Number and Polyhedra

In Fig. 3 are shown some ionic radii for M +, and M5+ oxidation states. Ionic radii vary, depending on choice of counter ion radius, model compounds, coordination number, etc. For a recent review see Shannon 

In 1948 Zachariasen 21), studying a number of alkali actinide fluorides, noted that the cell volumes of these compounds could be attributed to the volume of the fluoride ions and the larger alkali ions alone, with the actinide ions occup3ung interstices between the anions. In his examples he assigns a volume of 18 A for F, 7 A for Na+, and 21 A for K+, etc. This relationship is borne out in many structures of related com- 

Depolymerization by Fluoride Variable Valence in Fluoride Compounds 

With the metals of the first transition series, the maximum coordination number of higher oxidation states is six, and this is so firmly fixed that in their fluoride complexes the oxidation state of the metal in question can be fixed by controlling the mol fraction of alkali metal present 26). Thus, the fluorination of a vanadium salt in the presence of a one, two, or three mol ratio of potassium ion, yields KV F6, KgWVFe, or KsV Fe. The same tendency is shown, but to a lesser degree, by metals of the second transition series, as exemplified by KRuFe and KgRuFe. For an unusual example in the third series, note that OsFe is known, OsFt is not stable, but heptavalent osmium is found as six-coordinated OsOFs (27). 

With the actinides this just mentioned tendency is minimized. Actinide fluoride complex structures can display the same coordination number over a range of valence states, i.e. Na2U F8 and NasU Fs in which the actinide coordination is cubic. However, this behavior is unusual. The weak directional influence of the f-electrons, coupled with the larger size of actinide ions allows them to accommodate to the several different coordination polyhedra possible with higher coordination num- 

---

Table 2 Coordination numbers and polyhedra of zirconium and hafnium...

The structure of the KAu4Sn2 type is related to the CuAh type (Sinnen and Schuster 1978). An analysis of the structure (Skolozdra 1993) shows, that it can be obtained from the CuAh structure by filling the m symmetry tetrahedra in the CuAh structure with Au atoms. In the case of a compression of the structure, a change of coordination numbers and polyhedra in comparison with the structure type CUAI2 is observed. 

The electronic configuration (4d ) and size of cadmium(II) clearly favor its affinity for soft donor atoms as well as a certain variability in coordination numbers and polyhedra. The structures discussed here reveal the abihty of Cd(II) to exhibit coordination numbers firom three to eight, with the six-coordination (octahedral or trigonal prism) polyhedra being the most common. Examples have been reported in which Cd(II) shows two different coordination numbers in the same crystal. Moreover, high coordination numbers favour the formation of polymers with various dimensionalities, whereas the tridentate ability of soft S-thiolate atoms build nice clusters. 

The Number of Polyhedra with a Common Corner. The Electrostatic Valence Principle.—The number of polyhedra with a common corner can be determined by the use of an extended conception of electrostatic valence. Let ze be the electric charge of a cation and v its coordination number. Then the strength of the electrostatic valence bond going to each comer of the polyhedron of anions about it is defined as... 

The coordination conditions can be expressed in a chemical formula using a notation suggested by F. Machatschki (and extended by several other authors for recommendations see [35]). The coordination number and polyhedron of an atom are given in brackets in a right superscript next to the element symbol. The polyhedron is designated with a symbol as listed in Fig. 2.2. Short forms can be used for the symbols, namely the coordination number alone or, for simple polyhedra, the letter alone, e.g. t for tetrahedron, and in this case the brackets can also be dropped. For example ... 

The composition of a compound is intimately related to the way of linking the polyhedra. An atom X with coordination number c.n.(X) that acts as a common vertex to this number of polyhedra makes a contribution of l/c.n.(X) to every polyhedron. If a polyhedron has n such atoms, this amounts to n/c.n.(X) for this polyhedron. This can be expressed with Niggli formulae, as shown in the following sections. To specify the coordination polyhedra, the formalism presented at the end of Section 2.1 and in Fig. 2.2 (p. 5) is useful. 

Fig. 2.4-10. High coordination number (CN) polyhedra around alkali (alkaline earth) atoms in different mercu7 rich amalgams and in BaCdn (a) CN 15 and (b) CN 16 around K in...

In mercury rich alloys such as M Hg12 (M = K, Rb), a particular situation occurs due to high coordination number (CN) polyhedra (up to CN 22) in their structures. In these compounds the electropositive atoms are located in the centers of these polyhedra and are thus spatially separated from each other. The covalent Hg-Hg interaction, as discussed above, is of minor importance in these amalgams. 

General aspects of the coordination chemistry of the lanthanides with reference to the complexes of neutral oxygen donor ligands are dealt with in this section. For con venience, this section is subdivided into three parts 1. synthetic procedures, 2. stoichiometry, and 3. coordination numbers and coordination polyhedra. 

In layer structures, the number and availability of bonds to the nearest neighbours is decided by the electrons available as stipulated by valence theory. Layer structures based on linked polyhedra are the most common ones in catalysis. Some of these are described below. 

When the number and volume of the polyhedral compartments are given, the optimal structure of the foam is the one that creates the smallest total film area. This condition constitutes a formidable but straightforward mathematical optimization problem. Solution as an average, the polyhedra consist of 13.4 sides. Experimentally it was indeed found that the polyhedra most commonly found in foams have 14 sides, followed by 12 sides as a second choice. 

---