Madelung constant

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. [E. Madelung, "Phys. Zs." 19, (1918), 542]

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 = 1.6022e|−19 C:,4pi epsilon_o = 1.112e|−10 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: [Charles Kittel: "Introduction to Solid State Physics.", Wiley 1995, ISBN 0471111813]

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

However, this is wrong as this series diverges as was shown by Emersleben in 1951. [O. Emersleben: Math. Nachr 4 (1951), 468] [D. Borwein, J. M. Borwein, K. F. Taylor: Convergence of Lattice Sums and Madelung's Constant, "J. Math. Phys." 26 (1985), 2999-3009, DOI|10.1063/1.526675] 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 [H. M. Evjen: On the Stability of Certain Heteropolar Crystals, "Phys. Rev." 39 (1932), 675–687, http://link.aps.org/abstract/PR/v39/p675] ) or integral transforms, which are used in the Ewald method. [P. P. Ewald: Die Berechnung optischer und elektrostatischer Gitterpotentiale, "Ann. Phys." 64 (1921), 253-287, DOI|10.1002/andp.19213690304]

Links

* [http://mathworld.wolfram.com/MadelungConstants.html Mathworld article]
* [http://www.research.att.com/~njas/sequences/A085469 Decimal expansion of Madelung constant (negated) for simple cubic lattice]

References


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  • Madelung constant — noun a numerical constant used in determining the electrostatic energy of each ion in a crystal …   Wiktionary

  • Madelung — may refer to:*Erwin Madelung (1881 1972) German physicist *Otto Wilhelm Madelung (1846 1926) German surgeon *Wilferd Madelung (1930 ) scholar of Islam *Madelung constant chemical energy of an ion in a crystal …   Wikipedia

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  • Otto Wilhelm Madelung — (May 15, 1846, Gotha July 22, 1926, Göttingen) was a German surgeon who was a native of Gotha. In 1869 he received his medical doctorate from the University of Tübingen, and later worked in a military hospital during the Franco Prussian War.… …   Wikipedia

  • Erwin Madelung — (May 18, 1881 – August 1, 1972) was a German physicist. He was born in 1881 in Frankfurt/Main. His father was the surgeon Otto Wilhelm Madelung.He earned a doctorate in 1905 from the University of Goettingen, specializing in crystal structure,… …   Wikipedia

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