Vibrational dynamics of amorphous ice

Article abstract of DOI:10.1103/PhysRevB.59.3569

Using inelastic neutron scattering, we have measured a range of amorphous forms of ice, including high-density and low-density amorphous ices, in the energy transfer region from 2 to 500 meV (16-4033 cm^-1). The measured spectra show that the pressure-produced high-density amorphous ice is significantly different from the two low-density amorphous forms of ice (obtained by vapor deposition and by annealing at 120 K of the high-density amorphous ice), as we expected. However, considerable differences were also observed between the vapor deposits and the low-density amorphous ice obtained from annealing the high-density amorphous ice (the latter has a spectrum very similar to ice-Ih and ice-Ic), indicating that the hydrogen bonding in the two systems are markedly different.

Full text of DOI:10.1103/PhysRevB.59.3569

PHYSICAL REVIEW B
VOLUME 59, NUMBER 5
1 FEBRUARY 1999-I
Vibrational dynamics of amorphous ice
A. I. Kolesnikov and Jichen Li
Department of Physics, UMIST, PO Box 88, Manchester, M60 1QD, United Kingdom
S. F. Parker and R. S. Eccleston
ISIS Facility, Rutherford Appleton Laboratory, Chilton, Didcot, Oxon, OX11 OQX, United Kingdom
C.-K. Loong
IPNS, Argonne National Laboratory, Argonne, Illinois 60439
͑Received 16 July 1998͒
Using inelastic neutron scattering, we have measured a range of amorphous forms of ice, including high-
density and low-density amorphous ices, in the energy transfer region from 2 to 500 meV ͑16–4033 cmϪ1͒.
The measured spectra show that the pressure-produced high-density amorphous ice is significantly different
from the two low-density amorphous forms of ice ͑obtained by vapor deposition and by annealing at 120 K of
the high-density amorphous ice͒, as we expected. However, considerable differences were also observed
between the vapor deposits and the low-density amorphous ice obtained from annealing the high-density
amorphous ice ͑the latter has a spectrum very similar to ice-Ih and ice-Ic͒, indicating that the hydrogen bonding
in the two systems are markedly different. ͓S0163-1829͑99͒05605-2͔
I. INTRODUCTION
also be transformed to hda-ice by slow warming to about 120
K.20 It is still an open question, as to what degree these
pressure produced hda-ices differ from each other, because
the hda-ice from ice-VIII has been annealed above the glass
transition temperature.21
Since the discovery of the noncrystalline ͑amorphous͒
form of ice by Burton and Oliver in the early 1930s,1 the
understanding of the microscopic properties of amorphous
solid water has attracted considerable attention for many de-
cades. Comprehensive experimental investigations, including
x-ray,2,3 electron3,4 and neutron diffraction ͑ND͒,5–12 infrared
͑IR͒ absorption and Raman scattering13–18 were carried out
by various research groups worldwide in order to understand
the basic structural and dynamical properties of these amor-
phous forms of ice. The understanding of the mechanism
behind the complex formation and stability of the different
kinds of amorphous ice is important to the interpretation of a
wide range of unusual behavior related to water in aqueous
matter, in living organism and in interstellar objects like
comets.
Previous studies have found that there are many different
forms of amorphous ice. Their structures and densities vary
with the formation conditions and the sample preparation
procedures. These ices can be categorized into two types, the
high-density amorphous ͑hda͒ and low-density amorphous
͑lda͒ ice. There are mainly two ways to produce the hda-ice.
One is by a pressure treatment, which was realized by two
different thermobaric processing methods ͑through different
paths in the P-T phase diagram of ice͒: ͑i͒ pressurizing a
crystalline hexagonal ice ͑ice-Ih͒ to about 12 kbar at tem-
peratures below 120 K results in an amorphous structure19
with a density of 1.31 g/cm3. It is believed that the pressure
crushes the open-tetragonal structure of ice-Ih and gives rise
to the higher density which is otherwise not obtainable in
normal circumstances. This form of hda-ice can be recovered
as a metastable state after releasing the pressure at low tem-
peratures ͑Ͻ120 K, i.e., below the glass transition tempera-
ture͒ and its density at ambient pressure reduces to 1.17
g/cm3. ͑ii͒ The recovered high-pressure phase ice-VIII can
The other way of producing hda-ice is by vapor deposi-
tion at low temperatures ͑Ͻ30 K͒ and low vapor pressures
͑at 10Ϫ4 bars, yielding a deposition rate less than 100
␮m/h͒.2,6 The samples made by this method6 are in the form
of thin layers ͑0.05 ␮m͒ and the results may also depend on
the substrate materials. It was believed that at low tempera-
tures and vapor pressures, some water molecules do not have
sufficient energy to form the tetrahedrally coordinated struc-
ture. As a result, the water molecules can enter interstitial
positions in the open ice network ͑similarly to ice-Ih͒. The
oxygen-oxygen distances between these and surrounding
molecules lie between the first and the second coordination
spheres corresponding to lda-ice or ice-Ih, and hence the
density of this deposited ice is higher. A peculiar property of
this vapor deposited hda-ice is its unstable structure; it
gradually transforms to a ‘‘restrained’’ form of amorphous
ice at about 77 K ͑well below the ice glass transition tem-
perature of ϳ100 K͒ and then to a lda-ice at 120 K.6
The classification of the lda-ice is equally complex. It was
frequently reported that there are many different structures
among the lda-ice. It may be reasonable to classify them into
two categories: vapor deposited lda-ice3,6,13,14 and lda-ices
obtained by other technique dealing with bulk samples. The
latter ices could be obtained by ͑i͒ thermal treatment of hda-
ices described above;4,5 ͑ii͒ hyperquenching of ␮m-size wa-
ter droplets;12,22 ͑iii͒ electron or ion-bombardment of crystal-
line forms at low temperature.23,24 All of them have a similar
density of ϳ0.94 g/cm3. The fundamental difference of these
two categories of lda-ice is mainly due to the fact, that for
the vapor deposited ice, the deposition process produces a
structure with high porosity, resulting in large number of
0163-1829/99/59͑5͒/3569͑10͒/$15.00
PRB 59
3569
©1999 The American Physical Society

3570
KOLESNIKOV, LI, PARKER, ECCLESTON, AND LOONG
PRB 59
unconnected hydrogen bonds. On the other hand, the local
structures of lda-ice obtained by annealing hda-ice or by vit-
rification of water droplets are very similar to ice-Ih.
vibrational dynamics of different forms of ice is as strong as
ever.
Diffraction studies show that there are some similarities
between the different types of hda-ice, e.g., all of them have
the first diffraction peak at higher Q values: 2.15 ÅϪ1 for
the pressure-produced hda-ice5 and ϳ1.9 ÅϪ1 for vapor de-
posited ice.2,6 However, the corresponding Q values for the
lda-ice obtained by heating hda-ice and for lda hyper-
quenched water9 are 1.70 and 1.79 ÅϪ1, respectively.
II. INELASTIC INCOHERENT NEUTRON
SCATTERING TECHNIQUE
Inelastic scattering of thermal neutrons is a unique tech-
nique to study vibrational dynamics of molecular substances.
First, the energies of the thermal neutrons are comparable to
energies of vibrational modes ͑1–500 meV͒, the neutron
wavelength is of the same order as the interatomic distances
in condensed materials and the neutron mass is of the same
order as the mass of vibrating atoms, and therefore, the scat-
tering process is very sensitive to the space-dynamical char-
acteristics of the system. Secondly, inelastic incoherent neu-
tron scattering ͑IINS͒ spectrum is directly proportional to the
vibrational density of states weighted by the atomic mean-
square displacements associated with each vibrational mode
and by the neutron scattering cross sections of the constituent
atoms. Hence, IINS measures all vibrational modes simulta-
neously and the spectrum provides direct information about
intermolecular and intramolecular interactions.
The most obvious and direct way of examining the ob-
served structure of the amorphous ices is by using x-ray,
electron, and ND techniques. The use of x rays for the struc-
tural investigation of liquids and amorphous substances is
well established and has a long history. However, because of
the small charge, hydrogen atoms are almost invisible to x
rays ͑and electrons͒, but they can be detected by neutrons.
Development of the ND technique is relatively recent and
has rapidly advanced in condensed matter research in the last
twenty years. One of the promising ways is obtaining the
partial atom-atom structure factors for amorphous ice by
combining the different diffraction techniques and/or by us-
ing H2O/D2O mixtures in ND ͑H and D atoms have different
amplitudes of coherent neutron scattering͒. Hence, more
complete information on the partial correlation functions can
be derived. However, these experiments are difficult due to
the fact that H atoms have very large incoherent neutron
scattering cross section and the data analysis also requires
rather sophisticated corrections for inelasticity effect. For
this reason, the results are often not consistent, e.g., various
conclusions were given from different research groups con-
cerning the structure of liquid water.25–27
The physical properties of ice and water depend on the
interactions between the water molecules. Development of a
suitable pairwise interaction potential for water is essential to
understand the origin of the microscopic behavior of water/
ice, including the geometrical conformations of the amor-
phous ice. After decades of simulations based on a variety of
computational techniques, such as molecular dynamics,
many effective pairwise potentials were developed, such as
TIP4P ͑Ref. 28͒ and SK.29 Their success is still limited to
accounting for a few features, such as partial structure
factors27 and density maximum30 for water. However, none
of them are capable of reproducing the vibrational spectra of
ice measured by neutron spectroscopy.31,32 These spectra are
directly proportional to the vibrational density of states
which can be rigorously calculated by molecular or lattice
dynamics calculations using a suitable potential. Thus repro-
ducing the structural properties is not sufficient for the vali-
dation of the potentials. Further quantitative comparison of
the simulations with the measured vibrational spectra of
water/ice is a good criterion in determining the interatomic
potentials. This is because the frequencies of the vibrational
modes depend on interatomic forces, which are determined
by the double derivatives of the pairwise potentials ͑i.e.,
Ϫd2V/dr2͒, thus the vibrational frequencies are very sensi-
tive to the curvature of the pairwise potential and hence it
provides a stringent test of the suitability of the potential. In
order to get further progress in the description of the water-
water potential, the need for precise experimental data of the
The quantity measured by neutron scattering can be de-
scribed by the neutron scattering cross section per unit solid
angle d⍀ and per unit energy dE:33
d2␴
k
ϭ
S Q,E͒
͑
d⍀dE k0
ϱ
k
i͓Q„r ͑0͒Ϫr ͑t͒…͔
eiEt/បdt, ͑1͒
͘
i
j
*
ϭ
b b e
͗
͵
͚
i
j
k0
ij
Ϫϱ
where S(Q,E) is a dynamical structure factor, Q and E are
the neutron momentum and energy transfers, k and k0 are
neutron momenta after and before scattering, bi neutron scat-
tering amplitude for atom i, ri(t) the vector position of atom
i at time t, the summation is going over all atoms and ͗...͘
means averaging over the assemble. The incoherent neutron
scattering cross-section for hydrogen atom, ␴incϭ4␲(bH2
H
ϪbH2), is much larger ͑by factor of 10 or more͒ than for
other atoms. On this reason for hydrogen containing materi-
als the dynamical structure factor in Eq. ͑1͒ can be trans-
formed to the following expression for IINS contributions
S
l,kϪl(Q,E), due to annihilation of l and creation of (kϪl)
excitations,
inc
H
␴
S Q,E͒ϭ
S
Q,E͒ϭ
exp Ϫ2W Q͒
͓ ͑ ͔
͑
͑
͚
͚
l,kϪl
4␲
l,k
l,k
k
ប2Q2
2m
k
G ␻ ͒...G ␻ ͒
͑ ͑
1 k
ϫ
d␻ ...d␻
͵
ͩ
ͪ
1
k
␻ ...␻ kϪl͒!l!
͑
1
k
l
ϫ
n ␻ ͒ϩ1
͓ ͑
n ␻ ͒
͔
͑
͟
i
͟
j
iϭlϩ1
jϭ1
ϫ␦ EϪ
ប␻ ϩ
ប␻
.
͑2͒
͚
͚
i
j
ͩ
ͪ
iϭlϩ1,k
jϭ1,l
Here n(␻) is the population Bose factor, m is a neutron
mass, W(Q) is a Debye-Waller factor for hydrogen atom

PRB 59
VIBRATIONAL DYNAMICS OF AMORPHOUS ICE
3571
Q2G ␻͒
tional and librational vibrations, as well as the intramolecular
bending and stretching modes of the water molecule.
1
2
ប
͑
2
W Q͒ϭ
Qu͒
ϭ
͘
2n ␻͒ϩ1 d␻.
͓ ͑ ͔
͑
͑
͗
͵
12m
␻
͑3͒
III. EXPERIMENTAL
The G(␻) function introduced in Eqs. ͑2͒ and ͑3͒ is a gen-
eralized vibrational density of states for hydrogen atom,
which can be described by a following summation over all
normal modes
In order to produce a vapor deposited ice, a special center
stick for a liquid helium ‘‘orange’’ cryostat ͑OC͒ has been
constructed. The samples were prepared by condensing water
vapour ͑double distilled H2O or 99.9% purity D2O͒ on the
surface of an aluminium sample can ͑hollow cylinder, 60
mm long and 10 mm inner diameter͒, which was kept at a
temperature below 15 K. The temperature of water vapour at
the entrance to the sample can was kept at 288 K, in order to
prevent blockage of the tip of the vapor jet by ice. The flow
rates were monitored by a mass flow controller. To reduce
the thermal exchange between the vapor and the substrate
surface, low deposition rates ͑low vapour pressures͒ were
used. The deposition processes were carried out for 45 and
120 h with a flow-rate of 7 and 13 mg/h for H2O sample ͑for
IINS measurements at TFXA and HET spectrometers,46 re-
spectively͒, and 110 h at 20 mg/h for D2O sample. The esti-
mated sample thicknesses were 0.15 and 0.7 mm for H2O,
and 1.0 mm for D2O, which corresponded to the rates of
samples growth of 3, 6, and 9 ␮m/h, respectively, and were
similar to those used by other groups.6 After preparation in
the cryostat the entire assembly ͑sample plus cryostat͒ was
installed on the neutron spectrometer. After ND and IINS
measurements the samples of the as-prepared vapor depos-
ited ice were transformed to some intermediate states and
finally to ice-Ih by heating the sample to several high tem-
peratures ͑for more details, see the next section͒.
1
G ␻͒ϭ
͉
ej 2
͉
␦ ␻Ϫ␻ ͒,
͑
͑4͒
͑
͚
j
3M
j
where ej is the polarization vector of normal modes and M is
hydrogen atomic mass.
Equation ͑2͒ can be applied to estimate the multiphonon
neutron scattering ͑MPNS͒ contributions by using the mea-
sured data and an iterative technique.34–36 At the first itera-
tive step, G(␻) and W(Q) could be calculated from the
experimental data based on an assumption that the IINS
spectrum of ice is the one-phonon spectrum. At the second
and subsequent steps, the difference between the experimen-
tal spectrum and that resulting from the multiphonon pro-
cesses could be accepted as the new one-phonon spectrum.
For ice spectra the convergence usually can be reached in
few iterations.36
Thus the one-phonon part of dynamical structure factor
S(Q,E) for IINS spectrum
inc
H
បQ2G ␻͒
␴
͑
S Q,E͒ϭ
exp Ϫ2W Q͒
͓ ͑ ͔
͑
4␲
2m␻
ϫ n ␻͒␦ Eϩប␻͒ϩ„n ␻͒ϩ1…␦ EϪប␻͒
͓ ͑
͑
͑
͑
͔
The hda-ice samples, including H2O, D2O and their mix-
ture ͑D2O͒ ͑H2O͒0.1, were prepared by pressuring ice-Ih at
0.9
͑5͒
77 K to about 15 kbar. The samples were recovered when the
pressure was subsequently released at liquid nitrogen tem-
perature. While submerged in liquid nitrogen, each sample
was transferred into an aluminium container, which was at-
tached to the cryostat center stick. The center stick was
quickly inserted into closed cycle refrigerator ͑CCR͒ or OC
precooled at Ͻ100 K. The sample volume was evacuated to
ϳ20 mbar and then the sample was cooled to 5 or 15 K, in
OC or CCR, respectively. After IINS measurements the H2O
hda-ice was transformed into lda-ice by heating the sample
to 120 K at 1 K/min. The structure of the samples was mea-
sured by ND. No difference was observed between the IINS
spectra for lda-ice and ice-Ih samples, indicating that the
configuration and intermolecular interactions of the water
molecules in these ice forms, to high accuracy, are the same.
The IINS spectra of prepared ices were measured on dif-
ferent spectrometers, TFXA, HET,46 and HERMECS ͑Ref.
47͒ in order to cover the energy transfer range of 2–500
meV. The TFXA spectrometer ͑with a fixed scattering angle
of ϳ135° and a fixed final energy ϳ4 meV͒ at the ISIS
pulsed neutron source ͑Chilton, UK͒ was used to obtain high
quality data in the energy transfer region from 2 to 150 meV.
The use of time and energy focusing, in conjunction with a
pyrolytic graphite analyzer in an inverse geometry, allows us
to make IINS measurements with very high energy resolu-
tion of ⌬E/EϷ1%. The IINS spectrum of ice measured on
TFXA gives poorer information at energy transfers above
150 meV, because at these energies the Debye-Waller factor
is directly related to generalized vibrational density of states
G(␻).
This is a clear contrast to Raman and IR spectroscopies,
which are insensitive to intermolecular translational and li-
brational modes for ice crystals, because the dipole moments
and polarizability induced by these modes, are small. Knowl-
edge of this part of the vibrational spectrum of ice is impor-
tant in the understanding the nature of hydrogen bonding in
the system. For instance, neutron scattering observes two
strong peaks at 28 meV ͑226 cmϪ1͒ and 37 meV ͑298
cmϪ1͒,31,32 while IR and Raman see only one peak at 28
meV. A weak shoulder at 36 meV in the Raman spectrum
has been considered to be a combination mode or an
overtone.37 This observation has attracted considerable sci-
entific attention, and renewed efforts in computer simula-
tions have been directed to describe these features in the
measured IINS spectra.
In the past few years, we have made a systematic inves-
tigation of the phonon dynamics of many exotic crystalline
forms of ice using neutron spectroscopy21,31,32,38–45 and
shown that the IINS spectra of ice are very sensitive to their
structural changes. In this paper, we present a series of IINS
measurements for different amorphous ices: the pressure-
produced hda-ice, the lda-ice obtained by annealing of hda-
ice and the vapor deposited amorphous ice. The measure-
ments cover a wide range of neutron energy transfers, from
2 to 500 meV, which includes the intermolecular transla-

3572
KOLESNIKOV, LI, PARKER, ECCLESTON, AND LOONG
PRB 59
the as-prepared state ͑dep-ice͒, as well as for the deposited
D2O sample after its annealing to 185 K ͑cubiclike ice-‘‘Ic’’͒
and to 255 K ͑ice-Ih͒. Because of the large incoherent cross
section and the small coherent cross section from H2O ice
coh
͑␴incϭ80.26 b, ␴cohϭ1.7568 b, ␴incϭ0.0008 b, and ␴
H
H
O
O
ϭ4.232 b, 1 bϭ1ϫ10Ϫ24 cm2͒ in comparison with D2O
͑␴incϭ2.05 b and ␴cohϭ5.592 b͒, the scattered intensity for
D
D
H2O is dominated by the large, isotropic incoherent neutron
scattering background from hydrogen atoms. All the data
represent the superposition of structure factors from the ice
samples and polycrystalline aluminium ͑the material of the
sample-can, maximum d-spacing for aluminium is 2.34 Å͒.
The position of the right-most peaks for dep-ice D2O sample
has dϭ3.7 Å, which corresponds to the first peak in the
structure factor, S(Q), at neutron momentum transfer Q1
ϭ1.70Ϯ0.04 ÅϪ1. This value is in a good agreement with
the values for the position of the first peak in the structure
factor for lda-ice in the literature.2–5,7–10,12,19,20
The first peak in the S(Q) spectrum for amorphous sub-
stances with tetrahedrally coordinated molecules is identified
as a ‘‘first sharp diffraction peak’’ ͑FSDP͒ and the origin of
this peak can be ascribed to the ordering of interstitial voids
around the oxygen atoms comprising the tetrahedrally coor-
dinated continuous-random network structure.48 The position
of the FSDP is given by the simple relation: Q1ϭ3␲/2d,
where d is the average nearest-neighbor atom-void separa-
tion ͑which is equivalent to the average oxygen-oxygen near-
est neighbor distance in the case of ice͒. Thus the d value, or
estimated oxygen-oxygen separation in the dep-ice, is 2.77
Ϯ0.10 Å.
Annealing to 185 and 255 K results in the transformation
of deposited lda-ice to ice-‘‘Ic’’ and ice-Ih; that is seen from
the diffraction patterns in Fig. 1. The correlation length, ␰,
obtained from the expression ␰ϭ2␲/⌬Q, where ⌬Q, is the
full width at half maximum of the first peak in the structure
factor, is 26 Å for the D2O sample, and it is about 11 Å for
H2O dep-ice. The corresponding values for the annealed D2O
samples are 36 and 105 Å for ice-‘‘Ic’’ and ice-Ih, respec-
tively. The ␰ values for deposited ices are typical for amor-
phous samples.2 However, the correlation length for ice-
‘‘Ic’’, ␰ϭ36 Å, is much smaller than for normal crystalline
material, and we designated this intermediate quasicubic-like
state, as ice-‘‘Ic.’’ This state could correspond to the so-
called restrained amorphous ice-Iar, which was observed in
Ref. 6 in the sequence of transformations during annealing of
vapor deposited hda-ice ͑called as Iah͒ to lda-ice ͑at tem-
peratures between 40 and 70 K, called as Ial͒ and to Iar ͑at
Tϭ144 K͒.
FIG. 1. Powder ND patterns measured on HET at 5 K of the
as-prepared deposited ice D2O ͑solid curve͒ and H2O ͑dashed
curve͒, and after annealing of the deposited ice D2O to 185 K ͑ice-
‘‘Ic’’͒ and to 255 K ͑ice-Ih͒. The diffraction patterns also contain
peaks from the aluminium sample can which are indicated by ar-
rows at top.
significantly suppresses the intensity of the one-phonon part
of the spectrum ͑due to large value of the neutron momen-
tum transfer Q͒. HET at ISIS or HERMECS at IPNS
͑Argonne, IL͒, were used to record high quality spectra at
higher energy transfer ͑up to 600 meV͒. Both HET and
HERMECS are direct-geometry time-of-flight spectrometers;
the energy of incident neutrons, Ei , can be selected by a
rotating chopper. Hence the measurements of IINS spectra
can be made with the incident neutrons having energy very
close to the energy region of interest. With correctly chosen
conditions ͑Ei and scattering angle͒ the neutron momentum
transfers at high-energy transfers are much smaller than on
TFXA. This made it possible to measure good quality INS
spectra for different ice-phases up to the highest phonon en-
ergies, around 400–450 meV.13 In addition, the HET ͑after
removing the neutron chopper͒ and TFXA spectrometers can
perform ND measurements. This flexibility allowed us to
measure the structure and vibrational spectra of many pre-
pared ices using the same spectrometers without having to
load and reload the samples.
We did not observe the hda-ice from ND measurements;
all the vapor deposits were the lda-ice as observed by other
ND experiments.3,7 The plausible reason for this may be that
in the neutron scattering experiments bulk samples were
used ͑mass of ϳ1 g and thickness of ϳ1 mm͒, while in
transmission electron microscope measurements6 the hda-ice
Iah was in the form of thin layers ͑ϳ0.05 ␮m thick͒, and
thus it represents only the surface features. The formation of
Iah during vapor deposition can be due to the specific bond-
ing preference of water molecules on the surface.49,50
The IINS measurements in most cases were carried out at
temperatures at or below 15 K in CCR and OC. The purpose
of using the low temperature is to minimize the multiphonon
neutron scattering and the temperature effects. The IINS data
were transformed to the dynamical structure factor S(Q,E)
vs energy transfer using the standard programs for the spec-
trometers. The background spectra for the empty can in CCR
and OC were measured under the same conditions and sub-
tracted from the original data.
IV. NEUTRON DIFFRACTION
OF THE VAPOR DEPOSITED ICE
V. INTERMOLECULAR VIBRATIONS
FOR HDA AND LDA ICE
Figure 1 shows the ND patterns measured at 5 K for the
vapor deposited H2O and D2O samples ͑see also Ref. 44͒ in
All known ices ͑except for ice-X, which has an ioniclike
structure͒ are molecular substances, which consist of well-

PRB 59
VIBRATIONAL DYNAMICS OF AMORPHOUS ICE
3573
FIG. 3. The generalized vibrational density of states ͑transla-
tional bands͒ transformed from the S(Q,␻) spectra ͑Fig. 2, curves 5
and 6͒ for the as-prepared deposited ice H2O ͑full circles, curve 2͒
and after annealing to 215 K, ice-Ih ͑open circles, curve 1͒. The
curves are the polynomial-smoothed data. The inset shows these
curves and their difference ͑curve 3͒ in the energy transfer region
below 15 meV.
FIG. 2. The IINS spectra (H2O) in the range of intermolecular
translational vibrations measured at Tϭ5 K on TFXA for ice-VI
͑curve 1͒, hda-ice ͑curve 2͒, ice-Ih ͑curve 3͒, and dep-ice ͑Ref. 51͒
͑curve 4͒, and on HET (EIϭ45 meV) for dep-ice ͑curve 5͒ and
after annealing to 215 K, ice-Ih ͑curve 6͒. For clarity the plotted
spectra are shifted along the y axis. The solid curves are the
polynomial-smoothed data. The dashed line plotted over curve 5 is
the smoothed spectrum of the ice-Ih for better visual comparison.
at energy transfers below 15 meV. The difference spectrum
͑curve 3͒ clearly reveals the excess of vibrational states in
addition to acoustic phonons in the region 3 to 6 meV in the
deposited amorphous ice. This is the so called low energy
excitation ͑LEE͒ or ‘‘boson’’ peak,51 which is commonly
observed in the vibrational spectra of amorphous substances.
This result is in agreement with earlier work,52 which
showed the existence of the localized low frequency modes
in simulated deposited amorphous ice. The origin of this
peak may be associated with the distortion of the local struc-
ture due to the topological disorder in amorphous substances,
and it is believed that this peak is responsible for the excess
of specific heat in the region between 1 and 20 K. By nor-
malizing the translational part of the spectra to three ͑which
is the three degrees of freedom per water molecule͒ the in-
tegrated intensity under the difference curve of G(␻) in the
range of LEE ͑2–6 meV͒, or the number of LEE per water
molecule, is found to be 0.028 ͑or about 0.01 LEE/atom͒.
The value for deposited glassy propylene (CH2ϭCHCH3)
and 1-butene (CH2ϭCHCH2CH3) is equal to 1.64 and 1.4
LEE/molecule53 ͑or 0.18 and 0.12 LEE/atom͒, respectively,
and 0.02 and 0.007 LEE/atom for typical tetrahedrally-
coordinated glasses SiO2 and Se, respectively.54
The librational bands for the H2O samples and the trans-
lational and librational bands for the D2O samples for the
amorphous forms and crystalline phases40,41 are shown in the
Figs. 4 and 5, respectively. In the translational part of the
IINS spectra ͑see Fig. 2͒ for the as-prepared vapor deposited
ice H2O and after annealing to 215 K ͑ice-Ih͒, the position of
the first peak is observed at about the same energy value, ϳ7
meV. This value is in good agreement with the energy of the
acoustic peak in the low-density forms of ice, Ih, Ic, and
lda-ice in the literature.31,32,40–42,45 This indicates that their
bulk densities are very close to each other. This peak shifts
to higher energy for the high-pressure forms of ice ͑Figs. 2
and 5͒. Its position for hda-ice is at 9.6 meV for H2O and at
separated water molecules interacting with each other via
hydrogen bonds. The forces between atoms in the water mol-
ecule are much stronger than the hydrogen bond forces
͑the parameters corresponding to intramolecular O-H stretch-
ing force and O-H...O hydrogen bond used in LD
calculations31,32 were 36.1 and 2.1 eV/Å2, respectively͒.
Hence the vibrational spectra of ice can be subdivided into
intermolecular vibrations at less than 120 meV and intramo-
lecular vibrations between 200 and 450 meV without over-
lap. The intermolecular part could be further subdivided into
two bands due to translational ͑Ͻ40 meV͒ and librational
͑50–120 meV͒ vibrations of water molecules. The intramo-
lecular vibrations could be assigned to bending, ␯2 , at about
200 meV and stretching modes, ␯ and ␯3 , at about 420
1
meV, according to the internal degrees of freedom within a
water molecule.
Figure 2 shows the IINS spectra of two amorphous ice
forms, the hda-ice and dep-ice. The spectra of crystalline
phases ice-Ih and ice-VI ͑Refs. 40 and 41͒ are also shown in
the figure for comparison. The spectra were measured on the
TFXA ͑curves 1 to 4͒ and HET ͑curves 5 and 6͒ spectrom-
eters. The ice-Ih sample measured on HET ͑curve 6͒ was
obtained by heating the dep-ice to 215 K in the OC on the
HET spectrometer. Thus the measurements for dep-ice and
ice-Ih were carried out under exactly the same conditions
͑the temperature for both samples during the measurements
was 5 K͒. At energy transfers less than 6 meV, there is a
pronounced extra intensity in the IINS spectrum for the as-
prepared deposited ice when compared to ice-Ih. Figure 3
shows the translational band of the generalized vibrational
density of states G(␻) for these two ice forms, obtained
from the measured S(Q,␻) spectra.
The inset in Fig. 3 shows the G(␻) for dep-ice and ice-Ih

3574
KOLESNIKOV, LI, PARKER, ECCLESTON, AND LOONG
PRB 59
FIG. 5. The IINS spectra (D2O) measured at Tϭ5 K on TFXA
for hda-ice ͑curve 1͒, ice-VI ͑curve 2͒, and ice-Ih ͑curve 3͒, and on
HET (EIϭ130 meV) for the dep-ice ͑Ref. 51͒ ͑curve 5͒ and after
annealing of dep-ice to 215 K, ice-Ih ͑curve 4͒. The curves are the
polynomial-smoothed data. The dashed curve over the solid curve 4
is the spectrum for dep-ice ͑curve 5͒ plotted for visual comparison.
FIG. 4. The IINS spectra (H2O) in the range of intermolecular
librational vibrations measured at Tϭ5 K on TFXA for ice-VI
͑curve 1͒, hda-ice ͑curve 2͒, ice-Ih ͑curve 3͒, dep-ice ͑Ref. 51͒
͑curve 5͒, after annealing dep-ice to 185 K ͑curve 4͒ and ice-IX
͑Ref. 46͒ ͑curve 6͒, and on HET (EIϭ170 meV) for the dep-ice
͑dashed curve 7͒ and after annealing of dep-ice to 215 K, ice-Ih
͑solid curve 7͒. The curves are the polynomial-smoothed data. The
dashed curves over the solid curves 3, 4, and 6 are the spectrum for
dep-ice, curve 5, plotted for visual comparison.
also that the oxygen-oxygen nearest distances are longer in
these ices compared to ice-Ih.
It is seen from the comparison in Fig. 4 that the librational
part of the INS spectrum of the dep-ice resembles that of
ice-IX ͑curve 6͒ in many details, but not the spectrum for
normal ice-Ih or other forms of ice. This indicates that the
transverse intermolecular force constants between nearest-
neighbor water molecules, which mainly determine the be-
havior of the librational band, are very similar for the vapor
deposited ice and ice-IX. The spectrum for the deposited
sample annealed at 185 K, quasi-cubic-like ice-‘‘Ic’’ ͑or per-
haps relaxed amorphous͒, is also considerably different from
that for ice-Ih. This observation means that the vibrational
spectrum of the ice-‘‘Ic’’ is definitely different from the
spectra for low density ices ͑Ih, Ic, and lda-ice͒.32
ϳ9 meV for D2O samples, and the peak splits for ice-VI
with maxima at 9.0 and 11.0 meV for H2O and 8.5 and 10.5
meV for D2O.
It follows from the above data, that there is no long-range
order in the deposited ice, because the correlation length ob-
tained from ND is very short (␰ϭ11 Å) and the translational
part of the IINS spectrum above 15 meV is featureless ͑i.e.,
does not exhibit any peaks͒, but its structure is relaxed ͑the
number of LEE/atom is much less compared to other depos-
ited amorphous materials, such as propylene and 1-butene͒.
On the other hand the lda-ice, produced by heating the hda-
ice, has shown32 that its IINS spectrum was very similar to
those for low-density crystalline phases, ice-Ih and ice-Ic.
That means, the lda-ice may have extended-range or at least
medium-range order.
From the above analysis we can conclude, that the char-
acteristic features of the IINS spectrum for hda-ice ͑the en-
The librational part of the IINS spectra for H2O hda-ice,
dep-ice and intermediate state ice-‘‘Ic’’ and ice-Ih, obtained
by heating dep-ice to 185 and 230 K, respectively, are shown
in Fig. 4, together with ice-VI and IX for comparison pur-
poses. It was found that the characteristic value for the libra-
tional band in the IINS spectrum for different ice forms is the
position of its low-energy cut-off. As shown in Fig. 4, the
low-energy cut-off of the librational band for the as-prepared
deposited ice ͑curve 5͒ is shifted by about 3.5 meV to lower
energy compared to ice-Ih ͑curve 3͒. This shift is even higher
for hda-ice ͑curve 2͒ and ice-VI ͑curve 1͒, about 11 and 13
meV, respectively. Assuming the moment of inertia for wa-
ter molecules in these ices is about the same, the result indi-
cates that the transverse forces between the water molecules
are smaller for the higher density ices. The latter suggests
FIG. 6. The IINS spectra measured at Tϭ5 K on TFXA for
hda-ice H2O and D2O ͑the bottom curve and top points, respec-
tively͒. The curve plotted above points at the top is the librational
band of the IINS spectrum for H2O with energy scale reduced by
factor & ͑see the text͒.

PRB 59
VIBRATIONAL DYNAMICS OF AMORPHOUS ICE
3575
FIG. 7. The IINS spectra of (D2O)1Ϫx(H2O)x measured at T
ϭ5 K on HERMECS (EIϭ150 meV) for hda-ice ͑xϭ0.10, curve
1͒ and on HET (EIϭ130 meV) for dep-ice ͑Ref. 51͒ ͑xϭ0.05,
curve 3͒ and ice-Ih ͑curve 2͒. The curves are the polynomial-
smoothed data. The dashed curve over ice-Ih spectrum ͑curve 2͒ is
the spectrum of dep-ice plotted for visual comparison.
FIG. 8. The IINS spectra (H2O) measured at Tϭ5 K on HET
for dep-ice ͑points and solid curve 1͒ and ice-Ih ͑open points and
dashed curve 2͒ with EIϭ500 meV and for ice-VI ͑Ref. 47͒ ͑open
points and curve 4͒ with EIϭ600 meV, and on HERMECS (EI
ϭ500 meV) for the hda-ice ͑triangles and curve 3͒. The curves are
the polynomial-smoothed data. The solid curve over the spectrum
for ice-Ih is the spectrum of dep-ice plotted for visual comparison.
The spectra are shown in the energy transfer range: ͑a͒ 0–500 meV
͑the plotted spectra are shifted along the y axis͒; ͑b͒ 120–250 meV,
bending modes ͑lines are two Gaussian fits͒; and ͑c͒ 340–470 meV,
stretching modes ͑lines are one and two Gaussian fits for as-
prepared deposited ice and ice-Ih, respectively͒. The parameters of
the fits are in Table I.
ergy positions of the first ‘‘acoustic’’ peak, at about 9 meV,
and the low-energy cut-off of the librational band, at 40 and
56 meV, for D2O and H2O͒ are very similar to ice-VI. The
differences are: the spectrum for ice-VI shows well defined
splitting of the first acoustic peak, two optic peaks in the
translational part ͑ϳ21 and ϳ35 meV͒ and two peaks in the
range of the librational band. All these features are missing
from the hda-ice spectrum, which looks unstructured in the
optic part of the translational band and in the whole area of
the librational band. Extra intensity is seen in the acoustic
part ͑below ϳ8 meV͒ of the hda-ice spectrum, compared to
ice-VI, for both isotopic forms. This extra intensity has a
similar origin to that for the deposited amorphous ice dis-
cussed above. The peaks at around 20 and 28 meV in ice-Ih
are missing in the spectra of the amorphous hda-ice. Absence
of long range order in hda-ice structure could be responsible
for the broad featureless behavior in this part of the spectra.
Figure 6 shows a comparison of hda-ice spectra for H2O
and D2O samples. The first peak in both spectra are at the
same energy. Surprisingly, the spectra exhibit harmonic
isotope behavior for their librational parts. The libra-
tional band for H2O sample, energy scaled by a factor of
in the spectrum of dep-ice is at slightly lower energy, about
102 meV, but has much larger width ͓full width at half maxi-
mum ͑FWHM͒ Ϸ10 meV͔. The position of the local mode
peak for hda-ice is at the lowest energy, around 97 meV,
with FWHM of ϳ16 meV. We can conclude that ͑i͒ the
transverse forces acting on the water molecules in hda-ice
are much weaker than in dep-ice and ice-Ih, and ͑ii͒ the
distribution of these forces ͑which is proportional to the
FWHM of the local mode peak͒ in amorphous ices is much
greater than in ice-Ih.
ͱ
I(D2O)/I(H2O)Ϸ&, where I is the moment of inertia,
well describes the measured spectrum for D2O hda-ice. This
means that the potentials experienced for the two isotopic
forms of water in hda-ice are the same ͑within the detectable
error margin͒ and it very close to harmonic at low tempera-
ture.
The librational parts of the spectra for the mixture of D2O
and H2O hda-ice ͑curve 1, 10% H2O in D2O measured on
HERMECS͒ and dep-ice ͑curve 3, 5% H2O in D2O measured
on HET͒ are shown in Fig. 7. The spectrum for ice-Ih ͑curve
2, dep-ice heated to 255 K and then measured on HET͒ are
also plotted for comparison.44 The hydrogen local mode peak
in the spectrum of ice-Ih can be fitted with two narrow Gaus-
sians placed at 102.7 and 106.4 meV. The local mode peak
VI. INTRAMOLECULAR VIBRATIONS
The IINS spectra of the dep-ice, ice-Ih ͑produced by heat-
ing dep-ice to 255 K͒, hda-ice and ice-VI in the energy trans-
fer region below 500 meV are shown in Fig. 8. The peaks
around 200 and 420 meV are due to the bending and stretch-
ing modes, respectively. The stretching mode peaks for the
crystalline phases, ice-Ih and ice-VI, are clearly split into
two components, while those for the amorphous samples
look like single peaks. The positions of the bending and
stretching mode peaks determined by fitting to Gaussian
functions are listed in Table I. Lines in Figs. 8͑b͒ and 8͑c͒

3576
KOLESNIKOV, LI, PARKER, ECCLESTON, AND LOONG
PRB 59
show the fitted spectra for dep-ice and ice-Ih, measured at
exactly the same conditions. The bending mode peaks for all
samples can be fitted with two Gaussians. The positions of
the Gaussians for the bending mode peaks of ice-Ih, dep-ice,
and ice-VI are about the same, but they are more split
for hda-ice. For the stretching mode peaks, an increase of
the energy positions is observed in the following se-
quence: ice-Ih→dep-ice→ice-VI→hda-ice. This indicates
that the O-H covalent-bond distances in the water molecules
of the corresponding ice forms are decreasing. Moreover,
due to the ordering of the oxygen positions in the ice-Ih and
ice-VI and disordering in the amorphous ices, one should
expect a sharper splitting of the peaks in the spectrum of the
crystalline phases ͑due to the symmetric and asymmetric
stretching modes of water molecules͒ and a broad peak for
the amorphous ices, due to the broader distribution of forces
acting to the water molecules. This behavior was observed
for the spectra of ice-Ih and ice-VI ͑splitting of the stretching
mode peak͒ and dep-ice ͑the peak broadening͒, but not for
the hda-ice. The stretching mode peak for hda-ice is much
narrower. This case will be discussed below after consider-
ation of the MPNS processes and time-independent back-
ground corrections.
The peak seen at about 285 meV can be explained by
MPNS, mainly due to simultaneous excitation of intramo-
lecular bending modes and intermolecular librational modes.
The librational-librational two-phonon contributions are ex-
pected to be at energy transfers about 180 meV, that is in the
region of bending modes. To separate and remove these con-
tributions, which is important at high neutron momentum
transfers, we have calculated the MPNS spectrum ͑up to
four-phonon processes͒ in a harmonic isotropic approxima-
tion using the measured data and the iterative technique as
described earlier.
Figure 9͑a͒ shows the measured S(Q,␻) and calculated
MPNS spectra for dep-ice and ice-Ih. As we supposed, the
peak at about 285 meV is indeed due to the MPNS contri-
butions. The derived one-phonon spectra in the ranges of
bending and stretching modes are shown in parts ͑b͒ and ͑c͒
of Fig. 9. The MPNS contributions in the stretching mode
region are smooth and do not influence the peak position of
the derived one-phonon spectra ͓see Fig. 9͑c͒ and Table I͔.
The effect of the MPNS in the range of the bending modes is
very strong. Because the MPNS spectrum arises mainly from
the double convolution of the librational band, and the libra-
tional band for the deposited ice is about 3.5 meV softer than
that for ice-Ih, the calculated MPNS peak for the deposited
ice is by about 7 meV softer compared to that for ice-Ih. The
shape of the derived one-phonon bending peaks in Fig. 9͑b͒
differs from that of the experimental spectra, it is more sym-
metrical after the subtraction of the MPNS contribution.
Hence we believe that the left hand peak found in the Gauss-
ian fit of the bending mode peaks is an artifact due to the
MPNS processes.
FIG. 9. ͑a͒ The IINS spectra (H2O) measured at Tϭ5 K on
HET (EIϭ500 meV) for dep-ice ͑solid curve͒ and ice-Ih ͑dashed
curve͒. The curves are the polynomial-smoothed data from Fig. 8.
The bottom curves are the calculated MPNS contribution. The
curves in the parts ͑b͒ and ͑c͒ show the extracted one-phonon spec-
tra for dep-ice ͑solid points and solid curve 1͒ and ice-Ih ͑open
points and dashed curve 2͒ in the range of bending and stretching
modes, respectively. The long-dashed curve in part ͑c͒ shows an
estimated time-independent background ͓see the text and expres-
sion ͑7͔͒.
(L2). The time-of-flight spectra, F(t), are usually trans-
formed to representations of neutron energy transfer, F(E)
͓or S(Q,E), see previous͔. The relationship between the
time-of-flight, t, and the neutron energy transfer, E, and F(t)
and F(E) spectra are
L1
L2
t E͒ϭ
͑
ϩ
͑6͒
ͱ
ͱ
␣ EI ␣ EIϪE
and
dt
L2
F E͒ϭF t͒
ϭF t͒
,
3
͑7͒
͑
͑
͑
dE
ͱ
2␣ EIϪE͒
͑
where ␣ is a coefficient. It is seen from Eq. ͑7͒ that, if the
measured F(t) spectrum has any small time-independent
background, the corresponding F(E) spectrum will have a
singularity at EϭEI .
We have approximated this background by use of the ex-
pression ͑7͒ ͓shown as a long dashed curve in Fig. 9͑c͔͒
and subtracted it from the one-phonon spectra. The resulting
stretching modes data are shown in Fig. 10 ͑see also
Table I͒. The average position of the stretching
mode peak ͑obtained by fitting with one Gaussian͒ is 414.1
meV (FWHMϭ55 meV) for ice-Ih and 417.5 meV
(FWHMϭ36 meV) for ice-VI. Thus this value for ice-Ih is
about 4 meV softer than for the dep-ice and ice-VI. This is
about 1.5 times larger than the shift for the stretching mode
peaks for the dep-ice and ice-Ih (22 cmϪ1ϭ2.7 meV), ob-
served in the Raman spectra.13 The highest shift is 12 meV
observed for hda-ice. The shift can be attributed to the dif-
ference in oxygen-hydrogen covalent bond length, which is a
function of the oxygen-oxygen separation distance. Using an
empirical relation between the energy of the stretching mode
The background of IINS spectra measured on the direct-
geometry time-of-flight spectrometers at energy transfers
close to the energy of the incident neutrons (EI) increases
noticeably. The collected neutron scattering data are propor-
tional to the number of neutrons that arrive at the detectors,
and are a function of the time-of-flight from the moderator to
the sample position (L1) and sample position to detector

PRB 59
VIBRATIONAL DYNAMICS OF AMORPHOUS ICE
3577
TABLE I. The energy positions and full width at half maximum ͑in brackets͒ of the fits with Gaussian
functions for the IINS spectra of the as-prepard deposited ice ͑dep-ice͒, ice-Ih ͑produced by heating the
dep-ice to 255 K͒, ice-VI and hda-ice. The values are in meV. ͑ ͒ The time-independent backgound-
*
corrected data ͑see the text͒.
Bending modes
Stretching modes
Derived one-
phonon data
Experimental
data
Derived one-
phonon data
Experimental
data
Background-
corrected data
*
ice-Ih
189.2
͑35͒
208.0
͑27͒
181.7
͑21͒
203.6
͑27͒
392.3
͑14͒
413.1
͑36͒
392.7
͑36͒
413.5
͑36͒
393.5
͑12͒
415.7
͑51͒
dep-ice
ice-VI
hda-ice
189.9
͑44͒
188.7
͑39͒
185.0
͑50͒
209.0
͑23͒
208.2
͑24͒
211.8
͑27͒
194.1
͑31͒
209.5
͑22͒
414.9
͑39͒
418.7
͑33͒
420.9
͑30͒
415.7
͑38͒
418.5
͑50͒
418.7
͑33͒
426.0
͑51͒
401.2
͑12͒
401.2
͑12͒
and the oxygen-oxygen distances at different pressures in
ice,55 one can find that compared to ice-Ih the O—O dis-
tances are 0.03 Å longer in the dep-ice and ice-VI, and
ϳ0.08 Å longer in the hda-ice. Thus the estimated oxygen-
oxygen separations are ϳ2.78 Å in the dep-ice and ice-VI,
and ϳ2.83 Å in the hda-ice. The similar estimates from Ra-
man data for hda-ice made in Ref. 16 was 2.82 Å. The struc-
tural x-ray4 analysis for hda-ice gives a value of 2.8 Å. From
the ND study56 the oxygen-oxygen distances in ice-VI are
2.726 to 2.788 Å ͑the average is 2.774 Å͒.
behavior of the high energy part of the translational band.
The spectra for dep-ice compared to ice-Ih show softening of
the entire librational band by 3.5 meV, and a shift to higher
energy for the bending mode peak by 5.9 meV ͑difference
between the main peaks͒ and for the stretching mode peak by
4.4 meV. The librational bands in the IINS spectra of hda-ice
and ice-VI show similar behavior, but the bending and
stretching modes peaks for hda-ice are shifted to higher en-
ergy compared to ice-VI by about 3.6 and 8.5 meV, respec-
tively. The shifts to lower energy for the librational band and
to higher energy for the bending and stretching modes can be
explained by the fact that the oxygen-oxygen distances be-
tween nearest-neighbor water molecules are longer: ͑i͒ in
the dep-ice than in other low-density ices, such as ice-Ih,
ice-Ic and lda-ice; and ͑ii͒ in hda-ice than in ice-VI. The
peaks in the IINS spectra at about 285 meV and the left-hand
shoulder around 185 meV are of MPNS origin. For this
reason the bending mode peaks seen could be described
by a single Gaussian only ͑the higher energy component in
Table I͒.
VII. CONCLUSIONS
The measured IINS spectra for the amorphous H2O and
D2O dep-ice and hda-ice, compared to their crystalline ana-
logues, such as ice-Ih and ice-VI, show that the energy po-
sitions of the first peak in the acoustic part of the spectra are
about the same, but the data reveal additional low-energy
excitations below the first acoustic peak and ‘‘smoothed’’
The data obtained clearly indicate that the low-
temperature and low-rate water vapor deposition has pro-
duced a low-density amorphous ice, which differs from an-
other low-density amorphous ice, obtained by heating to
ϳ120 K the hda-ice at ambient pressure, and very strongly
differs from the hda-ice. The hda-ice, produced by pressuris-
ing the ice-Ih at 77 K to 15 kbar, has a lot of similarities with
ice-VI, obtained at the same pressure at room temperature
and then slowly cooled down to 77 K.
ACKNOWLEDGMENTS
This work was supported by a research grant from the
Engineering and Physical Sciences Research Council, and
we acknowledge the use of the neutron facility at the Ruth-
erford Appleton Laboratory ͑UK͒ and at Argonne National
Laboratory ͑supported by the U.S. DOE-BES under Contract
No. W-31-109-ENG-38͒.
FIG. 10. The stretching mode peaks for H2O dep-ice ͑solid
points and solid curve 1͒ and ice-Ih ͑open points and dashed curve
2͒ derived from the IINS spectra measured on HET after correction
for MPNS and time-independent background.

3578
KOLESNIKOV, LI, PARKER, ECCLESTON, AND LOONG
PRB 59
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