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5 Molecular Theory of Local Field
5.4 Solutions to Problems
5.4.1 Interfacial Dielectric Constant
[Problem 5.1] Let us derive ε in Eq. (5.33) on the basis of the Lorentz model
illustrated in Fig. 5.1a–d.
(a, b) Bulk First, suppose a spherical cavity of radius r embedded in the bulk of a
slab with a dielectric constant ε, and put the system in an electric field E ext
x or E ext
z .
Show that the local fields inside the cavity in the two cases are
E
loc
x = f x E
ext
x =
ε + 2
3
E
ext
x
and E
loc
z = f z E
ext
z =
ε + 2
3ε
E
ext
z .
(5.32)
(c, d) Surface Next, suppose a hemisphere cavity at the interface. Show that the
local fields E loc
x and E loc
z at the center of the hemisphere in the two cases are
E
loc
x = f
surf
x E
ext
x =
ε + 5
6
E
ext
x
and E
loc
z = f
surf
z E
ext
z =
2ε + 1
3ε
E
ext
z .
(5.34)
Take the ratio of the local field factors to derive ε = f surf
x /f surf
z .
Here we discuss the relation between the local field and the dielectric constant.
The relation is distinct in the bulk and at the interface, and the difference could be
used to define the dielectric constant at the interface. The local field inside the cavity
is evaluated by the Lorentz model [3].
(1) Bulk In case (a),
(a)
E
loc
x = E
ext
x +
sphere
dσ
P x cos θ
r 2
cos θ = E
ext
x +
4π
3
P x
=
ε + 2
3
E
ext
x = f x E
ext
x ,
(5.36)
where σ denotes the surface of the cavity, and the polarization P x =
ε − 1
4π
E
ext
x .
In case (b),
(b)
E
loc
z = E
ext
z +
sphere
dσ
P z cos θ
r 2
cos θ − 4πP z = E
ext
z −
8π
3
P z