First Principles Thermoelasticity of Minerals

1. 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 •

2. …``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)

3. First Principles VCS-MD(Wentzcovitch, Martins, Price, PRL 1993) Damped dynamics MgSiO3 P = 150 GPa

4. Lattice (a,b,c)th < (a,b,c)exp ~ 1% Tilt angles th - exp < 1deg Kth = 259 GPa K’th=3.9 Kexp = 261 GPa K’exp=4.0 (• Wentzcovitch, Martins, & Price, 1993) ( Ross and Hazen, 1989)

5. Lower Mantle Mineral sequence II + + (Mg(1-x-z),Fex, Alz)(Si(1-y),Aly)O3 (Mgx,Fe(1-x))O CaSiO3

6. Lower Mantle Mineral sequence II + (Mgx,Fe(1-x))SiO3 (Mgx,Fe(1-x))O

7. TM of mantle phases CaSiO3 (Mg,Fe)SiO3 5000 Mw Core T 4000 HA solidus T (K) 3000 Mantle adiabat 2000 peridotite 0 20 40 60 80 100 120 P(GPa) (Zerr, Diegler, Boehler, Science1998)

8. Thermodynamic Method • VDoS and F(T,V) within the QHA N-th (N=3,4,5…) order isothermal (eulerian or logarithm) finite strain EoS IMPORTANT: crystal structure and phonon frequencies depend on volume alone!!….

9. (Thermo) Elastic constant tensor  kl equilibrium structure re-optimize

10. Phonon dispersions in MgO (Karki, Wentzcovitch, de Gironcoli and Baroni, PRB 61, 8793, 2000) - Exp: Sangster et al. 1970

11. Phonon dispersion of MgSiO3 perovskite Calc Exp - Calc Exp 0 GPa - Calc:Karki, Wentzcovitch, de Gironcoli, Baroni PRB 62, 14750, 2000 Exp:Raman [Durben and Wolf 1992] Infrared [Lu et al. 1994] 100 GPa

12. Zero Point Motion Effect MgO F (Ry) - - Volume (Å3) Static300KExp (Fei 1999) V (Å3) 18.5 18.8 18.7 K (GPa) 169 159 160 K´ 4.18 4.30 4.15 K´´(GPa-1) -0.025 -0.030

13. MgSiO3-perovskite and MgO 4.8 (256) 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]

14. Elasticity of MgO (Karki et al., Science 1999)

15. table 10.97 (Wentzcovitch et al, Phys. Rev. Lett (in press))

16. Thermal expansivity and the QHA provides an a posteriori criterion for the validity of the QHA (10-5 K-1)    MgSiO3 Karki et al, GRL (2001)

17. Criterion: inflection point of (T) The QHA invalid MgO MgSiO3 Brown & Shankland’s T

18. …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

19. Comparison with Experiments (Ross & Hazen, 1989) o Calc. 77 K < T < 400K 0 GPa < P < 12 GPa o o

20. Comparison with Experiments (Ross & Hazen, 1989) o Calc. 77 K < T < 400K 0 GPa < P < 12 GPa o 1% o LDA +ZP Exp. LDA

21. Test: comparison with experiments 0.05% 0.003 (Ross & Hazen)

22. Predictions 4000 K 3000 K 2000 K 1000 K 300 K

23. 300 K 1000K 2000K 3000 K 4000 K Cij(P,T) cij (Oganov et al,2001) (Wentzcovitch et al, Phys. Rev. Lett. in press)

24. Velocities V (km/sec) &  (gr/cm3) (Wentzcovitch et al, in press)

25. Aggregate Moduli 38 GPa 88 GPa

26. Effect of Fe alloying (Kiefer,Stixrude, Wentzcovitch, GRL 2002) (Mg0.75Fe0.25)SiO3 || + + + 4

27. Pyrolite (20 V% mw) Perovskite 0.10<xFe<0.15 100 GPa 38 GPa aaaa Brown & Shankland T (Mg(1-x),Fex)SiO3 (Jackson,1998) aaaa Wentzcovitch et al, PRL, in press)

28. (Masters et al, 2000) 3D Maps of Vs and Vp Vs V Vp

29. Lateral variations in VS and VP (Karato & Karki, JGR 2001) (MLDB-Masters et al., 2000) (KWH-Kennett et al., 1998) (SD-Su & Dziewonski, 1997) (RW-Robertson & Woodhouse,1996)

30. Lateral variations in V and VP (MLDB-Masters et al., 2000) (SD-Su & Dziewonski, 1997) (Karato & Karki, JGR 2001)

31. Relations 0.42 ≤ A ≤ 0.37 with

32. Anderson Gruneisen parameters: s 

33. Lateral heterogeneity ratio: R/sRs/p 1/A MLDB (MLDB-Masters et al., 2000)

34. R/s and R/p R/sR/p FWD CF FDW FDW’ IT IT- Ishi & Tromp, 1999 CF-Cadek & Fleitout, 1999 FDW & FDW’, Forte at al., 1993 FWD, Forte at al., 1994

35. 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(20))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.

36. 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