2.3 Microwave Spectrum: Dielectric Relaxation
59
Fig. 2.3 The microwave
part of the real, , and
imaginary, , parts of the
dielectric function of water at
different temperatures from
5 to 90 ◦ C (see numbers near
the curves). Graphs for
temperatures above 5 ◦ C are
shifted up for clarity. Dots
are experimental data points,
lines are model (2.25). The
dielectric relaxation band
shows a blue shift with an
increase in temperature.
Collated on the data from [6]
10
9
10
10
10
11
10
12
Frequency (Hz)
ε'
ε''
0
80
60
40
20
0
40
30
20
10
5
25
20
15
10
30
70
50
40
35
90
T (°C)
where ∞ is the high-frequency limit of the dielectric constant, τ D is the relaxation
time, and = (0)– ∞ , where (0) is the static dielectric constant of water.
4 Dividing
the real and imaginary parts, one gets
ε
(ω) = ε ∞ +
1 + ω 2 τ
2
D
;
(2.23)
ε
(ω) =
· ωτ D
1 + ω 2 τ
2
D
.
(2.24)
Formula (2.22) fits the experimental data for water up to 100 GHz, but at higher
frequencies, the experimental dielectric loses exceed those predicted by the Debye
equation, demonstrating an additional contribution. This high-frequency wing of
Debye relaxation is discussed in Sect. 2.6.
Figure 2.3 shows the temperature dependence of the dielectric relaxation mode.
As the temperature increases, the relaxation band shifts in the direction of higher
frequencies (blue shift). The band intensity and the corresponding dielectric constant,
(0), decrease with an increase in temperature, showing the negative temperature
trend of the dielectric constant (see Fig. 2.12).
4 Equation (2.22) has an important consequence. It connects the dielectric relaxation band with the
static dielectric constant, which means that they both have the same microscopic nature discussed
in Sect. 4.5.