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