Madelung constant

In crystallography, the Madelung constants ( according to Erwin Madelung, the first derived ) is a dimensionless factor, which is defined as:

With

• , The average binding energy per ion in the crystal lattice and
• , The average binding energy per ion at a single ion pair.

The Madelung constant depends only on the structure type, but not of the ionic charge or the cell parameters.

Typical crystal structures, to which the Madelung constant is applicable, are the alkali metal halides, in which the bond is formed by Coulomb forces. Here, the alkali metal atom transfers an electron to the halogen atom, and each atom is formed a spherically symmetric charge distribution.

Because the Madelung constant by Coulomb's law is derived for point charges, it loses its validity at the non- point-like ions ( ions of covalent bonds, such as pyrite in the crystal) and ions of different polarity (e.g., in the series ZnS, TiO 2, CdCl2, CdI2 ).

Calculation of the binding energy in the lattice

The binding energy of an ion pair can be calculated using Coulomb's law as follows:

With

As in a crystal lattice, not only an ion pair is present in the room but other cations and anions in the crystal formation more energy is released, however, be needed again in order to bring the same ions. The following equation will illustrate this fact:

The stored energy in the grid EIG results here as the sum of vacant and needed in lattice formation energies for each ion. Where n is the number of how many times a particular ion, and c is a factor which represents the distance of the ion. These factors can lead to a dependent of the crystal factor α - are combined so that the following equation for the binding energy of an ion in the lattice - the Madelung constant:

This equation describes the binding energy of only one ion in the lattice. In order to get the energy that is released during the formation of a certain amount of substance, this equation has to be multiplied by the Avogadro constant and the amount of substance:

As can be seen, this value is negative, since the grating formation is exothermic. For a more precise calculation of lattice energies, the sole consideration of regularly spaced Coulomb point charges is not enough. An extension of the model leads to the Born- Landé equation.

Calculation of the Madelung constant on the example of NaCl

When ionic lattice of NaCl there is a face-centered cubic crystal structure, as is shown on the right (red and green are the anions, the cations ). The distance between the two ions is at d = 0.3 nm NaCl about

From the above equation now have the number of the respective adjacent ion n, whose relative distance c are multiples of the distance d as well as their charge, so whether they attract or repel determined.

If we start from an ion, for example, the red shown with the number 0, so we have the first n = 6 ions ( green shown with the number 1 ) at a distance of 1d, which are tightened. This is followed by 12 red ion ( 2), whose distance by means of the Pythagorean theorem gives the value and be rejected by the similar ion, and 8 green ion at a distance of, which are tightened. The following table sets these numbers continue:

Substituting this procedure continues, we arrive at the following, frequently more appropriate in the literature series representation of the Madelung constant:

However, this is wrong because this series diverges, as was first demonstrated in 1951. The sum of the points of the crystal lattice is caused convergent, that depends on the order of summands. The above series would be a summation of concentric spherical shells meet, which is not physically meaningful. The "correct" value is obtained by summing over the lattice points inside a cube with edge length and the limit for is formed. A mathematical grounds that this value is the " right" one was in 1985 by David Borwein, Jonathan Borwein and Keith F. Taylor: You define a function of a complex variable:

Where the dash means that the term is omitted. For sufficiently large real part of the series is absolutely convergent, the Madelung constant ( ) is obtained by analytic continuation.

The summation over cubes converges so slowly that it is unusable for practical calculations. With a little trick, however, much better approximations can be found: it is summed over all points within a cube with edge length, the points on the sides are only half counted at the edges to a quarter and at the corners to one-eighth. This method is known as Evjen method. Already delivers 1.7470 with a very good approximation; = 1.74750 and = 1.7475686. The physical motivation of this method arises from the requirement that the partial sums are to be formed on an electrically neutral (finite) crystal.

Even before Paul Peter Ewald Evjen published today known as the Ewald method method for calculating the Madelung constant. His method is a special case of the Poisson summation formula, which Ewald, however, was not known.

Today, many numerical methods and powerful computers available, so the calculation of Madelung constants for arbitrary lattice with high accuracy is no longer a problem.

Values ​​for some crystal structures

Notes: