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

BO approximation

• Born-Oppenheimer approximation (1927) Ions (RI ) + electrons (ri )

RRRERM I

II

)(2 22

22

rERR

ZZeRE VJI

JI

JI

2)(

2

II R

REF

)(

lmlm

RE

)( 0)(1

det 22

JIJIRR

RE

MM

IRR

Molecular dynamics Lattice dynamics

forces stresses phonons

Electronic Density Functional Theory (DFT) (T = 0 K)

• Hohemberg and Kohn (1964)

][)()(][][ nFrnrVdrnEE extvv r

• Kohn and Sham(1965) (auxiliary non-interacting system)

][)()(][][][ nErnrdrVnEnTnE xcextHartreev

i

ii pmnT 2

2

1][

)()()( rrrni

ii energy minimization...

DFT1

dft1

irr

)(rVSCF

• Kohn-Sham equations: (one electron equation)

)()()()()(2

22

rrrVrVrVm iiixcHartreeext

'

)('

)(

][)(

rr

rndr

rn

nErV Hartree

Hartree

)(

][)(

rn

nErV xc

xc

with and

• Local density approximation (LDA)

]([)( )rnrdrnExc

Quantum Monte CarloCeperley and Alder, 1980

dft2

Lattice Dynamics• Quasi-Harmonic Approximation

• Equation of motion Dynamical matrix

...2

1})({})({

''''''

0 ls sl

lsslslslls

lslslsls uCuuCRERE

)( 0lslsls uRR

00

ls

ls u

EC

0''

2

''slls

slsl uu

EC

''

''''sl

slslslls

lss uCu

EuM tiiqR

s

sls

leM

qvu

)( '

''2 )(

sssss vqDv

'

)(

''''

'

)(ss

RRiq

lslslss

MM

eCqD

ll

Lattice Dynamics and Linear Response

''

''''sl

slslsllss uCuM

Within Density Functional Theory

ionextRls EdrrVrnnFREls

)()(min})({ }{

0''

2

0''

2

0''

}{

00''

2

''

})({})({

slls

ion

slls

ls

sl

ionR

lsslls

lsslsl uu

Edr

uu

RVn

u

V

u

n

uu

REC ls

ls

ion

ls

extR

ls

ls

u

Edr

u

rVrn

u

REls

)(

)(})({ }{

Hellmann-Feynman theorem

Force constant matrix

Density Functional Perturbation Theory

)()()()(2

22

rrVPrrVm iSCFciiSCF

)()(4)( rrrnFi

i i

)()()()( rVrVrVrV xcHartreeextSCF

''

)'()( 2 dr

rr

rnerVHartree

)())((')( rnrnrV xcxc

(Baroni et al., PRL 1987; Giannozzi et al., PRB 1991)

qq

ZqZqeC lslsna

slsl ..

).().(4 '*'

*2

''

(non-analytic contribution to LO modes)

Pseudopotentials

NucleusCore electronsValence electrons

V(r)

1.0

0.5

0.0

-0.5

0

Radial distance (a.u.)

Troullier-Martins (1991)

rRl (

r)

1 2 3 4 5

3s orbital of Si

Real atom

Pseudoatom

r

Ion potential

Pseudopotential

1/2 Bond length

Fictitious molecular dynamicsH. C. Andersen (1978)

IJI

JII

II U

mL

,

2 ,2

1

2RRR VPUV

WmL ext

IJIJI

II

I ,

22 ,2

1

22RRs 3

2

V

(N,E,V) (N,H,P)

R V1

3s

h1

h2

hij(t) i=vector indexj=cart. index

VPUhWm

L extji

jiI

i ,

2,22

ii

T

sgs

r hs ghTh

Parrinello-Rahman MD (1980)

Variable Cell Shape MD (Invariant)

VPUKKL extLDALIVCS

ji

jiInv W

KL

,

2,2

)( 0ta

ji

jiPR h

WK

L,

2,2

a

a2WKL

2

2

3 WKL

0hh )1(

W(t)

AndInv LL (N,H,P)

(Wentzcovitch, 1991)

Typical Computational Experiment

Damped dynamics (Wentzcovitch, 1991)

)(~ PI),(~ int rffr

P = 150 GPa

Amorphization in Quartz under Pressure

quartz cristobalitetridymite

coesite

stishovite

Collaborators: C.R.S. da Silva (UMN), J. Chelikowsky (UMN), N. Binggeli (EPFL)

Hemley, Prewitt, Kingma, in Reviews in Mineralogy, 29 (1996)

(Hemley,1987)

Microstructure of -quartz during amorphization

Kingma, Maede, Hemley, Mao, & Veblen, Science (1993) Q – Quartz

Q’- Quartz-like

* - New peaks

Mechanical instability of quartz under pressure

Binggeli & Chelikowsky, PRL 1993 (shear instability)

Chapplot & Sikka, PRL 1993 (phonon softening)

quartz

-Quartz

ComparisonQuartz - 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)

New phase

New Phase

Nature of P induced coordination change

Stolper & Ahrens, GRL (1987)

1) Gradual increase in density

2) Occurs at room T

3) Changes are reversible

Polyhedra

Si-O distances (A)o

1.531 1.607 1.6241.683 1.673 1.6801.714 1.763 1.6831.752 1.768 1.7261.760 1.813 1.7972.030 1.817

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

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

Structural Transition in Ruby (Al2O3: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)

Suggestive X-ray diffraction pattern

•Experimental confirmation (Funamori and Jeanloz, Science (1997))

• Comparison with EDS (Jephcoat, Hemley, Mao, Am. Mineral.(1986))

175 GPa

corundum

Rh2O3 (II)

50/50% mixture

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)

Optical transitions in ruby

Intra-d transitions in Cr3+ (d3)

Ab initio calculation of Al2O3:Cr

(80 atoms/cell)

(Duan, Paiva, Wentzcovitch, Fazzio, PRL (1998))

Eigenvalue SpectraCorundum Rh2O3 (II)

Multiplet method for e-’s in X-tal field(Fazzio, Caldas, & Zunger, PRB (1984)

(Sugano, Tanabe, & Kamimura, 1962)

[ [

Deformation parameters

Racah parameters B and C

Orbital deformation parameters

Optical transitions X Pressure

(Duan, Paiva, Wentzcovitch,Fazzio, PRL (1998)

(Sugano, Tanabe, & Kamimura, 1962) (Fazzio, Caldas, & Zunger, 1984)

Phase transition in Cr2O3

• Corundum Rh2O3 (II) phase transition AFM at 14 GPa, PM at 18 GPa.

• Experimental confirmation: Rheki & Dubrovinsky (2002) unpublished PT = 30GPa, T= 1500 K.

Dobin, Duan, & Wentzcovitch, PRB 2000

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 Rh2O3 (II) transition in Cr2O3 near of below 30 GPa

• Need more experiments: Study of Y line above 30 GPa NEXAFS under pressure…

Thermoelasticity of LM minerals and the problem of LM temperature and composition

Core T

Mantle adiabat

solidus

HA

Mw(Mg,Fe)SiO3

CaSiO3

peridotite

P(GPa)0 4020 60 80 100 120

2000

3000

4000

5000

T (

K)

(Zerr, Diegler, Boehler, 1998)

Collaborators: B.B. Karki (UMN), S. de Gironcoli & S. Baroni (SISSA)

Phonon dispersion in MgO & MgSiO3 perovskite

Calc Exp 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]

Quasiharmonic approximation

qj B

qjB

qj

qj

Tk

VhTk

VhVUTVF

)(exp1ln

2

)()(),(

Volume (Å3)F

(ry

)

4th order finite strain equation of state

static zero-point

thermal

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

-

-

-

-

Thermal expansivity of MgO & MgSiO3-pv(Karki, Wentzcovitch, Gironcoli and Baroni, GRL in press)

(1

0-5 K

-1)

(1

0-5 K

-1)

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]

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

KS at Lower Mantle P-T

300 K 1000 K 2000 K 3000 K

LM Geotherms

1000

2000

3000

4000

5000

6000

500 1000 1500 2000 2500 3000

T (

K)

Depth (km)

Pv

Solidus

Isentropes

Pyrolite

CMB |

Tc

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)

Epilog

• Beyond QHA and beyond elasticity (rheology)

• Transition metal (Fe) bearing systems

• Alloy systems

• Press on to Gbars…