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. 792825
https://doi.org/10.1107/97809553602060000932 Chapter 3.4. Lattice complexes^{a}Institut für Mineralogie, Petrologie und Kristallographie, PhilippsUniversität, D35032 Marburg, Germany In Section 3.4.1, the concept of lattice complexes and limiting complexes is introduced and compared with the concept of orbit types and noncharacteristic orbits. To this end it is necessary to differentiate strictly between the two terms `point configuration' and `crystallographic orbit', both of which have often been used with two slightly different meanings: (1) for sets of points that are equivalent with respect to a given space group, i.e. in the mathematical sense of `orbit'; (2) for such sets of points, but detached from their generating space groups. A `lattice complex' is defined as a set of point configurations that may be generated within one type of Wyckoff set. Furthermore, the following items are introduced and illustrated by examples: Wyckoff position, Wyckoff set, type of Wyckoff set, limiting complex, comprehensive complex, Weissenberg complex, degrees of freedom of a lattice complex, and reference symbols of the lattice complexes. In Section 3.4.2, the concept of characteristic and noncharacteristic orbits is introduced and compared with the concept of lattice complexes and limiting complexes. In Section 3.4.3, the descriptive symbols of lattice complexes are introduced, their properties are described and their interpretation is demonstrated by numerous examples. Tables 3.4.3.2 and 3.4.3.3 give the explicit assignment of the Wyckoff positions of all plane groups and space groups, respectively, to Wyckoff sets and to lattice complexes. For each Wyckoff position, the reference symbol of the corresponding lattice complex is tabulated. In addition, a descriptive symbol is given that describes the arrangement of points in the corresponding point configurations. It refers directly to the coordinate description of the Wyckoff position. Section 3.4.4 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 limitingcomplex 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 transition, 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 lowsymmetry modification. (v) Many incorrect spacegroup 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. 
The term lattice complex (Gitterkomplex) was originally coined by P. Niggli (1919), but he used the term in an ambiguous manner. Later, Hermann (1935) modified and specified the concept of lattice complexes. The rigorous definition used in this chapter was proposed later still by Fischer & Koch (1974a) [cf. also Koch & Fischer (1978a)]. An alternative definition was given by Zimmermann & Burzlaff (1974) at around the same time.
In crystal structures belonging to different structure types and showing different spacegroup symmetries, some of the atoms may have the same relative locations (e.g. Cl in CsCl and F in CaF_{2}). The concept of lattice complexes can be used to reveal relationships between such crystal structures even if their space groups belong to different types.
The terms `point configuration' (Fischer & Koch, 1974a) and `crystallographic orbit' (Matsumoto & Wondratschek, 1979) have frequently been used as synonyms for sets of points in threedimensional space that are equivalent with respect to a space group . Such sets of points may be classified in two different ways: (1) according to the concept of lattice complexes (German: Gitterkomplexe) and of limiting complexes, which goes back to Hermann (1935) and has been defined more strictly by Fischer & Koch (1974a); (2) according to the concept of types of crystallographic orbits and of noncharacteristic orbits introduced by Wondratschek (1976). As the two approaches^{1} are strongly related but not identical, the classes originating from the two concepts will be compared and the differences worked out.
Both terms, `point configuration' and `crystallographic orbit', have been used with two slightly different meanings: (1) for sets of points that are equivalent with respect to a given space group, i.e. in the mathematical sense of `orbit'; (2) for such sets of points, but detached from their generating space groups. The second meaning is referred to, for example, if one speaks only of a primitive cubic point lattice. As within both concepts both meanings are required, one has to distinguish between them. In the following, therefore, the term `crystallographic orbit' is restricted to the first meaning and the term `point configuration' is restricted to the second meaning.
In mathematics, an orbit is a very general grouptheoretical term describing any set of objects that are mapped onto each other by the action of a group (cf. Section 1.1.7 ). In fact, orbits are always present in crystallography where equivalence classes are defined by means of a group action (e.g. a spacegroup type is the orbit of a space group in the set of all space groups under the action of the affine group). In the present context, however, the term (crystallographic) orbit will be used in a much more restricted sense, as proposed by Wondratschek (1976):
From any point of , the symmetry operations of a given space group generate an infinite set of symmetryequivalent points, called a crystallographic orbit with respect to or, for short, a crystallographic orbit (cf. Section 1.4.4 ). The space group is called the generating space group of the orbit.
Each point of a crystallographic orbit defines uniquely a largest finite subgroup of , which maps that point onto itself, its sitesymmetry group (cf. Section 1.4.4 ). Sitesymmetry groups that belong to different points out of the same crystallographic orbit are conjugate subgroups of .
Example
The points and ; and form an orbit of a given space group Pmna together with the infinitely many other points that can be generated from the first four by the translations of Pmna. The sitesymmetry group 2.. of each such point consists of the identity operation 1 and of a twofold rotation. The position of the twofold axis can easily be read from the corresponding coordinate triplet. The sitesymmetry groups of the first two points are and , respectively. They can be mapped onto another by conjugation e.g. with the glide reflection of Pmna. This glide reflection also interchanges the two twofold axes as can easily be learned by inspecting the spacegroup diagram.
The crystallographic orbits of a given space group subdivide the set of all points of into equivalence classes. It is also possible, however, to define equivalence of orbits on the set of all crystallographic orbits of :
Two crystallographic orbits of a space group belong to the same Wyckoff position (cf. Section 1.4.4 ) if and only if the sitesymmetry groups of any two points stemming from the first and the second orbit are conjugate subgroups of .^{2}
Example
The points and belong to different orbits of a given space group Pmna. Their sitesymmetry groups and are conjugate subgroups of Pmna (cf. the previous example). Therefore, the two orbits belong to the same Wyckoff position of Pmna, namely to 4e.
The following definition results in a coarser classification of crystallographic orbits:
Two crystallographic orbits of a space group belong to the same Wyckoff set (German: Konfigurationslage, cf. Fischer & Koch, 1974a) if and only if the sitesymmetry groups of any two points stemming from the first and the second orbit are conjugate subgroups of the affine normalizer of (cf. Section 1.4.4.3 ).^{3}
Accordingly, all orbits of a certain Wyckoff position belong to the same Wyckoff set. The assignment of orbits to Wyckoff sets, therefore, also defines an equivalence relation on the Wyckoff positions of a space group. The Wyckoff sets of the space groups were first tabulated by Koch & Fischer (1975).
Example
In space group Pmna, the sitesymmetry groups of the points and are and . There is no symmetry operation from Pmna that maps these sitesymmetry groups onto another by conjugation and hence the two corresponding orbits do not belong to the same Wyckoff position of Pmna. The Euclidian (and affine) normalizer of Pmna is a space group of type Pmmm with half the lattice parameters compared with those of Pmna (cf. Chapter 3.5 ). It contains e.g. the twofold rotation that maps by conjugation the two sitesymmetry groups onto another and also the two axes in the spacegroup diagram. Therefore, the two orbits belong to the same Wyckoff set even though they belong to the different Wyckoff positions 4e and 4f.
In analogy to the transition from a single space group to its type, it seems desirable to transfer also the terms `Wyckoff position' and `Wyckoff set' to the whole spacegroup type. For Wyckoff positions, however, such a generalization is not possible: two space groups of the same type can be mapped onto each other by infinitely many isomorphisms or affine mappings. Each isomorphism results in a unique relation between the Wyckoff positions of the two groups, but different isomorphisms may give rise to different relations so that the Wyckoff positions of the same Wyckoff set change their roles.
Such ambiguities, however, cannot occur for Wyckoff sets, because all Wyckoff sets of a certain space group differ in their grouptheoretical relations to that group. Therefore, Wyckoff sets may be classified as follows:
Two Wyckoff sets stemming from space groups of the same type belong to the same type of Wyckoff set if and only if they are related by an isomorphism (affine mapping) of the two space groups (German: Klasse von Konfigurationslagen, cf. Fischer & Koch, 1974a; Koch & Fischer, 1975). The 219 types of space group in give rise to 1128 types of Wyckoff set.
Example
Take, in a particular space group of type , the Wyckoff position 4l . The points of each corresponding orbit form squares that replace the points of the tetragonal primitive point lattice of Wyckoff position 1a. For all conceivable orbits of 4l, the squares have the same orientation, but their edges differ in their lengths. Congruent arrangements of squares but shifted by or by or by give the orbits of the Wyckoff positions 4m, 4n and 4o, respectively, in the same space group. The four Wyckoff positions 4l to 4o, all with site symmetry m2m., make up a Wyckoff set (cf. Table 3.4.3.3). They are mapped onto each other, for example, by the translations , and , which belong to the Euclidean (and affine) normalizer of the group. If one space group of type is mapped onto another space group of the same type, the Wyckoff set 4l to 4o as a whole is transformed to 4l to 4o. The individual Wyckoff positions may be interchanged, however, because of the different possible choices for the origin in each individual space group of type . All the Wyckoff sets 4l to 4o stemming from all different space groups of type constitute together a type of Wyckoff set.
For the comparison of crystal structures belonging to different types, another kind of equivalence relationship between crystallographic orbits may be useful:
One may consider the set of points belonging to a certain orbit without paying attention to the generating space group of the orbit. Such a bare set of points is called a point configuration. Two crystallographic orbits are called configuration equivalent if their point configurations are identical.
This definition uniquely assigns orbits to point configurations, but not vice versa.
Example
Let us consider a certain space group of type with lattice vectors a, b, c together with two of its nonmaximal subgroups, namely with index 4 and P432 with index 16, both with lattice vectors 2a, 2b, 2c. The orbit of belongs to Wyckoff position 1b of (site symmetry ), and the corresponding set of points, its point configuration, forms a primitive cubic point lattice. As both subgroups have doubled unitcell edges, the point turns to . The respective orbits belong to Wyckoff position 8c of (site symmetry 23.) and to 8g of P432 (site symmetry .3.), and both correspond to the original point configuration. Therefore, the three orbits 1b , 8c and P432 8g with are configuration equivalent (together with several other orbits from certain other subgroups of ). They all give rise to one and the same point configuration, a specific primitive cubic lattice of points. The generating space group, however, cannot be identified just by looking at the point configuration.
The eigensymmetry of a point configuration is the most comprehensive space group that maps this point configuration onto itself. Accordingly, exactly one crystallographic orbit out of each class of configurationequivalent orbits stands out because its generating space group coincides with the eigensymmetry of its point configuration. In the case of the example above, this specific orbit is 1b (as long as the origin of remains unchanged).
The concept of configuration equivalence may also be applied to types of Wyckoff set: two types of Wyckoff set are configuration equivalent if and only if for each crystallographic orbit belonging to the first type there exists a configurationequivalent crystallographic orbit belonging to the second type of Wyckoff set, and vice versa. All types of Wyckoff set differ in their crystallographic orbits, but configurationequivalent types of Wyckoff set result in the same set of point configurations.
A lattice complex is the set of all point configurations that correspond to the crystallographic orbits of a certain type of Wyckoff set.
There exist 402 classes of configurationequivalent types of Wyckoff set and, therefore, 402 lattice complexes in .
Example
Let us consider again the type of Wyckoff set 4l to 4o (the last example in Section 3.4.1.2). The set of all corresponding point configurations constitutes a lattice complex. Its point configurations may be derived as described above, but now – instead of starting from just a particular group – starting from all space groups of type with all conceivable positions of the origins and lengths and orientations of the basis vectors. Accordingly, the point configurations may differ in their relative position in space, their orientation, and in the distances between the centres and the size of their squares.
Just as all crystal forms of a particular type may be related to different pointgroup types, the same lattice complex may occur in different spacegroup types.
Example
The lattice complex `cubic primitive lattice' may be generated, among others, in , in and in with site symmetry , and , respectively. The type of Wyckoff set specified by 1a, b leads to the same set of point configurations as or . Each point configuration of this lattice complex can be generated by a properly chosen space group of each of these spacegroup types.
Configurationequivalent crystallographic orbits do not necessarily belong to configurationequivalent types of Wyckoff set.
Example
The orbits of the types of Wyckoff set and both refer to the set of all conceivable primitive cubic point lattices. Therefore, these two types of Wyckoff set are configuration equivalent and are associated with the same lattice complex. The type of Wyckoff set P432 8g , however, comprises apart from crystallographic orbits with also those with . The orbits with refer to the same set of point configurations as and , whereas those with give rise to point configurations with different properties. As a consequence, the type of Wyckoff set P432 8g is not configuration equivalent with and , and, therefore, belongs to another lattice complex.
As this example shows, lattice complexes do not form equivalence classes of point configurations, but a certain point configuration may belong to several lattice complexes.
As each type of Wyckoff set uniquely refers to a certain lattice complex, one can also assign all corresponding Wyckoff sets, Wyckoff positions and crystallographic orbits to that lattice complex. A certain lattice complex, however, is frequently related to different types of Wyckoff set.
Among the different types of Wyckoff set belonging to a certain lattice complex, one stands out because its crystallographic orbits show the highest site symmetry. This one is called the characteristic type of Wyckoff set of that lattice complex, and the corresponding spacegroup type its characteristic spacegroup type. All other types of Wyckoff set are referred to as noncharacteristic. The term `characteristic' may also be transferred to particular Wyckoff sets out of the characteristic type. The space groups of all the other types in which the lattice complex may be generated are subgroups of the space groups of its characteristic type.
Different lattice complexes may have the same characteristic spacegroup type, but then they differ in the oriented site symmetry of their Wyckoff positions within that spacegroup type.
The characteristic spacegroup type together with the oriented site symmetry expresses the common symmetry properties of all point configurations of a lattice complex and can be used for its identification. For the purpose of reference symbols of lattice complexes, however, instead of the site symmetry the Wyckoff letter of one of the Wyckoff positions with that site symmetry is arbitrarily chosen, as first done by Hermann (1935). This Wyckoff position is called the characteristic Wyckoff position of the lattice complex.
Example
is the characteristic spacegroup type for the lattice complex of all cubic primitive point lattices. The Wyckoff positions with the highest possible site symmetry are la and 1b , from which 1a has been chosen as the characteristic position. Thus, the reference symbol of this lattice complex is .
Example
is also the characteristic spacegroup type for a second lattice complex that corresponds to Wyckoff position 8g .3m . The reference symbol for this lattice complex is . Each of its point configurations may be derived by replacing each point of a cubic primitive lattice by eight points arranged at the corners of a cube.
All types of Wyckoff set (together with their Wyckoff sets and Wyckoff positions) that generate, as described above, the same set of point configurations are assigned to the same lattice complex. Accordingly, the following criterion holds: two Wyckoff positions are assigned to the same lattice complex if there is a suitable transformation that maps the point configurations of the two Wyckoff positions onto each other and if their space groups belong to the same crystal family (cf. Section 1.3.4.4 ). Suitable transformations are translations, proper or improper rotations, isotropic or anisotropic expansions or more general affine mappings (without violation of the metric conditions for the corresponding crystal family), and all their products.
By this criterion, the Wyckoff positions of all space groups (1731 entries in the spacegroup tables, 1128 types of Wyckoff set) are uniquely assigned to 402 lattice complexes. This assignment was first done by Hermann in Internationale Tabellen zur Bestimmung von Kristallstrukturen (1935). The corresponding information has also been given by Fischer et al. (1973).
The same concept has been used for the point configurations and Wyckoff positions in the plane groups. Here the Wyckoff positions (72 entries in the planegroup tables, 51 types of Wyckoff set) are assigned to 30 plane lattice complexes or net complexes (cf. Burzlaff et al., 1968). The complexes for the crystallographic subperiodic groups in threedimensional space, i.e. for the crystallographic point groups, rod groups and layer groups, have been derived by Koch & Fischer (1978a).
As has been shown above, lattice complexes define equivalence classes of orbits but not of point configurations. This property gave rise to the concept of limiting complexes and comprehensive complexes (Fischer & Koch, 1974a; Koch, 1974).
For morphological crystal forms an almost analogous situation exists. A certain tetragonal prism, for example, may be a general representative of the crystal form `tetragonal prism' on the one hand or it may be a special representative of the crystal forms `tetragonal pyramid' or `tetragonal disphenoid' on the other hand. In the first case the generating point group may belong to the types 4/mmm, 422, 4/m or (with site symmetry 2 for each face), in the second case the types of the generating point group are 4mm or 4 and (site symmetry m) or , respectively. The crystal form `tetragonal prism' is a limiting form of both crystal forms `tetragonal pyramid' and `tetragonal disphenoid'.
If a first lattice complex forms a true subset of a second one, i.e. if each point configuration of the first lattice complex also belongs to the second one, then the first one is called a limiting complex of the second one and the second complex is called a comprehensive complex of the first one (cf. Koch & Fischer, 1985).
Example
The cubic lattice complex 16c involves two limiting complexes, namely 2a and 16b . The orbits from 16c with and from 2a are configuration equivalent, and so are the orbits from 16c with and from 16b.
Example
The tetragonal lattice complex 4a is a comprehensive complex of the cubic complex 8a. Each orbit of 8a is configuration equivalent to a crystallographic orbit of a special space group of type with axial ratio .
Furthermore, two lattice complexes without a limitingcomplex relationship may have a nonempty intersection. Then the point configurations of the intersection result in one or, in very exceptional cases, in two or more other lattice complexes (cf. Koch, 1974).
Example
The intersection of the two lattice complexes 24g and 24g consists of all point configurations belonging to 24h, i.e. each point configuration out of this intersection refers to an orbit from 24h and, in addition, to an orbit from 24g with and to another one from 24g with z = 0.
Example
The intersection of the trivariant lattice complexes 192j and P432 24k consists of two bivariant limiting complexes, namely of 24k and of 24m .
Each point configuration of a given lattice complex is uniquely related to two space groups: (1) the space group that reflects its eigensymmetry, and (2) a space group that belongs to the characteristic spacegroup type of the lattice complex under consideration. In most cases the two groups coincide. Only when the point configuration under consideration belongs to a limiting complex is the first group a proper supergroup of the second one.
Complete lists of the limiting complexes of all lattice complexes are not available. Koch (1974) derived the limiting complexes of the cubic lattice complexes. The limiting complexes that refer to specialized coordinate parameters may be derived from a table by Engel et al. (1984), who listed the respective noncharacteristic orbits for all spacegroup types. The limiting complexes of the tetragonal and trigonal lattice complexes that are due to metrical specializations are tabulated by Koch & Fischer (2003) and by Koch & Sowa (2005), respectively.
Fischer & Koch (1978) tabulated the limiting complexes for the crystallographic point groups, rod groups and layer groups. As each type of plane group uniquely corresponds to a certain type of isomorphic layer group, information on the limiting complexes of the lattice complexes of the plane groups may easily be extracted from the respective table for the layer groups. This information may also be taken from a list of the noncharacteristic orbits of the plane groups by Matsumoto & Wondratschek (1987).
Each Wyckoff position shows a certain number of coordinate parameters that can be varied independently. For most lattice complexes, this number is the same for any of its Wyckoff positions. For the lattice complex with characteristic Wyckoff position 12j m.. , for instance, this number is two. The lattice complex has two degrees of freedom. If, however, the variation of a certain coordinate corresponds to a shift of the point configuration as a whole, the lattice complex has fewer degrees of freedom than the Wyckoff position that is being considered. Therefore, I4_{1} 8b is the characteristic Wyckoff position of a lattice complex with only two degrees of freedom, although position 8b itself has three coordinate parameters that can be varied independently. The lattice complex P4/m j has two degrees of freedom and refers to Wyckoff positions with two as well as with three independent coordinate parameters, namely to P4/m 4j m.. and to P4 4d 1 .
According to its number of degrees of freedom, a lattice complex is called invariant, univariant, bivariant or trivariant. In total, there exist 402 lattice complexes, 36 of which are invariant, 106 univariant, 105 bivariant and 155 trivariant. The 30 plane lattice complexes are made up of 7 invariant, 10 univariant and 13 bivariant ones.
Most of the invariant and univariant lattice complexes correspond to several types of Wyckoff set. In contrast to that, only one type of Wyckoff set can belong to each trivariant lattice complex. A bivariant lattice complex may either correspond to one type of Wyckoff set (e.g. ) or to two types (P4 d, for example, belongs to the lattice complex with the characteristic Wyckoff position ).
Depending on their sitesymmetry groups, two kinds of Wyckoff position may be distinguished:
A lattice complex refers either to Wyckoff positions exclusively of the first or exclusively of the second kind. Most lattice complexes are related to Wyckoff positions of the first kind.
There exist, however, 67 lattice complexes without point configurations with infinitesimally short distances between symmetryrelated points [cf. Hauptgitter (Weissenberg, 1925)]. These lattice complexes were called Weissenberg complexes by Fischer et al. (1973). The 36 invariant lattice complexes are trivial examples of Weissenberg complexes. The other 31 Weissenberg complexes with degrees of freedom (24 univariant, 6 bivariant, 1 trivariant) are compiled in Table 3.4.1.1. They have the following common property: each Weissenberg complex contains at least two invariant limiting complexes belonging to the same crystal family (see also Section 3.4.3.1.3).
Example
The Weissenberg complex Pmma 2e is a comprehensive complex of Pmmm a and of Cmmm a. Within the characteristic Wyckoff position, refers to Pmmm a and to Cmmm a.

Apart from the seven invariant plane lattice complexes, there exists only one further Weissenberg complex within the plane groups, namely the univariant rectangular complex p2mg c.
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.
3.4.3. Descriptive latticecomplex symbols and the assignment of Wyckoff positions to lattice complexes
For the study of relations between crystal structures, latticecomplex symbols are desirable that show as many relations between point configurations as possible. To this end, Hermann (1960) derived descriptive latticecomplex symbols that were further developed by Donnay et al. (1966) and completed by Fischer et al. (1973). These symbols describe the arrangements of the points in the point configurations and refer directly to the coordinate descriptions of the Wyckoff positions. Since a lattice complex, in general, contains Wyckoff positions with different coordinate descriptions, it may be represented by several different descriptive symbols. The symbols are further affected by the settings of the space group. The present section is restricted to the fundamental features of the descriptive symbols. Details have been described by Fischer et al. (1973). Tables 3.4.3.2 and 3.4.3.3 give for each Wyckoff position of a plane group or a space group, respectively, the multiplicity, the Wyckoff letter, the oriented site symmetry, the reference symbol of the corresponding lattice complex and the descriptive symbol.^{4} The comparatively short descriptive symbols condense complicated verbal descriptions of the point configurations of lattice complexes.
An invariant lattice complex in its characteristic Wyckoff position is represented by a capital letter (sometimes in combination with a superscript). The first column of Table 3.4.3.1 gives a complete list of these symbols in alphabetical order. The characteristic Wyckoff positions are shown in column 3. Lattice complexes from different crystal families but with the same coordinate description for their characteristic Wyckoff positions receive the same descriptive symbol. If necessary, the crystal family may be stated explicitly by a small letter (column 2) preceding the latticecomplex symbol: c cubic, t tetragonal, h hexagonal, o orthorhombic, m monoclinic, a anorthic (triclinic).
Example
D is the descriptive symbol of the invariant cubic lattice complex a as well as of the orthorhombic lattice complex Fddd a. The cubic lattice complex cD contains – among others – the point configurations corresponding to the arrangement of carbon atoms in diamond and of silicon atoms in βcristobalite. The orthorhombic complex oD is a comprehensive complex of cD. It consists of all those point configurations that may be produced by orthorhombic deformations of the point configurations of cD.

The descriptive symbol of a noncharacteristic Wyckoff position depends on the difference between the coordinate descriptions of the respective characteristic Wyckoff position and the position under consideration. Three cases may be distinguished, which may also occur in combinations.
In noncharacteristic Wyckoff positions, the descriptive symbols P and I may be replaced by C and F, respectively (tetragonal system), C by A or B (orthorhombic system), and C by A, B, I or F (monoclinic system). If the lattice complexes of rhombohedral space groups are described in rhombohedral coordinate systems, the symbols R, , M and of the hexagonal description are replaced by P, I, J and , respectively (preceded by the letter r, if necessary, to distinguish them from the analogous cubic invariant lattice complexes).
The descriptive symbols of lattice complexes with degrees of freedom consist, in general, of four parts: the shift vector, the distribution symmetry, the central part and the siteset symbol. Either of the first two parts may be absent.
Example
..2 C4xxz is the descriptive symbol of the lattice complex in its characteristic position: is the shift vector, ..2 the distribution symmetry, C the central part and 4xxz the siteset symbol.
Normally, the central part is the symbol of an invariant lattice complex. The shift vector and central part together should be interpreted as described in Section 3.4.3.1.2. The point configurations of the Wyckoff position being considered can be derived from that described by the central part by replacing each point by a finite set of points, the site set. All points of a site set are symmetryequivalent under the sitesymmetry group of the point that they replace. A site set is symbolized by a string of numbers and letters. The product of the numbers gives the number of points in the site set, whereas the letters supply information on the pattern formed by these points. Site sets replacing different points may be differently oriented. In this case, the distributionsymmetry part of the reference symbol shows symmetry operations that relate such site sets to one another. The orientation of the corresponding symmetry elements is indicated as in the oriented sitesymmetry symbols (cf. Section 2.2.12). If all site sets have the same orientation, no distribution symmetry is given.
Examples

In the case of a Weissenberg complex (cf. Section 3.4.1.5.2; Weissenberg, 1925; Fischer et al., 1973), the central part of the descriptive symbol always consists of two (or more) symbols of invariant lattice complexes belonging to the same crystal family and forming limiting complexes of the Weissenberg complex under consideration. The shift vector then refers to the first limiting complex. The corresponding siteset symbols are distinguished by containing the number 1 as the only number, i.e. each site set consists of only one point.
Example
In , each of the two points and , represented by , is replaced by a site set 1z containing only one point, i.e. the points of are shifted along the z axis. The shifts of the two points are related by a twofold rotation .2., i.e. are running in opposite directions. The point configurations of the two limiting complexes and B refer to the special parameter values and , respectively.
The central parts of some lattice complexes with two or three degrees of freedom are formed by the descriptive symbol of a univariant Weissenberg complex instead of that of an invariant lattice complex. This is the case only if the corresponding characteristic spacegroup type does not refer to a suitable invariant lattice complex.
Example
In , each of the two points and , represented by , is replaced by a site set 2y of two points forming a dumbbell. These dumbbells are oriented parallel to the y axis.
The symbol of a noncharacteristic Wyckoff position is deduced from that of the characteristic position. The four parts of the descriptive symbol are subjected to the transformation necessary to map the characteristic Wyckoff position onto the Wyckoff position under consideration.
Example
The lattice complex with characteristic Wyckoff position Imma 8h has the descriptive symbol for this position. Another Wyckoff position of this lattice complex is . The corresponding point configurations are mapped onto each other by interchanging positive x and negative y directions and shifting by . Therefore, the descriptive symbol for Wyckoff position Imma i is .
In some cases, the Wyckoff position described by a latticecomplex symbol has more degrees of freedom than the lattice complex (see Section 3.4.1.5.1). In such cases, a letter (or a string of letters) in brackets is added to the symbol.
Different kinds of relations between lattice complexes are brought out.
In many cases, limitingcomplex relations can be deduced from the symbols. This applies to limiting complexes due either to special metrical parameters (e.g. etc.) or to special values of coordinates (e.g. both P4x and P4xx are limiting complexes of P4xy). If the site set consists of only one point, the central part of the symbol specifies all corresponding limiting complexes without degrees of freedom that are due to special values of the coordinates (e.g. . for the general position of ).
In Tables 3.4.3.2 and 3.4.3.3, the Wyckoff positions of all plane and space groups, respectively, are listed. Each Wyckoff position is identified by its Wyckoff letter together with its oriented sitesymmetry symbol. It is assigned to its lattice complex by means of the reference symbol (cf. Section 3.4.1.3). Characteristic Wyckoff positions are marked by asterisks (e.g. 2e in ). If in a particular space group several Wyckoff positions belong to the same Wyckoff set (cf. Sections 1.4.4.3 and 3.4.1.2; Koch & Fischer, 1975), the reference symbol is given only once (e.g. Wyckoff positions 4l to 4o in ). To enable this, the usual sequence of Wyckoff positions had to be changed in a few cases (e.g. in ). For Wyckoff positions assigned to the same lattice complex but belonging to different Wyckoff sets, the reference symbol is repeated. In , for example, Wyckoff positions 4c and 4d are both assigned to the lattice complex . They do not belong, however, to the same Wyckoff set because the sitesymmetry groups .. of 4c and .. of 4d are different.

