# Evaluating the Madelung Constant

#### Question

The Madelung constant is used in determining the electrostatic potential of a single ion in a crystal by approximating the ions by point charges:

$$V_i = \frac{e}{4\pi\epsilon_0r_0}\sum_j\frac{z_jr_0}{r_{ij}} = \frac{e}{4\pi\epsilon_0r_0}M_i$$

where $r_0$ is the nearest neighbour distance, and the Madelung constant of the ith ion is given by the sum over all the other ions in the crystal:

$$M_i = \sum_j\frac{z_j}{r_{ij}/r_0}.$$

For the cubic crystal NaCl, the summation may be performed over three orthogonal co-ordinates:

$$M_\mathrm{Na} = \sum_{j,k,l=-\infty}^{\infty}' \frac{(-1)^{j+k+l}}{\sqrt{j^2+k^2+l^2}},$$

where the prime indicates that the term $(0,0,0)$ is excluded. Benson's formula provides a practical and efficient way of evaluating this sum as:

$$M_\mathrm{Na^+} = -12\pi\sum_{m,n=1,3,...}^{\infty}\mathrm{sech}^2\left(\frac{1}{2}\pi\sqrt{m^2+n^2}\right),$$

where the summation is performed over all positive odd integers $m$ and $n$.

Write a program to calculate the Madelung constant for $\mathrm{Na^+}$ ions in NaCl, -1.74756...

#### Solution

To access solutions, please obtain an access code from Cambridge University Press at the Lecturer Resources page for my book (registration required) and then sign up to scipython.com providing this code.