First Principles Thermoelasticity of Minerals: Insights into the Earths LM Seismic observations and the nature of the LM T and composition in the lower mantle Origin of lateral heterogeneities Origin of anisotropies What and how we calculate MgSiO 3 -perovskite MgO Geophysical inferences Renata M. Wentzcovitch U. of Minnesota Acknowledgements Bijaya B. Karki Stefano de Gironcoli (SISSA) Stefano Baroni (SISSA) Lars Stixrude (Ann Arbor) Shun Karato (U. of MN/Yale) Funding: NSF/EAR The Contribution from Seismology Longitudinal (P) waves Transverse (S) wave from free oscillations PREM (Preliminary Reference Earth Model ) (Dziewonski & Anderson, 1981) P(GPa) Pie 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 3D Maps of V s and V p V s V V p ( Masters et al, 2000) Description 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) Lattice Preferred Orientation (LPO) Shape Preferred Orientation (SPO) Mantle flow geometry LPOSeismic anisotropy slip system C ij Anisotropic Structures Slip system Zinc wire F Slip systems and LPO Mantle Anisotropy SH>SV SV>SH (Karato) 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) S&H(pv) Method Soft & separable pseudopotentials (Troullier-Martins) Structural optimizations First principles variable cell shape MD for structural optimizations xxxxxxxxxxxxxxxxxx(Wentzcovitch, Martins,& Price, 1993) Phonon thermodynamics Density Functional Perturbation Theory xxxxxxxxxxxxxxxxxx(Gianozzi, Baroni, and de Gironcoli, 1991) Typical Computational Experiment Damped dynamics (Wentzcovitch, 1991) P = 150 GPa 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 kl ij c ijkl (i,j) m 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 Cristoffels eq.: with is the propagation direction Wave velocities in perovskite (Pbnm) (Wentzcovitch et al.,1998) Voigt: uniform strain Reuss: uniform stress Voigt-Reuss-Hill averages: Poly-Crystalline aggregate Theory x PREM Quasi Harmonic Approximation Pressure: Bulk modulus: Isothermal, K T, and adiabatic, K S Coeff. of thermal expansion: 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 ( 3 ) F (Ry) 4 th order finite strain equation of state staticzero-point thermal MgO Static 300K Exp (Fei 1999) V ( 3 ) K (GPa) K K(GPa -1 ) Thermal expansivity of MgO (Karki, Wentzcovitch, Gironcoli and Baroni, Science 286, 1705, 1999) (10 -5 K -1 ) Thermal expansivity of MgSiO 3 -pv (Karki, Wentzcovitch, Gironcoli and Baroni, GRL in press) ( 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 zero pressure EoS: K = (c c 12 )/3 Tetragonal strain: c s = c 11 - c 12 Shear strain: c 44 (Karki, Wentzcovitch, de Gironcoli and Baroni, Science 286, 1705, 1999) Elastic moduli of MgO at high P and T (Karki et al., Science 1999) Comparison with seismic data Velocity (km s -1 ), Density (g cm -3 ) Pressure (GPa) 300 K, MgO 1000 K 2000 K 3000 K K S at Lower Mantle P-T 300 K 1000 K 2000 K 3000 K LM geotherms Me At depths greater than 1200 km, the rate of rise of the bulk modulus is too small for the lower mantle to consist of an adiabatic and homogeneous layer of standard chondritic or pyrolitic composition. Superadiabatic gradients, or continuous changes in chemical composition, or subtle transformations, or all are required to account for the relatively low bulk modulus of the deeper part of the LM,. (2001)