In the sodium chloride structure, the Na⁺ ion at the centre is surrounded by six Cl^- at a distance r. The attractive potential energy is:

The next nearest neighbours for the Na⁺ ion are the twelve Na⁺ ions on each edge of the unit cell, at a distance of (2)^1/2r. This gives rise to a repulsive potential energy term:

In the next shell, the Na⁺ ion experiences an attractive potential energy with the eight Cl- ions at the corner of the unit cell, at a distance of (3)^1/2r:

The net attractive energy between the Na⁺ ion and all other ions in the crystal is given by the infinite (and slowly converging series):

This summation is repeated for every ion in the crystal (i.e. 2N_A ions per mole of NaCl). As this counts every interaction twice, it is then necessary to divide by two giving the total energy as:

where the Madelung Constant, A, is the numerical value of the infinite series summation. The Madelung constant depends only on the crystal structure not on the size or charge of the ions. Every compound with the rocksalt structure has the same Madelung constant. The table below lists the Madelung constant for a number of common structural types.

Table 1. The Madelung constant for common MX and MX₂ structures. The final column shows the Madelung constant divided by the number of ions in the chemical formula.

Note that, for a given stoichiometry, the Madelung constant and, hence, the lattice energy, increase with the coordination number:

Solids adopt structures with higher coordination numbers to maximize the ionic bonding

One curious feature of the expression in equation (5) is that the lattice energy has the largest magnitude when r = 0! The model does not implicitly include any repulsion between unlike charges. The hard sphere model instead imagines that the repulsion between ions with unlike charges is zero until they touch and is then infinite. This turns out to be a poor approximation and equation (5) leads to lattice energies which are at least 10% greater than experimentally determined values.

A better treatment uses a repulsive force which is small at large separations but rises sharply when the ions come into contact. A possible function with this behaviour is B/rⁿ where n is large and both n and B are constants for each compound and can, in principle, be determined from measuring the compressibility of the crystal. The resulting lattice energy is the sum of the attractive term in equation (5) and this repulsive function:

At the equilibrium interatomic separation, r_eq, Eelectrostatic is minimized and:

Substituting this into equation (6) gives the Born-Landé equation:

The n value can be obtained from compressibility measurements and usually lies between 5 and 12. For ions with noble gas configurations, a reasonably accurate value for n can be obtained by taking the weighted average of the empirical constants in the table below.

Table 2. Approximate compressibility factors, n, for ions.  

For example, Na⁺ and Mg²⁺ both have the same configuration as Ne and Cl^- has the same configuration as Ar. The n values for NaCl and MgCl₂ are thus:

NaCl: n =

and

MgCl2: n =

More recently, the repulsive part has been represented by the function:

Use of this function yields the Born-Meyer equation:

The table below compares the experimental lattice energy for NaCl with that obtained using the hard-sphere model (equation (5)), the Born-Landé model (equation (8)) and the Born-Meyer model (equation (9)).

Table 3. Comparison of the experimental and calculated lattice energy for NaCl.

The hard-sphere, Born-Landé and Born-Meyer equations unfortunately require knowledge of the Madelung constant and hence the crystal structure. They therefore cannot be used to predict stabilities of unknown compounds. The Born-Landé and Born-Meyer equations also require knowledge of the compressibility variables, n and ρ respectively.

Kapustinskii noticed that the Madelung constant divided by the number of ions in the chemical formula, v, is almost constant for many crystal structures. Table 1 includes the value of A/v for various MX (v = 1), MX₂ (v = 3) and M₂X (v = 3) structures. It varies between 0.73 – 0.88. Kapustinskii also proposed using a universal value of ρ = 0.345 and sums of tabulated values of cation (r₊) and anion (r_- radii in place of the measured value of their separation in the structure. The result of these simplifications is, with all constants in SI units, the Kapustinskii equation:

As shown in Table 3, the Kapustinskii equation yields lattice energies which are very similar to those obtained using the more accurate equations. It has the distinct advantage that all it requires is knowledge of the chemical formula of the ionic compound.

3 Which Crystal Structure Type Is Adopted?

As noted above, for any given stoichiometry, the Madelung constant and hence the lattice energy increases with the coordination number of the cation and anion. This is the reason why higher coordination numbers are always favoured. However, as the coordination number around the cation increases, it becomes increasingly difficult to maintain cation-anion contact without the anions overlapping. This is the basis of the radius ratio rules.

4 Do The Radius Ratio Rules Work?

Figure 6. The relationship between the structures of the alkali metal halides (u NaCl and n CsCl) and ionic radii. The dotted lines showing the limiting radius ratios for 3, 4, 6 and 8 coordination.