bei48482_FM

of the Naion due to these six Clions is therefore

U 1 

The next nearest neighbors are 12 Naions, each one the distance  2 raway since

the diagonal of a square rlong on a side is  2 r. The potential energy of the Naion

due to the 12 Naions is

U 2 

When the summation is continued over all theandions in a crystal of infinite

size, the result is

Ucoulomb  6 .. .1.748

or, in general,

Coulomb energy Ucoulomb (10.1)

This result holds for the potential energy of a Clion as well, of course.

The quantity is called the Madelung constantof the crystal, and it has the same

value for all crystals of the same structure. Similar calculations for other crystal varieties

yield different Madelung constants. Crystals whose structures are like that of cesium

chloride (Fig. 10.6), for instance, have 1.763. Simple crystal structures have

Madelung constants that lie between 1.6 and 1.8.

The potential energy contribution of the repulsive forces due to the action of the

exclusion principle has the approximate form

Repulsive energy Urepulsive (10.2)

The sign of Urepulsiveis positive, which corresponds to a repulsion. The dependence

on rnimplies a short-range force that increases as the interionic distance rdecreases.

The total potential energy of each ion due to its interactions with all the other ions is

therefore

UtotalUcoulombUrepulsive (10.3)

How can we find the value of B? At the equilibrium separation r 0 of the ions, Uis

a minimum by definition, and so dUdr 0 when r r 0. Hence

rr 0 ^0

nB



r 0 n^1

e^2



4  0 r^20

dU



dr

B



rn

e^2



4  0 r

B



rn

e^2



4  0 r

e^2



4  0 r

12



 2 

e^2



4  0 r

12 e^2



4  0  2  r

6 e^2



4  0 r

340 Chapter Ten

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