A previous article showed that the electrostatic term of the lattice energy of a crystal contains a factor (A) that depends on the type of crystalline reticulum. This article will look more in detail how to calculate this term and its value for a simple ionic system.
The total Coulomb interaction energy of a crystal is given by the sum of the single pair interaction terms:
for ions with charges qA and qB and distance rAB. The sum extends to all pairs of ions present in the solid for any crystalline structure.
The sum converges very slowly because the first neighbours give a first substantial contribution with a negative summation term. Second neighbour atoms produce a slightly weaker positive term. It then continues to infinity with terms of alternating signs by smaller and smaller values. In this way, the resulting effect will be that the attraction between cations and anions predominates and provides a favourable negative contribution to the solid’s energy.
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:
(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:
In the case of one-dimensional lattice, 
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 
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 ion placed at the lattice’s centre and then moving radially until the first 
where 
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:
The potential energy for one mole in the crystalline structure is given by:
| 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 ( ) | 2.408 | 0.80 | (6,3) |
Curundum ( ) | 4.172 | 0.83 | (6,4) |
* This this the value of A divided by the number n of ions in the chemical formula of the salt.
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.
If you have found interesting and useful my article, do not forget to press “Like it” and subscribe for updates on new ones!
Very nice presentation to introduce someone to the topic. Well written, concise, and emphasizing both the fundamentals and numerical values. Compared to the “hair grind” summations on Wolfram or the precise but unclear discussion on Wikipedia this posting is where I would recommend a student start reading.
LikeLiked by 2 people