Elements of Modern Physics
308
8.8 EXAMPLES
Here, some examples that illustrate and extend the ideas about the solid state
are considered.
Example 1
The bulk modulus of a crystalline solid can be estimated from Eq. (8.5).
A pressure P produces a decrease in length, ∆l,
P = C l 0 ∆l
(8.141)
where C is a constant, so that the work done is
W = 3CV 0
0
x dx
∆
∫
=
2
0
3
( )
2
C V
l
∆
(8.142)
where V 0 is the volume. This causes a change in energy given by
∆E = N[E(R 0 – ∆R) – E(R 0 )]
≈
0
2
2
2
( )
2
=
∂
∆
∂
R R
N E R
R
(8.143)
N being the number of ion pairs, and where the fact that E is a minimum
at R 0 has been used. Equating W and ∆E gives
C =
0
2 2
0
2
0
0
3
R R
R
N
E
V l
R =
  ∂
  ∂
 
(8.144)
Therefore, the bulk modulus is
K =
0
2
2
0
2
0
0
1
/
9
R R
P
N
E
R
V V
V
R =
 
∂
−
=  
∆
∂
 
(8.145)
where N/V 0 is the number of ion pairs/unit volume, and E is given in Eq. (8.3)
with n ≈ 10. For NaCl, the above expression gives an estimate of about
3.5 × 10
10
J/m
3
(experimentally it is about 3 × 10
10
J/m
3
).
Example 2
It can be shown that when a, b and c are mutually orthogonal, the distance
between the planes with Miller indices (h, k, l) is given by