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First Principles Thermoelasticity of Minerals
Renata M. M. Wentzcovitch
Department of Chemical Engineering and Materials Science U. of Minnesota, Minneapolis
• First Principles Thermodynamics Method
• Thermoelasticity of Mg(,Fe)SiO3 Crystal structure(P,T)
Elasticity: comparison with calculations and experiments Elasticity: comparison with PREM Logarithm ratios and lateral variations
• Summary
•
…``First Principles’’…
• Density Functional Theory ( , , )
• Local Density Approximation (Kohn and Sham,1965; Ceperley-Alder, 1985)
• First Principles Pseudopotentials (Troullier-Martins, 1991)
• Born-Oppenheimer Variable Cell Shape Molecular Dynamics (Wentzcovitch, 1991-3)
• Density Functional Perturbation Theory for Phonons (Gianozzi et al., 1991)
EH )]([ rnE
iiin *
First Principles VCS-MD (Wentzcovitch, Martins, Price, PRL 1993)
Damped dynamics
)(~ PI),(~ int rffr
P = 150 GPa
MgSiO3
Lattice
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)
+
Mineral sequence II
Lower Mantle
(Mgx,Fe(1-x))O(Mg(1-x-z),Fex, Alz)(Si(1-y),Aly)O3
+
CaSiO3
+
Mineral sequence II
Lower Mantle
(Mgx,Fe(1-x))O(Mgx,Fe(1-x))SiO3
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, Science1998)
Thermodynamic Method
qj B
qjB
qj
qj
Tk
VTk
VVUTVF
)(exp1ln
2
)()(),(
• VDoS and F(T,V) within the QHA
PVTSFG TV
FP
VT
FS
N-th (N=3,4,5…) order isothermal (eulerian or logarithm) finite strain EoS
IMPORTANT: crystal structure and phonon frequencies depend on volume alone!!….
equilibrium structure
kl
re-optimize
(Thermo) Elastic constant tensor
Pji
Tij
GPTc
2
),(
V
jiTij
Sij C
VTPTcPTc
),(),(
Tii
S
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
--
Zero Point Motion Effect
Volume (Å3)
F (
Ry)
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
-
-
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]
4.8
(256)
Elasticity of MgO
(Karki et al., Science 1999)
table
10.97
(Wentzcovitch et al, Phys. Rev. Lett (in press))
Thermal expansivity and the QHA
(
10-5 K
-1)
provides an a posteriori criterion for the validity of the QHA
MgSiO3
Karki et al, GRL (2001)
The QHA
Criterion: inflection point of (T)
Brown & Shankland’s T
invalid MgO
MgSiO3
…IMPORTANT: structural parameters and phonon frequencies depend on volume alone!!
• Structures at high P are determined at T= 0
P(V,0)
• P’(V,T’) within the QHA
• At T 0… V(P’,T’)=V(P,0) structure(P’,T’) = structure(P,0)
Corresponding States
Comparison with Experiments(Ross & Hazen, 1989)
77 K < T < 400K
0 GPa < P < 12 GPa
o
o
o
Calc.
Comparison with Experiments(Ross & Hazen, 1989)
77 K < T < 400K
0 GPa < P < 12 GPa
o
o
o
Calc.
LDALDA+ZPExp.
1%
Test: comparison with experiments
(Ross & Hazen)
0.003
0.05%
Predictions4000 K3000 K2000 K1000 K 300 K
cij
(Wentzcovitch et al, Phys. Rev. Lett. in press)
300 K1000K2000K3000 K4000 K
(Oganov et al,2001)
Cij(P,T)
Velocities
V (
km/s
ec)
&
(g
r/cm
3 )
(Wentzcovitch et al, in press)
Aggregate Moduli
38 GPa 88 GPa
Effect of Fe alloying
(Kiefer,Stixrude, Wentzcovitch, GRL 2002)
(Mg0.75Fe0.25)SiO3
4
+ + +
||
Pyrolite (20 V% mw)Perovskite
Brown & Shankland T
38 GPa 100 GPa
0.10<xFe<0.15
aaaa
aaaa41
Fepv
Femw
x
x
(Mg(1-x),Fex)SiO3
(Jackson,1998)
Wentzcovitch et al, PRL, in press)
3D Maps of Vs and Vp
Vs V Vp
(Masters et al, 2000)
RS / P lnVs
lnVP P
(MLDB-Masters et al., 2000)(KWH-Kennett et al., 1998)(SD-Su & Dziewonski, 1997)(RW-Robertson & Woodhouse,1996)
Lateral variations in VS and VP
(Karato & Karki, JGR 2001)
R / S lnV
lnVS P
(MLDB-Masters et al., 2000)(SD-Su & Dziewonski, 1997)
Lateral variations in V and VP
(Karato & Karki, JGR 2001)
Relations
RS / P 1
(1 A)R / S AP
A 4VS
2
3VP2
0.42 ≤ A ≤ 0.37with
R / S (S 1)
( 1)P
S lnKS
lnP
lnG
lnP
Anderson Gruneisen parameters:
P
Ss
K
ln
ln
P
G
ln
ln
s
Lateral heterogeneity ratio:
(MLDB-Masters et al., 2000)
MLDBR/
s
R
s/p
1/A
R/s and R/p
R/
s
R/
p
CF
FWD
FDW’
FDW
ITIT- Ishi & Tromp, 1999CF-Cadek & Fleitout, 1999FDW & FDW’, Forte at al., 1993FWD, Forte at al., 1994
Summary
• We are building a consistent body of knowledge about lower mantle phases using adequate methods.
• Inferences about LM based on current knowledge:
- Homogeneous LM based on (Mg(1-x),Fex)SiO3 and (Mg(1-y),Fex)O alone cannot explain PREM’s elastic gradients
- CaSiO3, (Mg(1-x-z) Alz,Fex,Alz)SiO3
- (Mg(1-y),Fey(20))O and y/x
• Anelasticity is less important in the LM than Karato estimated.
• Bonus: crystal structure of MgSiO3 at high P,T. Easiest way to test our predictions.
Acknowledgements
Bijaya B. Karki (LSU)Stefano de Gironcoli (SISSA)Stefano Baroni (SISSA)Matteo Cococcioni (MIT)
Shun-ichiro Karato (U. of MN/Yale)
Funding: NSF/EAR, NSF/COMPRES