Page 251 - Electrical Properties of Materials

The Debye equation 233

So by eqns (10.24) and (10.25), The LHS is the quantity defined
as the dipole moment, μ,in
3
Zed =4πr 0 E . (10.26) eqn (10.6).
By Ze = q and μ = qd, hence eqn (10.26) may also be written as
3
μ =4πr 0 E (10.27)
We can now return to eqn (10.7), P = N m μ, to find the induced polarization.
The density of atoms per unit volume is
1 1
N m = = , (10.28)
a 3 (2r) 3
leading to
1 3 π
P = 4πr 0 E = 0 E . (10.29)
(2r) 3 2
From eqns (10.4) and (10.5) it follows then that
P π
χ = = , (10.30)
0 E 2
whence
~
r =1 + χ = 2.57, (10.31)
and
√
~
n = r = 1.6. (10.32)
Thus, our very approximate estimate is at the high end of our small sample
in Table 10.1. We could refine this model to give a different fit by remarking
3
that less close packing would give N m <1/(2r) , and we could also take into
account quantum orbits. This would change things slightly at the expense of
considerable calculations. But the main point of this aside is that a simple ap-
proach can sometimes give a reasonable answer and at the same time enhance
insight into the phenomenon.

10.9 The Debye equation

We have seen that frequency variation of relative permittivity is a complicated
∗
affair. There is one powerful generalization due to Debye of how materials ∗ Nobel Prize in Chemistry 1946
with orientational polarizability behave in the region where the dielectric po-
larization is ‘relaxing’, that is the period of the a.c. wave is comparable to the
alignment time of the molecule. When the applied frequency is much greater
than the reciprocal of the alignment time, we shall call the relative dielectric
constant ∞ (representing atomic and electronic polarization). For much lower
frequencies it becomes s , the static relative dielectric constant. We need to
find an expression of the form
(ω)= ∞ + f (ω), (10.33)
for which ω → 0 reduces to
f (0) = s – ∞ . (10.34)