Zinc Madelung constant

For compounds of the composition MX (M = cation, X = anion) the CsCl type has the largest Madelung constant. In this structure type a Cs+ ion is in contact with eight Cl-ions in a cubic arrangement (Fig. 7.1). The Cl- ions have no contact with one another. With cations smaller than Cs+ the Cl- ions come closer together and when the radius ratio has the value of rM/rx = 0.732, the Cl- ions are in contact with each other. When rM/rx < 0.732, the Cl- ions remain in contact, but there is no more contact between anions and cations. Now another structure type is favored its Madelung constant is indeed smaller, but it again allows contact of cations with anions. This is achieved by the smaller coordination number 6 of the ions that is fulfilled in the NaCl type (Fig. 7.1). When the radius ratio becomes even smaller, the zinc blende (sphalerite) or the wurtzite type should occur, in which the ions only have the coordination number 4 (Fig. 7.1 zinc blende and wurtzite are two modifications of ZnS). 

Because the Madelung constant has been computed by a summation over all lattice sites, it adopts characteristic values for all structure types [5,8]. To give a few examples, M arrives at (dimensionless) values of 1.6381 (zinc-blende-type), 1.7476 (sodium chloride-type), 1.7627 (caesium chloride-type), 5.0388 (fluorite-type), and 25.0312 (corundum-type) and does not scale with (= is independent of) the interionic distances. For the case of NaCl, the Madelung constant shows that the three-dimensional lattice surpasses the ionic pair in energy by almost 75%. This is what has made the formation of solid NaCl possible, a collective stabilization. 

Consequently, in cfc and bcc structures, in which the Madelung constants are practically equal, the factor AE prevails. In contrast, in the zinc-blende structure, the Madelung constant is lower and the stability arising from the delocalization is destroyed by the electrostatic term, which is too weak in absolute value. 

The geometric series in the parentheses is a constant that depends on the crystal structure. In other words, if the sodium cations and chloride anions assumed the CsCl, zinc blende, Avurtzite, or some other crystal structure, the series in the parentheses would be different from that given for NaCl. These unique series for each crystal structure are known as Madelung constants M) and are listed in Table 8.1. Using the symbol A/njCi for the Madelung constant unique to the sodium chloride... 

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