Lattice Energy and Madelung Constant

At first glance, calculation of the lattice energy of a crystal may seem simple just take every pair of ions and calculate the sum of the electrostatic energy between each pair, using the equation below. 

Zj = ionic charges in electron units To = distance between ion centers e = electronic charge = 1.602 X 10 C 

For simple compounds, p = 30 pm works well when tq is also in pm. Lattice enthalpies are twice as large when charges of 2 and 1 are present, and four times as large when both ions are doubly charged. Madelung constants for some crystal structures are given in Table 7-2. 

Source D. Quane, J. Chem. Educ., 1970, 47, 396, has described this definition and several others, which include all or part of the charge (2) in the constant. Caution is needed when using this constant because of the different possible definitions. 

Calculate the lattice energy for NaCl, using the ionic radii from Appendix B-1 

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An alternative to the Eyjen method is described by Ewald and formulated by Tosi. This method involves a summation over the reciprocal lattice and, unlike Eyjen s method, is valid over any point in the crystal, not just the lattice sites, and so is important for ealculations involving lattice defects or interstitial ions. Van Gool and Piken have developed a program for automatically calculating lattice energies and Madelung constants based on this method and have used this on compounds such as P-alumina. The Madelung term remains constant for a particular stmcture type. Some values are listed in Table 3.4. ... 

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