Madelung factor

The Ba2ZnFe structure type 299) (page 54), if built up of A+ and Me + according to A2+Me +Fe, pelds a much lower Madelung factor of 14.59. On the other hand the corresponding value is highest for this structure type, if compounds A2 +Me2+F6 are concerned. This readily... 

The constant B from formula (1) consists, like the factor for the Coulomb energy, of a factor of proportionality b for the repulsive energy of an ion pair b rn. In the lattice a kind of Madelung factor K also occurs. Since the repulsive energy decreases so rapidly with increasing distance this interaction is practically restricted to the nearest neighbours in the lattice, so that K can, therefore, be put practically equal to the coordination number. 

The same Madelung factors apply to the exact correlation energies, so for any lattice the low-density limit value obtained by dimensional... 

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. ... 

The series inside the parentheses converges to a sum that is 2 ln2 or 1.38629. This value is the Madelung constant for a hypothetical chain consisting of Na+ and Cl- ions. Thus, the total interaction energy for the chain of ions is —1.38629N0e2/r, and the chain is more stable than ion pairs by a factor of 1.38629, the Madelung constant. Of course NaCl does not exist in a chain, so there must be an even more stable way of arranging the ions. 

Details of the calculation of Madelung constants for all of the common types of crystals are beyond the scope of this book. When the arrangement of ions differs from that present in NaCl, the number of ions surrounding the ion chosen as a starting point and the distances between them may be difficult to determine. They will most certainly be much more difficult to represent as a simple factor of the basic distance between a cation and an anion. Therefore, each arrangement of ions (crystal type) will have a different value for the Madelung constant. The values for several common types of crystals are shown in Table 7.3. 

The principle that embodies this relationship can be stated as the products will be those in which the smaller ions will combine with oppositely charged ions of higher charge. In these cases, the Madelung constants may be different, so other factors may be involved. However, the principle correctly predicts reactions such as the following ... 

In Table 2 we present the shells which correspond to our best choice as well as the values of the scaling factors which result from the solution of the system of equations (1). The Madelung potential values on each cluster site calculated with the finite adjusted array are given in Table 3. When these values are compared to that of the Table... 

The geometry of the crystal introduces a factor multiplying the pairwise ionic interaction, which is the Madelung constant /t. It is a dimensionless constant, dependent on the geometry of the crystal under consideration. For an ionic binary crystal, consisting of N each positive and negative ions, fi is defined by... 

The overall stability of the NaF lattice is represented by the resultant of the many stabilizing attractions (Na+-F ) and destabilizing repulsions (Na+-Na+ and F -F ), which amount to a stabilization which is 1.74756 times that of the interaction between the individual Na+-F- ion pairs. The factor, 1.74756, is the Madelung constant, M, for the particular lattice arrangement and arises from the forces experienced by each ion. These are composed of six attractions at a distance r, 12 repulsions at a distance 2 mr, eight attractions at a distance 31/2r, six repulsions at a distance of 41/2r, 24 attractions at a distance 51/2r, and so on. 

It was noted in Chapter 4 that the Madelung constant of a structure may be expressed in various ways. The way that is conceptually simplest in terms of the Bom-Lande equation is the simple geometric factor. A, such that when combined with the true ionic charges, Z and Z, the correct electrostatic energy is formulated. It was noted that some workers have favored using another constant, A. combined with the highest common factor of Z and Z. Z -... 

As long as we do not neglect to understand each or the factors in the Born-Lande equation (4.13), we can simplify the calculations. It should be realized that the only variables in the Bom-Lande equation are the charges on the ions, the inlernuclear distance, the Madelung constant, and the value of n. Equation 4.13 may thus be simplified with no loss of accuracy by grouping the constants to give ... 

The factor A is a positive numerical constant called the Madelung constant its value depends on how the ions are arranged about one another. For ions arranged in the same way as in sodium chloride, A = 1.748. 

The formula is converted into dimensionless units by defining a dimensionless distance based on a characteristic spacing for each compound. It is noted that the closest interionic distance can be specified as the sum of two ionic radii, d = r + r+. Using the well established anionic radii for halide and chalconide ions [75], a conversion factor R = /r 1 r is calculated in each case, and used to define the dimensionless distance d = d/R, such that the Madelung energy,... 

In addition to Mg, other divalent cations, including Co, Ni, and Cu, can be incorporated in the spinel structure. A1 can be replaced by Ga, In, Cr, and other trivalent ions. In addition to normal spinels, so-called inverse structures are also possible in which half the trivalent cations occupy the octahedral sites, with the other trivalent cations and all the divalent cations in tetrahedral positions. Several factors influence whether a given spinel will adopt the normal or inverse structure, including the following (i) the relative ionic radii, (ii) the Madelung constants of the normal and inverse structures,... 

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