1. The role of first principles calculations in geophysics
Renata Wentzcovitch
University of Minnesota
Minnesota Supercomputing Institute
ASESMA’10

2. Acknowledgements
K. Umemoto (GEO, U of MN), Z. Wu (USTC, Hefei, PRC), Y. Yu (U of MN), T. Tsuchiya, J. Tsuchiya (Ehime U., Japan)
S. de Gironcoli (SISSA, Trieste), M. Cococcioni (U of MN)
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3. How well can we describe minerals by first principles?
What property?
Is it a solid solution or an end member?
Does it have iron or hydrogen bonds?
What is the PT range?
DFT within LDA, GGA (PBE), and DFT+U
Variable cell shape MD (VCS-MD)
Density functional perturbation theory
Quasiharmonic approximation (QHA)
(Quantum ESPRESSO)
We are making progress in all fronts. We test against experiments as much as we can but there are no experiments in many cases.
We develop intuition and extrapolate. This requires experience (a lot).
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4. Typical Computational Experiment
(Wentzcovitch, Martins, and Price, PRL 1993)
Damped dynamics
P = 150 GPa

5. Perovskite and the Earth’s mantle Perovskite and the Earth’s mantle
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6. The Contribution from Seismology
Longitudinal (P) waves
Transverse (S) wave
Bulk (Φ) wave
from free oscillations
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7. PREM (Preliminary Reference Earth Model) (Dziewonski & Anderson, 1981)
P(GPa) 24
135
329
364
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8. Lower Mantle + (Mgx,Fe(1-x))SiO3 (Mgx,Fe(1-x))O ASESMA’10
Mineral sequence II
Lower Mantle +
(Mgx,Fe(1-x))SiO3
(Mgx,Fe(1-x))O
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9. TM of lower mantle phases
CaSiO3
(Mg,Fe)SiO3
5000 Mw
Core T
4000 HA
solidus
T (K)
3000
Mantle adiabat
2000
peridotite 20 40 60 80
100
120
P(GPa)
(Zerr, Diegler, Boehler, Science1998)

10. Thermodynamics Method
• VDoS and F(T,V) within the QHA
N-th (N=3,4,5…) order isothermal (eulerian or logarithm) finite strain EoS
IMPORTANT: crystal structure and phonon frequencies
depend on volume alone!!….
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11. Validity of the QHA ASESMA’10
Tsuchiya et al., J. Geophys. Res., 110(B2), B02204/1-6 (2005).
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12. Thermoelastic constant tensor CijS(T,P)
kl
The isothermal elastic constants are determined by calculating the 2nd derivative of G w.r.t. strains. Then a standard thermodynamic correction gives the adiabatic cij’s.
equilibrium
structure
re-optimize
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13. cij Cij(P,T) ASESMA’10 (Oganov et al,2001) 300 K 1000K 2000K 3000 K
(Wentzcovitch, Karki, Cococciono, de Gironcoli, Phys. Rev. Lett. 2004)
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14. Effect of Fe alloying (Mg0.75Fe0.25)SiO3 || + + + 4 ASESMA’10
(Kiefer, Stixrude,Wentzcovitch, GRL 2002)
(Mg0.75Fe0.25)SiO3 || + + + 4
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15. Comparison with PREM 100 GPa 38 GPa Brown & Shankland T(r)
Pyrolite (20 V% mw)
Perovskite
100 GPa
38 GPa
(Wentzcovitch et al.
Phys. Rev. Lett. 2004)
Brown & Shankland T(r)
Wentzcovitch, Karki, Cococciono, de Gironcoli, Phys. Rev. Lett. 92, (2004)

16. What's Down There? ASESMA’10 9 January 2004
                                                                                        
Previous Story / Next Story / January - June 2004 Archive
Phys. Rev. Lett. 92, (issue of 9 January 2004)
  9 January 2004
What's Down There?
Like bats using echolocation to navigate through the night, geophysicists rely on seismic waves for information on the Earth's deep interior. Almost everything known about that inaccessible region is inferred from the speed of sound waves generated by earthquakes. In the 9 January PRL, however, a team describes a calculation of the properties of the Earth's lower mantle starting from basic physics principles. The results disagree slightly with seismic data and suggest that the structure of minerals in the Earth's lower mantle is more complex than geophysicists have assumed.
L.H. Kellogg et al., Science 283, 1881 (1999),
copyright AAAS
Lava lamp. A new calculation suggests geophysicists still don't know exactly what the Earth's mantle is made of. Other research suggests that there are slow but complex flows in the mantle, even though it's entirely solid.
The Earth has an iron core surrounded by a dense layer
called the mantle, which is capped with a thin rind of rocky
crust. Seismic measurements reveal the density and elasticity
of the mantle, but not much about its composition. Perovskite, the mineral that dominates the lower mantle, contains mainly magnesium, silicon, and oxygen, but researchers suspect that a lot of iron and aluminum are present as impurities. Exactly how much isn't known, nor how these impurities would affect the elasticity of the rock. To further complicate the mystery, minerals often behave in unexpected ways at the extreme pressures found 1000 kilometers underground. Iron, for example, becomes non-magnetic and may tend to migrate from perovskite toward another mineral called magnesiumwustite, as the pressure rises.
Thermoelasticity of MgSiO3 Perovskite: Insights on the Nature of the Earth's Lower Mantle R. M. Wentzcovitch, B. B. Karki, M. Cococcioni, and S. de Gironcoli Phys. Rev. Lett. 92, (issue of 9 January 2004)
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17. UNKNOWN PHASE Pbnm Perovskite
Drastic change in X-ray diffraction pattern around 125 GPa and 2500 K
UNKNOWN PHASE
…in situ diamond anvil experiments by Murakami at al. at the TIT had found a phase transition in this material at P similar to those expected at the D” discontinuity.
At that point the structure of the post-perovskite phase had not been identified and we got work.
Pbnm Perovskite
(M. Murakami and K. Hirose, private communication)
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18. MgSiO3 Perovskite ASESMA’10
----- Most abundant constituent in the Earth’s lower mantle
----- Orthorhombic distorted perovskite structure (Pbnm, Z=4)
----- Its stability is important for understanding deep mantle (D” layer)
MgSiO3 perovskite is the most abundant phase in the Earth’ s LM and its properties determine to a great extent the properties of this region. Until recently, experimental investigations of its stability field had not reached condition of the bottom of the lower mantle. This region, the D” region is right above the core mantle boundary has seismic signature very different from the rest of the lower mantle. Hypothesis about this region are abundant and I will mention some later but, it was with great interest that we received the news in the fall 2003 that …
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19. Ab initio exploration of post-perovskite phase in MgSiO3
- Reasonable polyhedra type and connectivity under ultra high pressure -
SiO4 chain
SiO3 layer
Perovskite
Among the families of structures that were being considered as likely candidates there was this one of layered structures. SiO4 stishovite like chains is a favorable structural unit at high pressures. However the stoichiometry of perovskite required them to be linked by the apices.
There is another reason why this layered structure is likely and I hope it will become clear in a few slides ahead. It has to do with the compressive behavior of perovskite itself. So the search included structures consisting of stacked layers intercalated by Mg ions.
SiO3 Mg
MgSiO3
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20. Crystal structure of post-perovskiteb c a Pt
After persistent work by Taku Tsuchiya this structure was found. It had lower energy than pv above 120 GPa and its X-ray d.p. matched pretty well that of the TIT group (except for this 110 peak).
At that point we did not know that other materials existed with this structure. Later we learned this is the structure of CaIrO3.
Lattice system: Bace-centered orthorhombic
Space group: Cmcm
Formula unit [Z]: 4 (4)
Lattice parameters [Å] a: (4.286)
[120 GPa] b: (4.575)
c: (6.286)
Volume [120 GPa] [Å3]: (123.3) ( )…perovskite
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21. Deformation of perovskite under shear strain ε6
Structural relation between Pv and Post-pv
Post-perovskite c’ a’ b’ θ
Perovskite a b c
The relationship between the structures of pv and ppv is quite striking.
This is the view of pv along the c axis. Under compression this angle theta closes faster than other equivalent ones.
Stishovite like chains can be produced if this angle theta vanishes.
This can happen if one of the O’s at the end of these 2 edges gets out of the way, i.e., if the connections between octahedra along these 2 chains break producing layers.
You can actually make this happen by forcing artificially the angle theta to close under application of shear stress e_6. This leads directly to the Cmcm ppv structure and this was the reason why we also investigated layered structures.
At the end the c axis of pv is // to the c axis of ppv, 110 // to b and 1-10 // to a.
Deformation of perovskite under shear strain ε6
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22. High-PT phase diagram 7.5 MPa/K Perovskite Post- perovskite D” 1000 K
Mantle adiabat
ΔPT~10 GPa
Hill top
Valley bottom
~8 GPa
~250 km
7.5 MPa/K
LDA
GGA
Perovskite
Post-
perovskite
Tsuchiya, Tsuchiya, Umemoto,
Wentzcovitch, EPSL 224, 241 (2004) D”
Sidorin, Gurnis, Helmberger (1998)
6 MPa/K
1000 K
Here is our thermodynamic phase boundary. A CS of 7.5 MPa/K. The thickness of the line is our estimate of the uncertainty. The boundaries correspond to LDA and GGA calculations. LDA usually underestimates Pt and GGA can overestimate. ZPM adds 3 GPa to the transition pressures. This PB encompasses the P,T conditions in which Murakami et al observed the transformation, this red star. The center of this band is our best estimated value. At 127 GPa, the pressure 300km above CMB, we have T = 2750 plus or minus 250 K, this green happy face. Our CS of 7.5 MPa/K is consistent with the value that Sidorin Helmberger and Gurnis quoted would be necessary for the D” discontinuity to be produced by a solid-solid phase transition, 6 MPa/K, represented here by this dashed line.
Let me make a quick comparison with other calculations.
Obviously there are many interesting coincidences suggesting that the D” discontinuity could result from this phase transition and that we should learn more about the properties of this phase and try to compare with seismic observations of D”.
It may not be straitghtforward though. As you heard this morning This region is the most incompletely probed in the lower mantle.
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23. D'' Layer Demystified http://sciencenow.sciencemag.org/archives.shtml
24 March 2004
MONTREAL--Deep within Earth, where hellish temperatures and pressures create crystals and structures like none ever seen on the surface, a strange undulated layer separates the mantle and the core. The composition of this region, called the d" layer (pronounced "dee double prime"), has puzzled earth scientists ever since its discovery. Now, a team of researchers believes they know what the d" layer is.
Three thousand kilometers deep in Earth, the solid rock of the mantle meets the liquid outer core. At this juncture, seismic waves from earthquakes traveling through Earth suddenly change speed, and sometimes direction. These sudden shifts trace the border of the d" layer, which rises and falls in ridges and valleys. Researchers suspected that the layer marks a change in the crystal structure of the rock, which might happen at different depths depending on the temperature. This would explain the rises and dips of the boundary. But what could account for the sudden speed shifts of the seismic waves?
The explanation may lie in an entirely new kind of crystal structure, according to presentations by Jun Tsuchiya and Taku Tsuchiya here 23 March at a meeting of the American Physical Society. They and colleagues at the University of Minnesota in Minneapolis collaborated with a team from the Tokyo Institute of Technology led by Motohiko Murakami. The Tokyo team used a diamond anvil to squeeze and heat a grain of perovskite, the dominant mineral deep within the earth. They then took an x-ray image to see what happened to the molecular structure of the mineral in conditions like those in the d" layer. The Minnesota group then analysed the x-ray. Only one crystal structure fit the x-ray data, and it was like nothing anyone had seen before.
The first news about this transition was broken in March 24 after our March Meeting presentation in Montreal. It came out with this infamous title of D” layer demystified. Probably this sounded offensive to many geophysicists who were thinking hard and coming up with complex explanations this region. So before I proceed, let me tell you about D”.
Strange stuff. Post-perovskite owes its odd crystal structure to the intense heat and pressure at the boundary between the mantle and core. CREDIT: RENATA WENTZCOVITCH

24. Deep mantle observables from regional studies
So, let me leave you with a general view of D”:
Because earthquakes and seismic detectors are not uniformly distributed on Earth some regions are simply not probed. It is still not known if the D” discontinuity is ubiquotous. This idea has been proposed and if it is really caused by the ppv transition then it might depend on the T at the CMB.
In a couple places it has been well studied:
a) in the CPR the D” discontinuity might not exist. The reflections above CMB might be caused by scatterers, i.e., heterogeneities of km lengths. The ULVZ is clear there. Anisotropy is observed and is more complicated than in a transversely isotropic medium. By consensus this region is hot and complicated.
b) Underneath CA: Clear D” discontinuity 300km above CMB. ULVZ is not observed and a simple transversely anisotropic medium explains shear wave splitting (SH faster than SV). The picture one has (or had) in mind is:
It is cold region where plates have been accumulating. Ponding plates generate high stress and lateral material flow. Anisotropy may be explained by LPO (low T high stress regime). This is one region where we can test a semi-naïve estimate of anisotropy caused by ppv.
c) Underneath Africa: large low velocity province and very tall (up to 1000km). This structure is referred to as superplume. Large ULVZ and not well defined anisotropic structure.
In summary: Structure and properties of D” vary as much as the geologic features on Earth’ surface, probably much more dramatically.
Yes, this is not simple, but I think that the ppv transition helps us to understand one major feature and points to a new way of thinking about this region.
Using seismology, addressing the issue of “flux” depends on inference from the static time snapshots of the structure we retrieve in our analyses.
“Super plume”
Large low velocity zone
D” discontinuity
D” anisotropy
Weak or no
anisotropy
Ultra-low velocity zones
D” anisotropy
Scatterers
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Lay, Garnero, Williams [PEPI, 2004, in press]

25. ASESMA’10
Lay, Garnero, Williams [2004, PEPI]

26. Lowermost mantle heterogeneity
Large-scale lengths:
Lowermost mantle heterogeneity
dVs: Grand
dVΦ: Sb10L18 4 2 -2 -4
dVs (%)
1.5
0.0
-1.5
dVΦ (%)
Here is a very interesting thing:
The anti-correlation between bulk and shear velocity.
This relationship exists only in D”.
Bulk velocity is fast where shear is slow and vice versa.
On the basis of hard core mineral physics data we know this cannot be caused by temperature effects alone. So, these variations have been attributed to some unknown large scale chemical heterogeneity.
What I want to show today is that the post-perovskite transition appears to explain this enigma.
From:
Lay, Garnero, AGU/IUGG Monograph (2004),
Lay, Garnero, Williams, PEPI (2004)
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27. Aggregate Elastic Moduli of Perovskite
Aggregate Elastic Moduli of Post-perovskite
Bppv ≈ Bpv
Gppv > Gpv
These are the aggregate moduli:
These are perovskite’s and these are post-perovskite’s.
The bulk modulus and its temperature dependence are quite similar to pv’s.
In contrast G is larger and has larger T dependence as well
(Wentzcovitch et al., PRL 2004)
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28. Seismic velocity of Perovskite Seismic velocity of Post-perovskite
Longitudinal
Shear
Bulk
These are velocities and density in pv and ppv.
Vs and Vp in ppv are larger and have larger T dependence because of G. Vf on the other hand is silightly smaller.
This is because of the density increase and unchanged value of K.
Contrast in S waves is larger than in P waves.
(Wentzcovitch et al., PRL 2004)
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29. Δ Δ Δ ΔV (%) Velocity discontinuity along the phase boundary
I am showing here the velocity discontinuities across the transition, along the phase boundary. The dark lines correspond to the jumps at the mid points in the band representing the PB. The shaded regions are the jumps throughout the entire range width of that band. It can be seen that the discontinuity in Vs is always larger than in Vp, and the discontinuity in Vf is always negative. This is the least uncertain prediction. Therefore, this transition can cause an anticorrelated anomaly between Vs and Vf, not only in certain places but everywhere it occurs. Δ
Wentzcovitch, Tsuchiya, Tsuchyia, Proc. Natl. Acad. 103, 543 (2006)
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30. ASESMA’10 Now let me bring back a picture I showed earlier.
The Vf structure in D”:
CPR is supposed to be hotter than average and CP colder than average (for more than one reason this interpretation makes sense), but Vf in CPR is faster and in CP it is slower than average.
This might be understood by assuming that in D” the average mineralogy consists predominantly of ppv.
In regions hotter than average, and it does not need to be much hotter according to our simple phase boundary, pv predominates and therefore Vf is larger than average while in colder regions ppv predominates and Vf is smaller. In fact according to this new interpretation, in the CPR D” should not be present. Reflections should be caused indeed by something else like scatterers, for instance.
The mysterious anti-correlation in Vf and Vs, and the magnitude of velocity jumps at the D” where it is real, are the facts that appear natural consequences of this phase transition. I am not going to touch on the question of anisotropy at high T because at high T anisotropy decreases considerably and I don’t believe it will be possible to explain these observations without learning much more about plastic deformation mechanisms in ppv. I think it will be necessary to invoke other sources of anisotropy to explain these observations.
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Lay, Garnero, Williams [2004, PEPI]

31. Ratio of VS and VP anomalies　　
P (GPa)
MLBS
MLBS – Masters et al., (2000)
Here are the ratios of the S and P velocity anomalies produced by lateral temperature variation in ppv and pv at several T’s as a function of pressure. The shaded lines are spherical averages of the ratios extracted from Masters et al. (2000) tomographic model. As you can see lateral T variation produces large anomalies ratio in ppv than in pv but still smaller than this parameter in the lowermost mantle.
Here I show the ratio of velocity anomalies produced by a phase change along the phase boundary. The shaded are are our uncertainties. Lateral variation in phase content can enhance this seismic parameter in the correct direction.
Wentzcovitch et al., Proc. Natl. Acad. 103, 543 (2006)
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32. Ratio of VΦ and VS anomalies
MLBS
P (GPa)
MLBS – Masters et al., (2000)
PPv-thermal Pv-thermal Pv-PPv Transition
Ratio of VΦ and VS anomalies
Now the anomalies in bulk and shear velocities. Here you see the anti-correlation between bulk and shear velocity anomalies at the bottom of mantle. Temperature variations in pv and ppv produce very different values. But lateral variation in phase abundance can produce the negative anomaly ratio observed.
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33. Lowermost mantle heterogeneity
Large-scale lengths:
Lowermost mantle heterogeneity
dVs: Grand
dVΦ: Sb10L18 4 2 -2 -4
dVs (%)
1.5
0.0
-1.5
dVΦ (%)
Here is a very interesting thing:
The anti-correlation between bulk and shear velocity.
This relationship exists only in D”.
Bulk velocity is fast where shear is slow and vice versa.
On the basis of hard core mineral physics data we know this cannot be caused by temperature effects alone. So, these variations have been attributed to some unknown large scale chemical heterogeneity.
What I want to show today is that the post-perovskite transition appears to explain this enigma.
From:
Lay, Garnero, AGU/IUGG Monograph (2004),
Lay, Garnero, Williams, PEPI (2004)
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34. Comparison with PREM 100 GPa 38 GPa Brown & Shankland T(r)
Pyrolite (20 V% mw)
Perovskite
100 GPa
38 GPa
(Wentzcovitch et al.
Phys. Rev. Lett. 2004)
Brown & Shankland T(r)

35. Summary
Post-perovskite transition has changed the way geophysicists look at the Earth
The crystal structure of post-perovskite and its properties were obtained by first principles and experiments confirm our V vs P relation and more
The computed elastic properties of perovskite and pos-perovskite help to interpret large scale velocity anomalies in the D” region
First principles theory has won the hearts and minds of geophysicists since then.
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36.
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37. Other resources on mineral physics:
My web pages:
COnsortium on Materials Properties Research
in Earth Sciences (COMPRES)
Please joint us!