98
2 The Interaction of Electromagnetic Waves with Water
dielectrics. The spaced-apart electronic and atomic contributions make it possible
to apply the rule defined by (2.62) separately for the atomic part only, or, in other
words, to apply the equation to a limited frequency range.
As the maximum level of the conductivity of the atomic part is observed in the IR
region (see dashed area in Fig. 2.28), the microwave part of the spectrum does not add
too much to the integral intensity. Thus, the atomic dynamics in the IR part, which
corresponds to the vibrationally averaged dynamics, roughly reduces to the dynamics
of protons in the frame of reference of oxygen atoms. Taking into account the fact
that the proton mass m p is an order of magnitude lower than that of oxygen m O the
effective mass m
∗ is equal to m p · m O /(m p +m O ) = 1·18/(1+18)·m p = 0.95·m p ≈ m p .
Thus, one can temporarily exclude oxygen atoms from the analysis and consider
only hydrogen atoms as the primary charge carriers. In other words, (2.62) can be
reformulated [71] for the charges in a unit volume of matter. From (2.61), one gets
n p =
2
π
m p
q 2
p
ω 0
0
σ (ω)dω,
(2.63)
where n p and q p are the concentration and the effective charge of protons, respectively.
Figure 2.29a shows the broadband conductivity spectra of water at different temperatures, supplemented by their partial integrals S=
ω co
0
σ (ω)dω, depending on the
cutoff frequency, ω co . The high-frequency value of integrals, S ∞ , corresponds to the
frequency ω co = ω O , where ω O has been defined above (see Fig. 2.28), and is equal to
S ≈ 5·10
17 S/m·Hz for all forms of water. Using this value, one gets from (2.62) that
n p ≈ 5·10
28 m
−3 , which is the concentration of all hydrogen atoms: n
total
p
≈ 6·10
28
m
−3 (2 · 55.5 = 111 mol/l). Thus, the conductivity sum rule, defined by (2.63), works
for the protonic subsystem (intramolecular and intermolecular), which is responsible
for the dielectric response of water and ice at frequencies below ω O ≈10
15 Hz.
Figure 2.29b shows the dynamic conductivity spectra σ (ω) of light (H 2 O) and
heavy (D 2 O) water, and their partial integrals S. The spectra differ in the IR part,
i.e., the part where the intramolecular dynamics occur, while the relaxation part
(below about 500 cm
−1 ) remains unaffected by the isotopic substitution. The main
IR peaks of heavy water are shifted by a factor of a =
√
2 = 1.4 (see Sect. 2.8 for
details). The S ∞ value for D 2 O is exactly half that for H 2 O, which confirms the
validity of (2.62), as the replacement of protons by the twice-as-heavy deuterons
affects the S value by a factor of b = 2. The S spectrum of D 2 O, normalized by
coefficients a and b, coincides with that for H 2 O (see the dashed line in Fig. 2.29b).
Thus, the partial sum rule works separately for protons/deutrons independently on the
electronic contribution. In other words, the left-hand side of the water conductivity
spectrum reflects the dynamics of protons in the frame of reference of oxygen atoms.
Table 2.11 contains the parameters of the sum rule, defined by (2.62), for the spectra
shown in Figs. 2.28 and 2.29.
2 The Interaction of Electromagnetic Waves with Water
dielectrics. The spaced-apart electronic and atomic contributions make it possible
to apply the rule defined by (2.62) separately for the atomic part only, or, in other
words, to apply the equation to a limited frequency range.
As the maximum level of the conductivity of the atomic part is observed in the IR
region (see dashed area in Fig. 2.28), the microwave part of the spectrum does not add
too much to the integral intensity. Thus, the atomic dynamics in the IR part, which
corresponds to the vibrationally averaged dynamics, roughly reduces to the dynamics
of protons in the frame of reference of oxygen atoms. Taking into account the fact
that the proton mass m p is an order of magnitude lower than that of oxygen m O the
effective mass m
∗ is equal to m p · m O /(m p +m O ) = 1·18/(1+18)·m p = 0.95·m p ≈ m p .
Thus, one can temporarily exclude oxygen atoms from the analysis and consider
only hydrogen atoms as the primary charge carriers. In other words, (2.62) can be
reformulated [71] for the charges in a unit volume of matter. From (2.61), one gets
n p =
2
π
m p
q 2
p
ω 0
0
σ (ω)dω,
(2.63)
where n p and q p are the concentration and the effective charge of protons, respectively.
Figure 2.29a shows the broadband conductivity spectra of water at different temperatures, supplemented by their partial integrals S=
ω co
0
σ (ω)dω, depending on the
cutoff frequency, ω co . The high-frequency value of integrals, S ∞ , corresponds to the
frequency ω co = ω O , where ω O has been defined above (see Fig. 2.28), and is equal to
S ≈ 5·10
17 S/m·Hz for all forms of water. Using this value, one gets from (2.62) that
n p ≈ 5·10
28 m
−3 , which is the concentration of all hydrogen atoms: n
total
p
≈ 6·10
28
m
−3 (2 · 55.5 = 111 mol/l). Thus, the conductivity sum rule, defined by (2.63), works
for the protonic subsystem (intramolecular and intermolecular), which is responsible
for the dielectric response of water and ice at frequencies below ω O ≈10
15 Hz.
Figure 2.29b shows the dynamic conductivity spectra σ (ω) of light (H 2 O) and
heavy (D 2 O) water, and their partial integrals S. The spectra differ in the IR part,
i.e., the part where the intramolecular dynamics occur, while the relaxation part
(below about 500 cm
−1 ) remains unaffected by the isotopic substitution. The main
IR peaks of heavy water are shifted by a factor of a =
√
2 = 1.4 (see Sect. 2.8 for
details). The S ∞ value for D 2 O is exactly half that for H 2 O, which confirms the
validity of (2.62), as the replacement of protons by the twice-as-heavy deuterons
affects the S value by a factor of b = 2. The S spectrum of D 2 O, normalized by
coefficients a and b, coincides with that for H 2 O (see the dashed line in Fig. 2.29b).
Thus, the partial sum rule works separately for protons/deutrons independently on the
electronic contribution. In other words, the left-hand side of the water conductivity
spectrum reflects the dynamics of protons in the frame of reference of oxygen atoms.
Table 2.11 contains the parameters of the sum rule, defined by (2.62), for the spectra
shown in Figs. 2.28 and 2.29.