Question & Answer: If we were to examine the NaCl crystal in three dimensions, we would find that each Na^+ ion…..

If we were to examine the NaCl crystal in three dimensions, we would find that each Na^+ ion has 6 Cl^- ions as nearest neighbors at a distance r. 12 Na^+ ions as second nearest neighbors at a distance r squareroot 2, 8 Cl^- ions as third nearest neighbors at a distance r squareroot 3, and so on. Show that the electrostatic potential energy of the Na^+ atom can be written as E(r) = -e^2/4 pi epsilon_0 r[6-12/squareroot 2 + 8/squareroot 3 – ….] = -e^2M/4 pi epsilon_0r where M, called the Madelung constant, is given by the summation in the square brackets for this particular ionic crystal structure (NaCl). Calculate M for the first three terms and compare it with M= 1.7476, its value had we included the higher terms. What is your conclusion?