1. First Principles Calculations in Mineral Physics Overview of methods Amorphization of quartz under pressure Structural transitions in ruby and the ruby pressure scale Thermoelasticity of LM minerals and the problem of LM temperature and composition Epilog Renata Wentzcovitch CEMS, U of MN and SISSA

2. BO approximation Born-Oppenheimer approximation (1927) Ions (R I ) + electrons (r i ) Molecular dynamicsLattice dynamics forces stresses phonons

3. Electronic Density Functional Theory (DFT) (T = 0 K) Hohemberg and Kohn (1964) Kohn and Sham(1965) (auxiliary non-interacting system) energy minimization... DFT1 dft1

4. Kohn-Sham equations : (one electron equation) with and Local density approximation (LDA) Quantum Monte Carlo Ceperley and Alder, 1980 df t2

5. Lattice Dynamics Quasi-Harmonic Approximation Equation of motion  Dynamical matrix

6. Lattice Dynamics and Linear Response Within Density Functional Theory Hellmann-Feynman theorem Force constant matrix

7. Density Functional Perturbation Theory (Baroni et al., PRL 1987; Giannozzi et al., PRB 1991) (non-analytic contribution to LO modes)

8. Pseudopotentials Nucleus Core electrons Valence electrons V(r) 1.0 0.5 0.0 -0.5 0 Radial distance (a.u.) Troullier-Martins (1991) rR l (r) 12345 3s orbital of Si Real atom Pseudoatom r Ion potential Pseudopotential 1/2 Bond length

9. Fictitious molecular dynamics H. C. Andersen (1978) (N,E,V) (N,H,P)

10. i=vector index j=cart. index Parrinello-Rahman MD (1980)

11. Variable Cell Shape MD (Invariant) a a (N,H,P) (Wentzcovitch, 1991)

12. Typical Computational Experiment Damped dynamics (Wentzcovitch, 1991) P = 150 GPa

13. Amorphization in Quartz under Pressure quartz cristobalite tridymite coesite stishovite Collaborators: C.R.S. da Silva (UMN), J. Chelikowsky (UMN), N. Binggeli (EPFL)

14. Hemley, Prewitt, Kingma, in Reviews in Mineralogy, 29 (1996) (Hemley,1987)

15. Microstructure of  -quartz during amorphization Kingma, Maede, Hemley, Mao, & Veblen, Science (1993) Q – Quartz Q’- Quartz-like * - New peaks

16. Mechanical instability of quartz under pressure Binggeli & Chelikowsky, PRL 1993 (shear instability) Chapplot & Sikka, PRL 1993 (phonon softening)

17. quartz  -Quartz

18. Comparison Quartz - 0 GPa (exp) Quartz - 0 GPa (calc) K-phase – 33 GPa (calc) New phase – 25.5 GPa (exp) New phase – 26 GPa (calc) New phase – 27.4 GPa (exp)

19. New phase New Phase

20. Nature of P induced coordination change Stolper & Ahrens, GRL (1987) 1)Gradual increase in density 2)Occurs at room T 3)Changes are reversible

21. Polyhedra Si-O distances (A) o 1.531 1.607 1.624 1.683 1.673 1.680 1.714 1.763 1.683 1.752 1.768 1.726 1.760 1.813 1.797 2.030 1.817

22. Conclusions Nature of the intermediate phase of silica seems to be understood Properties: produced by a soft mode structure consists of 6-, and 5-fold Si at 33 GPa it is 10% denser than quartz (  H ~ 0.1 eV/atom) Amorphous could be the result of a generalized phonon stability

23. Optical transitions in ruby across the corundum to Rh2O3 (II) phase transformation Collaborators: W. Duan (U. of MN), G. Paiva (USP), & A. Fazzio (USP) Support: NSF, CNPq, and FAPESP

24. Structural Transition in Ruby (Al 2 O 3 :Cr) PIB (Cynn et al.-1990 and Bukowinski – 1994). Between 4 and 148 GPa LAPW (Marton & Cohen – 1994) 90 GPa Pseudopotentials (VCS-MD) (Thomson, Wentzcovitch, & Bukowinski), Science (1996)

25. Suggestive X-ray diffraction pattern Experimental confirmation (Funamori and Jeanloz, Science (1997)) Comparison with EDS (Jephcoat, Hemley, Mao, Am. Mineral.(1986)) 175 GPa corundum Rh 2 O 3 (II) 50/50% mixture

26. The high pressure ruby scale Forman, Piermarini, Barnett, & Block, Science (1972) (R-line) Mao, Xu, & Bell, JGR (1986) Bell, Xu,& Mao, in Shock Waves in Condensed Matter, ed. by Gupta (1986)

27. Optical transitions in ruby Intra-d transitions in Cr 3+ (d 3 )

28. Ab initio calculation of Al 2 O 3 :Cr (80 atoms/cell) (Duan, Paiva, Wentzcovitch, Fazzio, PRL (1998))

29. Eigenvalue Spectra Corundum Rh 2 O 3 (II)

30. Multiplet method for e - ’s in X-tal field (Fazzio, Caldas, & Zunger, PRB (1984) (Sugano, Tanabe, & Kamimura, 1962) [ [

31. Deformation parameters Racah parameters B and C Orbital deformation parameters

32. Optical transitions X Pressure (Duan, Paiva, Wentzcovitch,Fazzio, PRL (1998) (Sugano, Tanabe, & Kamimura, 1962)(Fazzio, Caldas, & Zunger, 1984)

33. Phase transition in Cr 2 O 3 Corundum  Rh 2 O 3 (II) phase transition AFM at 14 GPa, PM at 18 GPa. Experimental confirmation: Rheki & Dubrovinsky (2002) unpublished P T = 30GPa, T= 1500 K. Dobin, Duan, & Wentzcovitch, PRB 2000

34. Conclusions Calculated P-induced optical shifts in ruby agree well with experiments Phase transformation should affect mainly the U and Y absorption lines New interpretation of observed anomalies in absorption lines Prediction and confirmation of corundum to Rh 2 O 3 (II) transition in Cr 2 O 3 near of below 30 GPa Need more experiments: Study of Y line above 30 GPa NEXAFS under pressure…

35. Thermoelasticity of LM minerals and the problem of LM temperature and composition Core T Mantle adiabat solidus HA Mw (Mg,Fe)SiO 3 CaSiO 3 peridotite P(GPa) 040206080100120 2000 3000 4000 5000 T (K) (Zerr, Diegler, Boehler, 1998) Collaborators: B.B. Karki (UMN), S. de Gironcoli & S. Baroni (SISSA )

36. Phonon dispersion in MgO & MgSiO 3 perovskite Calc Exp (Karki, Wentzcovitch, Gironcoli, Baroni, PRB 2000) 0 GPa - Exp: Sangster et al. 1970 Exp: Raman [Durben and Wolf 1992] Infrared [Lu et al. 1994]

37. 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 ) 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 - - - -

38. Thermal expansivity of MgO & MgSiO 3 -pv (Karki, Wentzcovitch, Gironcoli and Baroni, GRL in press)  ( 10 -5 K -1 )

39. 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]

40. Elastic moduli of MgO at high P and T (Karki et al. 1999, 2000)

41. K S at Lower Mantle P-T 300 K 1000 K 2000 K 3000 K

42. LM Geotherms Pv Solidus Isentropes Pyrolite CMB | TcTc

43. 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 phase, or all are required to account for the relatively low bulk modulus of the deeper part of the LM,….” (Wentzcovitch, 2001)

44. Epilog Beyond QHA and beyond elasticity (rheology) Transition metal (Fe) bearing systems Alloy systems Press on to Gbars…