PHYS3070 Physics of the Earth:
from seismic structure to geodynamics
‘Geophysics … has the rigour of physics and the
vigour of geology’
C. M. R. Fowler
Goal: to connect the seismological investigation of the
Earth’s internal structure (Hrvoje Tkalcic’s segment)
with Paul Tregoning’s component concerning geodetic
observations of surface deformations
Ian.Jackson@anu.edu.au
WebCT
http://rses.anu.edu.au/people/jackson_i/PHYS3070/
Part I: Elasticity, equations-of-state &
interpretation of seismological models
Tensor stress & strain
Constitutive law & elastic waves
Elasticity & interatomic forces
Geophysical thermodynamics
Lattice vibrations & thermal energy
Anharmonicity & thermal expansion
Finite strain & cohesive energy @ high pressure
Mie-Grüneisen equation-of-state & thermal pressure
Anelasticity & seismic wave attenuation
Interpretation of seismological models
The End
References
Fowler, C.M.R. (2005) The Solid Earth, Cambridge University Press, 2nd
edn. (Hancock: QC806.F625 2005 - 2 hr reserve)
Lowrie, W. (1997), Fundamentals of Geophysics, Cambridge University
Press (Hancock: QC806.L67 1997 - 2 hr reserve)
Stacey, F.D. (1992) Physics of the Earth, 3rd Edn., Brookfield Press (ES:
QC806.S65 1992)
Poirier, J.P. (1991) Introduction to the Physics of the Earth's Interior,
Cambridge University Press, (ES: QE509.P64 1991)
Davies, G. F. (1999) Dynamic Earth, Cambridge University Press (ES &
Hancock: QE509.4 .D38 1999)
Jackson, I. (ed.) (1998) The Earth's Mantle: Composition, Structure and
Evolution, Cambridge University Press.
Turcotte, D.L. & Schubert, G. (1982) Geodynamics: Applications of
continuum physics to geological problems. Wiley, 1982 (ES:QE501.T83 )
Gross Earth seismological models
Fowler Fig. 8.3
Fowler Fig. 8.1
Inversion of traveltime versus angular distance
& free-oscillation data for spherically averaged structure
Seismological models for the
transition zone of the Earth’s mantle
Bulk sound speed, V! / km s-1
9.0
8.5
pyrolite model
NJPB
8.0
7.5
7.0
TNA
Nth Atl
6.5
6.0
200
SNA
300
400
500
600
700
800
900
Depth / km
Seismological models for bulk sound speed Vφ = (KS/ρ)1/2 vs depth
Jackson & Rigden, In The Earth’s Mantle, 1998
Lateral variations of seismic
wave speeds in the upper mantle
Surface-wave tomographic model of Fishwick et al. (2005)
VS variations (%) at 200 km depth
Recap on elasticity: tensor strain
Displacement
gradient tensor
eij = ∂ ui/∂ xj (i, j = 1, 2, 3)
 ui = eijxj for
homogeneous
deformation
Rigid-body rotation:
e21 = -e12 = δθ ≠ 0
∴define strain as symmetrical part
of eij
εij = (1/2)(∂ ui/∂ xj+∂ uj/ ∂ x i)
Recap on elasticity: tensor stress
σij is the component of force parallel
to xi per unit area oriented normal to
xj (exerted on the infinitesimal
element by the surrounding medium;
tension +ve)
Rotational equilibrium requires σji = σij
Translational equilibrium  the wave equation
ρ∂ 2ui/∂ t 2 = ∂ σij/∂ x j
Elasticity: constitutive law & elastic waves
Generalisation of Hooke’s law: σij = cijkl εkl
For isotropic medium: σij = λδijεkk + 2µεij
Kronecker ‘delta’ δij = 1 (for i = j), 0 (for i ≠ j)
Einstein summation convention εkk = ε11+ ε22+ ε33 = Δ
(known as the dilatation)
The wave equation ρ∂ 2ui/∂ t 2 = ∂ σij/∂ xj becomes
ρ∂ 2ui/∂ t 2 = λδij∂ 2uk/∂ x j∂ x k + µ(∂ 2ui/∂ x j∂ x j + ∂ 2uj/∂ x j∂ x i)
Substit’n of trial plane-wave solution ui = ui0f(n.r - vt) = ui0f(nmxm- vt)
 system of 3 eqns. linear in ui0 for given nm
 3 eigenvalues given by cubic in ρv2
1 compressional mode ρvP2 = λ + 2µ = K +(4/3)G
2 orthogonal shear modes ρvS2 = µ = G
‘Geophysical’ thermodynamics: the internally
consistent framework for understanding the
Earth’s internal structure
First law of thermodynamics (conservation of energy embodying the
equivalence of heat and work and their relationship with internal energy E)
dE = dQ − PdV; substitute for entropy S defined by dS = dQ/T, to get
dE(S,V) = TdS − PdV
Similar expressions for the other thermodynamic potentials:
Enthalpy H = E + PV
dH(S,P) = TdS + VdP
Helmholz free energy F = -kT ln Z = E − TS
dF(V,T) = − PdV − SdT
Gibbs free energy G = H − TS = E + PV − TS dG(P,T) = VdP − SdT
Z is the partition function of statistical mechanics
References: Stacey, Appendix E; Poirier, Ch. 1
Useful thermodynamic identities
Tools for functional analysis:
For Z = Z(X,Y),
dZ = (∂ Z/∂ X)Y dX + (∂ Z/∂ Y)X dY,
∂ 2Z/∂ X∂ Y = ∂ 2Z/∂ Y∂ X, &
(∂ Z/ ∂ X)W = (∂ Z/∂ X)Y + (∂ Z/∂ Y)X (dY/dX)W
(∂ Z/ ∂ Y)X = (∂ Z/ ∂ W )X / (∂ Y/ ∂ W )X
Applications in deriving thermodynamic identities:
E(S,V): (∂ E/∂ S)V = T (∂ E/∂ V)S = −P
H(P,S): (∂ H/∂ S)P = T (∂ H/∂ P)S = V
F(V,T): (∂ F/∂ V)T = −P (∂ F/∂ T)V = −S
G(P,T): (∂ G/∂ P)T = V (∂ G/∂ T)P = − S
(∂ T/∂ V)S = −(∂ P/∂ S)V
(∂ T/∂ P)S = (∂ V/∂ S)P
(∂ P/∂ T)V = (∂ S/∂ V)T
(∂ V/∂ T)P = −(∂ S/∂ P)T
Applications of thermodynamic identities
Thermal pressure
(∂ P/∂ T)V = (∂ S/∂ V)T = αKT  (dPth)V = (γ/V) dEth
(Grüneisen parameter: γ = αKTV/CV)
Isobaric & isochoric derivatives (∂ Z[X,Y]/ ∂ X)W with X = T, Y = V & W = P:
(∂ Z/ ∂ T )P = (∂ Z/∂ T)V + (∂ Z/∂ V)T (∂ V /∂ T )P = (∂ Z/∂ T)V − αKT (∂ Z/∂ P)T
e.g., CP=(∂ Q/∂ T )P=T(∂ S/∂ T )P = T(∂ S/∂ T )V + VαT(∂ S/∂ V )T = CV[1 + αγT]
Isothermal & adiabatic derivatives (∂ Z[X,Y]/ ∂ X)W with X = V, Y = T & W = S:
(∂ Z/ ∂ V )S = (∂ Z/∂ V)T + (∂ Z/∂ T)P(∂ T /∂ V )S
e.g., (∂ P/ ∂ V )S = (∂ P/∂ V)T + (∂ P/∂ T)V(∂ T /∂ V )S hence KS = KT[1 + α γT]
Adiabatic temperature gradient
(∂ T/∂ P)S = (∂ V/∂ S)P = (∂ V/∂ T)P/ (∂ S/∂ T)P = γT/KS
(∂ lnT/∂ lnρ)S = − (∂ lnT/∂ lnV)S = γ
Stacey Appendix E, Poirier Ch. 1
Elasticity & interatomic forces
Turcotte & Schubert Figs.7.4 & 7.5
Volume per ion pair V = 2r3
Internal energy E = -C0/r +D0/rn = -C(V/V0)-1/3 + D (V/V0)-n/3
P = -dE/dV [strictly -(∂ F/∂ V )T with F = E - TS]
Bulk modulus KT = -V(∂ P/∂ V )T
Pressure derivative of KT’ = (∂ K T/∂ P)T
∴ K0V0 = (C/9)(n-1) ∝ d2E/dV2 ~ constant for isostructural
compounds & K’0 = (n+7)/3
Interplanar forces & lattice vibrations
Poirier, Fig. 3.1
Sequence of identical parallel planes of atoms
interacting with neighbours:
Fn = - K [un-un-1] + K [un+1-un] = M ∂ 2un/∂ t 2
Trial solution un = u0 sin(kxn - ωt) with xn= na
Dispersion relation for lattice vibrations
Poirier, Fig. 3.2
Condition for solution: ω = 2(K/M)1/2 |sin(ka/2)|
Phase speed for longitudinal vibrations c = fλ = ω/k
In general, group velocity u = dω/dk ≠ c  dispersion
Limit as k  0, u = c = a(K/M)1/2
Lattice vibrations: acoustic & optic branches
Redrawn afetr Kittel
Poirier Fig. 3.4
Redrawn after Kittel
p atoms per unit cell  3p modes of vibration
Acoustic modes: un and un+1 in-phase as k  0
Optic modes: un and un+1 out-of-phase as k  0
Quantisation & lattice vibrational energy
Periodic boundary conditions for crystal
of length L = Na require k = m(2π/L)
Poirier Fig. 3.4
with m = 0, ±1, ± 2,…., ± N/2)
Density of (k,ω) states in reciprocal (k)
space is g(k) = 1/dk = L/2π = Na/2π
such that ∫ g (k)dk = N
1st approx’n to crystal lattice: collection of independent harmonic
oscillators of frequency ν, energy quantum hν, & (equilibrium)
phonon occupancy quantum number
p(ν,T) = [exp(hν/kBT) - 1]-1 ~ kBT/ hν @ high T
(Bose-Einstein statistics)
 Evib(T) = Σ(i=1,N) hνi p(νi,T) = ∫ p(ν,T) hν g(ν) dν
The Debye model of lattice vibrations
Key assumptions: all modes acoustic & non-dispersive (ω = vDk)
& uniformly distributed within spherical BZ of radius kD to match
actual BZ volume  density of states g(ν) = 9mν2/νD3
Normalisation: (∫ ( 0,νD) g(ν)dν = 3m, m atoms per unit cell
Poirier
Fig. 3.9
Poirier, Fig. 3.5
ω∝k
g(ω) ∝ ω2
g(ω) for MgO
vs. Debye model
Debye model: thermal energy & specific heat
ED(T) = ∫ p (ν,T) hν gD(ν) dν
= (9nRT/x3) ∫ (0,x) ξ3 dξ / [exp(ξ) - 1] (per mol)
with n atoms per formula unit, x = θ/T
& Debye temperature θ = hνD/kB
Poirier, Fig. 3.6
CV(T) = (∂ Q/∂ T )V = (∂ E/∂ T )V
~ T3 as T  0, ~ 3R for T >> θD
~T3


θD
3R
Anharmonicity & thermal expansion
Harmonic
Poirier Fig. 3.12
Anharmonic
Poirier Fig. 3.13
Asymmetry of potential well results in time-averaged inter-atomic spacing
greater than static equilibrium value, reduced inter-planar stiffness
constants, K & reduced vibrational frequencies ν ~ (K/M)1/2
Quasi-harmonic approximation: νi = νi(V)
with γi = -dlnνi/dlnV = −dlnθD/dlnV = γD
Finite strain & cohesive energy
@ high pressure
KT = -V(∂ P/∂ V )T = K0, constant
integrates to P = - K0 ln (V/V0) but
incompressibility must increase with P:
e. g., K’0 = (∂ K T/∂ P)T0 = (n+7)/3
for rocksalt lattice
Eulerian finite strain εij = (1/2)(∂ui/∂Xj +∂uj/∂Xi)
- (1/2)∑k (∂uk/∂Xi)(∂uk/∂Xj) (Poirier, p. 60)
P  isotropic compressional finite strain ε: V0/V = ρ/ρ0 = (1-2ε)3/2
Taylor series expansion of Helmholz free energy F = E - TS:
F(V,T) = a0 + a1f + a2f2 + a3f3 + ….. with ai = ai(T) & f = -ε
3rd-order Eulerian finite strain isotherm P(V)
Now P = − (∂ F/∂ V )T = − (∂ F/∂ f )T/ (∂ V /∂ f )T
KT = − V(∂ P/∂ V )T = − V(∂ P/∂ f )T / (∂ V /∂ f )T
KT’ = (∂ K T/∂ P)T = (∂ K T/∂ V )T / (∂ P/∂ V )T
= (∂ K T/∂ f )T / [(∂ V /∂ f )T (∂ P/∂ V )T] = −(V/KT)(∂ K T/∂ f )T /(∂ V /∂ f )T etc.
∴ P = (1/3V0)(1 + 2f)5/2(2a2f + 3a3f2) (P = 0 for strain f = 0)
KT = (1/9V0)(1 + 2f)5/2[ 2a2 + (14a2+6a3)f + 27a3f2)
KT’ = (1/3)[24a2 + 6a3+ (98a2 + 96a3)f + 243a3f2)/
[2a2+ (14a2+6a3)f + 27a3f2]
Initial conditions: KT = KT0, KT’ = KT’0 
a2 = 9KT0V0/2, a3 = (9KT0V0/2)(KT’0 − 4)
Hence 3rd-order Eulerian (Birch-Murnaghan) isotherm
P = 3KT0 (1 + 2f)5/2 [f + (3/2)(KT’0 - 4) f2]
Mie-Grüneisen-Debye equation-of-state
Finite-strain P(V) principal isotherm + Debye model for E(T,V) with θ(V)
Construct F(V,T) = FBM(V,0) + FD(V,T)
FD= ED− TSD with S = −(∂ F/∂ T )V  FD− T (∂ F D/∂ T )V = ED
∴(∂[FD/T]/∂ T )V = −ED/T2 & FD = −T ∫(0,T) (ED/T2) dT = −T ∫(0,T) (ED/T2) dT
∫ by parts 
FD = 9nRT(θ/T)-3 ∫ (0,θ/T) ξ2 ln [1- exp(-ξ)] dξ
P(V,T) = −(∂ F/∂ V )T = −(∂ FBM/∂ V )T − (∂ FD/∂ V )T
∴ P(V,T) = P(V,0) + (γD/V) ED(V,T)
i.e., thermal pressure: δPTH(V,T) = (γD/V) δED(V,T)
c.f. δPTH = (γ/V) δETH from (∂ P/∂ T )V = αKT = (γ/V)CV
 γ = γD
The thermal pressure
P(V,T) = P(V,0) + PTH(V,T)
with PTH(V,T) = (γD/V) ED(V,T)
Mie-Grüneisen-Debye EoS: completeness
From F(V,T) = F(V,T0) + FD(V,T)
= a2f2 + a3f3 + 9nRT[θ(f)]/T)-3 ∫ (0,θ(f)/T) ξ2 ln [1- exp(-ξ)] dξ
[with θ(f) specified by γ0 = -(dlnθ/dlnV)0 & q0 = (dlnγ/dlnV)0]
we have it all:
P(V,T) = −(∂F/∂V)T, S(V,T) = −(∂F/∂T)V
Hence E = F + TS, H = E + PV, G = F + PV
CV = T(∂ S/∂ T )V
KT(V,T) = −V(∂P/∂V)T , (∂P/∂T)V = αKT(V,T) = (γ/V)CV
α, γ, KS = KT(1 + αγT), (∂ T /∂ P)S = γT/KS etc.
Extension to shear strain
(Stixrude & Lithgow-Bertelloni, Geophys. J. Int., 2005)
Modelling the seismic properties
of the Earth’s interior
P(V,T), KT(V,T) & hence KS(V,T), G(V,T)
from internally consistent finite-strain
expansions of both static and thermal parts of
the Helmholz free energy F
 ρ(z), VP(z), VS(z)
For each mineral need
F0, V0, KT0, KT’0, G0, G’0, θ0, γ0, q0, ηS0
constrained by experimental data and/or ab
initio quantum-mechanical calculations
Optimal finite-strain model constrained by
diverse experimental data for MgO
Kennett & Jackson, in prep.
Elastic behaviour: essential characteristics
Hookean elasticity:
(i) Linearity: stress σ ∝ strain ε
(ii) Instantaneity: strain appears(disappears) instantanously
when stress is applied (removed)
(iii) Recoverability: strain is fully recovered when stress is
removed
No dissipation: time-varying stress and strain in phase
No dispersion: wave speeds are frequency independent
Beyond elasticity: anelastic behaviour
Relax requirement of instantaneity  anelasticity
Stress-induced diffusion of defects or redistribution of fluid occurs
with characteristic timescale τ, typically thermally activated and
contributes well-defined, delayed component of strain
More strain for the same stress 
lower (relaxed) modulus = stress/strain
Strain energy dissipation
Delayed anelastic strain  phase lag between stress
σ(t) = σ0 sinωt & resulting strain ε(t) = ε0 sin(ωt − δ)
Energy dissipated per cycle
ΔE = ∫ (0,2π) σ dε = ωσ0ε0 ∫ (0,2π) sinωt cos(ωt − δ) dt
Using cos(ωt − δ) = cosωt cosδ + sinωt sinδ &
sin2ωt = 2sinωt cosωt and cos2ωt = 1 − 2sin2ωt obtain
ΔE = (σ0ε0/2) ∫ (0,2π) [sin2ωt cosδ + (1 − cos2ωt)sinδ] d(ωt) = πσ0ε0sinδ
Maximum energy stored
Emax = ∫ (0,π/2) σ dεin phase = σ0ε0cosδ ∫ (0,π/2) sinωt d(sinωt)
= (σ0ε0cosδ)/2
Quality factor Q
Q = 2πEmax/ΔE = 1/tanδ or Q-1 = tanδ
Seismic properties: laboratory methods
Complementary experimental
techniques probe a wide
range of frequencies
(c.f. mHz - Hz of teleseismic waves)
Elastic wavespeeds: ultrasonic methods
Mode-specific piezoelectric or
ferroelectric transducers
Generate & detect elastic
waves
Simple pulse transmission
(time-of-flight) &
interferometric methods
Ultrasonic wave-propagation methods:
representative results
MgSiO3 perovskite
Li & Zhang
PEPI (2005)
Silicate perovskite
analogue ScAlO3
Data fitted to F(f,T) model 
V0,K0,K’0,G0,G’0,θ0,γ0,q0,ηS0
Jackson & Kung, PEPI, 2008
Opto-acoustic methods: Brillouin scattering
Vi = Δω λ /2sin(θ/2) (i = P, S)
from Doppler shift Δω
Application in diamond-anvil apparatus
 G(P) to 100 GPa for MgSiO3 perovskite
 improved constraints on G'
Murakami et al., EPSL (2007)
0
Pressure, GPa
100
Forced-oscillation method for laboratory study
of anelasticity @ seismic frequencies
Implementation within
internally heated gas
apparatus (Jackson &
Paterson, PAGEOPH, 1993):
P = 200 MPa, T to 1300°C
oscillation periods 1-1000 s
shear strains < 10-5
Underlying principle
Specimen
& reference
assemblies
& T profile
Specimen
encapsulation
Seismic-frequency forced-oscillation data
for dry melt-free polycrystalline olivine
Jackson, Fitz Gerald, Faul & Tan, JGR, 2002
Modelling elasticity with interatomic potentials
Putnis (1992)
 K (quartz) = 39.7 GPa
c.f. 39.3 GPa (measured)
Newton et al., Phys. Chem. Minerals (1980)
Quantum chemistry: H atom to crystals
Schrödinger equation: (-h2/2m)∇2ψ + Vψ = Eψ
Hydrogen atom
Analytical solution: s, p, d, … orbitals
Energy levels consistent with observed line spectrum
Multi-electron atoms
Electron-electron interaction: no analytic solutions
Aufbau and Pauli exclusion principles: self-consistent field atomic orbitals (a.o.)
Small molecules
Molecular orbitals as linear combinations of a.o. - coefficients chosen to
minimise total energy; high e- density between atoms = chemical bonding
Crystalline solids
Zero K Density functional theory: ground-state energy a unique function of the
spatial distribution of electron density; High-T Quasi-harmonic approach –
lattice vibrational frequencies ν(V); anharmonic ab initio molecular dynamics.
1998 Nobel prize in chemistry to Kohn & Pople
Seismic properties: ab initio constraints
C11
C22
C33
C12
C13
C23
C44
C55
C66
Theory
493
546
470
142
146
160
212
186
149
Exp
482
537
485
144
147
146
204
186
147
MgSiO3 perovskite: singlecrystal elastic constants (GPa)
ΔE = (1/2)Cε2
Oganov et al. (2001)
Single-crystal elastic anisotropy: olivine
triangular clusters of edge-sharing MO6 octahedra
capped by SiO4 tetrahedra form
stiff columns  high VP along [100]
Webb & Jackson, Am Mineral. (1990)
Polyhedral structure
of olivine (Putnis, 1992)
Mainprice (2007)
Treatise on Geophysics
Bulk Earth composition & pressureinduced phase transformations
Phase proportions, %
50
0
100
200
.
px
olivine
SIO4
tetrahedron
Depth, km
400
Pyrolite model upper-mantle
composition = magma (basalt)
+residue (harzburgite) Green &
Ringwood (1960’s)
wadsleyite
garnet
600
ringwoodite
-------------SIO6
octahedron
Mg-perovskite
800
After Irifune, Nature (1994)
mw
Ca-pv
Crystal structures of high-pressure minerals
(Mg,Fe)SiO3 perovskite
(Mg,Fe)O magnesiowüstite
CaSiO3 perovskite
Gross Earth seismological models
Fowler Fig. 8.3
Fowler Fig. 8.1
Inversion of traveltime versus angular distance
& free-oscillation data for spherically averaged structure
Lateral variations of seismic wave speeds in
the Australasian upper mantle
Surface-wave tomographic model of Fishwick et al., EPSL, 2005
VS variations (%) at 200 km depth
Optimal geotherms and VS(z) profiles for
contrasting tectonic provinces
thermal boundary layer
of varying depth
-------------------------------------
common
mantle
adiabat
Lab-based model inclusive of anelastic relaxation (Faul & Jackson, EPSL, 2005)
Seismological models for the
transition zone of the Earth’s mantle
Vφ = (Ks/ρ)1/2
= [VP2-(4/3)VS2]1/2
Bulk sound speed, V! / km s-1
9.0
8.5
pyrolite model
NJPB
8.0
TNA
7.5
7.0
Nth Atl
6.5
SNA
6.0
200
300
400
500
600
700
800
Depth / km
Jackson & Rigden, Ringwood volume, CUP,1998
900
Composition, elasticity & temperature
of the lower mantle
Lower-mantle mineralogy for
pyrolite composition
G’ & dG/dT from ultrasonics: pyrolite & 1600 K
adiabat OK (Li & Zhang, PEPI, 2005)
Lower G’(Brillouin scattering): [SiO2] >
pyrolite Murakami et al., EPSL (2007)
Lower mantle: new developments
Fe2+:
3s23p63d64s0
d-orbital degeneracy removed by the octahedral
crystal field (Brown et al., Chemistry,1991)
Pressure-induced spin-pairing in Fe
Minerals Badro et al. , Science, 2003 & 2004
δρ/ρ ~1%
‘Post-perovskite’ CaIrO3 phase of
MgSiO3 @ P > 120 GPa, T ~ 2500 K
Murakami et al., Science, 2004
Oganov & Glass, J. Chem. Phys., 2006 etc.
Earth’s core: composition & temperature
Preferred hexagonal
close-packed structure
for pure Fe under innercore conditions
Inner core conditions
& properties
Fowler Fig. 8.12b
Core is significantly less
dense & somewhat more
compressible than pure Fe
Constraints on core composition &
temperature from ab initio calculations
Mol fraction in liquid Fe
Ab initio calculation of equations-of-state
V(P,T) & Gibb’s free energies G(P,T) for
both solid and liquid phases  melting
temperature Tm and element partitioning
Si
S
O
Can match densities of inner and outer
core with thermodynamic equilibrium
at inner-outer core boundary:
P = 330 GPa,Tm = 5600 K
Inner core: 8 mol% S/Si & 0.3% O
Outer core: 10 mol% S/Si & 8% O
Alfè et al., EPSL, 2002
Stable crystal structure for the inner core?
?
or
hexagonal close-packed
symmetric 12-coordination
body-centred cubic
split 8-6 coordination
with larger interstices
Thermal/compositional stabilisation of the bcc
phase? (Vocadlo et al., Nature, 2003)
Part I: Elasticity, equations-of-state &
interpretation of seismological models
Tensor stress & strain
Constitutive law & elastic waves
Elasticity & interatomic forces
Geophysical thermodynamics
Lattice vibrations & thermal energy
Anharmonicity & thermal expansion
Finite strain & cohesive energy @ high pressure
Mie-Grüneisen equation-of-state & thermal pressure
Anelasticity & seismic wave attenuation
Interpretation of seismological models
The End
Next: Part II Heat transport & geodynamics