In an ionic crystal lattice each cation will be attracted by anions next to it and repulsed by cations near it. Consequently the coulomb potential leading to the lattice energy depends on the type of crystal. To get the total lattice energy you must sum all of the electrostatic interactions on a given ion. The general form of the electrostatic potential is $$V=\frac{Q_{1} Q_{2} e^{2}}{d_{12}}$$ where \(Q_{1}\) and \(Q_{2}\) are the charges on ions 1 and \(2, d_{12}\) is the distance between them in the crystal lattice. and \(e\) is the charge on the electron. (a) Consider the linear "crystal" shown below. The distance between the centers of adjacent spheres is \(R .\) Assume that the blue sphere and the green spheres are cations and that the red spheres are anions. Show that the total electrostatic energy is $$V=-\frac{Q^{2} e^{2}}{d} \times \ln 2$$ (b) In general, the electrostatic potential in a crystal can be written as $$V=-k_{M} \frac{Q^{2} e^{2}}{R}$$ where \(k_{M}\) is a geometric constant, called the Madelung constant, for a particular crystal system under consideration. Now consider the NaCl crystal structure and let \(R\) be the distance between the centers of sodium and chloride ions. Show that by considering three layers of nearest neighbors to a central chloride ion, \(k_{M}\) is given by $$k_{M}=\left(6-\frac{12}{\sqrt{2}}+\frac{8}{\sqrt{3}}-\frac{6}{\sqrt{4}} \cdots\right)$$ (c) Carry out the same calculation for the CsCl structure. Are the Madelung constants the same?

Step by step solution

01

Linear Electrostatic Energy

To find the total harmonic potential \(V\), consider an arbitrary blue cation in the crystal and the contribution from all other ions. The ions on the same side contribute with the same sign while those on the opposite side contribute with the opposite sign. Therefore, let's assume that in a 1D crystal we sum from the center of blue cation and get for the electrostatic potential: \(V= \frac{Q_{1}Q_{2}e^{2}}{R} - \frac{2Q_{1}Q_{2}e^{2}}{2R} + \frac{Q_{1}Q_{2}e^{2}}{3R}\) ... This is a converging series that sums up to \(\frac{-Q_{1}Q_{2}e^{2}}{R}\ln 2\) as stated in the problem

02

Electrostatic Potential with Madelung Constant

The general form of the electrostatic potential in a 3D crystal structure includes the Madelung constant \(k_{M}\): \(V=-k_{M}\frac{Q^{2}e^{2}}{R}\). Here considering the NaCl structure, the chloride ion is surrounded by 6 sodium ions at distance R, 12 chloride ions at distance \(\sqrt{2}R\), 8 sodium ions at distance \(\sqrt{3}R\) and so on. Hence the Madelung constant \(k_{M}\) is: \(k_{M} = 6 -\frac{12}{\sqrt{2}} + \frac{8}{\sqrt{3}} - \frac{6}{\sqrt{4}} + ...\)

03

Electrostatic Potential for the CsCl structure

The same calculations are to be made for CsCl structure. However, since geometrical structure of CsCl is different than that of NaCl, the Madelung constant \(k_{M}\) could differ for these two structures.

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