4.5 The Microscopic Origin of the Electrodynamic Properties of Water and Ice
161
Fig. 4.12 a The equivalent
scheme for (4.20) explaining
the mechanics of the
oscillation of ionic species in
water in a soft surrounding
with friction and moving
suspension points. b The
corresponding potential
energy of the ionic species
(solid line) and the harmonic
parabolic potential
approximation (dashed line)
κ=mω 0
2
γ
Equilibrium
position (x 0 )
exp(-iωt)
x
x
U
(a)
(b)
Suspension point
m
½κx
2
x 0
and the real part of dynamical conductivity:
σ (ω) =
n ± q
2
γ 0
m
ω
2
τ r a + b
ω 2 a 2 + b 2 ,
(4.24)
where a = γ 0 (1 + γ 0 τ r ), and b(ω) = γ 0 τ r (ω
2
0 − ω
2
) + γ
2
0 − ω
2
0 . Note that (4.24)
satisfies the sum rule 2.61 (see Sect. 2.8).
Figure 4.13 shows the spectra calculated by (4.23) and (4.24) with different
parameters. In the limiting case, τ r = 0 (free particle) and ω 0 = 0 (oscillator with
a zero eigenfrequency), we have a frequency-independent conductivity plateau in
all spectral regions. In another limiting case, τ r → ∞ (strongly localized charge),
the function (4.24) coincides with the Lorentzian oscillator (see curve 7). This case
corresponds to a parabolic potential with infinite walls (Fig. 4.12b). In intermediate
cases, the solution combines the properties of an overdemped oscillator with static
conductivity in the low-frequency region. As the residence time τ r increases, the lowfrequency conductivity decreases, and the peak at ω 0 becomes more pronounced (see
Fig. 4.13).
In the case of large damping γ 0 ω 0 , (4.24) gives the curve with a widened
oscillatory peak (see curve 1 in Fig. 4.13). The function has two plateaus, highfrequency and low frequency, which correspond to the quasi-free motion and the
hindered motion, respectively. Figure 4.13 shows that at frequencies ω ≈ ω 0 , the
charge looks more like a harmonic oscillator at the bottom of the parabolic potential,
but at low frequencies, ω ω 0 , it overcomes the potential barrier (see Fig. 4.12b)
and shows a conductivity plateau:
161
Fig. 4.12 a The equivalent
scheme for (4.20) explaining
the mechanics of the
oscillation of ionic species in
water in a soft surrounding
with friction and moving
suspension points. b The
corresponding potential
energy of the ionic species
(solid line) and the harmonic
parabolic potential
approximation (dashed line)
κ=mω 0
2
γ
Equilibrium
position (x 0 )
exp(-iωt)
x
x
U
(a)
(b)
Suspension point
m
½κx
2
x 0
and the real part of dynamical conductivity:
σ (ω) =
n ± q
2
γ 0
m
ω
2
τ r a + b
ω 2 a 2 + b 2 ,
(4.24)
where a = γ 0 (1 + γ 0 τ r ), and b(ω) = γ 0 τ r (ω
2
0 − ω
2
) + γ
2
0 − ω
2
0 . Note that (4.24)
satisfies the sum rule 2.61 (see Sect. 2.8).
Figure 4.13 shows the spectra calculated by (4.23) and (4.24) with different
parameters. In the limiting case, τ r = 0 (free particle) and ω 0 = 0 (oscillator with
a zero eigenfrequency), we have a frequency-independent conductivity plateau in
all spectral regions. In another limiting case, τ r → ∞ (strongly localized charge),
the function (4.24) coincides with the Lorentzian oscillator (see curve 7). This case
corresponds to a parabolic potential with infinite walls (Fig. 4.12b). In intermediate
cases, the solution combines the properties of an overdemped oscillator with static
conductivity in the low-frequency region. As the residence time τ r increases, the lowfrequency conductivity decreases, and the peak at ω 0 becomes more pronounced (see
Fig. 4.13).
In the case of large damping γ 0 ω 0 , (4.24) gives the curve with a widened
oscillatory peak (see curve 1 in Fig. 4.13). The function has two plateaus, highfrequency and low frequency, which correspond to the quasi-free motion and the
hindered motion, respectively. Figure 4.13 shows that at frequencies ω ≈ ω 0 , the
charge looks more like a harmonic oscillator at the bottom of the parabolic potential,
but at low frequencies, ω ω 0 , it overcomes the potential barrier (see Fig. 4.12b)
and shows a conductivity plateau: