Dispersion Relation of an OH-Stretching Vibration from Inelastic X-Ray Scattering
Bjo¨rn Winkler,* Alexandra Friedrich, Dan J. Wilson, and Eiken Haussu¨hl
Geowissenschaften, Goethe-Universita¨t, Altenhoeferallee 1, D-60438 Frankfurt a.M., Germany
Michael Krisch and Alexei Bosak
European Synchrotron Radiation Facility, 6 rue Jules Horowitz, B.P. 220, F-38043 Grenoble Cedex, France
Keith Refson
Rutherford-Appleton Laboratory, Building R3, Chilton, Didcot, Oxfordshire OX11 0QX, United Kingdom
Victor Milman
Accelrys, 334 Cambridge Science Park, Cambridge CB4 0WN, United Kingdom
(Received 18 January 2008; revised manuscript received 30 April 2008; published 4 August 2008)
We show that recent advances now allow us to measure the wave vector dependence of OH-stretching
frequencies at energies around 400 meV by inelastic x-ray scattering using ID28@ESRF. We found a
large, unexpected dispersion when we measured the dispersion relations of the hydrogen stretching
frequencies of diaspore,
-AlOOH, where the hydrogen atoms participate in a hydrogen bond of
intermediate strength. We can account for this behavior with density functional perturbation theory
calculations and a simple model based on H-H interactions.
DOI: 10.1103/PhysRevLett.101.065501 PACS numbers: 63.20.D

Hydrogen bonds influence structure-property relations
in a very large variety of compounds. (For a definition of
hydrogen bonding and a review of earlier work, see [1,2].)
Consequently, numerous experimental techniques are em-
ployed to study the detailed atomic arrangement and the
dynamics of R
O
H . . . O
R0 groups, where R and R0
represent parts of a structure, O-H represents a signifi-
cantly covalent bond with a typical distance of dOH 
1
A, and dH . . . O represents the hydrogen bond between
the hydrogen and the acceptor atom, with a typical distance
of dH . . . O  1:5–1:72
A.
The dynamics of hydrogen bonds in crystals, melts,
and glasses are routinely characterized using infrared or,
to a lesser extent, Raman spectroscopy, where weak hydro-
gen bonds typically have O-H-stretching frequencies of
OH  3600 cm
1 450 meV  108 THz, interme-
diate hydrogen bonds have OH  2800–3100 cm
1,
and strong hydrogen bonds have OH< 2700 cm
1.
However, both infrared and Raman spectroscopy probe
the hydrogen dynamics in the long wavelength limit and
cannot provide information on the wave vector dependence
of the stretching vibration. This is a significant limitation,
as this prevents the elucidation of the coupling of the
hydrogen dynamics to the remainder of the lattice dynam-
ics. Inelastic neutron spectroscopy, INS, can, at least in
principle, provide such information, but at reactor sources
the flux for measurements of energy transfers above
150 meV is too low for such measurements. At spallation
sources, transitions at such high energy transfers can, in
principle, be observed, but there is an intrinsic limitation to
the energy resolution of E=E  5%. Also, the large
incoherent scattering of hydrogen will be problematic.
Hence, the only technique currently available to probe
the dispersion relation of OH-stretching vibrations is in-
elastic x-ray spectroscopy, IXS. However, an earlier at-
tempt to measure OH-stretching modes failed despite
extended measuring times [3].
An alternative to the experimental investigations men-
tioned above are studies based on quantum mechanical
model calculations. Specifically, the parameter-free predic-
tion of the lattice dynamics of crystals has been one of the
outstanding successes of density functional perturbation
theory (DFPT) ([4,5]). DFPT has been successfully used
to predict and interpret phonon dispersion curves of insu-
lators, semiconductors, and metals at ambient conditions
and at high pressures. However, as far as we are aware, the
experimental confirmation of DFT predictions of disper-
sion curves has been limited to relatively low frequencies
such as those probed by INS.
Here, we present a combined theoretical and experimen-
tal study of the OH-stretching vibrations in diaspore,
AlOOH, using density functional perturbtion theory and
inelastic x-ray scattering. Diaspore belongs to the family of
oxy-hydroxides and is isostructural to goethite, FeOOH,
and manganite, MnOOH [6]. The structure is represented
in Fig. 1.
Diaspore is a model system to study intermediate hydro-
gen bonds, as it has a comparatively small unit cell, simple
chemistry, and is of intermediate (orthorhombic) symme-
try. It has recently been studied by DFT calculations and by
high-pressure single crystal x-ray diffraction up to
0.5 Mbar [7–10]. The hydrogen bond is slightly kinked,
and the OH-stretching frequencies of the four symmetri-
cally inequivalent modes at ambient pressure are centered
PRL 101, 065501 (2008) P H Y S I C A L R E V I E W L E T T E R S week ending8 AUGUST 2008
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around 2950 cm
1. A detailed study on the band assign-
ment of phonons at the -point, based on DFT calculations
has recently been presented [9].
The phonon dispersion curves of the hydrogen bond of
diaspore (Fig. 2) have been predicted based on DFPT
calculations using the CASTEP code. These calculations
are extensions of those presented in earlier work for the
pressure-dependence of -point phonons [8], and details of
the computational procedure are given there. We see a
significant LO/TO splitting of 22 meV. For several di-
rections in the Brillouin zone, BZ, our calculations pre-
dicted a large dispersion of the OH-stretching frequencies.
Specifically, at reciprocal lattice points 
0:5 0:5 0 and

0:5 0 0, there are two doubly degenerate vibrations
with frequencies of 357 and 375 meV, respectively, while
at 
0:5 0 0:5, all four stretching vibrations have frequen-
cies of 358 meV (Fig. 2).
Predictive lattice dynamics calculations simplify IXS
measurements dramatically, as the computed dynamical
matrix can be used to calculate the dynamical structure
factor for a variety of wave vector transfers, and hence,
optimal scattering geometries can be chosen in advance.
Such calculations showed that the intensities of the hydro-
gen stretching vibrations were best measured using recip-
rocal lattice points at 2:5 0
1 and 2:5 0 0 in order to
obtain spectra dominated by either one or the other doubly
degenerate stretching mode. The expected spectra are
shown in Fig. 3.
The experiment was carried out on the IXS beam line II
(ID28) at the European Synchrotron Radiation Facility in
Grenoble, France. The instrument was operated using the
silicon 8 8 8 configuration, which provides a total in-
strumental energy resolution of 6 meV full-width-half-
maximum (FWHM). The momentum resolution was set
by slits in front of the analyzers to 0:8 nm
1. The trans-
verse dimensions of the focused x-ray beam at the sample
position were 270 80 m2 horizontally and vertically,
respectively. The spectrometer operated in the horizontal
scattering plane, and seven IXS spectra were recorded
simultaneously. The energy scans were performed by vary-
ing the monochromator temperature while the analyzer
temperature was kept fixed. Conversion from the tempera-
ture scale to the energy scale was accomplished by using
the known thermal expansion coefficient of silicon:

T 
0  T
T0 with
0  2:581 10
6 1=K
and   0:016 10
6 1=K2, T0  295:5 K [11]. The va-
lidity of this conversion has been checked by comparing
the measured diamond dispersion curve for longitudinal
acoustic phonons with well established inelastic neutron
scattering results. The overall experimental resolution
(6 meV FWHM) was experimentally determined by mea-
suring the scattering from a disordered sample of Plexiglas
at a Q-transfer of 10 nm
1, corresponding to the first
maximum in the static structure factor SQ, and at T 
10 K in order to maximize the elastic contribution to the
FIG. 2. Prediction of the dispersion relation of the OH-
stretching vibrations along high symmetry directions in the BZ
from DFPT calculations.
energy transfer [meV]
0 100 200 300 400
In
te
ns
ity
[a
rb
. u
nit
s]
102
103
104
105
106 (2.5 0 0)
(2.5 0 1)
FIG. 3. Based on DFPT calculations, dynamical structure fac-
tors and hence IXS spectra were predicted in order to optimize
the measurement strategy.
FIG. 1. The structure of diaspore. We use a setting in space
group Pbnm, with a  4:401
A, b  9:421
A, c  2:845
A [6].
Black, gray, and light gray spheres represent Al, O, and H-atoms,
respectively. Dashed lines indicate hydrogen bonds.
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scattering. Further experimental details can be found else-
where [12].
From a gem quality natural sample of unknown origin,
we have cut a 750 m
thick (001)-slice, with the [100]
and [010] directions parallel to the edges of the slice. The
sample was glued onto a standard goniometer head. The
measurements were performed at ambient conditions. Scan
times ranged from 60 to 2040 seconds per step.
Typical spectra over the whole energy range, composed
of several scans with varying counting times, are shown in
Fig. 4.
Because of recent instrumental improvements, we were,
in contrast to earlier measurements [3], able to clearly
identify the OH-stretching modes. The measured spectra
are in very close agreement to the predicted spectra. This
semiquantitative comparison shows that the dynamical
structure factors obtained from the DFPT calculations
generally give the correct frequencies and intensity ratios.
For a more detailed data evaluation, we used an in-house
fitting program at the ESRF. The interpretation of these
results will be given elsewhere, as here we will focus on the
OH-stretching frequencies only.
At high energies, we could clearly distinguish between
the two doubly degenerate OH-stretching modes, by mea-
suring the scattering at 2:5 0
1 and 2:5 0 0 (Fig. 5).
The high frequency mode has an energy of
374.0(8) meV, while the low frequency mode has an energy
of 357(1) meV. The splitting between these two modes is
therefore 17 meV. DFPT had predicted energies of 375 and
357 meV, respectively, and hence a splitting of 18 meV.
Both modes have a FWHM of about 34 meV. This is
significantly larger than the instrumental resolution of
about 6 meV. The origin of this broadening is not obvious
at the moment, as there are no indications for static or
dynamic disorder of the hydrogen atoms or of an unusually
anharmonic OH bond. It is, however, worthwhile to note
that the widths of the OH stretching vibrations measured
with infrared spectroscopy are unusually large (up to
100 meV according to [9]), and hence we conclude that
for currently unknown reasons, the OH-stretching frequen-
cies in diaspore are intrinsically very broad.
An analysis of the eigenvectors shown in Fig. 6 shows
that theses modes are nearly pure OH-stretching vibrations.
For the high frequency phonon, the H-H distances re-
main constant during the vibration at about 2.42 A˚ , while
for the antisymmetric low frequency phonon, the hydro-
gens are further part for most of the time. Hence, the H-H
repulsion energy is lowered as the mode amplitude in-
creases, lowering the potential and consequently the fre-
quency of this mode.
Energy transfer [meV]
-100 0 100 200 300 400 500
In
te
ns
ity
[a
rb
. u
nit
s]
102
103
104
105
106
(2.5 0 0 )
Energy transfer [meV]
-100 0 100 200 300 400 500
In
te
ns
ity
[a
rb
. u
nit
s]
103
104
105
106 (2.5 0 -1)
FIG. 4. Exprimentally determined (points) and predicted (lines) phonon spectra of diaspore at 2:5 0 0 and 2:5 0 1 from inelastic
x-ray scattering and DFPT, respectively.
energy transfer [meV]
300 320 340 360 380 400 420 440 460
In
te
ns
ity
[a
rb.

u
n
its
]
0
20
40
60
80
100
120
140
160
(2.5 0 -1)
Energy transfer [meV]
300 320 340 360 380 400 420 440 460
In
te
ns
ity
[a
rb.

u
n
its
]
0
10
20
30
40
50
60
70
(2.5 0 0)
FIG. 5. Left: Spectrum measured at 2:5 0
1, which is dominated by an OH-stretching vibration having an energy of 374 meV.
Total counting time was 2040 s per step. Right: Spectrum measured at 2:5 0 0, which is dominated by an OH vibration having an
energy of 357 meV. Total counting time was 1380 s per step.
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The scattered intensities measured with the remaining
six detectors give dispersion relations. These, however,
will generally not coincide with high symmetry directions.
Hence, we performed additional DFPT calculations. These
are compared to the experimental values in Fig. 7.
Because of the low symmetry, there are no degeneracies.
Without further measurements, it is not obvious if for the
specific scattering geometry we really have sampled the
lower branch, as would be implied by the excellent agree-
ment shown in Fig. 7. However, this is not of importance,
as irrespective of which branch we have actually observed,
the dispersion is clearly present.
In summary, the agreement of the DFPT results with
experiment for high frequency phonons is as good as it is
for low frequency phonons and thus clearly confirms the
predictive power of the DFPT calculations. The experi-
ments show that the OH-stretching frequencies of hydro-
gen bonds of intermediate strength can have a wave vector
dependence and that the dispersion relation can be ob-
served by inelastic x-ray scattering. However, appreciable
counting times are required. Hence, this technique will
generally not be applicable to study the pressure-
dependence of the dynamics of hydrogen bonds, as dia-
mond anvil cell experiments require thin samples, which
would lead to unreasonable counting times. This implies
that our prediction of a threefold increase of the dispersion
at 30 GPa in comparison to ambient pressure will remain
untested in the foreseeable future, but due to the predictive
power of the DFPT calculations demonstrated here, we
believe that this prediction is reliable. In contrast, tempera-
ture dependent measurements are unproblematic, and
hence IXS can be used to study the origin of phase tran-
sitions governed by the dynamics of the hydrogen bonds.
This study was partially funded by the German Science
Foundation and the European Science Foundation within
the framework of the ESF-Eurocores EuroMinScI, which
is funded from the EC 6th framework program under
Contract No. ERAS-CT-2003-980409. Additional funding
has been provided by the MaterialsGrid project (www.
materialsgrid.org). The computations have been performed
on computers of the ‘‘Center for Scientific Computing’’ of
the Goethe-Universita¨t Frankfurt.
*b.winkler@kristall.uni-frankfurt.de
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wave vector [A -1]
0.00 0.02 0.04 0.06 0.08 0.10 0.12 0.14 0.16
en
er
gy
[m
eV
]
355
360
365
370
375
380
(0.5, 0, 0) to (0, 0, -0.33)
FIG. 7. Dispersion of the OH-stretching frequencies along a
low symmetry direction. The left vertical line corresponds to
0:5 0 0, the right vertical line to 0 0
0:333.
FIG. 6 (color online). Eigenvectors of the phonons at the BZ
boundary at 0:5 0 0, showing that during the high frequency
vibration (left) the hydrogen remain at a constant distance, while
during the low frequency motion (right) the hydrogens move
further apart.
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