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5 Solid State Physics
5.9 The lattice of the NaCl structure which is face-centered is represented in
Fig. 5.3. The shortest inter ionic distance is represented by r . A given sodium
ion is surrounded by 6Cl
− ions at a distance r, 12Na
+ ions at a distance
r
√
2, 8Cl
− ions at a distance r
√
3, 6Na
+ ions at a distance r
√
4, 24Cl
−
ions at a distance r
√
5, etc. The coulomb energy of of this ion in the field of
all other ions is therefore
E c = −
e
2
r
6
√
1
−
12
√
2
+
8
√
3
−
6
√
4
+
24
√
5
− · · ·
where e is the charge per ion. Series of this sort which consists of pure numbers
depends on the crystal structure and is known as the Madelung constant.
Fig. 5.3 The lattice of the
NaCl structure
5.10 E r = −
a
r
+
b
r 7
(a) For equilibrium, E r must be minimum, so that
dE
dr
= 0
a
r 2 −
7b
r 8 = 0
Or, r = r o = (7b/a)
1/6
(b) Energy of attraction, E A =
−a
r o
Energy of repulsion, E R =
b
r
7
0
|E A |
|E R |
=
a
b
r
6
o =
a
b
·
7b
a
= 7
Force, F = −
dV
dr
Attractive force F A =
a
r 2
o
(r = r o )
Repulsive force F R =
−7b
r 8
o
=
−7b
r 2
o
·
a
7b
= −
a
r 2
o
Thus the two forces are equal in magnitude.