Madelung constant, effective

Many of the spinel-type compounds mentioned above do not have the normal structure in which A are in tetrahedral sites (t) and B are in octahedral sites (o) instead they adopt the inverse spinel structure in which half the B cations occupy the tetrahedral sites whilst the other half of the B cations and all the A cations are distributed on the octahedral sites, i.e. (B)t[AB]o04. The occupancy of the octahedral sites may be random or ordered. Several factors influence whether a given spinel will adopt the normal or inverse structure, including (a) the relative sizes of A and B, (b) the Madelung constants for the normal and inverse structures, (c) ligand-field stabilization energies (p. 1131) of cations on tetrahedral and octahedral sites, and (d) polarization or covalency effects. ... 

For a discussion on the effective coordination number , its relation to atomic size, bond strength, Madelung constant, etc., see also Simon (1983). For a computation of the heats of formation based on the so-called effective coordination, see a formula by Kubaschewski and Evans (1958), and for a discussion on its application and limits, see Borzone etal. (1993). 

If we now consider the effect of the crystal lattice, by introducing Madelung constant M and multiplying by Avogadro s number Nq (i.e., the number of atoms per weight formula unit), we get... 

The results for the three models are summarized in Table 9.2, which lists the effective Madelung constant n (t, defined by eiectrosuiic = — peff/a the effective Born coefficient, defined by U = — NAncff/a(l — l/neff) and the lattice energy U. 

The effect of coordination number on the Madelung constant, A, and therefore on the lattice energy, is clearly demonstrated in Table 3.4 when same-formula types are compared the larger the coordination number, the larger the value of A. This is due, of course, to the larger number of attractive interactions that are possible. On the whole, though, this variation amounts to only a few per cent of the lattice energy. 

Macrocyclic effect 83 Macrocyclic ligand 56 Madelung constant 45... 

The latter is less by a factor of M, the Madelung constant. Then (M — 1) must be the effect of the rest of the lattice. This suggests the corrected value 7 = nMjZ for ionic solids. 

Constant k is usually assigned a value of 1.75 to 2.00. Normally, the valence state ionization energies are corrected for charge effects and the calculations cycled to self-consistency (75). Madelung correction terms have also been included by some workers (109). 

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