Lattice complexes

A description which in some simple cases could be considered alternative to those exemplified in Table 3.2 is based on the lattice complex concept. Listing the symbols of the lattice complexes occupied by the different atoms in a structure (for instance, symbol P for the point 0, 0, 0 and its equivalent points), provides in fact 

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 . 

The signs + and — as superscripts in front of the letters (for instance +Y, Y) indicate enantiomorphic forms. The superscript ( star ) indicates a complex that may be built up as a combination of two complexes of the same kind shifted against each other by A, A, A (W = W + VifAfAW = W + W ) or as a combination of two enantiomorphic forms (see the examples shown in Fig. 3.15). 

Other representations may be obtained by subdividing the unit cell into a number of similar subcells. In the cubic system the subdivision is made along the three axes by the same factor which is used as a subscript in the new lattice complex 

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Lattice complex P (multiplicity 1 multiplicity is the number of equivalent points in the unit cell)... 

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

A schematic representation of cubic lattice complexes is given in Figs 3.14 and 3.15 this could also be useful as an indication of possible combinations and splitting . Such relations may be useful while comparing different structures and studying their interrelations and possible transformations (order-disorder transformations, etc.)... 

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.)... 

With reference to a description in terms of lattice-complex combination, the AuCu3-type structure corresponds to a combination of P and Jcomplexes (AuCu3 P + J). According to Hellner (1979), this structure may be considered as pertaining to an F-family as a consequence of the particular splitting, previously described, of the points of the F complex. 

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". 

A number of publications have appeared recently on super-lattice complexes which have enhanced conductivity, eg. "nazirpsio NaaPOif 2Zr02 2Si02 whose conductivity at room temperature is of the same order as that of an aqueous salt solution. Most of the super-lattices are unstable thermodynamically, and can be expected to collapse under chemical attack by the anodic and cathodic reactants. However, there may exist some thermodynamically stable structures, and the search should concentrate on the complicated phase-diagram studies of selected quatemarys. 

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... 

An unexpected acetylation troubled two laboratories where 4-amino-triazole-5-carboxamide, while being recrystallized from anhydrous acetic acid, deposited a crop that was reported as a diacetyl derivative [57JOC707 71JCS(C)2156]. However, this product, of which a 64% yield was obtained by 5 hr refluxing, turned out to be a 1 1-lattice complex of 4-acetamido-triazoIe-5-carboxamide with acetic acid [73JCS(P1)943]. No similar example has been reported. 

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