32
1 A Historical Review of the Structures of Water and Ice
sponds to the model of diffusion described above (see Sect. 1.4), where r is the radius
vector. Figure 1.25b shows the mean square displacement, one period of which has
two components. The first one, l 0 , corresponds to the drift of the equilibrium point of
the oscillatory state within time τ 0 . The second one, l 1 , is the proton displacement to
a new equilibrium position within time τ 1 . For the observation times where t τ 0
+ τ 1 , the diffusion coefficient is equal to the averaged coefficient D. For t ∼ τ 0 or
τ 1 , the diffusion coefficient becomes time dependent, which, at the first glance, can
be interpreted as anomalous diffusion,
24 but which reduces to the sum of the normal
diffusion processes on closer consideration.
Neutron scattering data allow one to obtain the characteristic diffusion lengths
and times shown in Fig. 1.25. The algorithm for the determination of these diffusion
parameters was suggested by Singwi. The differential scattering cross section is
determined by [83]
d
2
σ
ddω
=
κ
κ 0
b
2
c S(q, ω) + b
2
i S S (q, ω)
,
(1.14)
where κ and κ 0 are the wave vectors of scattered and incident neutrons, respectively;
b c is the coherent and b i is the incoherent scattering amplitudes; and E = ω and q =
(κ-κ 0 ) are the energy and angular momentum of the scattering process, respectively.
The general scattering function S(q, ω) is the Fourier transform of the space-time
correlation function G(r , t), and S s (q, ω) is the Fourier transform of the autocorrelation function G s (r , t). The latter determines the probability of finding a particle at
the point (r , t), if at time t = 0 the same particle was at the origin.
At the classical limit, and assuming that the translational and rotational degrees of
freedom are not correlated, the interpretation of the G(r , t) function is based on the
idea that an isolated hydrogen atom, being at the origin at the time t = 0, determines
the probability of finding an atom in a unit volume at point r at time t. The atom in
position (r , t) can be either the same that was at the origin at the time t = 0, or a
different hydrogen atom.
G s (r , t) is defined by the sum:
G s (r, t) =
i
F i (r, t),
(1.15)
where F i is given by
24 Anomalous diffusion is a diffusion process with a non-linear relationship between the mean
squared displacement (MSD) and time.
1 A Historical Review of the Structures of Water and Ice
sponds to the model of diffusion described above (see Sect. 1.4), where r is the radius
vector. Figure 1.25b shows the mean square displacement, one period of which has
two components. The first one, l 0 , corresponds to the drift of the equilibrium point of
the oscillatory state within time τ 0 . The second one, l 1 , is the proton displacement to
a new equilibrium position within time τ 1 . For the observation times where t τ 0
+ τ 1 , the diffusion coefficient is equal to the averaged coefficient D. For t ∼ τ 0 or
τ 1 , the diffusion coefficient becomes time dependent, which, at the first glance, can
be interpreted as anomalous diffusion,
24 but which reduces to the sum of the normal
diffusion processes on closer consideration.
Neutron scattering data allow one to obtain the characteristic diffusion lengths
and times shown in Fig. 1.25. The algorithm for the determination of these diffusion
parameters was suggested by Singwi. The differential scattering cross section is
determined by [83]
d
2
σ
ddω
=
κ
κ 0
b
2
c S(q, ω) + b
2
i S S (q, ω)
,
(1.14)
where κ and κ 0 are the wave vectors of scattered and incident neutrons, respectively;
b c is the coherent and b i is the incoherent scattering amplitudes; and E = ω and q =
(κ-κ 0 ) are the energy and angular momentum of the scattering process, respectively.
The general scattering function S(q, ω) is the Fourier transform of the space-time
correlation function G(r , t), and S s (q, ω) is the Fourier transform of the autocorrelation function G s (r , t). The latter determines the probability of finding a particle at
the point (r , t), if at time t = 0 the same particle was at the origin.
At the classical limit, and assuming that the translational and rotational degrees of
freedom are not correlated, the interpretation of the G(r , t) function is based on the
idea that an isolated hydrogen atom, being at the origin at the time t = 0, determines
the probability of finding an atom in a unit volume at point r at time t. The atom in
position (r , t) can be either the same that was at the origin at the time t = 0, or a
different hydrogen atom.
G s (r , t) is defined by the sum:
G s (r, t) =
i
F i (r, t),
(1.15)
where F i is given by
24 Anomalous diffusion is a diffusion process with a non-linear relationship between the mean
squared displacement (MSD) and time.