160
4 The Dielectric Properties and Dynamic Structure of Water and Ice
that in the gas [57]. IR peaks of the intramolecular dynamics of water are also stable,
showing no shift between vapor and liquid. The shifts 3,756 → 3,500 cm
−1 (7%)
and 3,652 → 3,600 cm
−1 (< 2%) [58] are too small to be responsible for the high
molecular polarizability required for dielectric constant interpretation in terms of the
local field. Another intermolecular polarization mechanism is required. A possible
mechanism was described above, and it finds an unambiguous interpretation in the
ionic model of water.
4.5.4 The Second Relaxation and Terahertz Spectrum
At short time intervals, we can neglect the dynamics of the ionic atmosphere and
write a single equation of motion for the central ion performing harmonic oscillations
in a soft parabolic potential:
m ¨
x + mγ ˙
x + mω
2
0 x = q E 0 exp(−iωt),
(4.20)
where ω 0 is the eigenfrequency of the oscillations, and E 0 is the amplitude of the
periodic external electric field. Figure 4.12a shows the equivalent mechanical scheme
for (4.20). The scheme and the corresponding equation coincide with the forced
oscillations with friction, considered in [59], and is similar to the dynamics of the
Kapitza pendulum [60, 61] in which the mathematical pendulum is replaced by a
spring oscillator. Introducing x 0 (t) = q E(t)/mω
2
0 , which represents the change of
the equilibrium coordinate of the oscillations by the external field E(t) = E 0 exp(iωt), we get a simple equation of a damped harmonic oscillator with a moving
equilibrium point:
m ¨
x + mγ ˙
x + mω
2
0 (x − x 0 (t)) = 0,
(4.21)
where to let the particle conduct, we assume the friction has the following form:
γ (t) = γ 0 exp(−t/τ r ), where τ r is the length of time an ion is in the oscillatory state.
In terms of the dynamic conductivity, σ (ω) = n ± q ˙
x(ω)/E(ω), (4.21) has the following solution [59]:
σ (ω) =
n ± q
2
· γ 0 (1 − iωτ r )
m
γ 0 τ r (ω
2
0 − ω 2 ) + γ
2
0 − ω
2
0 − iωγ 0 (1 + γ 0 τ r )
,
(4.22)
where n ± is the concentration of ionic species. Separating the real and imaginary
parts, and taking into account that = 1 + q ˙
x(ω)/E(ω), we obtain the real part
of the dielectric function:
ε
(ω) = 1 +
n ± q
2
γ 0
mε 0
τ r b − a
ω 2 a 2 + b 2 ,
(4.23)
4 The Dielectric Properties and Dynamic Structure of Water and Ice
that in the gas [57]. IR peaks of the intramolecular dynamics of water are also stable,
showing no shift between vapor and liquid. The shifts 3,756 → 3,500 cm
−1 (7%)
and 3,652 → 3,600 cm
−1 (< 2%) [58] are too small to be responsible for the high
molecular polarizability required for dielectric constant interpretation in terms of the
local field. Another intermolecular polarization mechanism is required. A possible
mechanism was described above, and it finds an unambiguous interpretation in the
ionic model of water.
4.5.4 The Second Relaxation and Terahertz Spectrum
At short time intervals, we can neglect the dynamics of the ionic atmosphere and
write a single equation of motion for the central ion performing harmonic oscillations
in a soft parabolic potential:
m ¨
x + mγ ˙
x + mω
2
0 x = q E 0 exp(−iωt),
(4.20)
where ω 0 is the eigenfrequency of the oscillations, and E 0 is the amplitude of the
periodic external electric field. Figure 4.12a shows the equivalent mechanical scheme
for (4.20). The scheme and the corresponding equation coincide with the forced
oscillations with friction, considered in [59], and is similar to the dynamics of the
Kapitza pendulum [60, 61] in which the mathematical pendulum is replaced by a
spring oscillator. Introducing x 0 (t) = q E(t)/mω
2
0 , which represents the change of
the equilibrium coordinate of the oscillations by the external field E(t) = E 0 exp(iωt), we get a simple equation of a damped harmonic oscillator with a moving
equilibrium point:
m ¨
x + mγ ˙
x + mω
2
0 (x − x 0 (t)) = 0,
(4.21)
where to let the particle conduct, we assume the friction has the following form:
γ (t) = γ 0 exp(−t/τ r ), where τ r is the length of time an ion is in the oscillatory state.
In terms of the dynamic conductivity, σ (ω) = n ± q ˙
x(ω)/E(ω), (4.21) has the following solution [59]:
σ (ω) =
n ± q
2
· γ 0 (1 − iωτ r )
m
γ 0 τ r (ω
2
0 − ω 2 ) + γ
2
0 − ω
2
0 − iωγ 0 (1 + γ 0 τ r )
,
(4.22)
where n ± is the concentration of ionic species. Separating the real and imaginary
parts, and taking into account that = 1 + q ˙
x(ω)/E(ω), we obtain the real part
of the dielectric function:
ε
(ω) = 1 +
n ± q
2
γ 0
mε 0
τ r b − a
ω 2 a 2 + b 2 ,
(4.23)