34
1 A Historical Review of the Structures of Water and Ice
Table 1.4 Proton diffusion parameters using neutron scattering data, defined by (1.22). D 0 , D 1 ,
and D are the local-continuous, discontinuous, and general diffusion coefficients, respectively. l 0 ,
l 1 , and l are the lengths of vibrations, of hops, and between neighboring oxygen atoms, respectively.
τ 0 and τ 1 are the waiting and hopping times, respectively. See also Fig. 1.25
D 0 (m 2 /s) D 1 (m 2 /s) D (m 2 /s) l 0 (Å)
l 1 (Å)
l (Å)
τ 0 (ps)
τ 1 (ps)
0.7·10 −9 5.8·10 −9 2.3·10 −9 0.31
1.6
2.8
3.7
1.7
where d is the displacement of the proton during the time of the complete elaboration
of the “heat cloud” during oscillatory motion. This model is better suited for the
trajectory shown in Fig. 1.25. If d
2
l
2 , (1.20) gives the relation between the
coefficients D 1 and D in the form:
D 0 τ 0 + D 1 τ 1 = D(τ 0 + τ 1 ),
(1.21)
showing that the averaged mean square displacement is an additive sum of drift and
hopping. If τ 1 τ 0 , the shape of the quasi-elastic scattering peak has a Lorentzian
shape with a width at half maximum of
E =
2
τ 0
1 + κ
2 D 0 τ 0 −
e
−2W
1 + κ 2 Dτ 0
,
(1.22)
where 2W = κ
2 Dτ 0 (d
2
/l
2
) is the Debye–Waller factor. Figure 1.24 shows that (1.22)
perfectly fits the experimental data on neutron scattering. The best-fit parameters are
given in Table 1.4.
Table 1.4 shows that neutron scattering data assume two characteristic times of
diffusion, which correspond to vibrational or translational motion. The elementary
period of diffusion is t = t 0 +t 1 ≈ 5 ps. The corresponding distance in which a proton
moves for time t is equal to 3.1 Å, and consists of two parts: l 0 ≈ 0.3 Å and l 1 ≈
2.8 Å (see Table 1.4 and Fig. 1.25). The latter coincide with the diameter of a water
molecule, or with the average distance between two neighboring molecules. As long
as the probability that all molecules can rotate on the same angle of 180
◦ is very small,
one can assume the distance l 1 corresponds to the intermolecular transfer of a proton.
This is the same as that observed in the NMR spectroscopy of water (see Sect. 1.3.3).
The time necessary for such a transfer is determined by the tracer diffusion coefficient
D tr = 2.3·10
−9 , because proton transfer is limited by self-diffusion (see Sect. 1.5).
D tr = D, as follows from the Table 1.4, which indirectly confirms the assumption
that neutron scattering line width is determined by intermolecular proton exchange,
rather than molecular reorientations, although the latter help molecules to adjust the
position that is suitable for proton transfer.
Neutron scattering shows that the protons of water oscillate around a drifting equilibrium position within t 0 ≈ 4 ps, and then switch to another equilibrium position
within t 1 ≈ 2 ps, showing a diffusion-oscillatory motion. Note that the vibrational
state residence time, t 0 , is several times longer than the period of oscillations. After
several collisions, the proton is transferred to a new position within time t 1 , after
1 A Historical Review of the Structures of Water and Ice
Table 1.4 Proton diffusion parameters using neutron scattering data, defined by (1.22). D 0 , D 1 ,
and D are the local-continuous, discontinuous, and general diffusion coefficients, respectively. l 0 ,
l 1 , and l are the lengths of vibrations, of hops, and between neighboring oxygen atoms, respectively.
τ 0 and τ 1 are the waiting and hopping times, respectively. See also Fig. 1.25
D 0 (m 2 /s) D 1 (m 2 /s) D (m 2 /s) l 0 (Å)
l 1 (Å)
l (Å)
τ 0 (ps)
τ 1 (ps)
0.7·10 −9 5.8·10 −9 2.3·10 −9 0.31
1.6
2.8
3.7
1.7
where d is the displacement of the proton during the time of the complete elaboration
of the “heat cloud” during oscillatory motion. This model is better suited for the
trajectory shown in Fig. 1.25. If d
2
l
2 , (1.20) gives the relation between the
coefficients D 1 and D in the form:
D 0 τ 0 + D 1 τ 1 = D(τ 0 + τ 1 ),
(1.21)
showing that the averaged mean square displacement is an additive sum of drift and
hopping. If τ 1 τ 0 , the shape of the quasi-elastic scattering peak has a Lorentzian
shape with a width at half maximum of
E =
2
τ 0
1 + κ
2 D 0 τ 0 −
e
−2W
1 + κ 2 Dτ 0
,
(1.22)
where 2W = κ
2 Dτ 0 (d
2
/l
2
) is the Debye–Waller factor. Figure 1.24 shows that (1.22)
perfectly fits the experimental data on neutron scattering. The best-fit parameters are
given in Table 1.4.
Table 1.4 shows that neutron scattering data assume two characteristic times of
diffusion, which correspond to vibrational or translational motion. The elementary
period of diffusion is t = t 0 +t 1 ≈ 5 ps. The corresponding distance in which a proton
moves for time t is equal to 3.1 Å, and consists of two parts: l 0 ≈ 0.3 Å and l 1 ≈
2.8 Å (see Table 1.4 and Fig. 1.25). The latter coincide with the diameter of a water
molecule, or with the average distance between two neighboring molecules. As long
as the probability that all molecules can rotate on the same angle of 180
◦ is very small,
one can assume the distance l 1 corresponds to the intermolecular transfer of a proton.
This is the same as that observed in the NMR spectroscopy of water (see Sect. 1.3.3).
The time necessary for such a transfer is determined by the tracer diffusion coefficient
D tr = 2.3·10
−9 , because proton transfer is limited by self-diffusion (see Sect. 1.5).
D tr = D, as follows from the Table 1.4, which indirectly confirms the assumption
that neutron scattering line width is determined by intermolecular proton exchange,
rather than molecular reorientations, although the latter help molecules to adjust the
position that is suitable for proton transfer.
Neutron scattering shows that the protons of water oscillate around a drifting equilibrium position within t 0 ≈ 4 ps, and then switch to another equilibrium position
within t 1 ≈ 2 ps, showing a diffusion-oscillatory motion. Note that the vibrational
state residence time, t 0 , is several times longer than the period of oscillations. After
several collisions, the proton is transferred to a new position within time t 1 , after