Chapter 12, Problem 8P

The Madelung constant for the NaCl structure may be found by summing an infinite alternating series of terms giving the electrostatic potential energy between an Na⁺ ion and its 6 nearest Cl⁻ neighbors, its 12 next-nearest N⁺ neighbors, and so on (see Fig. 12.1a). (a) From this expression, show that the first three terms of the infinite series for the Madelung constant for the NaCl structure yield α = 2.13. (b) Does this infinite series converge rapidly? Calculate the fourth term as a check.