Imagine a cubic crystal like NaCl, with a negative charge at the center of a Cartesian coordinate system with scale units equal to the interatomic distance.

(a) Show that an ion at a position r units along the x axis, s units along the y axis, and t units along the z axis has a charge of e(-1)^r . (-1)^s . (-1)^t = e(-1)^{r + s + t}, where e is the electron charge.
(b) Using Equation 10-2 as a guide, calculate the Madelung constant for a cube 2 units on a side. Do the same for cubes of sides 4, 6, 8, 10, 12, 16, and 20 units. (You will probably want to use a computer spreadsheet to write a program to do the calculations for the larger cubes.) Are your answers approaching the value a = 1.7476?

a = 6- 12 8 + V2 V3 6 20 + 2 V5 10-2