There occur as many Madelung constants M _ i in a crystal structure as ions occupy different lattice sites.
3.
These second order Madelung constants turned out to have significant effects on the lattice energy and other physical properties of heteropolar crystals.
4.
For example, for the ionic crystal NaCl, there arise two Madelung constants & ndash; one for Na and another for Cl.
5.
The nearest neighbour distance amounts to half the lattice parameter of the cubic unit cell r _ 0 = a / 2 and the Madelung constants become
6.
This graph demonstrates the non-convergence of the expanding spheres method for calculating the Madelung constant for NaCl as compared to the expanding cubes method, which is convergent.
7.
The polarisation can be calculated on the basis of higher-order Madelung constants and has to be included in the calculation of the lattice energy by using a generalised Born Haber cycle.
8.
The Ewald summation was developed by Paul Peter Ewald in 1921 ( see References below ) to determine the electrostatic energy ( and, hence, the Madelung constant ) of ionic crystals.
9.
The electrostatic interaction model of ions in solids has thus been extended to a point multipole concept that also includes higher multipole moments like dipoles, quadrupoles etc . These concepts require the determination of higher order Madelung constants or so-called electrostatic lattice constants.
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