Download Brillouin scattering at high pressure: an overview

JOURNAL OF RAMAN SPECTROSCOPY
J. Raman Spectrosc. 2003; 34: 633–637
Published online in Wiley InterScience (www.interscience.wiley.com). DOI: 10.1002/jrs.1031
Brillouin scattering at high pressure: an overview
Alain Polian∗
Physique des Milieux Condensés, CNRS–UMR 7602, Université Pierre et Marie Curie, B77, 4 Place Jussieu, 75252 Paris Cedex 05, France
Received 21 March 2003; Accepted 8 April 2003
Brillouin scattering allows the determination of acoustic velocities and adiabatic elastic moduli in matter.
These data are crucial in many areas of science, such as fundamental physics, geosciences and technology,
especially when measured as a function of the density (pressure). In this paper, we present a review of
the work performed on Brillouin scattering under high pressure in diamond anvil cells, emphasizing the
most recent results. Copyright  2003 John Wiley & Sons, Ltd.
KEYWORDS: Brillouin scattering—High pressure—High temperature—Elastic properties
INTRODUCTION
Sound waves propagate in matter, independently of its structure. Brillouin scattering consists in the inelastic scattering of
light by the sound waves, or thermal excitation in a material,
in particular acoustic phonons in a crystal. The energy of
light is therefore modified, increased in the case of the annihilation of excitation, or decreased in the case of creation of
excitation. The measurement of this energy difference gives
information on the energy of the phonons and therefore on
the interatomic potentials of the material.
The theory of Brillouin scattering has been extensively
developed in the literature,1 – 4 so only the main ideas
will be covered here. In the interaction process between
a photon and thermal excitation, the energy and momentum
are conserved:
h̄ D šh̄ωi ωs 1
qE D šEki Eks 2
where ω and k are the wavenumber and the wavevector
of the photon, the subscripts i and s refer to the incident
and scattered beams and and q are the wavenumber
and the wavevector of the phonon, respectively, the plus
sign corresponds to the creation of a phonon (Stokes) and
the minus sign to annihilation (anti-Stokes). It should be
noted that the equations of conservation hold for both
Brillouin scattering and Raman scattering because the
physical processes are identical. From the experimental point
of view, the differences are large, coming from the difference
in the energies involved in the processes: the wavenumbers of
Ł Correspondence to: Alain Polian, Physique des Milieux
Condensés, CNRS–UMR 7602, Université Pierre et Marie
Curie, B77, 4 Place Jussieu, 75252 Paris Cedex 05, France.
E-mail: [email protected]
the optical phonons giving rise to Raman scattering are of the
order of 10–1000 cm1 (1.2–120 meV), whereas the Brillouin
wavenumbers are of the order of 0.1–6 cm1 . Because of
this proximity with the elastically scattered Rayleigh line,
apparatus of much higher resolution has to be used to
discriminate the Brillouin scattered light compared with
Raman scattering where a grating spectrometer is well suited.
Such a high-resolution apparatus is the Fabry-Pérot (FP)
interferometer. Recent advances in Fabry-Pérot spectroscopy
have been reviewed5 and will be described here only briefly.
A Fabry-Pérot system consists basically of two partially
transmitting plane mirrors, separated by a distance d. A
parallel beam of light is incident along the mirror axis.
Owing to interferences between the two parallel mirrors, the
light is transmitted only when
nd D m
2
3
where n is the refractive index of the medium between the
two mirrors, is the wavelength of the light and m is an
integer. Therefore, the wavelength of the transmitted light
is selected by varying the optical path nd of the light, by
changing either the distance between the mirrors or the
refractive index of the inter-mirror medium.
Two parameters describe the quality of an FP system:
the first is the contrast (C), defined as the ratio of the
maximum intensity transmission to the minimum intensity
transmission, and this quantifies the ability of the set-up
to measure signals that are small in comparison with the
Rayleigh scattering. The second is the finesse (F), defined as
the ratio of the distance between two successive transmission
peaks to the full width at half-maximum of a transmission
peak, and provides the capability of measuring peaks close
to the Rayleigh peak. The most important parameter is the
contrast. For a good quality FP system, one typically obtains
Copyright  2003 John Wiley & Sons, Ltd.
634
A. Polian
contrasts of the order of a few hundred, which is far from
sufficient to measure Brillouin scattering in most materials.
The solution to increase considerably the contrast is the
multipass FP system, which was introduced by Sandercock6
and yields measured contrasts above 1010 .
One measurement is not sufficient to determine the
wavenumber of a phonon: this quantity is only determined
modulo one ‘free spectral range’ (FSR), i.e. a wavenumber
equal to the distance between two successive transmission
peaks of the FP. The solution was again given by Sandercock,7
with the tandem FP interferometer, where two interferometers with different spacings are scanned synchronously.
BRILLOUIN SCATTERING
Brillouin scattering provides information about the acoustic
branches at a given point of the dispersion curves of the
material under study. The measured wavenumber shift of the
radiation is equal to that of the phonon under consideration
[Eqn (1)], and its wavevector is deduced from Eqn (2), so the
sound velocity may be calculated by
V D /q
4
This relation may also be written as
 cm1 D
2nV
sin
c
2
5
where  is the measured wavenumber shift (in cm1 , n
is the refractive index at , the wavelength of the exciting
light, c is the speed of light in vacuum and is the angle
between ki and ks . The sound velocity is related to the elastic
constants by
X D V 2
6
where X is a combination of elastic constants depending on
the direction of propagation of the phonon with respect to
the crystallographic axis and is the density. For example,
in a cubic crystal, a longitudinal phonon propagating along
the [110] direction is related to X D C11 C C12 C 2C44 /2
and the two non-degenerate transverse phonons are related
to X D C11 C12 /2 and X D C44 . Therefore, by correctly
choosing the geometry of the experiment, it is possible to
determine the complete set of elastic constants.
Figure 1 shows scattering geometries commonly used
in order to determine the elastic constants. In Fig. 1(a), the
p
modulus of the phonon wavevector is equal to nk 2, where
k ³ ki ³ ks . In Fig. 1(b), jqj D 2nk. This is the geometry
that produces the largest wavenumber shift. Figure 1(c)
shows an interesting geometry for Brillouin scattering under
pressure (provided that an ad hoc cell is available), because
measurement of the refractive index is not required. Indeed,
in the crystal the modulus of the wavevector of light is equal
to nk. Owing to the incidence angle, jqj D 2nksin(r), where
Copyright  2003 John Wiley & Sons, Ltd.
ki
ki
kd
kd
q
i
q
q
i
r
kd
(a)
ki
(b)
(c)
Figure 1. Most commonly used geometries in Brillouin scattering, (a) 90° scattering, jqj D 21/2 nk; (b) backscattering geometry, jqj D 2nk; (c) platelet geometry, jqj D 2nksin i D 2ksin r.
r is the angle of the refracted photon inside the crystal with
respect to the normal to the surface. Using the Descartes law
of refraction, one obtains jqj D 2ksin(i), where i is the known
angle of incidence.
The knowledge of the relative orientation of ki and ks with
respect to the crystallographic axes allows the determination
of that of q, and the selections rules2 allow one to deduce the
related combination of elastic constants.
BRILLOUIN SCATTERING IN A DIAMOND
ANVIL CELL
With a ‘classical’ diamond anvil cell (DAC), i.e. with only
optical access along the axis of the diamond, it is basically
only possible to use the backscattering geometry (only one
experiment has been performed in the platelet geometry
with this type of cell).8 Unfortunately in the backscattering
geometry, the selection rules favor the longitudinal mode,
so most of the experiments performed in this geometry
only give information on these phonons. Another problem
which occurs is due to the diamond anvils which have also
acoustic phonons, and the difference in volume between the
anvils and the sample make the intensity of the Brillouin of
the former much more intense than that of the latter. The
problem of the longitudinal modes of diamonds may be
avoided by using a tandem FP set-up.
The ‘Swiss cheese’ DAC9 enables one to measure
in the platelet geometry, and then gives information
on the transverse modes. Nevertheless, the maximum
pressure is achieved, on average, in a backscattering
geometry.10
Once the spectrum has been recorded, the sound velocity
can be computed, which means that [Eqn (5)] the refractive
index at the wavelength of the laser and at the desired density
should be measured or computed. The measurement of the
pressure dependence of the refractive index of compounds
is not easy. Various techniques have been proposed and
applied in a few cases for H2 and D2 ,11,12 GaS,13 xenon,14
argon15 and helium.16 In case where n is not measured
with pressure, a good approximation is given by the
Clausius–Mossotti equation, provided that the equation of
state is known.
Once the sound velocity has been determined, Eqn (6)
gives the effective elastic modulus corresponding to the
J. Raman Spectrosc. 2003; 34: 633–637
Brillouin scattering at high pressure
direction of propagation of the measured phonon. In order
to determine the linear combination of elastic constants
involved, one has to know the crystallographic orientations
of the crystal inside the DAC. This may be done when the
sample is solid under ambient conditions. When the sample
is loaded as a fluid in the DAC, there is no control on
the crystal orientations, and the determination of individual
elastic constants is not straightforward using the classical
methods. The first attempt17 was made in backscattering
geometry. In this geometry, only the longitudinal acoustic
(LA) modes are detected. The pressure dependence of the
velocity was measured for unknown directions, without
identifying the crystal orientation. The envelope for all
experimental points corresponds to maximum LA velocities,
v2 D [C11 C C12 C 4C44 ]/ along the h111i direction and
to minimum velocities along h100i, v2 D C11 /. For the
determination of all the elastic moduli, the isothermal bulk
modulus B D C11 C 2C12 /3 is used. Two problems may
arise by using this technique. First, it is assumed that the
maximum and minimum values of the sound velocities are
really reached, and second, the ratio of the specific heats,
cp /cv , is supposed to be constant and equal to 1.
The problem was solved for cubic crystals by Shimizu
and Sasaki18,19 using the platelet geometry, a large aperture
cell and a method first proposed by Every.20 Rotating a
crystal with an unknown crystallographic orientation in the
platelet geometry results in the determination of a set of
curves of the angular dependence of the speed of sound. In
an arbitrary direction, the sound velocity is related to the
elastic moduli and the direction cosines n1 , n2 and n3 of the
wavevector relative to the axes of the cubic crystal by
v2j D fj C11 , C12 , C44 , n1 , n2 , n3 7
where is the density of the crystal. The wavevector direction
may be expressed as a function of the Euler angles of the
crystal, , ϕ, and hence:
vj D gj C11 /, C12 /, C44 /, , ϕ, 8
The rotation angle ϕ is varied, and vj is then determined
by a least-squares fit between the calculated gj ϕ and
the experimental elastic moduli. Using this technique, the
orientation of the crystal is also precisely obtained.
A similar, but somewhat simpler, procedure is now
commonly used when working on compounds which
are crystalline under ambient conditions, i.e. in platelet
geometry, and rotating the cell around the load axis. The
orientation of the sample is known from the beginning, and
in that case the full set of elastic moduli may be obtained for
samples of lower symmetry. Nevertheless, it should be noted
that the highest pressures are obtained in the backscattering
geometry.29,34
Table 1 presents an extensive review of the results
obtained at ambient temperature in a DAC. It is clearly
Copyright  2003 John Wiley & Sons, Ltd.
Table 1. Summary of the compounds studied by Brillouin
scattering in a DAC
Material
Scattering geometry
NaCl
H2 , D2
a-SiO2
a-SiO2
a-SiO2
GaS
Mg2 SiO4
Mg2 SiO4
H2 O, D2 O
H2 O-VII
a-B2 O3
a-GeO2
a-As2 S3
TiO2
N2
Ar
Ar
HF
He
CH4
CH4
CS2
Kr
Kr
BaTiO3
Platelet
Platelet
Backscattering
Platelet
Platelet
Backscattering
Backscattering
Platelet C rotation
Backscattering
Platelet C rotation
Backscattering
Backscattering
Backscattering
Backscattering
Backscattering
Backscattering
Combined
Platelet
Backscattering
Backscattering
Platelet C rotation
Platelet
Backscattering
Platelet C rotation
Backscattering and
platelet
Backscattering and
platelet
Backscattering
Platelet C rotation
Backscattering
Platelet
Platelet
Backscattering
Platelet
Platelet C rotation
Backscattering
Platelet C rotation
Platelet C rotation
Platelet C rotation
Backscattering
Platelet C rotation
Platelet C rotation
Platelet C rotation
Platelet C rotation
Platelet C rotation
Platelet C rotation
Platelet C rotation
SrTiO3
NH3
NH3
C6 H6
CH3 OH
ZnO
Xe
Pentane/isopentane
H2 S
NaCl, KCl
CO2
H2
N2 O
1-Propanol
(MgFe)SiO4
HCl
MgO
MgO
HBr
Quartz SiO2
CaF2
Max.
pressure
(GPa)
Ref.
4
20
17
14
57.8
18
4
16
74
8
16
20
4
14
20
35
70
12
20
32
5
7
33
8
11 and 8
21
22
23
24
25
26
27
28
29
30
31
33
33
32
33
17
34
8
35
36
37
38
39
40
41
36 and 9
42
20
3.5
30
8.4
4
31
12
7
17
6
24
4.5
8.3
32
8
20
50
7
20
9.3
43
44
45
46
47
10
48
49
50
51
52
53
54
55
56
57
58
59
60
61
J. Raman Spectrosc. 2003; 34: 633–637
635
A. Polian
seen that in recent years, most of the experiments have been
performed in the platelet geometry, so that the complete set
of elastic moduli is obtained up to the highest pressure.
Most of the results obtained to date at high pressure in a
DAC were obtained at ambient temperature. Nevertheless,
some preliminary results at high temperatures have been
published on liquid H2 O62 and low-temperature data were
obtained on H2 S.63 Figure 2 represents the Brillouin spectrum
obtained at high pressure and high temperature in water.
This study was undertaken up to 450 ° C and 9 GPa (F. Datchi
and A. Polian, to be published).
One application of the determination of the full elastic
moduli set is the establishment of a thermodynamic pressure
scale. The measurement of both the volume and the elastic
constants as a function of the same secondary pressure scale
8 ×104
H2O
P=0.9 GPa, T=450 °C
Intensity/counts
6×104
R2
B1
B2
0
-1.5
-1.0
1.4
-0.5
0.0
0.5
1.5
1.6
1.0
1.7
1.5
∆σ/cm-1
Figure 2. Brillouin scattering of liquid H2 O at 0.9 GPa and
450 ° C. The peaks labeled Ri are the Rayleigh lines and Bi are
the Stokes and anti-Stokes lines of water (F. Datchi and
A. Polian, to be published).
1400
1200
1000
800
600
MgO at 54.7 GPa,
phonon direction [100]
R
R
R
VL(s)
VL(s)
D
D
400
VT(s)
200
0
-1.5
-1.0
VT(s)
-0.5
0.0
0.5
1.0
1.5
Frequency/cm-1
Figure 3. Brillouin scattering spectrum of MgO at 54.7 GPa,
from Ref. 55. VL and VT are the longitudinal and transversal
sound velocities, respectively, from MgO, R the Rayleigh lines
and D the Brillouin lines from the diamond.
Copyright  2003 John Wiley & Sons, Ltd.
B
A
1
PGPa D
B
0
9
hold with A D 1904 GPa and B D 7.715, in good agreement
with the previous determination with A D 1904 GPa and
B D 7.665.
CONCLUSION
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R3
R1
-1.7 -1.6 -1.5 -1.4
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(Fig. 3). The primary pressure scale determined in such a
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4×104
2×104
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