Valence-electron concentrations

Lithium has been alloyed with gaUium and small amounts of valence-electron poorer elements Cu, Ag, Zn and Cd. like the early p-block elements (especially group 13), these elements are icosogen, a term which was coined by King for elements that can form icosahedron-based clusters [24]. In these combinations, the valence electron concentrations are reduced to such a degree that low-coordinated Ga atoms are no longer present, and icosahedral clustering prevails [25]. Periodic 3-D networks are formed from an icosahedron kernel and the icosahedral symmetry is extended within the boundary of a few shells. 

The Zintl-Klemm concept evolved from the seminal ideas of E. ZintI that explained the structural behavior of main-group (s-p) binary intermetaUics in terms of the presence of both ionic and covalent parts in their bonding description [31, 37]. Instead of using Hume-Rother/s idea of a valence electron concentration, ZintI proposed an electron transfer from the electropositive to the electronegative partner (ionic part) and related the anionic substructure to known isoelectronic elemental structures (covalent part), e.g., TK in NaTl is isoelectro-nic with C, Si and Ge, and consequenUy a diamond substructure is formed. ZintI hypothesized that the structures of this class of intermetallics would be salt-like [16b, 31 f, 37e]. 

In reality there are subtle deviations from this simple picture. The energy levels shift somewhat from element to element, and different structure types have different band structures that become more or less favorable depending on the valence electron concentration. Furthermore, in the COOP diagram of Fig. 10.13 the s-p, s-d and p-d interactions were not taken into account, although they cannot be neglected. A more exact calculation shows that only antibonding contributions are to be expected from the eleventh valence electron onwards. 

We define the valence electron concentration per anion, VEC(X), as the total number of all valence electrons in relation to the number of anionic atoms ... 

Table 13.1 Examples of polyanionic compounds which have integral valence electron concentrations per anion atom...

For low values of the valence electron concentration (VEC< 4 for main group elements), covalent 2c2e bonds are not sufficient to overcome the electron deficiency. We have the case of electron-deficient compounds . For these, relief comes from multicenter bonds. In a three-center two-electron bond (3c2e) three atoms share an electron pair. An even larger number of atoms can share one electron pair. With increasing numbers of... 

Hume-Rothery phases (brass phases, electron compounds ) are certain alloys with the structures of the different types of brass (brass = Cu-Zn alloys). They are classical examples of the structure-determining influence of the valence electron concentration (VEC) in metals. VEC = (number of valence electrons)/(number of atoms). A survey is given in Table 15.1. 

Because of the permitted composition ranges, alloys with rather different compositions can adopt the same structure, as can be seen by the examples in Table 15.1. The determining factor is the valence electron concentration, which can be calculated as follows ... 

A theoretical interpretation relating the valence electron concentration and the structure was put forward by H. Jones. If we start from copper and add more and more zinc, the valence electron concentration increases. The added electrons have to occupy higher energy levels, i.e. the energy of the Fermi limit is raised and comes closer to the limits of the first Brillouin zone. This is approached at about VEC = 1.36. Higher values of the VEC require the occupation of antibonding states now the body-centered cubic lattice becomes more favorable as it allows a higher VEC within the first Brillouin zone, up to approximately VEC = 1.48. 

It has been found by Will (2004) from X-ray scattering measurements that valence electrons concentrate along the lines connecting the boron atoms, confirming that the boron layer is a covalently bonded network. The titanium layers are metallic. However, the layers are not characteristic of either pure Ti, or pure B, so the bonding is quite complex. 

For example, consider the TIC and TiN pair. Their lattice parameters are 4.32 A, and 4.23 A, respectively the difference is only two percent. Together with their mutual solubility (Schwarzkopf and Kieffer, 1953) this suggests that they have the same number of bonding valence electrons, although atomic carbon has four valence electrons, and atomic nitrogen has five. The extra nitrogen electron must be in a non-bonding state. This contradicts the valence electron concentrations assumed by Jhi et al., 1999. 

For interstitial solid solutions, too, the criteria used historically for the degree of solid solubility relates to elastic and electronic interactions. Experimentally observed maximum interstitial solubilities of H, B, C and N in Pd are inversely proportional to the sum of the s and p electrons, and hence are controlled by the valence electron concentration. Thus the electronic interactions dominate the... 

The chemistry of octahedral metal clusters culminates in the center of the Periodic Table with the heavy transition metals Nb, Ta, Mo, W, and Re. There is a plethora of clusters where the M-M bonded core is surrounded (and shielded) by non-metal ligands. When moving to the left of the Periodic Table the decrease in valence electron concentration calls for a stabilization through incorporation of interstitial atoms into the cluster core. Actually, the stabilization of the cluster occurs... 

Of course, valence electron concentration is not only related to the metal atoms but also to the number and valence of the ligands. Ligand deficiency creates vacant coordination sites at metal atoms and results in cluster condensation, which is the fusion of clusters via short M-M contacts into larger units ranging from zero- to three-dimensional. The chemistry of metal-rich halides of rare earth metals comprises both principles, incorporation of interstitial atoms and cluster condensation, with a vast number of examples [22, 23]. 

In several compounds, the so-called Valence-Electron Concentration (VEC) proved to be a parameter relevant to their composition, structure and stability. 

Similar classification criteria may be made by using the total valence-electron concentration previously defined (see equation 4.27) and defining, according to Parthe (1995) the tetrahedral structure equation ... 

Figure 4.26. Optimum values, in accordance with Wade s rules and as summarized by Miller et al. (2002), of the valence electron concentration VEC (total number of electrons divided by the cluster atom number) as a function of the number of cluster atoms for closo and nido deltahedra. On the left the values computed for the main group elements and on the right those relevant to the transition metals.

Table 4.6. Structure types and valence electron concentration. Examples of intermetallic phases ...

Remarks on the alloy crystal chemistry of the 11th group metals. A selection of the phases formed in the binary alloys of Cu, Ag and Au and of their crystal structures is shown in Tables 5.54a and 5.54b. For a number of these phases, more details (and a classification in terms of Hume-Rothery Phases ) are given in 4.4.5 and in Table 4.5 (structure types, valence electron concentration, etc.). Table 5.54a and 5.54b show the formation of several phases having a high content... 

The Cu5Zn8 ( Cu5Zn6 9 — Cu5Zn9 7) phase is a classical example of a Hume-Rothery phase ( electron compounds , brass-type phases) that is of a phase in which there is a structure-determining influence of the VEC (valence electron concentration, see 4.4.5). 

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