Single-crystal elasticity of grossular-and almandine-rich garnets to 11 GPa by Brillouin scattering

Abstract

1] The high-pressure elasticity of grossular-rich Grs 87 And 9 Pyp 2 Alm 2 and almandine-rich Alm 72 Pyp 20 Sps 3 Grs 3 And 2 natural garnet single crystals were determined by Brillouin scattering to 11 GPa in a diamond anvil cell. The experiments were carried out using a 16:3:1 methanol-ethanol water mixture as pressure medium. The aggregate moduli as well as their pressure derivatives were obtained by fitting the data to Eulerian finite strain equations. The inversion yields K S0 = 165.0 ± 0.9 GPa, G 0 = 104.2 ± 0.3 GPa, (@K S /@P) T0 = 3.8 ± 0.2, and (@G/@P) 0 = 1.1 ± 0.1 for the grossular-rich composition and K S0 = 174.9 ± 1.6 GPa, G 0 = 95.6 ± 0.5 GPa, (@K S /@P) T0 = 4.7 ± 0.3, and (@G/@P) 0 = 1.4 ± 0.1 for the almandine-rich garnet. Both individual and aggregate elastic moduli of the two garnets define nearly linear modulus pressure trends. The elastic anisotropy of the garnets increases weakly in magnitude with compression. Isothermal compression curves derived from our results are generally consistent with static compression data under hydrostatic conditions, and the effects of nonhydrostaticity on previous diffraction data can be identified. The pressure derivatives obtained here are generally lower than those reported in high-pressure polycrystalline ultrasonic elasticity studies. In combination with earlier Brillouin scattering data for pyrope, our results allow us to constrain the effect on elastic moduli of Fe 2+ -Mg 2+ substitution in pyrope-almandine, Ca 2+ -Mg 2+ in pyrope-grossular, and Fe 3+ -Al 3+ substitution in andradite-grossular at high pressures. This new data set thus allows us to place improved constraints on the compositional dependence of seismic velocities in the rocks of the upper mantle.

Full text

Single-crystal elasticity of grossular- and
almandine-rich garnets to 11 GPa by Brillouin
scattering
Fuming Jiang, Sergio Speziale, and Thomas S. Duffy
Department of Geosciences, Princeton University, Princeton, New Jersey, USA
Received 11 March 2004; revised 23 June 2004; accepted 2 August 2004; published 28 October 2004.
[1]The high-pressure elasticity of grossular-rich Grs
87
And
9
Pyp
2
Alm
2
and almandine-
rich Alm
72
Pyp
20
Sps
3
Grs
3
And
2
natural garnet single crystals were determined by
Brillouin scattering to 11 GPa in a diamond anvil cell. The experiments were carried out
using a 16:3:1 methanol-ethanol water mixture as pressure medium. The aggregate
moduli as well as their pressure derivatives were obtained by fitting the data to Eulerian
finite strain equations. The inversion yields K
S0
= 165.0 ± 0.9 GPa, G
0
= 104.2 ±
0.3 GPa, (@K
S
/@P)
T0
= 3.8 ± 0.2, and (@G/@P)
0
= 1.1 ± 0.1 for the grossular-rich
composition and K
S0
= 174.9 ± 1.6 GPa, G
0
= 95.6 ± 0.5 GPa, (@K
S
/@P)
T0
= 4.7 ± 0.3,
and (@G/@P)
0
= 1.4 ± 0.1 for the almandine-rich garnet. Both individual and aggregate
elastic moduli of the two garnets define nearly linear modulus pressure trends. The
elastic anisotropy of the garnets increases weakly in magnitude with compression.
Isothermal compression curves derived from our results are generally consistent with
static compression data under hydrostatic conditions, and the effects of nonhydrostaticity
on previous diffraction data can be identified. The pressure derivatives obtained here are
generally lower than those reported in high-pressure polycrystalline ultrasonic elasticity
studies. In combination with earlier Brillouin scattering data for pyrope, our results
allow us to constrain the effect on elastic moduli of Fe
2+
-Mg
2+
substitution in
pyrope-almandine, Ca
2+
-Mg
2+
in pyrope-grossular, and Fe
3+
-Al
3+
substitution in
andradite-grossular at high pressures. This new data set thus allows us to place
improved constraints on the compositional dependence of seismic velocities in the rocks
of the upper mantle. INDEX TERMS:3909 Mineral Physics: Elasticity and anelasticity; 3924
Mineral Physics: High-pressure behavior; 3934 Mineral Physics: Optical, infrared, and Raman spectroscopy;
3919 Mineral Physics: Equations of state; KEYWORDS:elasticity, garnet, high pressure, diamond anvil cell,
sound velocity, Brillouin scattering
Citation: Jiang, F., S. Speziale, and T. S. Duffy (2004), Single-crystal elasticity of grossular- and almandine-rich garnets to 11 GPa
by Brillouin scattering, J. Geophys. Res.,109, B10210, doi:10.1029/2004JB003081.
1. Introduction
[2] Garnets are abundant minerals in the igneous and
metamorphic rocks of the Earth’s crust as well as important
constituents of the mantle. Natural garnets in peridotites at
pressures greater than 1.5 GPa (45 km depth) in the
Earth’s mantle are pyrope-rich with variable amounts of
grossular and almandine [Rickwood et al., 1968; Lee, 2003].
One of the major phase changes expected under upper
mantle conditions is the dissolution of pyroxene into the
garnet structure producing Al-deficient garnets (majorite)
that are stable to the base of the upper mantle [Ringwood,
1967, 1991; Fei and Bertka, 1999]. Mantle mineralogical
models such as pyrolite and piclogite contain garnet volume
fractions at low pressures of 15% and 22% respectively
and these increase to 40% or more vol. fraction of garnet-
majorite at transition zone conditions [Ringwood, 1991; Fei
and Bertka, 1999]. For MORB compositions, the garnet
fraction ranges from 25% at conditions corresponding to
the top of the upper mantle to 90% majorite-garnet at
transition zone conditions [Irifune and Ringwood, 1993].
Understanding the elastic properties of garnets is thus
essential to the interpretation of regional seismic profiles
of the upper 660 km of the Earth’s interior [Duffy and
Anderson, 1989; Weidner and Wang, 2000]. In addition, the
effect of compositional changes on elastic properties is
important for interpreting lateral variations in seismic
velocity imaged by seismic tomography. The effects of
Mg
2+
-Ca
2+
and Mg
2+
-Fe
2+
substitution on the elastic prop-
erties of mantle minerals at upper mantle conditions are not
well constrained [Karato and Karki, 2001]. Such data are
necessary, for example, in interpreting seismic and geo-
dynamic studies of the continental lithosphere (tectosphere)
in terms of thermal and chemical properties of the region
[Jordan, 1978; Forte and Perry, 2000; Lee, 2003]. The
JOURNAL OF GEOPHYSICAL RESEARCH, VOL. 109, B10210, doi:10.1029/2004JB003081, 2004
Copyright 2004 by the American Geophysical Union.
0148-0227/04/2004JB003081$09.00
B10210 1of10

elastic properties of garnets are also important for modeling
seismic velocities in the lower continental crust [Jackson et
al., 1990; Morozov et al., 2003].
[3] The silicate garnet group X
3
Y
2
(SiO
4
)
3
includes a
series of isostructural species with space group Ia

3d,
where the eight coordinated X-site houses cations such
as Ca
2+
,Mg
2+
,Fe
2+
or Mn
2+
and the six coordinated
Y-site incorporates Al
3+
,Fe
3+
and Cr
3+
among others. The
structure consists of alternating SiO
4
tetrahedra and YO
6
octahedra which share corners to form a three-dimensional
network. The elastic tensor for cubic garnets consists of
three independent elastic stiffness coefficients: C
11
,C
12
,
and C
44
. The bulk moduli of garnets at ambient pressure
can be described by elasticity systematics [Bass, 1986;
Wang and Ji, 2001], and can also be related to the relative
compressibility of different structural units, especially the
dodecahedral site [Milman et al., 2001]. However, a
complete picture of the compositional dependence on the
shear modulus, individual C
ij
s, and all pressure derivatives
is not yet available [e.g., O’Neill et al., 1989; Wang and
Ji, 2001].
[4] Owing to their wide compositional range and the
ready availability of good quality specimens, the elastic
properties of garnets have been studied far more extensively
than any other mineral group [e.g., Wang and Ji, 2001].
Numerous X-ray diffraction studies at high pressure have
reported the isothermal bulk modulus, K
T0
, and its pressure
derivative, (@K
T
/@P)
T0
[Knittle, 1995]. Measurements of the
elastic tensor (and corresponding aggregate bulk and shear
moduli) at ambient and high pressures have been reported
using ultrasonic interferometry or resonant ultrasound spec-
troscopy as well as light scattering methods such as Bril-
louin spectroscopy. At ambient pressure, there is good
agreement for both individual and aggregate elastic proper-
ties among various studies using different experimental
techniques. At high pressures, however, there are disagree-
ments of up to 50% or more in reported pressure derivatives
of the bulk and shear modulus for a given composition. For
example, reported pressure derivatives for the bulk modulus
of grossular range from 4.5 [Weaver et al., 1976] to 6.1
[Olijnyk et al., 1991]. Similarly values of K
0
T0
=(@K
T
/@P)
T0
and G
0
0
=(@G/@P)
0
for majorite garnets differ by 60% and
36% in two reported studies [Gwanmesia et al., 1998;
Sinogeikin and Bass, 2002]. The magnitude of these dis-
crepancies greatly hinders efforts to make geological inter-
pretations on the basis of seismic data [Sinogeikin and Bass,
2002].
[5] In this study, we have carried out single-crystal
elasticity measurements on grossular-rich (Ca
3
Al
2
Si
3
O
12
),
and Fe-rich almandine-pyrope ((Fe,Mg)
3
Al
2
Si
3
O
12
) garnets
to pressures in excess of 11 GPa by Brillouin spectroscopy.
We have paid careful attention to sources of systematic error
(e.g., vignetting due to the limited angular opening of DAC,
resulting in asymmetric Brillouin peak lineshape), compo-
sitional heterogeneity, and have maintained hydrostatic
conditions in the diamond anvil cell. Together with our
detailed study on andradite [Jiang et al., 2004] and a
previous Brillouin study on pyrope [Sinogeikin and Bass,
2000], we are able to constrain the effects of Fe
2+
-Mg
2+
,
Ca
2+
%-Mg
2+
,andFe
3+
-Al
3+
substitution on the elastic
properties of garnets at high pressures. Grossular and
almandine-rich garnets were chosen because these compo-
sitions have not been extensively studied using optical
spectroscopy techniques [cf. Chai et al., 1997; Conrad et
al., 1999]. Better characterization of such compositions is
needed as natural garnets from mantle peridotites contain
roughly 12– 26 mol % almandine and 2 –20 mol % gros-
sular, in addition to 60– 86 mol % pyrope [Rickwood et al.,
1968; Lee, 2003]. Furthermore, we evaluate the cause of the
wide variability of the reported pressure derivatives of the
elastic moduli among garnets, and derive a consistent set of
elastic properties based on optical spectroscopy (Brillouin
scattering) studies.
2. Experimental Procedure
[6] A grossular single crystal from Sierra de las Cruces,
Coahuila, Mexico and a natural almandine single crystal of
Table 1. Result of Microprobe and Powder X-Ray Diffraction Analysis at Ambient Conditions
a
Oxides, wt %
Lattice Parameter, A
˚
SiO
2
TiO
2
Al
2
O
3
FeO MnO MgO CaO
Grossular-rich garnet: Grs
87
And
9
Pyp
2
Alm
1
;
other trace; r= 3.605 g/cm
3
40.67 0.39 21.35 3.69 0.08 0.67 36.93 11.886
Almandine-rich garnet: Alm
72
Pyp
20
Sps
3
Grs
3
And
2
;
other trace; r= 4.132 g/cm
3
37.76 0.01 21.69 33.53 1.39 4.99 1.79 11.536
a
All iron assumed to be FeO.
Figure 1. (a) Brillouin spectra of grossular-rich garnet at
room pressure and 3.2 GPa. (b) Brillouin spectra of
almandine-rich garnet at room pressure and 10.8 GPa.
Brillouin peaks from P-mode, S-mode, pressure medium
(M-E-W), and diamond are labeled.
B10210 JIANG ET AL.: ELASTICITY OF GARNETS BY BRILLOUIN SCATTERING
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B10210

unknown origin were used. Both showed characteristic
dodecahedral habits. Platelets parallel to the natural {110}
faces were cut and then double-side polished with succes-
sively finer grits down to a final diamond paper of 1 mm
particle size. The final sample thickness was approximately
30 mm. The polished samples were checked using a polar-
izing microscope, and there was no evidence of zoning. The
compositions of the samples used for our Brillouin measure-
ments were determined by electron microprobe analysis.
Ten points were analyzed across the samples, and they have
equivalent compositions within experimental uncertainties.
The densities at ambient conditions were determined
by powder X-ray diffraction and chemical analysis (see
Table 1).
[7] The polished platelets of grossular- and almandine-
rich garnets were loaded into a modified Merrill-Basset
diamond anvil cell with angular opening of 96and
compressed up to 11–12 GPa using a 16:3:1 methanol-
ethanol-water mixture as a pressure-transmitting medium.
In all experiments more than four ruby chips were placed
around the sample as pressure calibrants. Pressure deter-
mination was performed by measuring the ruby fluores-
cence shift [Mao et al., 1986]. The fluorescence peaks did
not show significant broadening over the whole pressure
range. In order to allow for possible stress relaxation after
each compression step, Brillouin measurements were car-
ried out at least one day after pressure increase. The
differences between pressure measured from different ruby
chips around the samples never exceeded ±0.2 GPa.
Pressures measured before and after each Brillouin data
set collection were always equivalent within mutual
uncertainties.
[8] The samples were excited with a single mode verti-
cally polarized neodymium vanadate laser (l= 532.15 nm)
with a power of 150 mW. Brillouin spectra were measured
using a six-pass Sandercock tandem Fabry-Perot interfer-
ometer in a forward symmetric scattering geometry, in
which acoustic velocities, V, can be determined without
knowledge of the sample refractive index [Whitfield et al.,
1976]:
V¼DnBl0
2 sin q=2ðÞ
;ð1Þ
where, l
0
is the incident laser wavelength, Dn
B
is the
measured Brillouin frequency shift and qis the scattering
angle external to the diamond cell (70in this study).
Details of the experimental setup are reported elsewhere
[Speziale and Duffy, 2002].
[9] Special attention was paid to determine the scattering
angle to within a few minutes of a degree using a reference
laser beam. Therefore experimental errors due to scattering
angles are neglected in our data analysis. The accuracy and
reproducibility of our system was tested on standard single
crystals of MgO and SrTiO
3
with known velocities. A
Table 2. Best-Fit Density, Elastic Constants, and Aggregate Moduli
a
P, GPa R, g/cm
3
C
11
, GPa C
12
, GPa C
44
, GPa K
S
, GPa G, GPa RMS, m/s
Grossular-Rich Garnet: Grs
87
And
9
Pyp
2
Alm
1
b
0.0001 3.605 314.5(4) 91.5(5) 99.7(2) 165.8(5) 104.3(3) 21
1.6 3.640 323.5(12) 95.1(9) 100.7(5) 171.2(10) 105.9(8) 41
3.2 3.674 332.7(6) 99.9(6) 103.1(3) 177.5(6) 108.2(5) 27
4.3 3.697 337.4(7) 101.7(7) 102.1(3) 180.3(7) 108.1(6) 28
6.5 3.740 352.4(10) 108.6(9) 104.9(5) 189.8(9) 111.4(8) 35
7.9 3.767 354.4(12) 110.7(8) 105.8(4) 191.9(9) 112.0(7) 33
9.4 3.798 370.6(13) 116.6(9) 107.1(5) 201.3(10) 114.7(8) 37
10.7 3.823 377.3(11) 121.3(8) 107.9(5) 206.7(9) 115.5(8) 32
Almandine-Rich Garnet: Alm
72
Pyp
20
Sps
3
Grs
3
And
2
c
0.0001 4.132 302.3(4) 110.6(4) 94.4(2) 174.5(4) 94.9(8) 16
1.0 4.155 307.0(10) 109.5(8) 94.4(4) 175.3(9) 96.1(7) 36
2.4 4.187 321.6(11) 121.4(8) 97.8(4) 188.1(9) 98.7(7) 34
3.0 4.200 329.3(8) 124.6(6) 99.5(8) 192.9(7) 100.6(5) 30
5.8 4.263 343.8(9) 130.1(6) 102.8(8) 201.3(7) 104.4(6) 27
7.7 4.303 354.6(9) 137.6(7) 104.9(4) 209.9(8) 106.3(7) 31
9.2 4.333 360.1(10) 140.8(7) 105.6(4) 213.9(8) 107.2(7) 29
10.8 4.364 374.9(8) 150.7(5) 107.7(8) 225.4(6) 109.4(5) 21
12.0 4.388 379.2(17) 152.6(12) 108.6(6) 228.1(14) 110.4(11) 46
a
Numbers in parentheses are 1-sdeviations in last digit. RMS, root mean square of the difference between observed and calculated velocities.
b
Average (q
0
,j
0
,c
0
)(135± 3, 109±3,85± 3); (hkl)(0.32, 0.95, 1.00).
c
Average (q
0
,j
0
,c
0
)(43±2,2± 2, 103± 2); (hkl)(0.22, 0.91, 1.00).
Table 3. Thermodynamic Parameters Used for Adiabatic to
Isothermal Conversion
Parameters Value References
Grossular
Thermal expansion a
0
19.2 10
6
K
1
Isaak et al. [1992]
Gru¨neisen parameter g
0
1.22 calculated
a
Specific heat C
P
325.5(7) J/(mol K) Isaak et al. [1992]
(@K
T
/@T)
P
0.02 GPa K
1
Isaak et al. [1992]
Almandine
Gru¨neisen parameter g
0
1.22 Soga [1967]
(@K
S
/@T)
P
0.0201 GPa K
1
Soga [1967]
(@a/@T)
P
5.24 10
8
K
2
Skinner [1966]
(@K
T
/@T)
P
0.0277 GPa K
1
calculated
b
a
Gru¨neisen parameter obtained as g
0
=a
0
K
S0
/(r
0
C
P
) using K
S0
and r
0
of
this study.
b
Calculated using (@K
T
/@T)
P
(@K
S
/@T)
P
/(1 + agT)K
S
/(1 + agT)
2
[ag+(@a/@T)
P
gT].
B10210 JIANG ET AL.: ELASTICITY OF GARNETS BY BRILLOUIN SCATTERING
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significant error in velocity determination can be caused by
vignetting due to the limited aperture of the DAC
[Sinogeikin and Bass, 2000]. To prevent this, a diaphragm
was added in front of the collection lens. A half-wave plate
in the incident light path was used to maximize the intensity
of the Brillouin peaks [Jiang et al., 2004].
3. Results
[10] For all Brillouin spectra, one quasi-longitudinal (P)
and one quasi-transverse (S) acoustic mode were observed.
Typical Brillouin spectra at ambient and high pressure are
shown in Figure 1. The measured frequency shifts have
been converted to velocity along the horizontal axis using
equation (1). For both samples at each pressure, Brillouin
measurement was performed for 36 directions at 5 degree
intervals. Eulerian angles (q
0
,j,c
0
) relating the crystallo-
graphic coordinate system to the laboratory coordinate
system were used to specify the crystal orientation and
acoustic wave-vector direction. The azimuthal angle, j, was
the only one varied during measurement.
[11] The velocity data were fitted to the Christoffel’s
equation [Every, 1980] to retrieve the elastic constants
(C
11
,C
12
, and C
44
) and the three Eulerian angles describ-
ing the crystal orientation. A starting model based on
literature values of C
ij
and our measured density at
ambient conditions was used for the room pressure veloc-
ity fitting, and the initial values for q
0
,j
0
,c
0
were
systematically varied until a satisfactory agreement be-
tween calculated and experimentally obtained velocities
was attained. The recovered orientation indicated that the
crystal plane was close to {110} as expected (see Table 2).
Despite the very low elastic anisotropy of these garnets
(A = 2C
44
/(C
11
-C
12
) = 0.98) at ambient pressure for the
almandine-pyrope garnet), we found that the crystal ori-
entation could be readily recovered to within ±3 degrees.
Densities at high pressures were initially estimated by
using the Birch-Murnaghan equation of state, with ambient
pressure values of the bulk modulus, K
T0
, and an initial
guess for the pressure derivative of the bulk modulus,
(@K
T
/@P)
T0
. High-pressure elastic constants were obtained
by fitting each velocity curve using the calculated density
and the C
ij
values from the previous pressure as initial
guesses. Errors in pressure determination in turn cause
uncertainties in elastic moduli at high pressures by intro-
ducing errors in density. Our estimations show that the
errors in elastic moduli would be 1.5 times the standard
deviation 1-sgiven in this study if this error in pressure
determination is considered.
[12] After the first round of fitting, the adiabatic aggregate
bulk modulus and the aggregate shear modulus, G, obtained
from Voigt-Reuss-Hill average, were calculated. In the
inversion of high-pressure moduli, an iterative procedure
[Zha et al., 1996; Speziale and Duffy, 2002] was adopted.
The calculated adiabatic bulk moduli K
S
were fit to third-
order Eulerian finite strain equation [Birch, 1978] to obtain
K
S0
and (@K
S
/@P)
T0
. These parameters were then converted
to isothermal K
T0
and (@K
T
/@P)
T0
, by applying thermody-
namic relations:
KT0¼KS0=1þagTðÞ ð2Þ
@KT=@PðÞ
T01þagTðÞ
1@KS=@PðÞ
T0

gT=KT0@KT=@TðÞ
P0;ð3Þ
where ais the volume thermal expansion coefficient and
gis the Gru¨neisen parameter (Table 3). The isothermal
K
T0
and (@K
T
/@P)
T0
were then used to construct
improved isothermal compression curves and the velo-
cities at each pressure were refit. The above procedure
was repeated and it converged after four iterations. It
should be mentioned here that the initial values of K
T0
and (@K
T
/@P)
T0
, which were used to calculate the initial
densities at high pressures, did not affect final results, but
only affect the number of iterations needed to achieve
convergence.
Figure 2. (a) Velocity data (symbols) and calculated
velocity curves (lines) of grossular at room pressure and
4.3 GPa. (b) Velocity data and calculated velocity curves of
natural almandine at room pressure and 10.8 GPa.
Approximate directions [111], [100], and [110] are marked.
Table 4. Individual Elastic Constants and Pressure Derivatives of Grossular at Ambient Conditions
a
Study
b
Composition
C
11
,
GPa
C
12
,
GPa
C
44
,
GPa (@C
11
/@P)
0
(@C
12
/@P)
0
(@C
44
/@P)
0
P
max
,
GPa
a Grs
87
And
9
Pyp
2
Alm
1
313.6(16) 90.7(7) 99.5(4) 6.1(2) 2.8(1) 0.9(1) 11
b Grs
99
And
1
321.7(8) 91.4(9) 104.6(4) 10
4
c Grs
80
And
14
Alm
3
Pyp
2
306.1(38) 88.7(38) 98.8(4) 10
4
d Grs
97
And
2
Pyp
1
318.8(8) 92.1(7) 102.9(2) 10
4
a
Numbers in parentheses are 1-sdeviations in last digit.
b
Studies are as follows: a, this study; b, Bass [1989]; c, Babuska et al. [1978]; d, Isaak et al. [1992].
B10210 JIANG ET AL.: ELASTICITY OF GARNETS BY BRILLOUIN SCATTERING
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[13] The robustness of our fitting results was confirmed
on a standard sample of MgO and also demonstrated by low
root mean square (RMS) differences between observed and
calculated velocities (Table 2). Figures 2a and 2b show the
observed velocities (symbols) and calculated velocity
curves (lines) for grossular-rich and almandine-rich crystals
at room and high pressure.
4. Discussion
4.1. Elastic Stiffness Constants, Aggregate Moduli, and
Their Pressure Derivatives
[14] The adiabatic aggregate bulk moduli, K
S
, were fit to
Eulerain finite strain equation to obtain K
S0
and K
0
S0
=
(@K
S
/@P)
T0
. They were converted to isothermal values of
K
T0
and K
0
T0
using equations (2) and (3). Pressure deriva-
tives of individual elastic constants C
ij
and the shear
modulus, G, were obtained by fitting experimental values
to the third-order finite strain equations [Davies, 1974]:
Cijkl ¼1þ2fðÞ
7=2C0
ijkl þa1fþ::
hi
PDijkl;ð4Þ
where, f= 1/2[(r/r
0
)
2/3
1] is the Eulerian finite strain, C
ijkl
0
is the value of elastic constant at ambient conditions, D
ijkl
=
d
ij
d
kl
d
ik
d
jl
d
il
d
jk
, and P is pressure. a
1
=3K
0T
(@C
ijkl
0
/
@P+D
ijkl
)7C
ijkl
0
, and @C
ijkl
0
/@P is the pressure derivative
of the elastic constant at ambient conditions. The relation
between the full notation of C
ijkl
and its contracted notation
C
ij
is:
ij;kl 11 22 33 23;32 31;13 12;21
i;j123 4 5 6
:
[15] The fitted values of individual elastic constants,
aggregate moduli, and their pressure derivatives are
reported in Tables 4– 7 together with values from the
literature. Figures 3 – 6 show the plots of elastic constants,
and aggregate moduli as a function of pressure. The
individual elastic constants and aggregate moduli for gros-
sular-rich and almandine rich garnets define nearly linear
trends with pressure. The elastic anisotropy of a cubic
material can be described by the anisotropy factor, A =
2C
44
/(C
11
-C
12
), where A = 1 corresponds to elastic isotropy.
The almandine-rich garnet has very low anisotropy (anisot-
ropy factor, A = 0.98) and becomes only slightly more
anisotropic with pressure (A = 0.96 at 12.0 GPa). The
grossular garnet also has relatively low anisotropy (A =
0.89) at ambient conditions and the anisotropy increases in
magnitude modestly (A = 0.84) at the highest pressure.
Increasing temperature decreases the elastic anisotropy of
garnets [Isaak et al., 1992], and so under upper mantle
conditions the elastic anisotropy of garnets should be small
and close to that at ambient conditions. For comparison, the
ambient pressure anisotropy factor for other cubic minerals
Table 5. Aggregated Moduli and Pressure Derivatives of Grossular at Ambient Conditions
a
Study
b
Composition
K
S0
,
GPa
K
T0
,
GPa
G
0
,
GPa (@K
S
/@P)
T0
(@K
T
/@P)
T0
(@G/@P)
0
P
max
,
GPa Method
c
a Grs
87
And
9
Pyp
2
Alm
1
165.0(9) 163.8(5) 104.2(3) 3.8(2) 3.9(2) 1.1(1) 11 BS
b Grs
99
And
1
168.4(7) 108.9(4) 10
4
BS
c Grs
80
And
14
Alm
3
Pyp
2
161.2(5) 120.6(4) 10
4
U
d Grs
97
And
2
Pyp
1
167.8(7) 106.9(2) 10
4
R
e Grs
100
166.8
d
108.9
d
5.46 1.10 10 BS
f Grs
90
Pyp
1
Alm
2
And
6
173(2) 4.25
d
25 X
f 162(3) 5.45
d
25 X
f Grs
97
Sps
1
Alm
1
175(4) 4.25
d
25 X
f 165(4) 5.45
d
25 X
g Grs
100
168(25) 6.1(15) 18.4 X
h Grs
97
Alm
3
169.3(12) 5.92(14) 36.9 X
i Grs
100
175(1) 4.4
d
11.6 X
a
Numbers in parentheses are 1-sdeviations in last digit.
b
Studies are as follows: a, this study; b, Bass [1989]; c, Babuska et al. [1978]; d, Isaak et al. [1992]; e, Conrad et al. [1999]; f, Weaver et al. [1976];
g, Olijnyk et al. [1991]; h, Pavese et al. [2001]; i, Zhang et al. [1999].
c
BS, Brillouin scattering; R, resonance method; U, polycrystalline ultrasonic method; X, X-ray diffraction.
d
Fixed value.
Table 6. Individual Elastic Constants and Pressure Derivatives of Almandine-Rich Garnet at Ambient Conditions
a
Study
b
Composition
C
11
,
GPa
C
12
,
GPa
C
44
,
GPa (@C
11
/@P)
0
(@C
12
/@P)
0
(@C
44
/@P)
0
P
max
,
GPa
aAlm
72
Pyp
20
Sps
3
Grs
3
And
2
304.4(18) 110.3(15) 94.6(5) 6.6(3) 3.6(2) 1.3(1) 12
bAlm
81
Pyp
14
Grs
4
Sps
1
304.8 112.3 94.4 10
4
cAlm
46
Sps
54
308.5 112.3 94.8 7.15 3.85 1.29 0.5
dAlm
76
Pyp
21
Grs
3
306.2 112.5 92.7 7.48 4.41 1.31 0.3
eAlm
52
Sps
46
Grs
1
306.5 111.2 94.4 6.69 3.54 1.26 1
a
Numbers in parentheses are 1-sdeviations in last digit.
b
Studies are as follows: a, this study; b, Verma [1960]; c, Wang and Simmons [1974]; d, Soga [1967]; e, Isaak and Graham [1976].
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such as MgO is 1.53 [Sinogeikin and Bass, 2000] and
MgAl
2
O
4
is 2.46 [Suzuki et al., 2000].
[16] For our grossular-rich garnet, Grs
87
And
9
Pyp
2
Alm
1
,
the fit value of the adiabatic bulk modulus K
S0
at ambient
condition is 165.0 ± 0.9 GPa, which is about 3 GPa lower
than previous values for grossular garnets [Bass, 1989;
Isaak et al., 1992]. Assuming a linear dependence of the
elastic moduli on composition expressed in mole fraction of
the end members, correcting for the larger andradite content
of our sample reduces this difference by about 1 GPa. For
the shear modulus, our value also remains 1.4 – 3.0 GPa
below the other studies after correction for andradite con-
tent. This level of uncertainty (1 – 3%) is typical of those
found when comparing multiple measurements on the same
or similar compositions for minerals such as fayalite [Isaak
et al., 1993] and fluorite [Speziale and Duffy, 2002]. We
note that the RMS deviations for the Brillouin measure-
ments of Bass [1989] are about 50% larger than this study.
[17] The fit values of K
0
T0
and G
0
0
of grossular are 3.9 ±
0.2 and 1.1 ± 0.1 and are compared with previous values in
Table 5. Conrad et al. [1999] reported a Brillouin scattering
study on grossular, andradite, and pyrope at high pressures
using a very similar experimental technique as this study. In
previous work [Jiang et al., 2004], we found large discrep-
ancies between our results and those of Conrad et al. [1999]
for andradite. For grossular, while the pressure derivative of
the shear modulus is in excellent agreement, we obtain a
much lower value for K
0
S0
.Conrad et al. [1999] measured
two crystal directions at each pressure compared with 36 in
this study. The large differences may result from orientation
errors in the previous study.
[18] The high-pressure X-ray diffraction studies listed in
Table 5 report a wide range of pressure derivatives. The
determination of the bulk modulus and its pressure deriva-
tive from static compression relies on fits to the slope of the
measured P-V curve, and hence is less direct than Brillouin
scattering measurements. Static compression studies also
suffer from a well-known tradeoff between fit values for
K
T0
and K
0
T0
. Furthermore, if measurements are restricted to
purely hydrostatic conditions, the compression range is very
limited making it difficult to constrain the elastic moduli.
On the other hand, data which covers a broad pressure range
are subjected to variable degrees of nonhydrostatic stress
depending on the nature and amount of the pressure
transmitting medium. Figure 7 shows a comparison of the
pressure-compression curve from our Brillouin data with
previous X-ray diffraction studies. Below 8 GPa, our
Table 7. Aggregated Moduli and Pressure Derivatives of Almandine-Rich Garnet at Ambient Conditions
a
Study
b
Composition
K
S0
,
GPa
K
T0
,
GPa
G
0
,
GPa (@K
S
/@P)
T0
(@K
T
/@P)
T0
(@G/@P)
0
P
max
,
GPa Method
c
aAlm
72
Pyp
20
Sps
3
Grs
3
And
2
174.9(16) 173.6(16) 95.6(5) 4.7(3) 4.7(3) 1.4(1) 12 BS
bAlm
81
Pyp
14
Grs
4
Sps
1
176.5 95.1 10
4
U
S
cAlm
46
Sps
54
177.7 96.1 4.95 1.44 0.5 U
S
d Alm
76
Pyp
21
Grs
3
177 94 5.43 1.40 0.3 U
S
eAlm
52
Sps
46
Grs
1
176.3 95.6 4.59 1 U
S
fAlm
73
Pyp
27
175 99 10
4
U
S
gAlm
100
175.1(9) 92.1 6.2(5) 1.6(2) 3 U
P
h Alm
76
Pyp
21
Grs
3d
173(6) 5.4
e
26 X
hAlm
100
168(5) 5.4
e
26 X
iAlm
100
175(7) 1.5(1.6) 10 X
jAlm
100
178
e
4.9(1) 21 X
a
Numbers in parentheses are 1-sdeviations in last digit.
b
Studies are as follows: a, this study; b, Verma [1960]; c, Wang and Simmons [1974]; d, Soga [1967]; e, Isaak and Graham [1976]; f, Chen et al. [1997];
g, Wang and Ji [2001]; h, Takahashi and Liu [1970]; i, Sato et al. [1978]; j, Zhang et al. [1999].
c
BS, Brillouin scattering; U
S
, single crystal ultrasonic method; U
P
, polycrystalline ultrasonic method; X, X-ray diffraction.
d
Same sample as in d.
e
Fixed value.
Figure 3. Pressure dependencies of the elastic stiffness
constants for grossular-rich garnet; solid lines are fits using
Eulerian finite strain equations.
Figure 4. Grossular aggregate adiabatic bulk modulus K
S
and G (symbols) and fits (lines) to Eulerian finite strain
equations.
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results are in very good agreement with earlier studies. The
results of Pavese et al. [2001] and Zhang et al. [1999]
indicated that their samples became increasingly less com-
pressible than ours at higher pressures above 8 GPa. Olijnyk
et al. [1991] performed compression studies on a synthetic
grossular using both methanol-ethanol (M-E) and N
2
as
pressure transmitting media. Their P-V curve obtained using
M-E as pressure medium is in very good agreement with
ours to 10 GPa, while their P-V curve above 9 GPa derived
using N
2
as pressure medium deviates from ours. The
methanol-ethanol mixture remains strictly hydrostatic up
to 12 GPa, while N
2
and Ne freeze above 3 GPa and
4.5 GPa, respectively, and generate nonhydrostatic stress
conditions at higher pressures.
[19] For almandine-rich garnet Alm
72
Pyp
20
Sps
3
Grs
3
And
2
,
both individual elastic constants C
ij
, aggregate moduli
K
S0
and G
0
are in agreement (Tables 6 and 7) with
those of a natural almandine with very similar compo-
sition Alm
76
Pyp
21
Grs
3
[Soga, 1967], and they also agree
well with other reported values (Table 7) after correcting for
compositional differences. In general, there is a good agree-
ment among various studies for @C
44
/@Pand(@G/@P).
However, there is a wide range of reported values for the
pressure derivatives of C
11
,C
12
, and K
S0
. Our result for K
0
S0
is
consistent with the low-end values of the range for K
0
S0
in
previous ultrasonic studies.
[20] A comparison of the isothermal compression curve
derived from the Brillouin measurements with static X-ray
diffraction data is shown in Figure 8. As with almandine,
our compression curve agrees well with earlier compres-
sion studies at low pressure, and is also consistent until
10 GPa with data obtained using M-E as pressure trans-
mitting medium [Sato et al., 1978]. Compression curves
using Ne and He pressure medium [Zhang et al., 1999]
deviate from ours above 6 GPa and 11 GPa, respectively.
It is somewhat surprising (Figure 8) that X-ray data
obtained with a He medium yields a similar compression
curve to those obtained using Ne and NaCl since helium is
expected to provide the closest approach to hydrostaticity.
However, deviations from the hydrostatic compression
curve can result for any pressure medium if an insufficient
Figure 5. Pressure dependencies of the elastic stiffness
constants C
ij
for natural almandine garnet; solid lines are fits
using Eulerian finite strain equations.
Figure 6. Almandine aggregate adiabatic bulk modulus
K
S
and G (symbols) and fits (lines) using Eulerian finite
strain equations.
Figure 7. Isothermal compression curve of grossular
calculated from present study (solid line) and its extrapola-
tion to higher pressures (dotted line). Previous compression
studies of X-ray diffraction are plotted together for
comparison. Filled and open circles are from the work of
Olijnyk et al. [1991] using M-E and N
2
as pressure medium,
respectively. From the work of Weaver et al. [1976], open
squares are 90% Grs, filled squares are 98% Grs, and the
pressure medium is NaCl.
Figure 8. Isothermal compression curve of almandine
calculated from the present study (solid line) and its
extrapolation to higher pressures (dotted line). Previous
compression studies of X-ray diffraction are plotted
together for comparison. From the work of Zhang et al.
[1999], open triangles are Ne as pressure medium and filled
triangles are He as pressure medium. From the work of
Takahashi and Liu [1970], NaCl is the pressure medium,
open squares are natural almandine, and filled squares are
synthetic almandine.
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amount of the medium is loaded together with the sample,
resulting in partial bridging and generation of deviatoric
stresses. The large range of K
T0
and especially K
0
T0
values
reported for X-ray studies in Tables 5 and 7 despite relatively
modest differences in pressure-volume behavior exhibited in
Figures 7 and 8 demonstrates the difficulty in reliably
constraining these parameters from X-ray diffraction data.
4.2. Implications for High-Pressure Elasticity
of Garnets
[21] In general, ambient pressure elastic moduli deter-
mined by different techniques are in reasonable agreement
(see Tables 5 and 7), but the pressure derivatives are
highly discrepant as seen in Figure 9. Compression studies
using X-ray diffraction can not give information about G
and its pressure derivative, and provide less reliable value
for K
T0
and K
0
T0
because of parameter tradeoffs and
nonhydrostatic stresses as discussed above. Ultrasonic
methods provide a direct measurement of elasticity and
acoustic velocities through travel time measurements, but
are subject to uncertainties when polycrystalline samples
are used, especially when combined with a limited pres-
sure range [e.g., Wang and Ji, 2001]. The presence of
pores and cracks in the sample could lead to an overes-
timate of pressure derivatives. Single-crystal ultrasonic
studies were instead generally limited to very low pres-
sures (<3 GPa). If the precision is adequate, slopes at
lower pressures can be well constrained, but may not be
applicable to higher pressures. Figure 9 shows pressure
derivatives of garnets measured by Brillouin and ultrasonic
methods. It is evident that pressure derivatives derived by
ultrasonic methods are more scattered and in general
higher than those by Brillouin scattering. The results of
Conrad et al. [1999] probably suffered systematic errors,
as discussed above and are not included. Brillouin scat-
tering has advantages over other methods in that measure-
ments on a large number of directions on high-quality
single crystals can be performed under strictly hydrostatic
conditions.
[22] The aggregate elastic properties of garnets at high
pressures have also been investigated theoretically using
density functional theory [Akhmatskaya et al., 1999]. The
pressure derivatives of the bulk modulus obtained theoret-
ically for grossular (K
T0
=166GPa,K
0
T0
=4.3)and
almandine (K
T0
= 177 GPa, K
0
T0
= 4.2) are within 10%
of the experimental values obtained here (which is the
expected accuracy of the DFT calculations [Milman et al.,
2001]). The theoretical values for pyrope (K
T0
= 170 GPa,
K
0
T0
= 4.3) are also consistent with spectroscopic measure-
ments [Sinogeikin and Bass, 2000] (Figure 9) rather than the
higher values reported in ultrasonic studies. A related
theoretical study [Milman et al., 2001] reported values
for the bulk modulus and pressure derivative of andradite
(K
T0
= 147 GPa, K
0
T0
= 4.4) that are 57% lower than
recent Brillouin scattering results [Jiang et al., 2004].
Nevertheless, it is notable that these theoretical studies
yield bulk modulus pressure derivatives in the range of
3.9– 4.8 over a wide range of garnet compositions, consis-
tent with the overall range reported in optical spectroscopic
determinations (Table 8) and considerably lower than the
upper range of values reported in static compression and
ultrasonic studies.
Table 8. End-Member Aggregate Elastic Properties of Garnets From Spectroscopic (Brillouin) Data
a
Formula a (A
˚)K
S0
(GPa) G
0
(GPa) (@K
S
/@P)
T0
(@G/@P)
0
Reference
b
Almandine Fe
3
Al
2
Si
3
O
12
11.531 175(2) 96(1) 4.9(2) 1.4(1) a
Pyrope Mg
3
Al
2
Si
3
O
12
11.452 171(3) 94(2) 4.1(3) 1.3(2) b
Grossular Ca
3
Al
2
Si
3
O
12
11.845 168(1) 109(4) 3.9(2) 1.1(1) a
Andradite Ca
3
Fe
2
Si
3
O
12
12.058 157(2) 90(1) 4.7(1) 1.3(1) c
Majorite Mg
4
Si
4
O
12
11.494 166(3) 85(2) 4.2(3) 1.4(2) d
a
Lattice parameters from Smyth and McCormick [1995].
b
References are as follows: a, this study, extrapolated to end-member composition using a linear mixing model; b, Sinogeikin and Bass [2000]; c, Jiang et
al. [2004]; d, Sinogeikin and Bass [2002].
Figure 9. (left) Pressure derivatives K
0
S0
and G
0
0
of various
garnets obtained by Brillouin scattering, impulsively
stimulated scattering (ISS), and ultrasonic methods. (right)
Pressure derivatives of pyrope-majorite compositions.
References are as follows: for Brillouin, this study, Jiang
et al. [2004], and Sinogeikin and Bass [2000, 2002]; for
ISS, Chai et al. [1997]; for ultrasonic, Chen et al. [1999],
Isaak and Graham [1976], Soga [1967], Wang and Ji
[2001], Wang and Simmons [1974], Liu et al. [2000],
Rigden et al. [1994], Gwanmesia et al. [1998], and Webb
[1989]. Values of K
0S0
for andradite and values of G
0
0
for
pyrope-almandine have been offset slightly in composition
for clarity.
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[23] Table 8 provides values for the aggregate elastic
moduli of end-member garnets derived from this work and
other Brillouin scattering studies [Bass, 1989; Sinogeikin and
Bass, 2000, 2002; Chai et al., 1997; Jiang et al., 2004]. No
compositional trends are evident in the pressure derivatives
except for an increase in K
0
S0
upon substitution of Fe
2+
for
Mg
2+
in the pyrope-almandine series and a similar increase in
K
S0
0
upon substitution of Fe
3+
for Al
3+
in the grossular-
andradite series. The value of K
0
S0
for pure almandine has
been extrapolated from our measured value assuming a linear
dependence on mole fraction. It is notable that a similar
increase in K
0
S0
with Fe
2+
-Mg
2+
substitution was observed for
the forsterite-fayalite series (S. Speziale et al., manuscript in
preparation, 2004).
[24] Figure 10 shows the bulk and shear moduli versus
composition for the pyrope-almandine series and the pyrope-
grossular series at selected pressures from 0 to 14 GPa using
the data of Table 8 and assuming a linear dependence of both
moduli and their pressure derivatives on composition
expressed in mole fraction of the end members. At ambient
pressure, the bulk modulus varies by 2% across the pyrope-
grossular and pyrope-almandine systems. Owing to the
higher K
0
S0
value for almandine, the increase in bulk modulus
from pyrope to garnet becomes 5.2% by 14 GPa. Ca
2+
-Mg
2+
substitution has a large effect on the shear modulus: the
rigidity of grossular is 16% greater than pyrope at ambient
pressure. This is mostly maintained across the upper mantle
pressure interval as the shear modulus of grossular is 11%
greater than pyrope at 14 GPa. In contrast to other silicates
such as olivine, the effect of Fe
2+
-Mg
2+
is modest for garnets.
The shear modulus of almandine is 2 – 3% larger than that of
pyrope at 0–14 GPa. The implications of these results for
assessing compositional variation in the upper mantle will be
the subject of a forthcoming publication.
5. Conclusions
[25] Brillouin scattering measurements have been carried
out under hydrostatic conditions to 11 GPa on a grossular-
rich and an almandine-rich garnet. Individual, and aggregate
elastic moduli, and their pressure derivatives have been
determined by fitting Eulerian finite strain equations. All the
elastic constants, and aggregate moduli define nearly linear
trends with pressure. The adiabatic bulk, and Voigt-Reuss-
Hill average shear moduli and their pressure derivatives
were constrained to K
S0
= 165.0 ± 0.9 GPa, G
0
= 104.2 ±
0.3 GPa, K
0
S0
= 3.8 ± 0.2 and G
0
0
= 1.1 ± 0.1 for the
grossular-rich garnet and K
S0
= 174.9 ± 1.6 GPa, G
0
=
95.6 ± 0.5 GPa, K
0
S0
= 4.7 ± 0.3 and G
0
0
= 1.4 ± 0.1 for the
almandine-rich garnet.
[26] A compression curve derived from our results is in
good agreement with high-pressure X-ray diffraction data,
especially those data recorded under hydrostatic conditions.
Elasticity data for almandine- and grossular-rich garnets
were compiled and evaluated. The ambient pressure moduli
show good agreement but reported pressure derivatives
exhibit wide variations that show little correlation with
composition, but rather appear to be related to experimental
technique. By restricting attention to high-pressure optical
spectroscopy (Brillouin scattering) studies, a set of self-
consistent aggregate elastic properties and their pressure
derivatives were derived for the major upper mantle garnet
end-member compositions.
[27] These results (Table 8) demonstrate that pressure
derivatives of elastic moduli for mantle-relevant garnets fall
in a relatively narrow range and are considerably lower than
inferred from some earlier studies. Several recent mineral-
ogical models for the upper mantle [Goes et al., 2000;
Cammarano et al., 2003; Lee, 2003] have adopted elastic
properties for garnets, especially grossular garnets and Mg-
rich majorites outside the range of the optical spectroscopic
data set. Our development of an experimentally consistent
data set of end-member garnet properties will result in
improved mineralogical models for the mantle, and also
allow better constraints on the lateral variation of seismic
velocities due to local chemical variations in the mantle.
[28]Acknowledgments. This work was supported by the National
Science Foundation and the David and Lucile Packard Foundation. We are
grateful to J. Delaney, Rutgers University, for the microprobe analysis.
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series at selected pressures from 1 bar to 14 GPa using
the data of Table 8. Numbers next to each line are
pressures in GPa.
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T. S. Duffy, F. Jiang, and S. Speziale, Department of Geosciences,
Princeton University, Princeton, NJ 08544, USA. (duffy@princeton.edu;
fuj2@lehigh.edu; speziale@uclink.berkeley.edu)
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