Madelung constant

In crystallography , the Madelung constant (after Erwin Madelung , who first derived it) is a unitless factor , which is defined as: ${\displaystyle \alpha \,\!}$

${\displaystyle \alpha ={\frac {E_{IG}}{E_{IP}}}}$

With

• ${\displaystyle E_{IG}}$, the average binding energy per ion in the crystal lattice and
• ${\displaystyle E_{IP}}$, the average binding energy per ion for a single ion pair.

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

Typical crystal structures to which the Madelung constant can be applied are the alkali halides , in which the bond is created by Coulomb forces . The alkali atom donates an electron to the halogen atom , and a spherically symmetrical charge distribution is created on each atom .

Because the Madelung constant is derived from Coulomb's law for point charges , it loses its validity for non-point ions (ions with covalent bonds such as in pyrite crystals) and for ions with different polarity (e.g. in the Series ZnS, TiO ₂ , CdCl ₂ , CdI ₂ ).

Calculation of the binding energy in the lattice

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

${\displaystyle E_{IP}=\int _{\infty }^{d}F\,\mathrm {d} r=\int _{\infty }^{d}{\frac {1}{4\pi \varepsilon _{0}}}\cdot {\frac {z^{2}e^{2}}{r^{2}}}\,\mathrm {d} r=-{\frac {1}{4\pi \varepsilon _{0}}}\cdot {\frac {z^{2}e^{2}}{d}}}$

With

z - number of charges of the ions

e - elementary charge

d - (smallest) distance between the ions

ε ₀ - dielectric constant of the vacuum

Since there is not only one pair of ions in a crystal lattice, but also other cations and anions in space, further energy is released during crystal formation, but it is also required again to approach ions with the same charge. The following equation should explain this:

${\displaystyle E_{IG}=n_{1}\cdot {\frac {1}{c_{1}}}\cdot E_{IP}-n_{2}\cdot {\frac {1}{c_{2}}}\cdot E_{IP}+n_{3}\cdot {\frac {1}{c_{3}}}\cdot E_{IP}-\ldots }$

The energy E _IG stored in the grid is the sum of the energies released and required for each ion during the grid formation. Here n is the number of times a certain ion occurs and c is a factor that indicates the distance between the ion. These factors can be combined to form a crystal-dependent factor α - the Madelung constant - so that the following equation results for the binding energy of an ion in the lattice:

${\displaystyle E_{IG}=\alpha \cdot E_{IP}=-\alpha {\frac {1}{4\pi \varepsilon _{0}}}\cdot {\frac {z^{2}e^{2}}{d}}}$

This equation describes the binding energy of only one ion in the lattice. To get the energy that is released when a certain amount of substance is formed, this equation has to be multiplied by Avogadro's constant and the amount of substance : ${\displaystyle n}$ ${\displaystyle N_{\mathrm {A} }}$ ${\displaystyle n}$

${\displaystyle E_{G}=-\alpha \cdot {\frac {N_{\mathrm {A} }}{4\pi \varepsilon _{0}}}\cdot {\frac {z^{2}\cdot e^{2}\cdot n}{d}}}$

As can be seen, this value is negative because the lattice formation is exothermic . For a more precise calculation of lattice energies, it is not sufficient to just consider regularly arranged Coulomb point charges. An extension of the model leads to the Born-Landé equation .

Calculation of the Madelung constant using the example of NaCl

The ion lattice of NaCl is a face-centered cubic crystal structure as shown on the right (the anions are red and the cations are green). The distance between the two ions in NaCl is approximately d = 0.3 nm.

From the above equation, the number of the respective neighboring ions n , their relative distance c as a multiple of distance d and their charge, i.e. whether they attract or repel each other, must now be determined.

When we talk about an ion, e.g. For example, if we run out of the number 0 shown in red, we first have n = 6 ions (shown in green with the number 1) at a distance of 1d that are attracted. This is followed by 12 red ions (2), the distance between which gives the value using the Pythagorean theorem and which are repelled by the ion of the same type, and 8 green ions at a distance of which are attracted. The following table continues these numbers: ${\displaystyle {\sqrt {2}}d}$ ${\displaystyle {\sqrt {3}}d}$

No.	n	c	charge
1	6th	${\displaystyle {\sqrt {1}}=1}$	+
2	12	${\displaystyle {\sqrt {2}}}$	-
3	8th	${\displaystyle {\sqrt {3}}}$	+
4th	6th	${\displaystyle {\sqrt {4}}=2}$	-
5	24	${\displaystyle {\sqrt {5}}}$	+
...	...	...	...

If one continues this procedure, one arrives at the following series representation of the Madelung constant , which is often found in the literature :

${\displaystyle \alpha =\sum _{k=1}^{\infty }(-1)^{k+1}{\frac {n_{k}}{c_{k}}}={\frac {6}{1}}-{\frac {12}{\sqrt {2}}}+{\frac {8}{\sqrt {3}}}-{\frac {6}{2}}+{\frac {24}{\sqrt {5}}}\pm \ldots \approx 1{,}7476}$

However, this is wrong because this series diverges, as was only proven in 1951. The sum over the points of the crystal lattice is conditionally convergent, i.e. it depends on the order of the summands. The above series would correspond to a summation over concentric spherical shells, which is also not physically meaningful. The "correct" value is obtained by adding up over the grid points within a cube with edge length and forming the limit value for . A mathematical justification that this value is the "correct" one was given in 1985 by David Borwein, Jonathan Borwein and Keith F. Taylor: They define a function of a complex variable : ${\displaystyle 2n+1}$ ${\displaystyle n\to \infty }$ ${\displaystyle s}$

${\displaystyle b(2s)=\sideset {}{^{~\prime }}\sum _{i,j,k=-\infty }^{\infty }{\frac {(-1)^{i+j+k}}{(i^{2}+j^{2}+k^{2})^{s}}}}$,

where the dash means that the term is to be omitted. For a sufficiently large real part of the series is absolutely convergent, the Madelung constant ( ) results from analytical continuation . ${\displaystyle i=j=k=0}$ ${\displaystyle s}$ ${\displaystyle s=1/2}$

The summation over cubes converges so slowly that it is useless for practical calculations. With a little trick, however, much better approximations can be found: It is totaled over all points within a cube with edge length , the points on the sides are only counted half, those at the edges to a quarter and those at the corners to an eighth. This procedure is known as the Evjen method. Already provides a very good approximation with 1.7470; = 1.74750 and = 1.7475686. The physical motivation of this procedure results from the requirement that the partial sums should be formed over an electrically neutral (finite) crystal. ${\displaystyle \alpha _{n}}$ ${\displaystyle 2n+1}$ ${\displaystyle \alpha _{3}}$ ${\displaystyle \alpha _{5}}$ ${\displaystyle \alpha _{10}}$ ${\displaystyle \alpha _{n}}$

Even before Evjen, Paul Peter Ewald published what is now known as the Ewald method for calculating the Madelung constant. His method is a special case of Poisson's molecular formula , which Ewald was not aware of.

Today numerous numerical methods and powerful computers are available, so that the calculation of the Madelung constants for any grid with high accuracy is no longer a problem.

Values ​​for some crystal structures

Structure type	Madelung constant
NaCl (1)	1.747564594633 ...
CsCl (2)	1.762675 ...
Zinc cover (3)	1.633806 ...
Titanium (IV) oxide	2.40 ...
Calcium fluoride	2.52 ...
Copper (II) oxide	4.12 ...
sapphire	4.17 ...

Remarks:

⁽¹⁾ face-centered cubic

⁽²⁾ cubic primitive

⁽³⁾ face-centered cubic with two atoms per unit cell

Individual evidence

1. ↑ E. Madelung: Phys. Zs. 19, (1918), p. 542.
2. ↑ Charles Kittel: Introduction to Solid State Physics. Oldenbourg, Munich and Vienna 1999, ISBN 3-486-23843-4 .
3. ^ O. Emersleben: Math. Nachr. 4 (1951), p. 468.
4. ^ ^{A b} D. Borwein, JM Borwein, KF Taylor: Convergence of Lattice Sums and Madelung's Constant. In: J. Math. Phys. 26 (1985), pp. 2999-3009, doi: 10.1063 / 1.526675 .
5. ↑ HM Evjen: On the Stability of Certain Straight Polar Crystals. In: Phys. Rev. 39 (1932), pp. 675-687, doi: 10.1103 / PhysRev.39.675
6. ↑ PP Ewald: The calculation of optical and electrostatic grid potentials. In: Ann. Phys. 64 (1921), pp. 253-287, doi: 10.1002 / andp.19213690304
7. ^ Rudolf Gross, Achim Marx: Solid State Physics . 1st edition. Oldenbourg Verlag, Munich 2012, ISBN 978-3-486-71294-0 , p. 119 .