Transcript Thermoelastic properties of ferropericlase R. Wentzcovitch Dept. of Chemical Engineering and Materials Science, Minnesota Supercomputing Institute J.

Thermoelastic properties of
ferropericlase
R. Wentzcovitch
Dept. of Chemical Engineering and Materials Science,
Minnesota Supercomputing Institute
J. F. Justo, C. da Silva, Z. Wu
Dept. of Chemical Engineering and Materials Science
T. Tsuchiya
Ehime University, Japan
Outline

Ab initio calculations of Fe in (Mg1-xFex)O

Thermodynamics of the spin transition

Thermoelastic properties of (Mg1-xFex)O

Geophysical implications
Motivation: Earth’s Minerals




Lower Mantle: Ferrosilicate Perovskite + ferropericlase
Low iron concentration (< 0.20)
High-temperatures and high pressures
Elasticity
(Mg1-xFex)O
ferropericlase
(Mg1-yFey)SiO3
perovskite
+
First Principles Calculations

Density Functional Theory (LDA+U)
(Cococcioni and de Gironcoli, PRB, 2005)

Plane waves + Pseudopotential
(Troullier-Martins, PRB, 1991, Vanderbilt, PRB, 1990)

Structural relaxation in all configurations

Density Functional Perturbation Theory
(Baroni et al., RMP, 2001)
Optimized Hubbard U
7
XFe=3.125%
XFe=12.5%
XFe=18.75%
Hubbard U (eV)
HS
6
5
LS
4
14
16
3
V (Å /molec)
FeO (Cococcioni, 2005)
18
20
First Principles Calculations: HS-LS transition
(Tsuchiya et al., PRL, 2006)
H=HLS-HHS (kJ/mol)
40
3.125%
12.5%
18.75%
20
0
-20
0
50
100
P (GPa)
PT = 32±3 GPa
No systematic dependence on XFe
H (kJ/
0
Equation of State (Mg0.81Fe0.19)O
-10
-20
12
B
(Tsuchiya et al., PRL, 2006)
100
50
0
n=0
n=1/3
n=2/3
n=1
P (GPa)
V (cm3/mol)
11
 nLS

n

n

n
HS 
 LS
10
9
C
∆V ~4%
8
0
20
40
60
80
100
P (GPa)
Experimental: + (J.F.Lin et al., Nature, 2005)
17% Fe and room temperature
Temperature Effects: n(P,T)
(Tsuchiya et al., PRL, 2006)
1) Magnetic entropy
2) HS/LS configuration entropy
3) Fe/Mg configurational entropy is insensitive to spin state
4) Vibrational energy and entropy are insensitive to spin state
5) Minimization of G(P,T,n) with respect to n:
n( P, T ) 
1
 H HS  LS 
1  m(2S  1) exp 

X
k
T
 Fe B 
LS fraction n(P,T)
(Tsuchiya et al., PRL, 2006)
XFe=18.75%
Exp
Geotherm (Boehler, RG, 2000)
Elasticity
of
Ferropericlase
Volume of the mixed spin state V(P,T,n)
 Mixed spin configuration was described by the
Vegard’s rule:
V ( P, T , n)  nVLS ( P, T )  (1  n)VHS ( P, T )
where n = low spin fraction
 Iron-iron interaction is not significant for
xFe=18.75%
High temperature elasticity
V ( P, T , n)  nVLS ( P, T )  (1  n)VHS ( P, T )
 Compressibility:
VLS
VHS
V (n)
n
n
 (1  n)
 (VLS  VHS )
K (n)
K LS
K HS
P T
 Compliances:
Sij (n)V (n)  nS VLS  (1  n) S VHS
LS
ij
HS
ij
1
n
 ij (VLS  VHS )
9
P T
11  12  1  44  0
Static +vibrational free energy
 VDoS and F(T,V) within the quasiharmonic approximation
F (V , T )  U (V )  
qj
 F 
P   
 V T
 qj (V )
2

  qj (V )  


 k BT  ln1  exp


k
T
qj
B



 F 
S   
 T V
G  F  TS  PV
IMPORTANT: crystal structure and phonon frequencies
depend on volume alone!!
Thermoelastic Constant Tensor Cijpure(P,T)
(Wentzcovitch et al., PRL, 2004)
Eulerian Strain
2


G 
T
cij ( P, T )  

  i  j  P
kl
cij ( P, T )  cij ( P, T ) 
S
equilibrium
structure
re-optimize
T
S
i 
 i
T
i  jVT
CV
“Approximate” Virtual Crystal model
Replace Mg mass by the average cation mass of the alloy
0.06
MgO
mgo with 18.75% iron mass
pure mgo
0.05
(Mg0.8125Fe0.1875)O
0.04
DoS
0.03
0.02
0.01
0.00
0
100
200
300
400
500
-1
frequency (cm )
ω(V) = ωLS(V) = ωHS(V)
600
700
Procedure to obtain Cij(P,T,n):
 Compute CijLS(P,T) and CijHS(P,T)
 SLS(P,T) = [CLS(P,T)]-1 and SHS(P,T) =[CHS(P,T)]-1
 Calculate
n( P, T ) 
1
st  vib
 GHS

 LS
1  m(2S  1) exp

X
k
T
 Fe B 
 Compute V(P,T,n) and Sij(P,T,n)
 C(P,T,n) = [S(P,T,n)]-1
 Compute K(P,T,n) and G(P,T,n)
Volume V(P,T,n(P,T)) for xFe= 18.75%
xFe= 18.75%
+ 300K (exp.)
+ Experiments (Lin et al., Nature, 2005) (xFe=17%, RT)
Elastic Constants (xFe= 18.75%)
Isotropic Elastic Constants
Experiments:
○ (Lin et al., GRL, 2006)
xFe = 25% (NRIXS, RT)
● (Lin et al., Nature, 2005) xFe= 17% (X-ray diffraction, RT)
□ (Kung et al., EPSL, 2002) xFe = 17% (RUS, RT)
Sound Wave Velocities
VP 
VS 
xFe= 18.75%
Experiments: ○ (Lin et al., GRL, 2006)
xFe = 25% (NRIXS, RT)
□ (Kung et al., EPSL, 2002) xFe = 17% (RUS, RT)
3
K G
4

G

Geophysical
Implications
Elasticity Along Mantle Geotherm
1150 km
1580 km
Geotherm (Boehler, Rev. Geophys. 2000)
-15%
6%
Wave Velocities Along Mantle Geotherm
1150 km 1580 km
-9%
-15%
6%
3%
Geotherm (Boehler, GRL,2000)
Seismic Parameters (Mantle Geotherm)
R / S 
 ln V
 ln VS
P
(Karato, Karki, JGR, 2001)
Geotherm (Boehler, RG, 2000)
(
Wave Velocities Along Mantle Geotherm
1150 km 1580 km
-9%
-15%
6%
3%
Geotherm (Boehler, GRL,2000)
Summary

HS-LS transition in (Mg1-xFex)O is well reproduced theoretically
 There is a strong softening in the bulk modulus across the spin
transition. This effect broadens and decreases with temperature
 Along a lower mantle geotherm this softening is more pronounced
between 45-70 GPa, i.e., 1150-1580 km
 The shear modulus increases monotonically in the same region
 Transition can produce negative values of R/s in the upper part of the
lower mantle
 The softening will likely occur also in ferrosilicate perovskite
 The Si/(Mg+Fe) ratio in the lower mantle should increase from pyrolitic
values because of the spin transtions in ferropericlase and ferrosilicate
perovskite
Acknowledgements
NSF/EAR 0135533
NSF/EAR 0230319
NSF/ITR 0428774
Japan Society for the Promotion of Science (JSPS)
Brazilian Agency CNPq
Computations performed at the MSI-UMN