Madelung constant Table

There are various algorithms which can be used to calculate the Madelung constant. Table 1.2 shows the values obtained for cubic lattices. 

Table 5.1 MADELUNG constants for some structure types...

Table 7.3 Madelung Constants for Some Common Crystal Lattices.

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 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. 

TABLE 1.14 Madelung constants for some common ionic lattices... 

The series 6 - 12/21/2 + 8/31/2 - 6/41/2 + 24/51/2 -. .. eventually becomes convergent and gives the value for the Madelung constant for the sodium chloride lattice (the standard description of lattices which have the same form as that adopted by sodium fluoride). The values of Madelung constants for some common crystal lattices are given in Table 7.5. 

Fortunately, the Madelung constant may be obtained mathematically from a converging series, and there are computer programs that converge rapidly. However, we need not delve into these procedures, but may simply employ the values obtained by other workers (Table 4.1). The value of the Madelung constant is determined... 

Using Fig- 4.7 generate the first five terms of the senes for the Madelung constant for NaCL How dose is the summation of these terms to the limiting value given in Table 4.1 ... 

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. 

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