Sp-valent elements

Fig. 3.5 The Pauling (left-hand panel) and Mulliken (right-hand panel) electronegativities for the sp-valent elements. Note that the electronegativities scales run vertically downwards in order to emphasize the similarity with the free-atom energy levels in Fig. 2.16 and negative inverse core sizes in Fig. 2.13.

The most famous example of the crystal structure correlating with the average number of valence electrons per atom or band filling, N, is the Hume-Rothery alloy system of noble metals with sp-valent elements, such as Zn, Al, Si, Ge, and Sn. Assuming that Cu and Ag have a valence of 1, then the fee -phase is found to extend to a value of N around 1.38, the bcc / -phase to be stabilized around 1.48, the -phase around 1.62, and the hep e-phase around 1.75, as illustrated for the specific case of Cu-Zn alloys in Fig. 6.15. In 1936 Mott and Jones pointed out that the fee and bcc electron per atom ratios correlate with the number of electrons required for a free-electron Fermi sphere first to make contact with the fee and bcc Brillouin zone faces. The corresponding values of the Fermi vector, fcF, are given by... 

The ratio of the bond integrals for sp-valent elements was found by Harrison (1980) by fitting a nearest-neighbour model to the first principles band... 

The free-electron gas model is a good starting point for the sp-valent metals where the loosely bound valence electrons are stripped off from their ion cores as the atoms are brought together to form the solid. However, bonding in the majority of elements and compounds takes place through saturated... 

We see from the values of the s-d separation in Fig. 2.17 that we expect the bonding and structural influence of the d states to be much more marked for the divalent alkaline earths Ca and Sr at the beginning of the transition series than for the divalent elements Zn and Cd at the end. Thus it is not unreasonable that purely sp-valent Be and Mg are found to be grouped in Fig. 1.8 with Zn, Cd and Hg rather than Ca, Sr and Ba. 

The variation in equilibrium bulk properties between one sp-valent metal and the next cannot be understood within the jellium model, since it has obscured the chemical behaviour of the elements by smearing out the ion... 

We have seen in the previous chapter that the total binding energy per atom of an elemental sp-valent system may be written within the approximation as the sum of three terms, namely... 

This applies for instance to the boron (Ill)-halides for which nearly the same Db-b values are obtained (Table 1). With dH/=dH/ assumed to be valid, Eq. (12) is applicable. For BFsand BCI3 with JHB2(St- g) = 0 and /JHcpd (St - -g) = 0, the result is = 0, and for BBra with f dHB2 (St- g) = 11.1 and dHcp(j(St g) =7.3, one obtains dHm= —3.8. This negative value for appears unreasonable. Values dHnj = 0 do exist in case of the diamond modification of C, Si, Ge and Sn for the tetra-valent sp -hybridized state of these elements. For sp -hybridized boron, dH = 0 is improbable in view of the comphcated structures of crystalline boron. 

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