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A detailed discussion of bonding and the manner in which it determines atomic coordination numbers and polyhedra is beyond the scope of this chapter. Although of very limited practical use, the 8‐N rule provides a very simple illustration of an approach to bonding. It states that the coordination number of a covalently bonded atom with N valence electrons is 8‐N. Now, the electronic configurations of silicon and oxygen are [Ne] 3s² 3p² and [He] 2s² 2p⁴, respectively, so that with 4 and 6 valence electrons these elements should have coordination numbers of 4 and 2, respectively, as actually observed in v‐SiO₂ and in silicate glasses at zero pressure. Silicon atoms with a higher coordination number do occur, but usually under high pressure, the only known exception being alkali phosphosilicate glasses (e.g. Na₂O–P₂O₅–SiO₂), in which the presence of some six‐coordinated silicon has been established [17].

Figure 9 Deviations of the relative abundance of Qⁿ ‐species in lithium silicate glasses (points) from an idealized binary distribution (continuous lines) as a function of the Li₂O/SiO₂ ratio, J [16]. Shading of the points indicating the original ²⁹Si NMR experimental data (see key on the figure): ▽ – Q ⁴, ○ – Q ³, ⋄ – Q ², □ – Q ¹, △ – Q ⁰.

For other glass formers, a higher cation coordination can in contrast occur with ease. For example, in pure B₂O₃ glass, all boron atoms are three‐coordinated, as depicted for a fragment of the network in Figure 10a. However, there are two alternative ways in which a modifier such as M₂O may be accommodated in a borate network. The additional oxygen from a M₂O unit can be incorporated into the network either by conversion of one BO to two NBOs (Figure 10b), as occurs in silicates, or by conversion of two borons from three‐ to four‐coordination (Figure 10c). Because ¹¹B NMR is sensitive to the presence of four‐coordinated boron, the average coordination number, n_BO, can be measured over very wide composition ranges. For xLi₂O·(1 − x)B₂O₃ glasses (Figure 11), n_BO increases with x, from 3 for B₂O₃ itself, to a maximum value of 3.44 ± 0.01 at 35–38 mol % Li₂O, and then falls for further increases in the Li₂O content [18].

Figure 10 The two differing effects of the addition of a network modifier cation M⁺ on the borate network. (a) Fragment of the network of pure B₂O₃ glass (small spheres are B atoms, and large spheres are O atoms). (b) Formation of non‐bridging oxygens. (c) Formation of four‐coordinated boron atoms.

This variation can be reasonably accounted for by a simple charge‐avoidance model [19], in which additional oxygen is incorporated into the borate network by the conversion of BO₃ units to BO₄ provided that centers of negative charge are not directly connected. These centers are not only NBOs but also BO₄ units since these have a net negative charge. For small amounts of modifier, the formation of BO₄ units causes n_BO to be equal to 3 + x/(1 − x). At larger contents, however, NBOs form instead so that n_BO falls back toward a value of three. Similarly, the thermophysical properties of borate glasses show a maximum (or minimum) as modifier is added to the glass known as the borate anomaly.

Figure 11 Boron‐oxygen coordination number, n_BO, for lithium borate glasses, Li₂O–B₂O₃, as determined by ¹¹B NMR (points) [18], compared with the prediction from a charge‐avoidance model [19].

Pure germania glass, GeO₂, forms a tetrahedral network, similar to that of silica, but with a smaller average Ge─Ô─Ge bond angle. As for borates, however, the addition of a modifier is accompanied by a growth and then a decline in the value of the average coordination number, n_GeO, and the thermophysical properties also show a maximum (or minimum). Originally, this germanate anomaly was ascribed to the formation of octahedral (i.e. six‐coordinated) germanium atoms. Compared to the borate anomaly, however, evidence concerning the structural aspects of the germanate anomaly is much less plentiful because germanium nuclei are not usefully accessible to NMR. For cesium germanate glasses, ND measurements [20] show that as Cs₂O is added to GeO₂ glass, n_GeO increases up to a maximum value of 4.36 for 18 mol % Cs₂O, and then falls for further increases in the Cs₂O content (Figure 12). The additional oxygen from an M₂O unit can lead to a growth in n_GeO either by converting one GeO₄ into a GeO₆ unit, or by converting two GeO₄ units into GeO₅ units, and both mechanisms are possible in principle. However, the variation of n_GeO for cesium germanates is much better predicted by a charge‐avoidance model if the higher germanium coordination number is five, rather than six. Nevertheless, evidence is beginning to emerge that the preferred higher Ge coordination in germanate glasses may depend on the modifier cation.

Figure 12 Germanium‐oxygen coordination number, n_GeO, for cesium germanate glasses, Cs₂O–GeO₂, as determined by neutron diffraction [20], compared with the predictions of charge‐avoidance models in which the higher GeO_n coordination is either 5 or 6.