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 MgSiO 3 -perovskite MgO Geophysical inferences Renata M. Wentzcovitch U. of Minnesota (USA) and SISSA (Italy)

The Contribution from Seismology Longitudinal (P) waves Transverse (S) wave from free oscillations

PREM (Preliminary Reference Earth Model) (Dziewonski & Anderson, 1981) P(GPa)

Mantle Mineralogy SiO MgO 37.8 FeO 8.1 Al 2 O CaO 3.6 Cr 2 O Na 2 O 0.4 NiO 0.2 TiO MnO 0.1 (McDonough and Sun, 1995) Pyrolite model (% weight) Depth (km) P (Kbar) V % Olivine perovskite  -phase spinel MW garnets opx cpx (Mg 1--x,Fe x ) 2 SiO 4 (‘’) MgSiO 3 (Mg,Al,Si)O 3 (Mg,Fe) (Si,Al)O 3 (Mg 1--x,Fe x ) O (Mg,Ca)SiO 3 CaSiO 3

Mantle convection

Temperature and Composition of LM

Lateral Heterogeneities

3D Maps of V s and V p V s V  V p ( Masters et al, 2000)

Anisotropy   isotropic transverse azimuthal V P V S1 = V S2 V P (  ) V S1 (  )  V S2 (  ) V P ( ,  ) V S1 ( ,  )  V S2 ( ,  )

Anisotropy in the Earth (Karato, 1998)

Mantle Anisotropy SH>SV SV>SH

Slip system Zinc wire F Slip systems and LPO

Lattice Preferred Orientation (LPO) Shape Preferred Orientation (SPO) Mantle flow geometry LPOSeismic anisotropy slip system C ij Anisotropic Structures

+ Mineral sequence II Lower Mantle 410 km Transition Zone (520 km (?)) 670 km (Mg x,Fe (1-x) )O (Mg x,Fe (1-x) )SiO 3 (Mg x,Fe (1-x) ) 2 SiO 4 (Olivine) Upper Mantle (Spinel)

T M of mantle phases Core T Mantle adiabat solidus HA Mw (Mg,Fe)SiO 3 CaSiO 3 peridotite P(GPa) 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 phonons xxxxxxxxxxxxxxxxxx(Gianozzi, Baroni, and de Gironcoli, 1991) + Quasiharmonic approximation (QHA) for thermal properties (e.g., , C P, S, K T, C ij ’s). Soft & separable pseudopotentials (Troullier-Martins)

abcxP K th = 259 GPa K’ th =3.9 K exp = 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)

Crystal ( Pbnm ) equilibrium structure  kl re-optimize Elastic constant tensor 

Yegani-Haeri, 1994 Wentzcovitch et al, 1995 Karki et al, 1997 within 5% S-waves (shear) P-wave (longitudinal) n propagation direction Elastic Waves

Cristoffel’s eq.: with is the propagation direction Wave velocities in perovskite (Pbnm)

Anisotropy P-azimuthal: S-azimuthal: S-polarization:

Voigt: uniform strain Reuss: uniform stress Voigt-Reuss averages: Poly-Crystalline aggregate

Polarization anisotropy in transversely isotropic medium High P, slip systems MgO: {100} ? (c 44 < c 11 -c 12 ) MgSiO 3 pv: {010} ? (soft c 55 ) Seismic anisotropy Isotropic in bulk LM 2% V SH > V SV in D’’ SH/SV Anisotropy (%) (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 (Karki, Wentzcovitch, de Gironcoli and Baroni, PRB 61, 8793, 2000) -

Phonon dispersion of MgSiO 3 perovskite 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 Volume (Å 3 ) F (Ry) 4 th order finite strain equation of state staticzero-point thermal MgO Static 300K Exp (Fei 1999) V (Å 3 ) K (GPa) K´ K´´(GPa -1 )

Thermal expansivity of MgO and MgSiO 3 (Karki, Wentzcovitch, de Gironcoli and Baroni, Science 286, 1705, 1999)  (10 -5 K -1 )

MgSiO 3 -perovskite and MgO 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 MgSiO 3 at LM Conditions

Adiabatic bulk modulus at LM P-T (Karki, Wentzcovitch, de Gironcoli and Baroni, GRL, 2001 )

LM Geotherms

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 MgSiO 3 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