First Principles Thermoelasticity of Minerals:Insights into the Earth’s LM
• Problems related with seismic observations T and composition in the lower mantle Origin of lateral heterogeneities Origin of anisotropies
• How and what we calculate MgSiO3-perovskite MgO
• Geophysical inferences
Renata M. Wentzcovitch U. of Minnesota (USA) and SISSA (Italy)
The Contribution from Seismology
VP K
4
3G
VS G
Longitudinal (P) waves
Transverse (S) wave
from free oscillations
PREM (Preliminary Reference Earth Model)
(Dziewonski & Anderson, 1981)
0 24 135 329 364P(GPa)
Mantle Mineralogy
SiO2 45.0MgO 37.8FeO 8.1Al2O3 4.5CaO 3.6Cr2O3 0.4Na2O 0.4NiO 0.2TiO2 0.2MnO 0.1
(McDonough and Sun, 1995)
Pyrolite model (% weight)
De
pth
(km
)
P (
Kba
r)
V %
8
4
12
16
20
6020 40 80 1000
100
300
500
700
Olivine
perovskite
-phase
spinel
MW
garnets
opx
cpx(Mg1--x,Fex)2SiO4
(‘’)
(‘’)
MgSiO3
(Mg,Al,Si)O3
(Mg,Fe) (Si,Al)O3
(Mg1--x,Fex) O
(Mg,Ca)SiO3
CaSiO3
Mantle convection
Temperature and Composition of LM
Lateral Heterogeneities
3D Maps of Vs and Vp
Vs V Vp
(Masters et al, 2000)
Anisotropy
isotropic
transverse
azimuthal
VP
VS1= VS2
VP () VS1 () VS2 ()
VP (,) VS1 (,) VS2 (,)
(VSH VSV)%
VS
Anisotropy in the Earth
(Karato, 1998)
Mantle AnisotropySH>SVSV>SH
Slip system
Zinc wire
F
Slip systems and LPO
Lattice Preferred Orientation (LPO) Shape Preferred Orientation
(SPO)
Mantle flow geometry
LPO Seismic anisotropy
slipsystem Cij
Anisotropic Structures
+
Mineral sequence II
Lower Mantle
410 kmTransition Zone
(520 km (?))
670 km
(Mgx,Fe(1-x))O
(Mgx,Fe(1-x))SiO3
(Mgx,Fe(1-x))2SiO4 (Olivine)
Upper Mantle
(Spinel)
TM of mantle phases
Core T
Mantle adiabat
solidusHA
Mw
(Mg,Fe)SiO3
CaSiO3
peridotite
P(GPa)0 4020 60 80 100 120
2000
3000
4000
5000
T (
K)
(Zerr, Diegler, Boehler, 1998)
Method
• Structural optimizations
• First principles variable cell shape MD for structural optimizations xxxxxxxxxxxxxxxxxx(Wentzcovitch, Martins,& Price, 1993)
Self-consistent calculation of forces and stresses (LDA-CA)
• Phonon thermodynamics
• Density Functional Perturbation Theory for phononsxxxxxxxxxxxxxxxxxx(Gianozzi, Baroni, and de Gironcoli, 1991)
+ Quasiharmonic approximation (QHA) for thermal properties
(e.g., , CP, S, KT, Cij’s).
• Soft & separable pseudopotentials (Troullier-Martins)
abcxP
K Vo
dP
dV
Kth = 259 GPa K’th=3.9
Kexp = 261 GPa K’exp=4.0
(a,b,c)th < (a,b,c)exp ~ 1%
Tilt angles th - exp < 1deg
( Wentzcovitch, Martins, & Price, 1993)
( Ross and hazen, 1989)
c11 c12 c13 * * *
c21 c22 c23 * * *
c31 c32 c33 * * *
* * * c44 * *
* * * * c55 *
* * * * * c66
• Crystal (Pbnm)
equilibrium structure
kl
re-optimize
Elastic constant tensor
jiTijcPTGPTG
2
1),,0(),;(
V
jiTij
Sij C
VTPTcPTc
),(),(
Tii
S
Yegani-Haeri, 1994Wentzcovitch et al, 1995Karki et al, 1997
within 5%
S-waves (shear)
P-wave (longitudinal)
n propagation direction
Elastic Waves
VI2 iI ikkICristoffel’s eq.: with ik cijkln jnl
n is the propagation direction
Wave velocities in perovskite (Pbnm)
Anisotropy
P-azimuthal:
S-azimuthal:
S-polarization:
avP
PPP
V
VVV
minmax
avS
SSS
V
VVV
minmax
avS
SS
SV
VVV
max
21
C11 C12 C12 * * *
C12 C11 C12 * * *
C12 C12 C11 * * *
* * * C44 * *
* * * * C44 *
* * * * * C44
C44G
C12 C11 2C44
C11 2C12
3K•Voigt: uniform strain
MN
MN ij
ij
MN CMNRSRS
MN SMNRSRS
CMNRS niMnj
NnkRnlagg S cijklf(n)
SMNRS niMnj
NnkRnlagg S sijklf(n)
•Reuss: uniform stress
•Voigt-Reuss averages:
CMNRS CMNRS SMNRS
1
2
• Poly-Crystalline aggregate
Polarization anisotropy in transversely isotropic medium
High P, slip systems
MgO: {100} ? (c44 < c11-c12)
MgSiO3 pv: {010} ? (soft c55)
Seismic anisotropy
Isotropic in bulk LM2% VSH > VSV in D’’
SH
/SV
Ani
sotr
opy
(%)
(Karki et al. 1997; Wentzcovitch et al1998)
-
-
-
Theory x PREM
Acoustic Velocities of Potential LM Phases
(Karki, Stixrude, Wentzcovitch,2001)
Phonon dispersions in MgO
Exp: Sangster et al. 1970
(Karki, Wentzcovitch, de Gironcoli and Baroni, PRB 61, 8793, 2000)
-
Phonon dispersion of MgSiO3 perovskite
Calc Exp
Calc Exp
Calc: Karki, Wentzcovitch, de Gironcoli, Baroni PRB 62, 14750, 2000
Exp: Raman [Durben and Wolf 1992] Infrared [Lu et al. 1994]
0 GPa
100 GPa
--
Quasiharmonic approximation
qj B
qjB
qj
qj
Tk
VTk
VVUTVF
)(exp1ln
2
)()(),(
Volume (Å3)
F (
Ry)
4th order finite strain equation of state
static zero-point
thermal
MgO
Static 300K Exp (Fei 1999)V (Å3) 18.5 18.8 18.7K (GPa) 169 159 160K´ 4.18 4.30 4.15K´´(GPa-1) -0.025 -0.030
-
-
-
-
Thermal expansivity of MgO and MgSiO3
(Karki, Wentzcovitch, de Gironcoli and Baroni, Science 286, 1705, 1999)
(
10-5 K
-1)
MgSiO3-perovskite and MgO
(gr/cm-3)
V (A3)
KT
(GPa) d KT/dP d KT
2/dP2
(GPa-1) d KT/dT (Gpa K-1)
10-5 K-1
3.580 18.80 159 4.30 -0.030 -0.014 3.12 Calc. MW
3.601 18.69 160 4.15 ~ -0.0145 3.13 Exp. MW
4.210 164.1 247 4.0 -0.016 -0.031 2.1 Calc. Pv
4.247 162.3 246 | 266
3.7 | 4.0
~ -0.02 | -0.07
1.7 | 2.2
Exp. Pv
Exp.: [Ross & Hazen, 1989; Mao et al., 1991; Wang et al., 1994; Funamori et al., 1996; Chopelas, 1996; Gillet et al., 2000; Fiquet et al., 2000]
Elastic moduli of MgO at high P and T(Karki et al., Science 1999)
Elasticity of MgSiO3 at LM Conditions
Adiabatic bulk modulus at LM P-T(Karki, Wentzcovitch, de Gironcoli and Baroni, GRL, 2001)
LM GeothermsLM Geotherms
1000
2000
3000
4000
5000
6000
500 1000 1500 2000 2500 3000
T (
K)
Depth (km)
Pv
Solidus
Isentropes
Pyrolite
CMB|
Tc
Stratified Lower Mantle(Kellogg, Hager, van der Hilst, 1999)
Summary• Building a consistent body of knowledge obout LM phases
• QHA is suitable for studying thermal properties of minerals at
LM conditions
• A homogeneous and adiabatic LM model appears to be incompatible with PREM.
• LPO in aggregates of MgO and MgSiO3 can exhibit strong anisotropy at LM conditions.
• We have all ingredients now to re-examine what has been said about lateral variations.
Acknowledgements
Bijaya B. Karki (U. of MN/LSU)
Shun-ichiro Karato (U. of MN/Yale)Boris Kiefer (U. of MI)Lars Stixrude (U. of MI)
Stefano Baroni (SISSA)Stefano de Gironcoli (SISSA)
Funding: NSF/EAR