Structure, elasticity, and wave-velocities of (Mg,Fe)SiO3perovskite at lower mantle conditions
Renata M. M. Wentzcovitch
Department of Chemical Engineering and Materials Science
U. of Minnesota, Minneapolis
• First Principles Thermodynamic Method
…``First Principles’’…
Previous results and the QHA at LM conditions
Crystal structures at high P,T’s
• Elasticity and wave-velocities
Thermoelasticity of Mg(,Fe)SiO3
Comparison with the British results
Comparison with PREM
• Summary
…``First Principles’’…
• Density Functional Theory ( E [n( r )] , H
 E , n   i * i )
(Hohemberg-Kohn-Sham, 1964-5)
i
• Local Density Approximation (Ceperley-Alder, 1985)
• First Principles Pseudopotentials (Troullier-Martins, 1991)
• Born-Oppenheimer Variable Cell Shape Molecular Dynamics
(Wentzcovitch, 1991-3)
• Density Functional Perturbation Theory for Phonons
(Gianozzi et al., 1991)
TM of mantle phases
CaSiO3
(Mg,Fe)SiO3
5000
T (K)
Mw
4000
HA
Core T
solidus
3000
Mantle adiabat
2000
peridotite
0
20
40
60
P(GPa)
80
100
120
(Zerr, Diegler, Boehler, Science1998)
Thermodynamic Method
• VDoS and F(T,V) within the QHA
F (V , T )  U (V )  
qj
 qj (V )
2

  qj (V )  


 k BT  ln 1  exp 

k BT  
qj


N-th (N=3,4,5…) order isothermal (eulerian or logarithm) finite strain EoS
 F 
P   
 V T
 F 
S   
 T V
G  F  TS  PV
IMPORTANT: crystal structure and phonon frequencies
depend on volume alone!!….
Phonon dispersions of MgSiO3 perovskite
Calc Exp
-
Calc Exp
0 GPa
-
Calc: Karki, Wentzcovitch, de Gironcoli, Baroni
PRB 62, 14750, 2000
Exp: Raman [Durben and Wolf 1992]
Infrared [Lu et al. 1994]
100 GPa
Karki et al, PRB (2000)
Thermal expansivity and the QHA
 (10-5 K-1)
 provides an a posteriori criterion for the validity of the QHA



MgSiO3
Karki et al, GRL (2001)
The QHA
Criterion: inflection point of (T)
invalid
MgO
MgSiO3
Brown & Shankland’s T
MgSiO3-perovskite and MgO

(gr/cm-3)
V
(A3)
KT
(GPa)
d KT/dP
d K T2/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
4.8
-0.016
-0.031
2.1
Calc.
Pv
3.7
|
4.0
~
162.3
246
|
266
-0.02
|
-0.07
1.7
|
2.2
Exp.
Pv
4.247
~

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]
…IMPORTANT: crystal structure and phonon frequencies
depend on volume alone!!
 Structures at high P are determined at T= 0
P(V,0)
 P’(V,T’) within the QHA
 At T 0…
V(P’,T’)=V(P,0)

structure(P’,T’) = structure(P,0)
• Is P isotropic?
(Thermo) Elastic constant tensor 
2


G 
T
cij (T , P )  

  i  j 
kl
cij (T , P)  cij (T , P) 
S
equilibrium
structure
re-optimize
T
S
i 
 i
T
i  jVT
CV
c
i
j
300 K
1000K
2000K
3000 K
4000 K
Cij(P,T)
(Oganov et al,2001)
(Wentzcovitch, Karki, & Coccociono, 2002)
V (km/sec) &  (gr/cm3)
Velocities
(Wentzcovitch et al, 2002)
Aggregate Moduli
38 GPa
88 GPa
Effect of Fe alloying
(Kiefer, Stixrude,Wentzcovitch, GRL 2002)
(Mg0.75Fe0.25)SiO3
||
+
+
4
+
Comparison with PREM
Pyrolite (20 V% mw)
Perovskite
100 GPa
38 GPa
Brown & Shankland T(r)
Me
“…At depths greater than 1400 km, the rate of rise of the bulk and shear moduli are too small and
too large respectively for the lower mantle to consist of a homogeneous isotropic layer of pure
perovskite or pyrolite composition. It seems that changes in chemical composition, or subtle phase
changes, or anisotropy, or a combination of all, are required to account for the elastic moduli of the
deeper part of the LM ,….” (2002)
Summary
• Building a consistent body of knowledge obout LM phases
• We have adequate methods (DFT, QHA) to examine elasticity
of major mantle phases
• The objective is to interpret seismic observations (1D, 3D,
anisotropy) in terms of composition, temperature, ``flow’’…
Acknowledgements
Bijaya B. Karki (LSU)
Stefano de Gironcoli and Matteo Coccocioni
(SISSA, Italy)
table
10.97