Solid State Physics
305
radiation. Actually the singularity at ω = ω j0 is displaced by the fact that the
state j is an unstable state which essentially requires a replacement of
(ω j0 – ω)
–1
by the real part of (ω j0 – ω – i/2 τ j )
–1
, i.e.,
0
1
ω + ω
j
→
0
2
2
0
(
) (2 )
−
ω − ω
ω − ω + τ
j
j
j
(8.130)
where τ j is the lifetime of state j [see Eq. (6.76)]. With this modification,
Eq. (8.129) provides a qualitative explanation of the polarizability illustrated in
Fig. 8.17(a). It may also be noted that when ω j0 ≈ ω, there is a significant
probability for transition to state j, as seen from the expression in Eq. (6.55) for
a j , which leads to absorption of radiation [Fig. 8.17(b)]. While Eq. (8.129) is
valid for a general charged system, it is practically useful mainly for electronic
polarizability for which z j0 can be calculated with some reliability. A rough
order of magnitude estimation for this electronic polarizability gives
α =
19 2
10 2
19
(1.6 10 ) (10 )
(1.6 10 )
−
−
−
×
×
≈ 1.6 × 10
–39
F.m
2
(8.131)
Ionic Polarizability
The ionic polarizability is due to the displacement of ions with respect to each
other. If it is assumed that the forces near equilibrium are simple harmonic, the
displacement in the presence of an electric field is given by
k ∆ x ≈ eE
(8.132)
where k is the force constant. This leads to a polarizability
α ≈ e
2
/k
(8.133)
Since k ≈ 20 N/m, α ionic ≈ 10
–39
F.m
2
The ionic contribution is important at low frequencies (ω j0 in Eq. (8.129) is
small). This explains the fact that NaCl has ε ≈ 5.6 at low frequencies whereas
at optical frequencies ε ≈ 2.25. The difference may be ascribed to the ionic
contribution to polarizability (see Fig. 8.17).
Orientational Polarizability
Molecules with permanent electric dipole moment align themselves in the
presence of an external electric field giving rise to an orientational polarizability.
The energy of a dipole p in an electron field E is
V = – p E cos θ
(8.134)
where θ is the angle between the dipole and the field. Therefore, the average
dipole moment (using Boltzmann distribution) is
305
radiation. Actually the singularity at ω = ω j0 is displaced by the fact that the
state j is an unstable state which essentially requires a replacement of
(ω j0 – ω)
–1
by the real part of (ω j0 – ω – i/2 τ j )
–1
, i.e.,
0
1
ω + ω
j
→
0
2
2
0
(
) (2 )
−
ω − ω
ω − ω + τ
j
j
j
(8.130)
where τ j is the lifetime of state j [see Eq. (6.76)]. With this modification,
Eq. (8.129) provides a qualitative explanation of the polarizability illustrated in
Fig. 8.17(a). It may also be noted that when ω j0 ≈ ω, there is a significant
probability for transition to state j, as seen from the expression in Eq. (6.55) for
a j , which leads to absorption of radiation [Fig. 8.17(b)]. While Eq. (8.129) is
valid for a general charged system, it is practically useful mainly for electronic
polarizability for which z j0 can be calculated with some reliability. A rough
order of magnitude estimation for this electronic polarizability gives
α =
19 2
10 2
19
(1.6 10 ) (10 )
(1.6 10 )
−
−
−
×
×
≈ 1.6 × 10
–39
F.m
2
(8.131)
Ionic Polarizability
The ionic polarizability is due to the displacement of ions with respect to each
other. If it is assumed that the forces near equilibrium are simple harmonic, the
displacement in the presence of an electric field is given by
k ∆ x ≈ eE
(8.132)
where k is the force constant. This leads to a polarizability
α ≈ e
2
/k
(8.133)
Since k ≈ 20 N/m, α ionic ≈ 10
–39
F.m
2
The ionic contribution is important at low frequencies (ω j0 in Eq. (8.129) is
small). This explains the fact that NaCl has ε ≈ 5.6 at low frequencies whereas
at optical frequencies ε ≈ 2.25. The difference may be ascribed to the ionic
contribution to polarizability (see Fig. 8.17).
Orientational Polarizability
Molecules with permanent electric dipole moment align themselves in the
presence of an external electric field giving rise to an orientational polarizability.
The energy of a dipole p in an electron field E is
V = – p E cos θ
(8.134)
where θ is the angle between the dipole and the field. Therefore, the average
dipole moment (using Boltzmann distribution) is