ELSEVIER
Earth and Planetary Science Letters 164 (1998) 371–378
High pressure elastic anisotropy of MgSiO3 perovskite and
geophysical implications
R.M. Wentzcovitch a , B.B. Karki a,Ł , S. Karato b , C.R.S. Da Silva a
a
Department of Chemical Engineering and Materials Science, Minnesota Supercomputing Institute, University of Minnesota,
Minneapolis, MN 55455, USA
b Department of Geology and Geophysics, University of Minnesota, Minneapolis, MN 55455, USA
Received 29 April 1998; revised version received 17 August 1998; accepted 8 September 1998
Abstract
Using plane wave pseudopotential method within the local density approximation (LDA), we calculate single-crystal
elastic constants .ci j / of orthorhombic MgSiO3 perovskite, generally accepted to be the major component of the lower
mantle, as a function of pressure up to 150 GPa. Our results are in excellent agreement with experimental data at zero
pressure and compare favorably with other pseudopotential predictions over the pressure regime studied. Here we use our
elastic constants to calculate anisotropy of seismic wave velocities as a function of pressure (depth). MgSiO3 perovskite is
shown to be highly anisotropic in all portions of the lower mantle and the nature of anisotropy changes significantly with
depth. The absence of significant seismic anisotropy in most of the lower mantle suggests that MgSiO3 perovskite assumes
nearly random orientation in most of this region. Anisotropy at the topmost lower mantle suggested by some studies can
be attributed to the preferred orientation of perovskite. However, anisotropy in the D00 layer is difficult to be attributed
to preferred orientation of perovskite. Some other mechanisms including the presence of the aligned melt pockets and=or
lattice preferred orientation of magnesiowustite are needed.  1998 Elsevier Science B.V. All rights reserved.
Keywords: high pressure; elasticity; anisotropy; lower mantle
1. Introduction
It is generally accepted that the lower mantle —
the largest single region of the Earth’s interior, is
composed of an assemblage of silicates and oxides
dominated by MgSiO3 perovskite. High-pressure
elasticity of the relevant minerals is of substantial
importance since our most precise and informative
observations of lower mantle region are related to
its elastic properties as derived from seismology.
Therefore, the combination of mineral physics and
Ł Corresponding
author. Fax: C1 612 626 7246.
seismology has become an efficient method to study
the deep Earth. Comparisons between rich seismic
data and the elastic properties of potentially relevant
minerals and assemblages are the only way to extract
information regarding the composition of this region.
The elastic anisotropy of these minerals are vital to
understand the seismic anisotropy and the geometry
of mantle flow.
Experimental studies in understanding high-pressure behavior of elastic properties of relevant phases
are still lacking. For example, no measurement of
single-crystal elastic constants of MgSiO3 perovskite
exists for any pressure other than ambient conditions
0012-821X/98/$ – see front matter  1998 Elsevier Science B.V. All rights reserved.
PII: S 0 0 1 2 - 8 2 1 X ( 9 8 ) 0 0 2 3 0 - 1
372
R.M. Wentzcovitch et al. / Earth and Planetary Science Letters 164 (1998) 371–378
[1]. Bulk modulus over a wide pressure range is
known from measured equation of state [2]. Sinelnikov et al. [3] have recently measured the shear
modulus as s function of pressure up to 8 GPa
and temperature up to 800 K which correspond to
the pressure and temperature conditions well below
the realistic conditions of the lower mantle. The
shear modulus which represents one third of seismic observations is still unknown at lower mantle
pressures. Without single-crystal elastic constants,
it is impossible to study elastic anisotropy and its
pressure dependence. As counterpart to experiments,
first-principles computer simulations have been increasingly popular in exploring various properties of
the Earth’s materials at the geophysically relevant
conditions. Both plane-wave pseudopotential (PWP)
and linearized augmented plane (LAPW) methods
[4,5] have already been used to study the structural properties of MgSiO3 perovskite throughout the
pressure regime of lower mantle. Recently, Karki et
al. [6] have performed pseudopotential calculations
to predict nine single-crystal elastic constants .ci j /
of MgSiO3 orthorhombic perovskite as a function of
pressure up to 140 GPa.
In this paper, we present first-principles predictions of full elastic constant tensor .ci j / of MgSiO3
perovskite at several pressures up to 150 GPa. Our
results are comparable with the previous study of the
high-pressure elasticity from a similar method [6]
which used different pseudopotentials (generated by
Q C tuning method [7]) than our Troullier–Martins
pseudopotentials [8]. The calculated ci j are used to
make a detailed investigation of pressure variation of
elastic anisotropy of MgSiO3 perovskite. We study
the single-crystal anisotropy and also the anisotropy
of a transversely isotropic aggregate of MgSiO3 perovskite. Finally, predicted wave velocity anisotropy
is used to discuss some geophysical implications.
such that forces and stresses are fully converged. The
Brillouin zone is sampled on a 4 ð 4 ð 2 Monkhorst–
Pack k-point mesh [10], requiring 4 special k-points
for the orthorhombic perovskite structure and higher
number (up to 8 k-points) for strained lattices. Further computational details have been given elsewhere
[11].
Determination of the elastic constants requires
knowledge of equilibrium structure at a given pressure. We first fully optimize a 20-atom unit cell of
MgSiO3 perovskite at five different pressures up to
150 GPa [4,11]. The structural optimization technique uses damped molecular dynamics with variable cell shape [4]. The elastic constants are then
determined from direct computation of the stresses
generated by small deformations of the equilibrium
unit cell [12]. Strains of different amplitudes ( 0.01
to 0.01) are used and the elastic constants are derived
from resulting linear stress–strain relation. Since
strains couple to the vibrational modes, the internal
parameters are re-optimized in the strained configuration.
It is more direct and convenient to calculate the
elastic constants using stress–strain relations than
using strain-energy density. There is no need of
an extra correction for the pressure contribution to
the elastic constants in using the stress–strain relations (so no volume conserving strains are required
unlike the case of the strain energy) [13]. Simple
stress–strain relations involving elastic constants individually can be obtained for simple strains so a
small number of computations are required to extract
the full elastic constant tensor. The elastic constants
are more precise since the calculations involve the
stresses that change to first order in strain, rather
than second order in the case of the strain energy.
3. Calculated elastic constants
2. Method
Computations are based on the density functional
theory (DFT) using the local density and pseudopotential approximations [9]. The soft and separable
Troullier–Martins pseudopotentials [8] are used. A
plane wave basis set with cutoff of 70 Rydbergs is
used to expand the valence electronic wave functions
MgSiO3 perovskite is stable in orthorhombic
.Pbnm/ phase throughout the lower mantle pressure regime. Nine elastic constants of MgSiO3 perovskite are determined as a function of pressure up to
150 GPa from stress–strain relations. The calculated
athermal (0 K) elastic constants at zero pressure are
found to compare favorably with experiment [1] and
previous pseudopotential calculations [6] (Table 1).
R.M. Wentzcovitch et al. / Earth and Planetary Science Letters 164 (1998) 371–378
373
Table 1
Nine elastic constants (ci j ), and bulk (K ) and shear (G) moduli in GPa of orthorhombic MgSiO3 perovskite compared with previous
studies
P (GPa)
c11
Zero pressure
This study
485 (17)
Cal [6]
487
Exp [1]
482
High pressure
30
632
60
753
100
926
150
1100
c22
560 (17)
524
537
735
922
1160
1416
c33
474 (17)
456
485
653
835
1056
1274
c44
c55
c66
c12
c13
c23
K
G
200 (7)
203
204
176 (7)
186
186
155 (7)
145
147
130 (10)
128
144
136 (10)
144
147
144 (10)
156
146
259
258
264
179
175
177
250
300
360
427
204
234
265
296
212
268
330
401
225
320
460
606
209
278
380
478
233
306
406
525
372
478
623
773
223
266
314
365
The small differences from experiments can be attributed to the fact that the calculations are athermal
(at 0 K) whereas experimental data are obtained at
300 K. Much of the differences between present
elastic constants and the pseudopotential results of
Karki et al. [6] can be accounted for the differences
in the pseudopotentials used.
No experimental data for single-crystal elastic
constants of MgSiO3 perovskite are presently available for any elevated pressure. Our elastic constants
at high pressures (up to 150 GPa), shown in Fig. 1
and Table 1, compare favorably with other pseudopotential calculations [6]. We find c22 > c11 > c33 ;
c44 > c55 > c66 and c23 > c13 > c12 at 0 GPa
which change respectively to c22 > c33 > c11 ;
c44 > c66 > c55 and c12 > c23 > c13 at pressures
around 30 GPa. This indicates that the b-axis is
the least compressible at all pressures, and a- and
c-axial compressibilities are very similar at low pressures (consistent with experiments [14,15]) but the
a-axis becomes most compressible at high pressures
(above 30 GPa). Elastic constants associated with
shear strains (such as c44 , c55 , c66 ) also change with
pressure differently. At high pressure, c55 becomes
much smaller than c44 and c66 , indicating that the
shear along the (010) plane becomes easy relative to
the shear along (100) and (001) planes.
4. Elastic anisotropy
It is important to understand the elastic anisotropy
and its pressure dependence for the relevant oxide
and silicate minerals in interpreting the seismologi-
Fig. 1. Pressure variation of elastic constants (ci j ) of MgSiO3
perovskite.
cal observations to extract information regarding the
geometry of the mantle flow. Using recent first-principles results on mineral properties, there have been
made some attempts in understanding the seismic
anisotropy of the deep mantle [16,17].
In order to study elastic anisotropy of MgSiO3
perovskite, we solve the Christoffel equation [18] to
determine the single crystal elastic wave velocities
in different directions, as shown in Fig. 2 at three
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R.M. Wentzcovitch et al. / Earth and Planetary Science Letters 164 (1998) 371–378
Fig. 2. Variation of (a) compressional and (b) shear wave velocities of MgSiO3 perovskite with propagation direction at three pressures
of 0, 30 and 130 GPa.
pressures of 0, 30 and 130 GPa. The elastic wave velocities vary significantly with propagation direction
and as well as with polarization direction suggesting
that the perovskite exhibits strong anisotropy in both
compressional (P) and shear (S) wave velocities [6].
A measure of the azimuthal anisotropy for compressional .AP / and shear .AS / waves is given by the
following relations
VPmax VPmin
ð 100I
hVP i
VSmax VSmin
ð 100
AS D
hVS i
AP D
where
hVP i D
s
K C 43 G
I hVS i D
²
(1)
s
G
²
(2)
are the isotropic aggregate velocities, and ² is the
density. The isotropic bulk .K / and shear .G/ moduli are taken from the Voigt–Ruess–Hill averaging
scheme [19]. There are two shear waves propagat-
ing with different velocities (i.e., S-wave birefringence). The wave velocity anisotropy of MgSiO3
perovskite depends on pressure in a complicated
manner (Fig. 3a,b).
With increasing pressure the azimuthal anisotropy
at first decreases up to about 30 GPa and then increases for both P and S waves (Fig. 3a). Karki et
al. [6] also found that the azimuthal anisotropy decreases up to pressure of 20 to 40 GPa and thereafter
increases continuously. Some directions of fastest
and slowest propagation also change with pressure,
as can be seen from Fig. 2. At zero pressure, P
wave is slowest and fastest in the [001] and [010]
directions respectively, as determined by ordering
c22 > c11 > c33 . The fastest S shear waves propagate
in [010] and [001] crystallographic directions which
correspond to c44 (i.e., polarization along [001] and
[010] directions respectively), whereas slowest S
waves propagate in [100] and [010] directions, consistent with the smaller value of c66 (i.e, the waves
are polarized along [010] and [100] directions respectively). At high pressure (above 30 GPa), P
R.M. Wentzcovitch et al. / Earth and Planetary Science Letters 164 (1998) 371–378
375
Fig. 3. Pressure variation of (a) azimuthal and (b) polarization (for S wave) anisotropy of elastic wave velocities of MgSiO3 perovskite.
wave velocity shows a minimum in the [100] direction, consistent with the smaller value of c11 , and
its fastest direction remains unchanged. For S waves,
the fastest directions remain unchanged due to relatively large value of c44 , however, one of the slowest
directions changes to [001] at 30 GPa with slowest waves polarized parallel to the [001] and [100]
directions (i.e, c55 ) respectively.
Fig. 3b shows the pressure dependence of the
polarization anisotropy of shear waves for three
crystallographic directions, namely, [100], [010] and
[001]. For [100] direction, the anisotropy at first decreases and then increases with increasing pressure,
whereas the anisotropy decreases for [010] direction and increases for [001] direction throughout the
pressure regime studied. This indicates that the polarization anisotropy of shear waves depends strongly
on propagation direction. The direction of maximum
velocity difference due to polarization for S waves
changes from [010] to [001] at about 30 GPa because
c55 becomes smaller than c66 at higher pressures
as discussed in previous paragraph. It is important
from the point view of seismology to estimate the
anisotropy of a transversely isotropic aggregate of
MgSiO3 perovskite [20,21]. A transversely isotropic
medium with vertical symmetry axis (c-axis) can be
characterized by five elastic moduli which can be
determined from single-crystal elastic constants .ci j /
using the following relations,
A D 38 .c11 C c22 / C 14 c12 C 12 c66
(3)
C D c33
(4)
F D 12 .c13 C c23 /
(5)
N D 18 .c11 C c22 /
1
c
4 12
L D 12 .c44 C c55 /
C 12 c66
(6)
(7)
The velocities of P waves propagating in perpendicular (horizontal) and parallel (vertical) to the axis of
transverse symmetry, and the velocities of S waves
polarized horizontally and vertically are given by
s
s
A
C
I VPV D
(8)
VPH D
²
²
s
VSH D
N
I VSV D
²
s
L
²
(9)
where ² is density. By simply interchanging coordinate axes, we can modify the formulae above to
calculate wave velocities of a transversely isotropic
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R.M. Wentzcovitch et al. / Earth and Planetary Science Letters 164 (1998) 371–378
moduli .c22 C c33 / and c44 which remain relatively
large at all pressures. For [010] symmetry axis, both
ATP and ATS are negative and become more negative
(more rapidly for S wave) with increasing pressure.
The opposite behavior of anisotropy compared to the
a-axis occurs because the elastic moduli .c11 C c33 /
and c55 are relatively small. Finally for [001] symmetry direction, ATP remains positive at all pressures
whereas ATS is negative up to 100 GPa and thereafter
it becomes positive, although anisotropy is small
for this orientation. As in the case of single-crystal
anisotropy, the pressure variations of the transverse
anisotropy predicted by two pseudopotential calculations are very similar.
5. Geophysical implications
Fig. 4. Pressure dependence of wave velocity anisotropy for
a transversely isotropic aggregate of MgSiO3 perovskite with
symmetry axis along [100], [010] and [001] directions.
medium with a- or b-axis as symmetry axis. The
degree of transverse anisotropy in P and S waves can
be determined using
ATP D
VPH VPV
ð 100I
hVP i
ATS D
VSH VSV
ð 100
hVS i
(10)
Fig. 4 shows the anisotropy factors ( ATP and ATS )
as a function of pressure for a transversely isotropic
composite of MgSiO3 perovskite with each crystallographic axis, in turn, as a symmetry axis. The choice
of the symmetry directions that are coincident with
crystallographic axes is made because the lattice preferred orientations of many orthorhombic materials
show such symmetry (olivine, orthopyroxene and
orthorhombic perovskite [22]. For [100] direction
(i.e., a-axis) as the axis of transverse isotropy, the
anisotropy factors are positive and increase (more
rapidly for P wave) with pressure. The P waves
propagate faster in a direction perpendicular to than
along the [100] direction. Horizontally polarized S
waves (SH) propagate faster than vertically polarized S waves (SV). This is determined by the elastic
As shown earlier, MgSiO3 perovskite exhibits
very strong single crystal anisotropy throughout the
lower mantle pressures. Single crystal anisotropy
gives the upper limit on the realistic anisotropy of
any polycrystalline aggregates. The magnitude of
anisotropy due to the lattice preferred orientation
of a polycrystalline aggregate is, in general, much
smaller (by a factor of 2 to 3) than that of a perfectly
oriented single crystal. This has been found to be the
case for several minerals including perovskite and
olivine [22,23]. But LPO of MgSiO3 polycrystals
can still produce seismically detectable anisotropy in
the lower mantle. The elastic anisotropy of MgSiO3
perovskite has important implications for seismic
anisotropy. However, some assumptions must be
made to infer seismic anisotropy from the present
calculation. First, the present calculations are performed at T D 0 K. However, temperatures in the
lower mantle are considered to be high (2000 to 4000
K) [24]. Therefore, corrections for the temperature
effects must be considered. We use a simplifying
assumption that the temperature effects can be accounted for by assuming the Birch’s law [25] of
corresponding state, namely, the notion that elastic constants depend only on density. Under this
assumption, elastic constants at high pressures and
temperatures can be estimated from those at high
pressure and zero temperature (i.e., from the elastic constants we calculated) by shifting pressure by
PTH D ÞK T , where Þ is thermal expansivity and
R.M. Wentzcovitch et al. / Earth and Planetary Science Letters 164 (1998) 371–378
K is our calculated bulk modulus. It is worth to
mention that Birch’s law predicts an increase in
anisotropy of MgO at low pressure with increasing
temperature in agreement, at least, qualitatively, with
recent elasticity measurements at combined pressure
(up to 8 GPa) and temperature (1600 K) conditions
[26]. With this assumption, elastic constants in the
shallow (700 km) and the deep (2500 km) lower
mantle correspond, respectively, to those at 10 and
100 GPa at T D 0 K. Second, the effects of Fe
are assumed to be negligible. This has been well
established for the bulk modulus [14], and for the
lack of other data, we also assume that this applies
to other elastic constants. Third, effects of anelastic
relaxation is not important for anisotropy. Many of
these assumptions can be challenged, but the main
conclusions as to the seismic anisotropy are believed
to not be significantly affected by these assumptions.
On the basis of these assumptions, we can draw
the following conclusions from this work. First, the
seismological observations of absence of anisotropy
in most of the lower mantle [27,28] suggests a nearly
random orientation of MgSiO3 perovskite grains.
The most likely mechanism to cause random orientation is deformation by superplasticity [22]. This conclusion must hold also for another important component of the lower mantle, namely magnesiowustite
[16]. Second, there is some hint as to the presence of
VSV > VSH anisotropy in the topmost lower mantle
[28]. If the (010) plane is assumed to be nearly horizontal for horizontal flow as shown by Karato et al.
[22], this can be attributed to the preferred orientation of MgSiO3 perovskite (Fig. 4). Note, however,
that the magnitude of anisotropy is rather small for
perovskite. Contribution from other anisotropic minerals, e.g., magnesiowustite, may also be required
[29]. Third, the D00 layer, bottom 200–300 km of
the lower mantle, appears to show significant but
spatially variable anisotropy [30]. Strong VSH > VSV
anisotropy is reported in the in circum Pacific. This
observation is difficult to reconcile with the lattice
preferred orientation of perovskite unless the dominant glide plane becomes (100) in the D00 layer
(Fig. 4). However, we consider that this is unlikely
because c44 =c55 ratio increases with pressure (Fig. 1),
which favors glide on the (010) plane relative to glide
on the (100) plane under high pressures. More likely
explanation is therefore the significant contribution
377
from magnesiowustite whose elastic anisotropy becomes very large under the D00 layer conditions
[12,29].
6. Concluding remarks
Our calculated elastic constants of orthorhombic
MgSiO3 perovskite over the pressure range 0 to 150
GPa are in excellent agreement with existing ambient pressure experimental data [1] and high pressure
results from a previous study of Karki et al. [6]. We
use our elastic constants to make detailed investigations of the elastic wave velocity anisotropy for the
single crystal and transversely isotropic aggregate
of MgSiO3 perovskite. MgSiO3 perovskite exhibits
strong anisotropy throughout the lower mantle and
the nature of anisotropy changes significantly with
depth. Some propagation directions of fastest and
slowest P and S waves, and also the direction of
maximum polarization anisotropy change at about
30 GPa and T D 0 K which is equivalent to the
conditions at 1500 km. Possible implications for the
seismic anisotropy of the lower mantle are discussed.
Our study has, however, some limitations including
(1) the use of Birch’s law to estimate the temperature
effects on elastic constants and (2) the ignorance
of the effects of Fe. Further studies on these issues
are needed to better understand the seismic wave
velocities in silicate perovskite in the lower mantle.
Acknowledgements
R.M.W. acknowledges support from the NSF
(EAR-9628199). S.K. acknowledges support from
the NSF (EAR-9505451, 9526239, 9706329).
C.R.S.D.S. is supported by CNPq from Brazil. [CL]
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