Madelung constant

The Madelung constant is used in determining the energy of a single ion in a crystal. It is named after Erwin Madelung, a German physicist.

Because the anions and cations in an ionic solid are attracted to each other by virtue of their opposing charges, separating the ions requires a certain amount of energy. This energy must be given to the system in order to break the anion-cation bonds. The energy required to break these bonds for one mole of an ionic solid under standard conditions is the lattice energy.

The lattice energy describes the amount of energy for a whole crystal but it doesn't describe the energy contribution of a single ion in the crystal. The energy of a specific ion would depend on how close it was to other ions, how many had the same charge, and how many had the opposite charge.

Since charges in ions are based on the electric energy of an electron, we can use the energy of the electron and then modify it to see how close the other electrons are.


 * $$E = -\frac{z^2 e^2 M}{4 \pi \epsilon_o r_0}$$

where
 * $$\, z$$ = charge of ions
 * $$\,e$$ = 1.6022 C
 * $$\,4\pi \epsilon_o$$ = 1.112 C²/(J m)
 * $$\, M$$ = Madelung's constant, which is representative of the distance of the ions from one another in a specific type of crystal.

The Madelung's constant depends on the geometric arrangement of the constituent ions in the crystal structure.


 * $$M = \sum_{i} (\pm)_{i} \frac{r_0}{r_i}$$

where
 * $$\, r_0$$ = distance to closest ion.

For example, for the ionic crystal NaCl:


 * $$M = \sum_{i,j,k=-\infty}^{\infty}\prime {{(-1)^{i+j+k}} \over { (i^2 + j^2 + k^2)^{1/2}}} $$,

the prime indicating that the term $$i=j=k=0$$ is to be left out. Since this sum is conditionally convergent it is not suitable as definition of Madelung's constant unless the order of summation is also specified. There are two "obvious" methods of summing this series, by expanding cubes or expanding spheres. The latter, though devoid of a meaningful physical interpretation (there are no spherical crystals) is rather popular because of its simplicity. Thus, the following expansion is often found in the literature:


 * $$M = -6 +12/ \sqrt{2} -8/ \sqrt{3} +6/2 - 24/ \sqrt{5} + \dotsb = -1.74756\dots$$

However, this is wrong as this series diverges as was shown by Emersleben in 1951. The summation over expanding cubes converges to the correct value. An unambiguous mathematical definition is given by Borwein, Borwein and Taylor by means of analytic continuation of an absolutely convergent series.

There are many practical methods for calculating Madelung's constant using either direct summation (for example, the Evjen method ) or integral transforms, which are used in the Ewald method.

Links

 * Mathworld article
 * Decimal expansion of Madelung constant (negated) for simple cubic lattice