## The calculation of the Madelung constant

The total Coulomb interaction energy of a crystal is given by the sum of the single pair interaction terms:

$\displaystyle V_{AB} = \frac{e^2}{4\pi\epsilon_0} \frac{Z_AZ_B}{r_{AB}} \hfill (1)$

for ions with charges qA and qB and distance rAB.  The sum is extended to all pairs of ions present in the solid for any crystalline structure.

The sum converges very slowly because the first neighbors contribute with an important negative term, the second neighbors with a positive term slightly weaker, and so on. In this way, the resulting effect will be that the attraction between cations and anions predominates and provides a favorable negative contribution to the energy of the solid.

Infinite one-dimensional reticulum.

In a mono-dimensional reticulum with alternate cations and anions at intervals of constant of length d, and having charges qA=+Z and qB=-Z, the interaction of one ion with all others is given by the series:

$V = -\frac{2Z^2}{d} + \frac{2Z^2}{2d} - \frac{2Z^2}{3d} + \frac{2Z^2}{4d} - ...$

$= -\frac{2Z^2}{d} \left(1 -\frac{1}{2} +\frac{1}{3} -\frac{1}{4} + ... \right)$

$= -\frac{2Z^2}{d} \ln 2$

(the factor 2 derives from the fact that each ion in the reticulum had on both sides identical ions). The sum depends only on the type of reticulum and by the distance d between the centres of adjacent particles:

$\displaystyle V = -\frac{e^2}{4\pi\epsilon_0} \frac{Z^2}{d} (2 \ln 2) \hfill (2)$

In the case of one-dimensional lattice, $\displaystyle V$ contain the term ${\frac{Z^2}{d}}$ depending on the charge type and reticulum, and the factor 2ln 2 (= 1,3862944), called constant of Madelung (A), that characterizes the symmetry of the reticulum.

## Infinite two-dimensional reticulum

In the case of a two-dimensional reticulum, the cations and the anions are disposed at regular intervals on contiguous squares of length d and with ${q_A = +Z}$ e ${q_B = -Z}$. Considering as centre the ion (1) in the above diagram and moving radially from it, the interaction of this ion with all the others is given by:

$V = -\frac{4Z^2}{d} + \frac{4Z^2}{\sqrt{2} d} - \frac{8Z^2}{\sqrt{5} d} + \frac{4Z^2}{2\sqrt{2} d} + \frac{4Z^2}{2 d} ...$

$= -\frac{4Z^2}{d} \left(1 -\frac{1}{\sqrt{2}} +\frac{2}{\sqrt{5}} -\frac{1}{2\sqrt{2}} + \frac{1}{2 } ... \right)$

## Infinite three-dimensional reticulum

This formulation can be easily extended to a three-dimensional lattice.

In a simple solid, like the NaCl in the figure, the Madelung constant depends on the crystal type and by inter-ionic distances. The Madelung constant for a three-dimensional lattice is calculated by starting from an opportune ion placed at the center of the lattice, then by moving radial until the first n1 neighbors, having charges of opposite value, at distance d1, by continuing to move radial to the reference ion, we will encounter the second n2 neighbors (having the same charges) at the distance d2 and so on. The Madelung constant, in this case, will be defined by the summation:

$\displaystyle A = \sum_i \left(-sgn(q_A q_B)\right) n_i \frac{1}{(d_i/d)} \hfill (3)$,

where ${-sgn(q_A q_B)}$ indicates the sign of each term of the sum and it is positive if the ions have opposite charges (attraction) and negative if they have equal charge (repulsion), and ${r^+}$ + ${r^-}$ is the sum of ionic radii of the ions in the lattice.

Example

As an example, we consider the Na+ in NaCl. It has 6 first neighbour’s Cl (n1 = 6 at a distance) d1=d; then there are 12 second Na+ as second neighbours (n2=12) at a distance ; 8 third neighbours   Cl (n3=8) at a distance ; etc. Therefore, the resulting series is given by the following expression:

$\displaystyle A = + 6 - \frac{12}{\sqrt 2} + \frac{8}{\sqrt 3} - ... \hfill (4)$

The potential energy for one mole in the crystalline structure is given by:

$\displaystyle V = - A N_A \frac{e^2}{4\pi\epsilon_0} \frac{Z_AZ_B}{d} \hfill(5)$

 Table 1: Constants of Madelung for some solids Structure A A/n* Coordination CsCl 1.763 0.88 (8,8) Halite (NaCl) 1.748 0.87 (6,6) Fluorite (CaF2) 2.519 0.84 (8,4) Wurtzite (ZnS) 1.641 0.82 (4,4) CdCl2 2.244 0.75 (6,3) CdI2 2.191 0.73 (6,3) Rutile ($TiO_2$) 2.408 0.80 (6,3) Curundum ($Al_2O_3$) 4.172 0.83 (6,4)

* n it is the number of ions in the composition formula

As shown for the NaCl (6,6) and  CsCl (8,8) structures, the constant of Madelung increases with the coordination number as the most important contribution comes from the first neighbours.