120
3 The Interaction of Electromagnetic Waves with Ice
arrow 2, where S(ω 0 ) = S ∞ = const, and N e f f ≈ 1 (one conducting proton per H 2 O
molecule). At frequencies above arrow 1, the spectra are dominated by intramolecular
vibrations, as evident from a comparison of the H 2 O and D 2 O (see Sect. 2.7). On the
contrary, the intermolecular charge motion dominates below arrow 1, or in the region
where dielectric relaxation occurs. The transition point between intramolecular and
intermolecular dynamics lies near the 5 THz vibration mode (see Sect. 2.6.1), where
N e f f ≈ 0.02 (see Fig. 3.9), or, in other words, corresponds to 1 proton charge per
50 H 2 O molecules. Such a high concentration of charge carriers assumes a strong
electrostatic interaction between them, because the average distance between charges
of about 2 nm [25] is shorter than the corresponding Debye screening length.
6 Thus,
the smallest “quantum” of the intermolecular conductivity of ice and water is a proton
(or proton hole) charge shared between 50 water molecules. Excess protons, or in
other words ions of H 3 O
+ and OH
− obey Brownian motion in the potential of mutual
screening, caused by electrostatic interaction.
3.5 Protonic Transport as a Fundamental Mechanism
of the Dielectric Response of Ice and Water
Let us now consider the oscillatory-diffusion motion of a single excess proton (an
excited third proton on the water molecule that briefly forms an H 3 O
+ ion) of a
mass m in a time-dependent potential (or in a polarizable atmosphere), taking into
account its electrostatic interaction with other excess protons and proton holes (the
missing proton on the water molecule that briefly forms an OH
− ion). Hereafter,
we do not distinguish between excess protons and proton holes, unless otherwise
stated, because the mechanism of their diffusion is nearly the same. Both H 3 O
+ and
OH
− obey Brownian-like diffusion as a whole between spontaneous proton-transfer
events by the Grotthuss mechanism (see Sect. 1.3), and electrostatically interact
with each other. Figure 3.11 shows a schematic snapshot of the corresponding water
structure. Each charge has two polarization spheres. The first sphere (the dashed
black circle) is formed by polar H 2 O molecules. The second one (the dashed red
circle) is made of charges of the opposite sign.
7 Each excess proton can move inside
the ionic atmosphere formed by the surrounding negative charges (proton holes) and
together with its center. The latter dynamics are expected to be responsible for static
conductivity, while the former is responsible for the high-frequency polarization
effects. Such an approach to the microscopic description of the dielectric response
allows one to reduce the problem of polarization in liquid water and ice to a singleparticle task.
6 The Debye length is the distance over which the electric field of a separate charge, placed in
a quasi-neutral medium containing free positively and negatively charges, becomes significantly
screened.
7 The optimal distribution of charge in an electrically neutral system of charges of two types is when
positively charged ions are surrounded by negatively charged ones and vice versa.
3 The Interaction of Electromagnetic Waves with Ice
arrow 2, where S(ω 0 ) = S ∞ = const, and N e f f ≈ 1 (one conducting proton per H 2 O
molecule). At frequencies above arrow 1, the spectra are dominated by intramolecular
vibrations, as evident from a comparison of the H 2 O and D 2 O (see Sect. 2.7). On the
contrary, the intermolecular charge motion dominates below arrow 1, or in the region
where dielectric relaxation occurs. The transition point between intramolecular and
intermolecular dynamics lies near the 5 THz vibration mode (see Sect. 2.6.1), where
N e f f ≈ 0.02 (see Fig. 3.9), or, in other words, corresponds to 1 proton charge per
50 H 2 O molecules. Such a high concentration of charge carriers assumes a strong
electrostatic interaction between them, because the average distance between charges
of about 2 nm [25] is shorter than the corresponding Debye screening length.
6 Thus,
the smallest “quantum” of the intermolecular conductivity of ice and water is a proton
(or proton hole) charge shared between 50 water molecules. Excess protons, or in
other words ions of H 3 O
+ and OH
− obey Brownian motion in the potential of mutual
screening, caused by electrostatic interaction.
3.5 Protonic Transport as a Fundamental Mechanism
of the Dielectric Response of Ice and Water
Let us now consider the oscillatory-diffusion motion of a single excess proton (an
excited third proton on the water molecule that briefly forms an H 3 O
+ ion) of a
mass m in a time-dependent potential (or in a polarizable atmosphere), taking into
account its electrostatic interaction with other excess protons and proton holes (the
missing proton on the water molecule that briefly forms an OH
− ion). Hereafter,
we do not distinguish between excess protons and proton holes, unless otherwise
stated, because the mechanism of their diffusion is nearly the same. Both H 3 O
+ and
OH
− obey Brownian-like diffusion as a whole between spontaneous proton-transfer
events by the Grotthuss mechanism (see Sect. 1.3), and electrostatically interact
with each other. Figure 3.11 shows a schematic snapshot of the corresponding water
structure. Each charge has two polarization spheres. The first sphere (the dashed
black circle) is formed by polar H 2 O molecules. The second one (the dashed red
circle) is made of charges of the opposite sign.
7 Each excess proton can move inside
the ionic atmosphere formed by the surrounding negative charges (proton holes) and
together with its center. The latter dynamics are expected to be responsible for static
conductivity, while the former is responsible for the high-frequency polarization
effects. Such an approach to the microscopic description of the dielectric response
allows one to reduce the problem of polarization in liquid water and ice to a singleparticle task.
6 The Debye length is the distance over which the electric field of a separate charge, placed in
a quasi-neutral medium containing free positively and negatively charges, becomes significantly
screened.
7 The optimal distribution of charge in an electrically neutral system of charges of two types is when
positively charged ions are surrounded by negatively charged ones and vice versa.