Thermoelastic properties of ferropericlase

Thermoelastic properties of ferropericlase

R. WentzcovitchDept. of Chemical Engineering and Materials Science,

Minnesota Supercomputing Institute

J. F. Justo, C. da Silva, Z. WuDept. of Chemical Engineering and Materials Science

T. Tsuchiya Ehime University, Japan

OutlineOutline

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 MineralsMotivation: Earth’s Minerals

(Mg1-yFey)SiO3

perovskite

(Mg1-xFex)O ferropericlase

+

Lower Mantle: Ferrosilicate Perovskite + ferropericlase

Low iron concentration (< 0.20) High-temperatures and high pressures Elasticity

First Principles CalculationsFirst 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)

14 16 18 20

4

5

6

7

V (Å 3/molec)

Hubbard U (eV)

XFe=3.125% XFe=12.5% XFe=18.75%

FeO (Cococcioni, 2005)

Optimized Hubbard U

HS

LS

PT = 32±3 GPaNo systematic dependence on XFe

(Tsuchiya et al., PRL, 2006)

0 50 100

-20

0

20

40

P (GPa)

Δ=H H

LS

-HHS

( / )kJ mol

3.125% 12.5% 18.75%

First Principles Calculations: HS-LS First Principles Calculations: HS-LS transitiontransition

0 50 100

-20

-10

0

10

20

30

P (GPa)

Δ ( / )H kJ mol

XFe=0.03125 XFe=0.125 XFe=0.1875

0 50 100

-20

-10

0

10

20

30

P (GPa)

Δ ( / )H kJ mol

n=0n=1/3n=2/3n=1A

B

C

0 20 40 60 80 1008

9

10

11

12

(V cm

3/ )mol

P (GPa)

n=0 n=1/3 n=2/3 n=1

Experimental: + (J.F.Lin et al., Nature, 2005)

17% Fe and room temperature

Equation of State (MgEquation of State (Mg0.810.81FeFe0.190.19)O)O(Tsuchiya et al., PRL,

2006)

∆ V ~-4%

⎥⎦

⎤⎢⎣

⎡

+=

HSLS

LS

nn

nn

Temperature Effects: n(P,T)Temperature Effects: n(P,T)

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:

(Tsuchiya et al., PRL, 2006)

1( , )

1 (2 1)exp HS LS

Fe B

n P TH

m SX k T

−

=⎡ ⎤Δ

+ + ⎢ ⎥⎣ ⎦

Exp

LS fraction n(P,T)LS fraction n(P,T)

XFe=18.75%

Geotherm (Boehler, RG, 2000)

(Tsuchiya et al., PRL, 2006)

Elasticity of

Ferropericlase

Elasticity of

Ferropericlase

Volume of the mixed spin state Volume of the mixed spin state V(P,T,n)V(P,T,n) Mixed spin configuration was described by

the Vegard’s rule:

where n = low spin fraction

Iron-iron interaction is not significant for xFe=18.75%

( , , ) ( , ) (1 ) ( , )LS HSV P T n nV P T n V P T= + −

High temperature elasticityHigh temperature elasticity

( , , ) ( , ) (1 ) ( , )LS HSV P T n nV P T n V P T= + −

Compressibility:

1( ) ( ) (1 ) ( )

9LS HS

ij ij LS ij HS ij LS HST

nS n V n nS V n S V V V

Pα ∂

= + − − −∂

Compliances:

11 12 1α α= = 44 0α =

THSLS

HS

HS

LS

LS

P

nVV

K

Vn

K

Vn

nK

nV

∂∂

−−−+= )()1()()(

∑∑ ⎟⎟⎠

⎞⎜⎜⎝

⎛⎥⎦

⎤⎢⎣

⎡−−++=

qj B

qjB

qj

qj

Tk

VTk

VVUTVF

)(exp1ln

2

)()(),(

ωω hh

PVTSFG +−=TV

FP ⎥⎦

⎤⎢⎣⎡∂∂

−=VT

FS ⎥⎦

⎤⎢⎣

⎡∂∂

−=

IMPORTANT: crystal structure and phonon frequencies depend on volume alone!!

Static +vibrational free energyStatic +vibrational free energy

VDoS and F(T,V) within the quasiharmonic approximation

equilibrium structure

kl

re-optimize

Pji

Tij

GTPc

⎥⎥⎦

⎤

⎢⎢⎣

⎡

∂∂∂

=

2

),(

V

jiTij

Sij C

VTTPcTPc

λλ+= ),(),(

Tii

S

λ

∂∂

=

(Wentzcovitch et al., PRL, 2004)

Thermoelastic Constant Tensor Thermoelastic Constant Tensor CCijij

purepure(P,T)(P,T)

Eulerian Strain

““Approximate” Virtual Crystal Approximate” Virtual Crystal modelmodel

MgO

(Mg0.8125Fe0.1875)O

Replace Mg mass by the average cation mass of the alloy

ω(V) = ωLS(V) = ωHS(V)

Compute CijLS(P,T) and Cij

HS(P,T)

SLS(P,T) = [CLS(P,T)]-1 and SHS(P,T) =[CHS(P,T)]-1

Calculate

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)

Procedure to obtain CProcedure to obtain Cijij(P,T,n):(P,T,n):

⎥⎦

⎤⎢⎣

⎡Δ++=

+−

TkXG

Sm

TPn

BFe

vibstLSHSexp)12(1

1),(

+ Experiments (Lin et al., Nature, 2005) (xFe=17%, RT)

xFe= 18.75%

Volume V(P,T,n(P,T)) for xVolume V(P,T,n(P,T)) for xFeFe= 18.75%= 18.75%

+ 300K (exp.)

Elastic Constants (xElastic Constants (xFeFe= 18.75%)= 18.75%)

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)

Isotropic Elastic ConstantsIsotropic Elastic Constants

Experiments: ○ (Lin et al., GRL, 2006) xFe = 25% (NRIXS, RT) □ (Kung et al., EPSL, 2002) xFe = 17% (RUS, RT)

3

4P

K GV

ρ

+=

S

GV

ρ=

xFe= 18.75%

Sound Wave VelocitiesSound Wave Velocities

Geophysical ImplicationsGeophysical Implications

Elasticity Along Mantle GeothermElasticity Along Mantle Geotherm

Geotherm (Boehler, Rev. Geophys. 2000)

6%

-15%

1150 km 1580 km

Geotherm (Boehler, GRL,2000)

Wave Velocities Along Mantle Wave Velocities Along Mantle GeothermGeotherm

6%

-9%

-15%

3%

1150 km 1580 km

Seismic Parameters (Mantle Geotherm)Seismic Parameters (Mantle Geotherm)

/

ln

lnSS P

VR

Vφ

φ

∂=∂

Geotherm (Boehler, RG, 2000)

(Kara

(Karato, Karki, JGR, 2001)

Geotherm (Boehler, GRL,2000)

Wave Velocities Along Mantle Wave Velocities Along Mantle GeothermGeotherm

6%

-9%

-15%

3%

1150 km 1580 km

SummarySummary

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

AcknowledgementsAcknowledgements

NSF/EAR 0135533 NSF/EAR 0230319NSF/ITR 0428774Japan Society for the Promotion of Science (JSPS)Brazilian Agency CNPq

Computations performed at the MSI-UMN