Crystal Madelung

For the case of ionic solids, the method of Ewald summations [44] is employed to take into account the long-range Coulomb potential of the crystal (Madelung potential). 

For a complex in a crystal, Madelung effects and effects due to the bonding of a ligand with generally more than one metal ion will be superimposed on the simple scheme for an isolated complex, and must be accounted for in predictive calculations. The model as discussed above, however, is adequate for the qualitative interpretation of, for example, NQR (see section IV. C) and spin density (see Sections VI. A and VI. B) data. In such cases it is frequently... 

Many-body problems wnth RT potentials are notoriously difficult. It is well known that the Coulomb potential falls off so slowly with distance that mathematical difficulties can arise. The 4-k dependence of the integration volume element, combined with the RT dependence of the potential, produce ill-defined interaction integrals unless attractive and repulsive mteractions are properly combined. The classical or quantum treatment of ionic melts [17], many-body gravitational dynamics [18] and Madelung sums [19] for ionic crystals are all plagued by such difficulties. 

Madelung constant (A) A number that appears in the expression for the lattice energy and depends on the type of crystal lattice. Example A = 1.748 for the rock-salt structure. 

For simple monovalent metals, the pseudopotential interaction between ion cores and electrons is weak, leading to a uniform density for the conduction electrons in the interior, as would obtain if there were no point ions, but rather a uniform positive background. The arrangement of ions is determined by the ion-electron and interionic forces, but the former have no effect if the electrons are uniformly distributed. As the interionic forces are mainly coulombic, it is not surprising that the alkali metals crystallize in a body-centered cubic lattice, which is the lattice with the smallest Madelung energy for a given density.46 Diffraction measurements... 

We have already mentioned that for sodium chloride approximately 1.78 times as much energy is released when the crystal lattice forms as when ion pairs form. This value, the Madelung constant (A) for the sodium chloride lattice, could be incorporated to predict the total energy released when 1 mole of NaCl crystal is formed from the gaseous Na+ and Cl- ions. The result would be... 

We have already defined the Madelung constant as the ratio of the energy released when a mole of crystal forms from the gaseous ions to that released when ion pairs form. In order to understand what this means, we will consider the following example. Suppose that a mole of Na+ and a mole of Cl... 

Table 7.3 Madelung Constants for Some Common Crystal Lattices.

In this series, the terms neither lead to a recognizable series nor converge very rapidly. In fact, it is a rather formidable process to determine the sum, but the value obtained is 1.74756. Note that this is approximately equal to the value given earlier for the ratio of the energy released when a crystal forms to that when only ion pairs form. As stated earlier, the Madelung constant is precisely that ratio. 

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. 

Each of the compounds shown in Eq. (9.101) has the same crystal structure, the sodium chloride structure, so the Madelung constant is the same for all of them. The term containing 1/n is considered to be a constant for the two pairs of compounds (reactants and products). Actually, an average value of n... 

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... 

A finite array of charges is built taking into account the symmetry elements of the crystal. The charges of the outermost ions are adjusted in order to provide the correct value of the Madelung potential on each cluster site as well as the electrical neutrality... 

Table 1 Madelung potential on the sites of the YBaf u Oj unite cell obtained with Gupta-Gupta [14] charges, in a.u. (infinite crystal)...

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