Invariant lattice complexes

A short list of invariant lattice complex symbols is reported in the following. For a complete list, for a more systematic description and formal definition, see the International Tables of Crystallography, Hahn (2002). 

The coordinates indicated in the reported partial list of invariant lattice complexes correspond to the so-called standard setting and to related standard representations. Some of the non-standard settings of an invariant lattice complex may be described by a shifting vector, defined in terms of fractional coordinates, in front of the symbol. The most common shifting vectors also have abbreviated symbols P represents 14, A,AP (that is the coordinates which are obtained by adding A, Vi, Ai to those of P, that is coordinates 14, 14, A), J represents A, A, A J (coordinates A, 0, 0 0, A, 0 0, 0, A) F" represents A,A,AF (coordinates At, A, A A, 3A, A 3A, A, 3A 3A, A, A) and F" represents A, /, 3A F. It can be seen, moreover, that the complex D corresponds to the coordinates F + F". 

Figure 3.15. An example of relations between invariant lattice complexes with the symbols of their representations.

A short list of examples of structure descriptions in terms of combination of invariant lattice complexes is here reported. (Compare with the structure descriptions in Chapter 7.)... 

This group of atomic positions corresponds to the so-called invariant lattice complex D see 3.7.1 and Fig. 3.14. 

In conclusion, notice also that in terms of combinations of invariant lattice complexes, the positions of the atoms in the level X can be represented by 2A, A, A G, and those in the level % by A, A, M G (where G is the symbol of the graphitic net complex, here presented in non-standard settings by means of shifting vectors). 

LiZn, LiCd, LiAl, Naln have this structure. This structure may be regarded as a completely filled-up face-centred cubic arrangement in which each component occupies a diamond-like array of sites. The structure may thus be presented as NaTl D + D (see the descriptions in terms of combination of invariant lattice complexes reported in 3.7.1). 

In terms of a combination of invariant lattice complexes the sphalerite structure may therefore be described as ZnS F + F". 

In many of the structures we have described, the metal atoms are in high symmetry positions, often at the positions of invariant lattice complexes. In the latter cases the problem of completely specifying the structure then becomes one of locating the anions... 

A Bauverband , the points of which can be described by an invariant point position, will be characterised by the symbol of the corresponding invariant lattice complex (capital letter) (Hermann I960 Donnay et al., 1966). The coordinates of the points of the invariant cubic lattice complexes, referred to their characteristic point position, aie listed in Table 1. The projections of the corresponding point pattern are presented in Fig. 1 with the z coordinates given in units of c/8. 

Fig. 2. Some relations between invariant lattice complexes...

The point configurations of point positions with degrees of freedom in general may be described by symbols of invariant lattice complexes plus those for coordination polyhedra following the lattice-complex symbol in parentheses. These polyhedra... 

With the knowledge of invariant lattice complexes and coordination polyhedra it is possible to describe homogeneous and heterogeneous Bauverbande and the positions of occupied voids with their coordination polyhedra. The symbols are easy to understand and the stmctures can be reconstructed from them. Examples are given in Tables 3 and 4 by Hellner (1965), Niggli (1972) and Koch and Fischer (1974). The question of iso-, homeo- and heterotypic structures and the classification principle have still to be discussed, however. 

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