Geoscience Reference
In-Depth Information
acoustic wave velocities (one compressional wave
velocity and two shear wave velocities) along
various directions using the Christoffel equation
(Musgrave
et al
., 1970):
λ
θ
|
C
ijkl
n
j
n
l
−
ρV
2
δ
ik
|=
det
0
(6.2)
sample
where
C
ijkl
is the elastic modulus tensor,
n
j
and
n
i
are the direction cosines of the phonon,
ρ
is
the density,
V
is the phonon velocity, and
δ
ik
is
the Kronecker delta.
There is no unique solution for determining the
aggregate shear properties from the single-crystal
properties; however, there are well-established
semi-empirical bounding procedures, and the ag-
gregate properties such as
K
S
,
G
,
V
S
,and
V
P
are
conventionally calculated from
C
ij
's using the
Voigt-Reuss-Hill averaging scheme (Watt
et al
.,
1976). When the elastic anisotropy is not large (in
a single phase aggregate) or when the difference
in elastic constants is not large (in a multi-phase
aggregate), the choice of averaging scheme does
not have a serious effect on the result.
Major advantages of this technique are that
a large sample volume is not required and the
results are not significantly affected by porosity,
micro-cracks, inclusions in the sample, and by the
uncertainties of sample dimensions. In addition,
the elastic anisotropy can be easily determined
from the single-crystal samples with this method
by rotating the sample perpendicular to the inci-
dent probe light.
Stokes
Anti-Stokes
Brillouin peaks
Brillouin peaks
LA
LA
TA
TA
Frequency shift (GHz)
Fig. 6.1
Schematic drawings of the experimental
configuration for the Brillouin scattering spectroscopy
in a symmetric scattering geometry from Murakami
et al
. ( 2009), and of the representative Brilluoin
spectrum. TA and LA respectively indicates the
transverse and longitudinal acoustic modes of the
Brillouin peaks from the sample. Reproduced with
permission of Elsevier.
where
λ
is the probe laser wavelength and
θ
is
the external scattering angle (Figure 6.1). Unlike
Raman shifts, which are typically tens to thou-
sands wave numbers (cm
−
1
) from Rayleigh line
derived from the elastic scattering of the incident
probe laser, Brillouin frequency shifts are usually
less than 1 cm
−
1
, which corresponds to the fre-
quency range from a few GHz to tens of GHz. To
observe such small frequency differences of
ω
,
the scattered light from the sample is normally
analyzed by a multi-pass tandem Fabry-Perot in-
terferometer to obtain extremely high resolution
and to reject the background caused by Rayleigh
scattering of the incident beam.
For seismological applications, acoustic wave
velocities are all we need, but one can also cal-
culate the elastic constants if one measures three
6.2.1 Challenges in high-pressure
measurement
Since the Brillouin spectroscopic technique is
based on the scattered light transmitted through
the sample, the diamond anvil cell high-pressure
apparatus is perfect match for Brillouin scattering
measurements under pressures. Diamond anvil
cell high-pressure apparatus (DAC) can stably
generate extremely high static pressures (Mao
et al
., 1989, 1990), and the recent advancement of
infrared laser heating technique in a DAC com-
bined with synchrotron radiation have allowed
us to explore successfully the phase equilibria
