4n, l = 4n + 2; h, k, l permutable), which of course hold for all orbits of that type, i.e. also for the special orbits from f described above. As geometrical structure factors are independent of metrical parameters, these reflection conditions are even valid for crystallographic orbits from f with if the coordinates are restricted to or to .
In general, the following statement holds: if a lattice complex causes special reflection conditions then exactly these conditions are also valid for any crystallographic orbit that refers to a comprehensive complex of that lattice complex if, in addition, this crystallographic orbit may be described by the same coordinate triplets as an orbit of the lattice complex under consideration.
References
Engel, P. (1983). Zur Theorie der kristallographischen Orbits. Z. Kristallogr. 163, 243–249.Engel, P., Matsumoto, T., Steinmann, G. & Wondratschek, H. (1984). The noncharacteristic orbits of the space groups. Z. Kristallogr. Supplement issue No. 1.
Koch, E. & Fischer, W. (1985). Lattice complexes and limiting complexes versus orbit types and noncharacteristic orbits: a comparative discussion. Acta Cryst. A41, 421–426.
Lawrenson, J. E. & Wondratschek, H. (1976). The extraordinary orbits of the 17 plane groups. Z. Kristallogr. 143, 471–484.
Matsumoto, T. & Wondratschek, H. (1979). Possible superlattices of extraordinary orbits in 3dimensional space. Z. Kristallogr. 150, 181–198.
Matsumoto, T. & Wondratschek, H. (1987). The noncharacteristic Gorbits of the plane groups G. Z. Kristallogr. 179, 7–30.
Wondratschek, H. (1976). Extraordinary orbits of the space groups. Theoretical considerations. Z. Kristallogr. 143, 460–470.