4.5 The Microscopic Origin of the Electrodynamic Properties of Water and Ice
153
two fragments. The first fragment is “in-cage” dynamics with the mobility μ ac , which
manifests at high frequencies. The corresponding AC conductivity σ ac is equivalent
to that observed in part (a). The second fragment is the low-frequency dynamics
with effective mobility, μ dc , reduced by the interaction of the ionic species with the
stationary lattice. The corresponding low-frequency conductivity σ dc is lower than
the high-frequency conductivity.
Finally, part (c) represents the case of water, in which the ionic species obey
ambipolar diffusion. As in the previous case, there are two characteristic mobilities.
The high-frequency mobility μ ac corresponds to the dynamics inside the ionic atmosphere (see dashed circles), while the low-frequency mobility, μ dc , is a result of the
dynamics of the same particle slowed-down by the ionic atmosphere. Note that the
transition regions between the high- and the low-frequency plateaus in parts (b) and
(c) are different. For the stationary lattice the conductivity σ (ν) is proportional to ν
1
(see Fig. 4.8), while ambipolar diffusion gives σ (ν) ∼ ν
2 . The latter is exactly the
same as for the Debye relaxation (see Sect. 2.3).
Ambipolar diffusion is described by the system of equations:
mγ 1 ˙
x = κ(x − X ) − f 1 (t),
Mγ 2 ˙
X = κ(X − x) − f 2 (t),
(4.7)
where m and M, x and X are the masses and coordinates of the central ion and its ionic
atmosphere, respectively, γ is a damping, and f 1 and f 2 are stochastic forces. The
first term on the right-hand side represents the electrostatic coupling with the elastic
spring constant κ, which connects the ion with the center of its ionic atmosphere,
both of which obey Brownian motion. System (4.7) corresponds to the simplified
case considered in Sect. 3.5.
The dynamic conductivity in the whole frequency range can be found from 4.7
by the Fourier transform of the velocity–velocity correlation function:
σ (ω) =
q
2 n ±
kT
∞
0
˙
x(0) ˙
x(t)e
−iωt dt,
(4.8)
which gives for the frequency-dependent dynamic conductivity:
σ (ω) =
σ dc + ω
2
τ
2
D1 σ ac
1 + ω
2
τ
2
D1
,
(4.9)
where σ dc = qn ± μ dc , σ ac = qn ± μ ac , and the relaxation time τ D1 is
τ D1 =
γ 1 γ 2 m M
(mγ 1 + Mγ 2 )κ
,
(4.10)
and the low- and high-frequency mobilities are
153
two fragments. The first fragment is “in-cage” dynamics with the mobility μ ac , which
manifests at high frequencies. The corresponding AC conductivity σ ac is equivalent
to that observed in part (a). The second fragment is the low-frequency dynamics
with effective mobility, μ dc , reduced by the interaction of the ionic species with the
stationary lattice. The corresponding low-frequency conductivity σ dc is lower than
the high-frequency conductivity.
Finally, part (c) represents the case of water, in which the ionic species obey
ambipolar diffusion. As in the previous case, there are two characteristic mobilities.
The high-frequency mobility μ ac corresponds to the dynamics inside the ionic atmosphere (see dashed circles), while the low-frequency mobility, μ dc , is a result of the
dynamics of the same particle slowed-down by the ionic atmosphere. Note that the
transition regions between the high- and the low-frequency plateaus in parts (b) and
(c) are different. For the stationary lattice the conductivity σ (ν) is proportional to ν
1
(see Fig. 4.8), while ambipolar diffusion gives σ (ν) ∼ ν
2 . The latter is exactly the
same as for the Debye relaxation (see Sect. 2.3).
Ambipolar diffusion is described by the system of equations:
mγ 1 ˙
x = κ(x − X ) − f 1 (t),
Mγ 2 ˙
X = κ(X − x) − f 2 (t),
(4.7)
where m and M, x and X are the masses and coordinates of the central ion and its ionic
atmosphere, respectively, γ is a damping, and f 1 and f 2 are stochastic forces. The
first term on the right-hand side represents the electrostatic coupling with the elastic
spring constant κ, which connects the ion with the center of its ionic atmosphere,
both of which obey Brownian motion. System (4.7) corresponds to the simplified
case considered in Sect. 3.5.
The dynamic conductivity in the whole frequency range can be found from 4.7
by the Fourier transform of the velocity–velocity correlation function:
σ (ω) =
q
2 n ±
kT
∞
0
˙
x(0) ˙
x(t)e
−iωt dt,
(4.8)
which gives for the frequency-dependent dynamic conductivity:
σ (ω) =
σ dc + ω
2
τ
2
D1 σ ac
1 + ω
2
τ
2
D1
,
(4.9)
where σ dc = qn ± μ dc , σ ac = qn ± μ ac , and the relaxation time τ D1 is
τ D1 =
γ 1 γ 2 m M
(mγ 1 + Mγ 2 )κ
,
(4.10)
and the low- and high-frequency mobilities are