first principles calculations in mineral physics overview of methods amorphization of quartz under...
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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)
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…