Chapter 37, Problem 61P

The Madelung constant (Section 37.3) is notoriously difficult to calculate because it’s the sum of an alternating series of nearly equal terms. But it can be calculated for a hypothetical one-dimensional crystal consisting of an evenly spaced line of alternating positive and negative ions (Fig. 37.25). Show that the potential energy of an ion in this “crystal” can be written as

$U=-\alpha \frac{k{e}^2}{r_0}$

where the Madelung constant α has the value 2 ln2.

FIGURE 37.25 Problem 61