Download Earth`s core and the Geodynamo

Earth’s core and the Geodynamo
(i) Structure and physical properties
of Earth’s core
Lecture 2:
Mineral physics and phase changes
in the deep Earth
Lecture 2: Mineral physics and phase
changes in the deep Earth
2.1 Introduction to Mineral Physics
2.2 Mineral Physics: Theory
2.3 Mineral Physics: Experimental Techniques
2.4 Mineral Physics: Ab-initio Computations
2.5 Structure of Iron alloy in Earth’s Core
2.6 Physical Properties of Earth’s Core
2.7 Summary
Homework for Lecture 2
• Exercise 2.1: Prepare a short presentation (5 mins) on
the mineral physics paper you have been
allocated.
Your should discuss:
(1) The goal of the study
(2) The methods used (overview)
(3) Summary of the results
(4) Implications / Limitations
• Presentations should be ready for next week.
You should bring your presentation in .pdf format.
2.1.1 Role of mineral physics
MINERAL = “Application of physics in order to understand
PHYSICS
and predict properties of Earth materials”
2.1.1 Role of mineral physics
• Studying Earth’s core we need mineral physics to:
(i) Determine those minerals responsible for observed core properties.
(ii) Determine the relevant crystal structure at core P,T.
(iii) Determine Elastic constants and Bulk Modulus of core materials,
in order to understand detailed seismic observations.
(Density vs Pressure, Inner core anisotropy, layering etc)
(iv) Determine the Heat Capacity, Viscosity, Thermal and Electrical
Conductivity of outer and inner core for geodynamical studies.
(v) Determine the melting temperatures of core material to constrain
the thermal structure of the core needed to understand the
evolution of Earth.
2.1.2 Determination of crystal structure
• Minerals with the same composition can occur in a number
of different PHASES which have different properties.
• For example, Fe at high P,T can occur in the following
arrangements:
fcc
hcp
(γ- Fe)
("-Fe)
(!-Fe)
bct, dhcp and determining
orthorhombically distorted
hcp experiments
• Mineral physicsAlsoinvolves
from
and calculations which phase is stable.
bcc
2.1.3 Prediction of physical properties
Calculations on Fe, FeS, FeSi
•• Experiments
Quality of calculations:
on Iron alloy
! EOS, phonon spectrum, DOS
•
samples at high P, T:
Stable phase of Fe at core conditions
! via lattice dynamics
- Multi-anvil
cells
! via molecular
dynamics
- Diamond anvil cells
•- Laser
Effect of
light elements
heating
- Shock wave experiments
•
Elastic constants >> VP, VS
Ab-initio
(Quantum Mechanical)
•• calculations
Birch’s Law
•
Anisotropy
• Extrapolation of lower P, T
!"#$%&#'()*&#+,((-.#,(#%/0%#1.-'').-'#
• behaviour
Melt in the inner
core?
using
simple theories.
,2*#(-+1-.,().-'3
2.1.4 Pressures and temperatures
in the deep Earth
• Pressures in the deep Earth:
~ 135 GPa at CMB
~ 330 GPa at ICB
~ 360 GPa at centre
=77>
42/('5
677#89,#:#6#;<,.#:#
(Courtesy of Prof. A. Oganov)
• Temperatures in the deep Earth:
~ 4000K at CMB
~ 5500 K at ICB
~ 6000 K at centre
But, greater uncertainties with T as can’t directly determine.
2.1.5 Example: Pioneering work of Birch
(From Fowler, 2005
after Birch, 1968)
• Francis Birch analyzed
experimental results of
seismic wave speed as
a function of density.
• Demonstrated that Fe must be alloyed with light elements
to explain observed density and wave speeds in the core.
2.2.0 Useful Book For Mineral Physics Theory
• Introduction to the Physics of the Earth’s interior by J.-P. Poirier
(Cambridge University Press), 2nd Edition, 2000.
2.2.1 Thermodynamics: Introduction
• Thermodynamics is a theory (i.e. a self-consistent set of propositions)
describing the response of a material to changes in the environmental
conditions, for example in temperature T, pressure P.
• Macroscopic concept of Temperature (T):
“If 2 systems are each in thermal equilibrium with a third system, then they
are in thermal equilibrium with each other and have the same temperature T.’’
• Macroscopic concept of Entropy (S):
“Entropy S is a function of state conserved during the reversible transfer of
heat. ”
Entropy is a measure of how close a thermodynamic system is to
equilibrium. It quantifies how much energy is available to do work
and that can potentially be manifest as heat.
2.2.1 Thermodynamics: Introduction
• Definitions: dU = Change in internal energy of closed system
dQ = Heat absorbed by closed system
dW = Work done on closed system
• Consideration of (1) Conservation of energy (taking into
account heat transfer and performing work) & (2) Heat transfer
and changes in the temperature of a system leads to:
3 Laws of Thermodynamics:
(1) dU = dQ + dW = dQ − P dV
(2) dQ ≤ T dS
(3) S → 0
as
dQ
for reversible a process
where dS =
T
T →0
Rudolf Clausius, Professor at ETH Zurich 1855-1867
Proposed the concept of Entropy
and stated the 2nd law of thermodynamics.
2.2.2 Thermodynamic Potentials
• We wish to describe changes in the energy of the system,
but sometimes don’t know all of dP, dV, dT,dS.
• To get round this, we define the following types of energy:
U
Internal Energy
H = U + PV
Enthalpy
F = U − TS
Helmholtz Free Energy
G = U + PV − TS
Gibbs Free Energy
• Then looking at changes in these energies we have,
dU = T dS − P dV
dH = dU + P dV + V dP = T dS + V dP
dF = dU − T dS − SdT = −SdT − P dV
dG = dU + P dV + V dP − T dS − SdT = V dP − SdT
• Note, at equilibrium (P,T) Gibbs Free Energy, G, is constant .
2.2.3 Thermodynamic variables
• Taking derivatives of the potentials while holding 1 variable
constant leads to expressions for the other physical variables:
!
"
!
"
∂U
∂U
=T
= −P
∂S V
∂V S
!
∂H
∂S
"
!
∂F
∂V
"
!
∂G
∂T
"
!
=T
P
!
= −P
T
∂H
∂P
"
∂F
∂T
"
= −S
"
=V
!
= −S
P
∂G
∂P
=V
S
V
T
2.2.4 Maxwell’s relations
• By taking one further derivative, useful relations between
variables are established:
"
"
!
!
∂2U
∂T
∂P
=
=−
∂V S
∂S∂V
∂S V
"
"
!
!
∂2H
∂T
∂V
=
=
Maxwell’s
∂P S
∂S∂P
∂S P
"
"
!
!
Relations
∂2F
∂S
∂P
=−
=
∂V T
∂T ∂V
∂T V
"
"
!
!
∂2G
∂S
∂V
=
−
=
∂P T
∂T ∂P
∂T P
• Relations btw variables can also be found by the chain rule,
!
" !
" !
"
for example:
∂V
∂S
∂P
∂S
P
∂P
V
∂V
= −1
S
2.2.5 Physical Observables
• Expressions for observables that can be measured, can
also be found in terms of thermodynamic variables, e.g.
!
"
!
"
∂S
∂U
• Heat Capacity at
=T
CV =
constant volume:
∂T V
∂T V
• Adiabatic (Isentropic)
Bulk Modulus:
KS = −V
• Thermal expansivity
at constant P:
1
α=
V
!
!
∂P
∂V
∂V
∂T
"
S
"
P
• Thermodynamics thus provides a framework whereby
measurable quantities can be related and even calculated
if we are able to work out the necessary potentials.
2.2.6 Equations of state
• EQUATION OF STATE (EOS) describes how the volume
(or density) of a material varies as a function of P, T.
e.g. For an ideal gas:
P V = RT where R = 8.31JK−1 mol−1
• For solids, the simplest isothermal (constant T) EOS is the
definition of bulk modulus:
dP
dP
dP
=ρ
=
K = −V
dV
dρ
dlnρ
• And the simplest isobaric (constant P) EOS is the
definition of thermal expansion coefficient:
α=
dlnV
1 dV
=
V dT
dT
2.2.6 Other examples of Equations of state
• Integration of isothermal EOS, allowing K to vary linearly
with pressure yields the Murnaghan Integrated Linear EOS:
!
"1/K0!
!
K
ρ = ρo 1 + 0 P
K0
with K = K0 + K0! P
K0 = K(P = 0, T = const)
ρ0 = initial density
• Using finite strain theory (see Poirier, 2000) to account for
variation of K’ with P gives, by 2nd order expansion of F, the
Birch-Murnaghan Integrated
EOS:
$
!
3K0
P =
2
"
ρ
ρ0
#7/3
−
"
ρ
ρ0
#5/3
• More complex EOS involve also involves thermal effects.
• EOS can be tested against experiments, but simple versions
such as those above often don’t work well at high P,T.
202 Multianvil Cells and High-Pressure Experimental Metho
(a)
2.2.1 Multi Anvil Cells
• Piston-cylinders used to
compress samples to high P
of up to 30GPa.
F
2.08.2
(b)
• Consist of 4, 6 or 8 anvils
allowing compression of
large samples (1-5 cm^3)
• Heating to high T up to
~3000K carried out by
heaters set into the
pressure medium.
can genera
der and be
is because
the bore is
imum atta
F
(c)
F
Figure 5 Conceptual drawings for compression of the
polyhedral pressure media. All surfaces of the regular
polyhedra are thrusted by equivalent normal forces. (a), (b),
and (c) show the tetrahedral, cubic, and octahedral
As mentio
tions have
types, corr
8, respecti
than the s
insulating
anvils. On
central pr
high press
fined by th
the preset
into the g
than 90%
compressio
For fur
polyhedral
dron with
anvils. Wit
the stroke
increases a
pressed an
medium be
the anvil fr
the massiv
hedral com
of Osaka U
breakage o
MAAs is an
the initial s
my knowl
largest num
compromi
the massiv
As the
compresse
the genera
2.2.1 Multi Anvil Cells
• For example, Multi-Anvil Press
at ETH:
2.2.2 Diamond Anvil Cells
• Small pressure bearing tip
expands to large base at
other end.
• Diamond used as can
resist deformation up to
high P.
• 300-550 GPa can be reached
but sample sizes small.
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D
Structure
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2.2.2 Diamond AnvilCore
Cells
muterials
# = 0°
X
#
A
#=
90°
R
7.875
Iron, body-centered cubic, Fe
7.884,
Iron-nickel (Im), bodyDiamond anvil
7.86
centered cubic, FE,.,,Ni,.,
4.603
Pyrrhotite, Fe,,,S
!3
Primary
4.61
Pyrite, FeS,
X-ray beam
5.50
Wiistite,
!1A Fe,,.,O
! 1A
Be
2"
7.0 16
3Fe,Si! 1B
+ FeSi !1B
Diffracted
7.646
Fe,Si + FeSi
X-ray5.12
beam
!3
Magnetite, Fe,O,
5.00
Hematite, Fe,O,
Diamond anvil
Hexagonal close-packed
Hexagonal close-packed
?
?
?
Hexagonal close-packed
Hexagonal close-packed
FeO (LS) + F~,o,(LS)I
Corundum (LS)
(From Mao et al., 1998)
Mantle muterials
3.58 �
None
Periclase, MgO
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3.98
None
Corundum,
Al,O,
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Laser heating can be carried out to reach high T ~5000K.
8?%
Rutile
Quartz,
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���SiO,
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3.34
Hypothetical perovskite
Bronzite,
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X-ray
diffraction
measurements,
T
measurements
andperovskite ?@
3.32
Hypothetical
Forsterite,
Mg,Si04
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can
carried
in situ
to determine
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Dunite,
(Mgn,,,,Fe,.,,),Si04
3.55
Hypothetical perovskite
mineral
properties.
Computation
is a brid
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4.18
• Provides?insight unob
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Spinel,����
MgAI,04
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Dunite, (Mgll.4s,~ell.55),~i04 3.85
perovskite
Shock
Wave
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• Hypothetical
Computational
scienc
rock salt
Calcia, CaO
3.35
CsCl
+
+
• Exponential
increase
• Since mid-1950’s
P,.T- have been
obtained
-.in
. very high-.
shock
wave
experiments.
tLS, Fez+ assumed to be in a low-spin orhital co
*Values
are for
1 bar and 25°C.
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• Simplest setup described by Ahrens (1980) is:
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a
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ublishers Ltd 1998
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• Allows set of Pressure vs Density (‘Hugoniot’) measurements,
but difficult to interpret as T hard determine.
2.2.3 Shock Wave Experiments
• Using nuclear explosions, pressures up to 400GPa could
be reached, but no longer allowed.....
N u c l e a r explosion
Prompt neutron flux
Fe
(From Ahrens, 1980)
wave propagates through the base plate into the sample (19). In aluminum, shock pre
plane-wave detonated explosive pad accelerates the metal flyer plate to -4.5 kmise
plate, pressures up to 140 GPa (I .4 Mbar) are achieved in copper base plates. (c) P
irradiate a 2"U plate, causing internal fission-induced heating. The resulting rapid exp
into the base plate and sample. Pressures of 2 TPa (20 Mbar) have been induced in m
gun (Fig. 3), 20-gram projectiles are launched at 7 kmlsec and impact samples I cm i
5 GPa (5 Mbar) (21).
2.2.3 Shock Wave Experiments
-
1036
• The old shock wave apparatus - a 21cm navy gun!
• 40mm gun on left hand side propels missiles at
2.5km/sec and can generate 50GPa (into lower
mantle)
2.2.3 Shock Wave Experiments
• Laser heating can also be used to induce shock waves
in pre-pressurized (e.g. by Diamond Anvil Cells) samples.
• P and T at2.2.3
centreSummary
of Earth ~ 360
andP,~ T
6000 K
ofGPa
High
• P and T at Experimental
core/mantle
~ 135 GPa and ~ 3000 to 4000 K
Techniques
•
•
•
•
Piston cylinder
Multi-anvil
Diamond-anvil
Shock guns
~ 4 GPa
~ 30 GPa (higher with sintered diamonds)
~ 200 GPa (temperatures uncertain, gradients high)
~ 200 GPa (temperatures extremely uncertain)
High P/T experiments are hard and can have large uncertainties.
=> Need theoretical/computational approach as well......
2.4.1 Ab-initio: Quantum mechanics
• Material properties are fundamentally governed by
Quantum Mechanics that determines atomic energy
levels via solutions of Schrödinger’s equation:
HΨ = EΨ where H = Ti + Te + Vii + Vee
Many body
wave-function
Energy
KE of ions
KE electrons
ion-ion repulsion
electron-electron
repulsion
• Since electrons much less massive than ions, their motion
follows ions, so they can be decoupled from ions and
Schrödinger’s equation solved for E as a function of ions
position.
• Energy minimisation-> equilibrium energy structure of system
& P.E. profile determining ions motions
2.4.2 Ab-initio: Density functional theory
• Must make approximations in order to solve QM equations:
-Density Functional Theory
- Local Density Approximation
- Generalised Gradient Approximation
- Pseudo-Potential Approximation
- Projected Augmented wave method
}
(For technical details
see Alfè, 2007)
• Changes in pressure implemented by changing the volume
of the simulation cell.
• Approximations work well for materials at core pressures.
• Compute Internal Energy (U) and Enthalpy (H=U +PV) by
minimizing with respect to ion positions. This determines:
- crystal structures
- elastic constants (hence anisotropy, bulk modulus etc.)
2.4.3 Ab-initio: Molecular dynamics
• But, so far no account has been taken of temperature
(i.e. 0K: only the energetics of bonding considered.)
• To take into account finite temperature effects the
equations of motion of the particles are also solved by
calculating the forces acting and time-stepping
(i.e.Molecular dynamics).
• Stable structures and Thermodynamic properties are
now found via the Gibbs Free Energy (G=F+pV) from the
Helmholtz free Energy F which is directly computed.
• Particle diffusion and hence VISCOSITY can also be
directly calculated.
2.4.4 Parallel computations on
supercomputers
+,%-+.$#/%*0'$/0$%1
$2&$#'-$/34%
• Core properties can therefore
be computed, but the
calculations are very HUGE.
"$+#54%0+-&63)3'+/
• Require large amounts of time on super-computers.
D%=)E$*%
7+.$#/%*6&$#0+-&63$#
• And experimental confirmation needed to ensure
F2GBF C?)4%+#%
8/+%('-'3*%+/%&#$**6#$93$-&$#)36#$:
BB%C?): approximations made are valid......
2.4.5 An example: Post-Perovskite
(Classic phase
transitions of Olivine
(Mg2SiO4 -Fe2SiO4)
in the mantle
after McKenzie,1983)
2.4.5 An example: Post-Perovskite
!"#$%"&'%()#*)$#(&+$%"#,(-'&%)%.$/0'1)23 /04%")
• Ab-initio molecular dynamics simulations predicted the
015)'6$/4)'&()7"#8&9)8'&9)&'6%
existence of a new
stable form of perovskite at lowermost
mantle pressures and temperatures.
2
(SiO6 octahedra and Mg atoms
(spheres) of post-perovskite.
Courtesy of Prof. A. Oganov)
!90(%)5'07"06
=.$/0'1()23 5'(;#1&'1<'&4)015
!"%5';&()7"#8&9)#*)23 5<%)&#)
@1;"%0(%()=0"&9A();##/'17)"0&
23 /04%")(9#</5)1#&)%.'(&)'1)H
:1'(#&"#$';)(&"<;&<"%)#*)$#(&+$%"#,(-'&%
=.$/0'1()23 01'(#&"#$4)015)%/%;&"';0/);#15<;&','&4
2.4.5 An example: Post-Perovskite
(Molecular dynamics
simulation of
post-perovskite.
Courtesy of
Prof. A. Oganov)
2.4.5 An example: Post-Perovskite
• Post-perovskite enables many unexplained properties of
D’’ (e.g. seismic anisotropy, heterogeneity) to be understood.
2.5.1 Possible crystal structure of Fe
fcc (γ- Fe)
("-Fe)
hcp
bcc (!-Fe)
Also bct, dhcp and orthorhombically distorted hcp
2.5.1 Summary of
structureDynamics
of pure Fe303
Inner-Core
• From Laser Heated Diamond Anvil Cell and Shock wave
experiments and ab-initio calculations:
ε (h.c.p.) Fe
Liquid
5000
c-Axis
", α?
Temperature (K)
solid
pre4000
Basal plane
d the
3000
ed in
β?
δ
wide),
2000 γ
ICB
CMB
than
1000
ε
α
ell be
300
100
200
1996).
Pressure (GPa)
alcu"-Fe) has lowest free energy at core P,T for pure Fe,
solid • h.c.p.(
Figure 3 Summary of the phase boundaries of iron. The
BUT b.c.c. (!-Fe) has only slighter higher energy and could
uch a occur
uncertainty
of the
increases
withStemperature
at v. high
P inboundaries
alloys with large
enough
or Si content.
and pressure, and the existence of the double h.c.p. $ and
diffib.c.c. !9 phases is uncertain. To the right is the crystal
t the
(From Sumita and Bergman, 2007)
ulate the
Fe/Si or Fe/O can explain the seismic data, and we propose
ar from
an Earth’s core composition based on ternary and
ystem is
2.5.2 Melting T of pure Fe
ty of the
provided
oal is to
mize the
Shock wave
ulations
measurements
ntegrand
Ab-initio
~6000K
calculations
at ICB
thermal
pressures
(From
required
Alfe
et al.
rinciples
2007)
DAC
measurements
DF as a
is again
ations.
ur that a
a simple
here r is
are two• Ab-initio calc. help resolve DAC and shock wave results.
5. Comparison
of likely
melting
Fe from
DFT
arameter But,Figure
influence
of impurities
to curve
lowerofmelting
T to
~5500K.
calculations and experimental data: black solid firstates that
principles results of [51] (plus or minus 600 K); black
nsemble
chained and maroon dashed curves: diamond anvil cell
2.5.3
Co-existence of BCC and HCP Fe in
measurements of [8,11]; green diamonds and green filled
inner
core? of [10,13]; black
square: diamond anvil
cell measurements
open squares,
black
open
circle Fe
andcould
magenta diamond:
• Randomly
oriented
h.c.p.
or b.c.c
shockisotropic
experiments
of [15]. Error
bars are those quoted in
nown, it explain
near-surface
layer.
original references.
s, and in
• h.c.p. and b.c.c. Fe have different
anisotropy, their co-existence could
help explain deeper heterogeneity.
• b.c.c. phase likely richer in light
element than h.c.p. phase.
• Mechanism producing alignment unknown.....
(After Song and
Helmberger, 1998)
• Remaining uncertainty about light element (S, Si, O, H or C?),
precise structure and hence temperatures makes detailed
knowledge of other properties difficult->only have ESTIMATES.
2.6.1 Estimates of physical properties of
the core
Property
Density Jump at ICB(#$)
Specific Heat (Cp)
Thermal Expansivity(!)
Kinematic viscosity(%)
Estimated Values
IC
OC
700±200 kgm-3
850±20 Jkg-1 K-1
1.4±0.5 x10-5 K-1
1 x10-5±2 m2 s-2
1 x1010±3 m2 s-2
Thermal diffusivity(&)
5±3 x10-6 m2 s-2
Magnetic diffusivity(!)
1.5±0.5 m2 s-2
(Taken from Olson, 2007)
2.7 Summary: self-assessment questions
(1) What is the role of mineral physics in deep Earth studies?
(2) What are the 3 laws of thermodynamics?
(3) Can you derive and use Maxwell’s relations?
(4) How are ab-initio computations used to determine the
(5) What are the likely stable phases of Fe in Earth’s core?
(6) Can you summarize the physical properties in Earth’s core?
Next time: Thermal structure of the core, inner core growth
and power sources for the geodynamo
References
- Ahrens, T.J., (1980) Dynamic compression of Earth materials. Science, Vol
207, pp.1035-1041.
- Alfè, D., (2007) Theory and practice- The Ab Initio Treatment of High Pressure
and Temperature Mineral Properties and Behavior. In Treatise on Geophysics,
Vol 2 Ed. G.D. Price, Chapter 2.13, pp. 359- 387.
- Ito, E., (2007) Theory and practice- Multi Anvil Cells and High Pressure
Experimental Methods. In Treatise on Geophysics, Vol 2 Ed. G.D. Price, Chapter
2.08, pp.198-230.
- Mao, H.K and Mao, W.L., (2007) Theory and practice- Diamond Anvil Cells for
High P-T Mineral Physics Studies. In Treatise on Geophysics, Vol 2 Ed. G.D.
Price, Chapter 2.09, pp.231-267.
- Olson, P., (2007) Overview of Core Dynamics. In Treatise on Geophysics, Vol 8
Ed. P. Olson, Chapter 8.01, pp.1-30.
- Poirier, J.P., (2000) Introduction to the Physics of Earth’s Interior. Cambridge
University Press.
-Sumita, I. and Bergman, M.I., (2007) Inner-core dynamics. In Treatise on
Geophysics, Vol 8 Ed. P. Olson, Chapter 8.10, pp.299-318.
-Vocadlo, L., (2007) Mineralogy of the Earth- The Earth’s core: Iron and Iron
Alloys. In Treatise on Geophysics, Vol 2 Ed. G.D. Price, Chapter 2.05, pp.91-121.