Next-nearest neighbours

Figure 7.10 Structure of yellow Til (a) showing its relation to NaCI (b). Tl has 5 nearest-neighbour I atoms at 5 of the vertices of an octahedron and then 21 + 2T1 as next-nearest neighbours there is one I at 336 pm. 4 at 349 pm, and 2 at 387 pm, and the 2 close Tl -Tl approaches, one at 383 pm. InX (X = Cl, Br, I) have similar structures in their red forms.

The reverse transition from a—involves a structural distortion along the c-axis and is remarkable for the fact that the density increases by 26% in the high-temperature form. This arises because, although the Sn-Sn distances increase in the a—transition, the CN increases from 4 to 6 and the distortion also permits a closer approach of the 12 next-nearest neighbours ... 

Wheland and Pauling (1959) tried to explain the inductive effect in terms of ar-electron theory by varying the ax and ySxY parameters for nearest-neighbour atoms, then for next-nearest-neighbour atoms and so on. But, as many authors have also pointed out, it is always easy to introduce yet more parameters into a simple model, obtain agreement with an experimental finding and then claim that the model represents some kind of absolute truth. 

Figure 4.6 (a) Schematic view of the two degenerate helical arrangements of the spins of a ID frustrated structure with antiferromagnetic next-nearest-neighbour (NNN) interactions. The angle is the one formed between the projections of two adjacent spins, drawn with different colours to... 

However, with the improved resolution of modem XPS instruments, BE shifts as small as 0.1 eV can be detected and may be significant. These shifts can be interpreted, to a first approximation, by changes in the atomic charge (an intraatomic effect), but to account for more subtle differences, as seen in the mixed-metal phosphides M aM P and mixed arsenide phosphides MAsi -VP>, the role of next-nearest neighbours cannot be neglected. These interatomic effects, as incorporated into the charge potential model, help explain the unusual trends in BE observed in these series. 

The next nearest neighbours to the central M+ are 12 M+ at distance V2r. The repulsive cation-cation interaction term is given as... 

The regular model for an ionic solution is similarly analogous to the regular solution derived in Section 9.1. Recall that the energy of the regular solution model was calculated as a sum of pairwise interactions. With two sub-lattices, pair interactions between species in one sub-lattice with species in the other sub-lattice (nearest neighbour interactions) and pair interactions within each sub-lattice (next nearest neighbour interactions), must be accounted for. 

Let us first derive the regular solution model for the system AC-BC considered above. The coordination numbers for the nearest and next nearest neighbours are both assumed to be equal to z for simplicity. The number of sites in the anion and cation sub-lattice is N, and there are jzN nearest and next nearest neighbour interactions. The former are cation-anion interactions, the latter cation-cation and anion-anion interactions. A random distribution of cations and anions on each of... 

The equations for the regular solution model for a binary mixture with two sublattices are quite similar to the equations derived for a regular solution with a single lattice only. The main difference is that the mole fractions have been replaced by ionic fractions, and that while the pair interaction is between nearest neighbours in the single lattice case, it is between next nearest neighbours in the case of a two sub-lattice solution. 

The reciprocal lattice model as derived above is the basis for many different variants. For simplicity we have assumed the interactions between the next nearest neighbours A+ -B+ andC- I) to be independent of composition, even though experiments have shown that this is often not the case. It is relatively simple to introduce parameters which allow the interaction energy, for example between A+ and B+, to depend on the concentration of C and D [14], One may also include other terms that take into account excess enthalpies that are asymmetric with regard to composition and the effects of temperature and pressure. 

Fig. 10. The degree of association into nearest- and next-nearest-neighbour complexes, p, versus concentration, c, at 500°C for manganese ions and cation vacancies in sodium chloride. Filled circles represent the simple association theory, open circles the Lidiard association theory, and crosses the present theory using Eq. (173) when the first term only has been retained in the virial appearing in the equation for the defect distribution function (Eq. (168)). The point of highest concentration represented by a cross may be in error due to the neglect of higher terms in the virial series, and the dotted curve has not been extended to include it.

In between these extremes lie a large number of CVM treatments which use combinations of different cluster sizes. The early treatment of Bethe (1935) used a pair approximation (i.e., a two-atom cluster), but this cluster size is insufficient to deal with fhistration effects or when next-nearest neighbours play a significant role (Inden and Pitsch 1991). A four-atom (tetrahedral) cluster is theoretically the minimum requirement for an f.c.c. lattice, but clusters of 13-14 atoms have been used by de Fontaine (1979, 1994) (Fig. 7.2b). However, since a comprehensive treatment for an [n]-member cluster should include the effect of all the component smaller (n — 1, n — 2...) units, there is a marked increase in computing time with cluster size. Several approximations have been made in order to circumvent this problem. 

These two structures have been described above (Sect. 2.6.3 and Sects. 2.8.1, 3.2) and are depicted in Figs. 16 and 29. We have already suggested (Sect. 4) that in forsterite (a-Mg2Si04, olivine type) the repulsive forces between O... O (d 2.6 A) are less than those between Mg... Mg (d = 3.0 A) which, in turn, are less than those between Mg... Si (d 2.7 A). One would therefore expect that in the phase transition a-Mg2Si04 (olivine) — y-Mg2Si04 (spinel) the principal result would be an increase in the last of these distances, in order to relieve the largest non-bonded repulsions . And this is indeed the case the mean non-bonded (next-nearest neighbour) distances are as shown... 

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