Picosecond acoustics for studying matter at extreme conditions Simon AYRINHAC, Michel GAUTHIER, Frédéric DECREMPS Institut de Minéralogie, de Physique des Matériaux et de Cosmochimie (Paris)
Son et lumière : from microphotonics to nanophononics Les Houches, France, February 16-27th, 2015
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This talk is based on the review article published in Ultrasonics 56 (2015) « Picosecond acoustics method for measuring the thermodynamical properties of solids and liquids at high pressure and high temperature » F. Decremps, M. Gauthier, S. Ayrinhac, L. Bove, L. Belliard, B. Perrin, M. Morand, G. Le Marchand, F. Bergame, J. Philippe Ultrasonics 56 129–140 (2015)
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Acknowledgements
Daniele Antonangeli Livia Bove Robert Pick
Laurent Belliard Bernard Perrin
Marc Morand Gilles Le Marchand Frédéric Bergame Julien Philippe
Livia Bove
Son et lumière : from microphotonics to nanophononics Les Houches, France, February 16-27th, 2015
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Outline 1. Why studying matter at extreme conditions ? 2. Combining diamond anvil cell with ps acoustics 3. Some achievements
Son et lumière : from microphotonics to nanophononics Les Houches, France, February 16-27th, 2015
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Pressure scale
Diamond synthesis
Abyssal depths
Industrial application
Atmosphere
10-4
10-3
10-2
0.1
1
Jupiter's interior Earth core
10
102
103
104
P (GPa)
Condensed matter under extreme conditions High pressures conditions are widely spread in universe Matter exists mainly at extreme conditions 5
Why increase P is interesting ?
- decreases the interatomic distances - explores the repulsive term of the interatomic potential - increases electronic orbitals overlapping - changes structure stability new phases • rich polymorphism of water • polyamorphism new properties • metal-isolant transition • supra enhanced new materials • superhard materials
- reach high P is essential in the study of the earth - planetary interiors are currently accessible only in laboratories 6
Physics of H-bonding Negative melting line : ice floats over water → open structure due to the hydrogen bonding
You are here
Why this rich polymorphism ? → the H bonding remains under HP and it is responsible of the structure distorsions
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Polyamorphism Polyamorphism is the existence of different amorphous phases • amorphous-amorphous transitions - water : HDA LDA O.Mishima et al, Nature 314 76 (1985)
• liquid-liquid transitions - elemental phosphorus - yttria alumina liquid (Y2O3-Al2O3) - in liquid Ga ?
Y.Katayama et al, Nature 403 170 (2000) S.Aasland and P.F.McMillan, Nature 369 633 (1994)
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Synthesis of superhard materials from graphit to diamond (8 GPa and 2000 K)
Vickers hardness (in GPa) :
sapphire
Industrial diamond
BC5
Natural diamond
25
70
78
96
T. Irifune et al, Nature 421, 599 (2003)
P=20 GPa, T=2400 °C
hardness : 130 GPa (hardness of diamond single crystal : 60-120 GPa ) 9
Superconductivity Tc increases with the application of high P • Ca @ 216 GPa, Tc=29 K
M.Sakata et al, PRB 83, 220512(R) (2011)
http://www.spring8.or.jp/en/news_publications/press_releas e/2011/110705/
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Electronic properties of « simple metals » Simple metals : alkali metals, only one valence electron - Na and Li are liquids at very high P
http://www.esrf.eu/UsersAndScience/Pub lications/Highlights/2011/dynamics/dyn2
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Electronic properties of « simple metals » Simple metals : alkali metals, only one valence electron - Na and Li are liquids at very high P
- Na : complex phase diagram at high P Sodium
http://www.esrf.eu/news/spotlight/spotlight69
http://www.esrf.eu/UsersAndScience/Pub lications/Highlights/2011/dynamics/dyn2
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Electronic properties of « simple metals » Simple metals : alkali metals, only one valence electron - Na and Li are liquids at very high P
- Na : complex phase diagram at high P Sodium
http://www.esrf.eu/news/spotlight/spotlight69
- Na : transparent at 150 Mbar and 300K
http://www.esrf.eu/UsersAndScience/Pub lications/Highlights/2011/dynamics/dyn2 http://www.sciencedaily.com/releases/2009/03/090312180838.htm
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Geophysics : preliminary earth reference model (PREM) Dziewonski and Anderson (1981) Physics of the Earth and Planetary interiors, 25, 297.
• inform us about the physical properties of the different layers • Wave velocities depend on pressure and temperature but also on the crystal structure and chemical composition of the constitutive minerals
→ Indoor seismology (laboratory measurements) is needed.
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Density of pure-Fe vs PREM
Density jump at ICB
~10% density difference for the liquid outer core ~6% density difference for the solid inner core
Light elements in the core (Si, S, O, C …), but how much ?
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Planetology and exoplanets Understand the structure, composition and formation of planets
Super-earths
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Diamonds in the sky ? M.Ross, Nature 292, 435 (1981) L.R.Benedetti et al, Science (1999) J.H.Eggert et al, Nature Physics 6, 40 (2010)
Diamond formation in giant planet interiors ?
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Outline 1. Why studying matter at extreme conditions ? 2. Combining diamond anvil cell with ps acoustics 3. Some achievements
Son et lumière: from microphotonics to nanophononics Les Houches, France, February 16-27th, 2015
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High pressure techniques
Piston-cylinder, Tmax ~2000 K
Paris-Edinburgh press, Tmax~2000 K
Diamond anvil cell, Tmax~4000 K
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The diamond anvil cell (DAC) He
Weir et al, J. Res. Natl. Bur. Stand. 63A 55 (1959) Jamieson et al, RSI 30 1016 (1959)
membrane filled with He
J.-C.Chervin et al, RSI 66 2595 (1995)
Pressure transmitting medium PTM (hydrostaticity)
advantages of diamond : • very hard • transparent
Hole drilled In the gasket
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Ruby as a P calibrant
ruby luminescence : P as a function of Raman line shift
m ruby spheres Al2O3:Cr3+
R1 ( P) 5 P(GPa) 380.8 1 1 694 . 2 Mao, Xu & Bell, Journal of Geophysical Research: Solid Earth 91 4673 (1986)
Chervin et al, High.Press.Res. 21 305 (2001)
disadvantages : • limited at high T due to the strong increase of linewidth R1 ( P, T ) R1 ( P) R1 (T ) • the ruby shift is P and T dependent 21
Measuring P and T in-situ Datchi et al, JAP 81 3333 (1997) Datchi et al, High Press. Res. 27 447 (2007)
Two pressure calibrants are used
S.V.Raju et al, JAP 110 23521 (2011)
T is measured • by ruby (at low T) • by an external power calibration (at high T)
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Physical acoustics Acoustics elasticity
ij cijklkl
kl sklmn mn
Elastic constants are directly related to the free energy derivative
cijkl
2G ij kl
Rich information, with high sensitivity to the variations of the interaction potential Equation of state without modelisation
1 V siihh V p
1 l l sijhhli l j l p
Knowledge of the equilibrium state (instabilities : Born criterium)
G 0
c 0 23
Elasticity and visco-elasticity at HP and HT : State of the art and objectives Classical ultrasonic (piezo MHz) in Multi Anvil • Need large volume sample (mm) • P max ≈ 20 GPa
Ultrasonic interferometry (piezo GHz) in DAC • No signal as soon as non-hydrostatic pressure induces cracks • P max ≈ 15 GPa
Brillouin scattering in DAC • Sample need to be transparent • P max ≈ 100 GPa
Inelastic X-ray scattering in DAC • Need to apply for beam time • P max ≈ 150 GPa 24
picosecond acoustics combined with DAC C.Thomsen et al, PRB 34 4129 (1986) B.Perrin et al, Physica B 263 571 (1999) Y.Sugawara et al, Phys. Rev. Lett. 88, 185504 (2002)
Ti:sapphire laser
100fs, 80 MHz, Tlaser = 12.55 ns
/2
PBS1
A.O.M.
pump
delay line (1m×4=13.5 ns)
probe
pol.
A
/2 pol. /2
sample PBS2
B PBS3 /4
/4
/4
/4
surface imaging Michelson interferometer + detection ref.mirror
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picosecond acoustics combined with DAC F. Decremps et al, PRL 100, 3550 (2008)
Ti:sapphire laser /2
PBS1
A.O.M.
pump probe
pol.
A
/2
DAC
pol. /2 PBS2
B PBS3 /4
/4
/4 ref.mirror
/4
measurements of P in-situ resistive heating furnace up to ~500°C
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Probe
Pump
e0 < 50 µm diamond
PTM
diamond sample
Probe
Pump
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Time resolved reflectivity measurement in DAC F. Decremps et al, PRL 96 35502 (2008)
Quasicrystal AlPdMn, P=9.8 GPa
T
t
Brillouin oscillations are due to the transparent PTM (gives sound velocity and attenuation at a frequency f)
d v t/2
f
1 2 n( ) v T 28
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Temporal method - relative variations of the signal with P - pump and probe may be perfectly collinear (difficult for an opaque material !) l-Hg 193°C
t0
gives v knowing thickness
v(P)
e0 t0 ( P)
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Surface imaging Y. Sugawara et al PRL 88, 185504 (2002)
in liquid Hg
duration : 13.2 ns flaser=80MHz Tlaser =12.55 ns (color scale adapted for each image) 100 m parallel and undeformed diamond culets homogeneous liquid sample
100 m 30
Movie analysis S.Ayrinhac et al J.Chem.Phys. 105 041906 (2014)
r/r
l-Hg, p=1 GPa, T=30°C 100
center (µm) DistanceRadius from the (µm)
integrated intensity profile @ t=5.2 ns
(a.u.)
90
4
80 70
2.8
60
1.5
50 40
0.25
30 20
-1
10 0 0
5
10
15
20
Time (ns)
Tlaser 12.55 ns
25
30
35
Time (ns)
This picture shows all the acoustic informations provided by this experiment How to explain the evolution of the ripples radius ?
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The origin of circles at the surface
diam.
liq.
diam.
e(t) R(t) pump
e0
probe
acoustic diffraction • spot : ~ 3 m • ~ 0.6 GHz ac~ 1 m
R 2 (t ) e 2 (t ) e02
e0 v(t0 pTlaser ) e(t ) v(t pTlaser ) v : sound velocity t0: time when the perturbation reach the surface R(t=t0)=0 : pump-probe coincidence delay (fixed) p : integer (fixed)
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r/r
l-Hg, P=1 GPa, T=30°C 100
(a.u.)
L T
90
4
Radius (µm)
80 70
2.8
fit R(t) : free parameters v and t0 thickness e0
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1.5
50 40
0.25
30
- reflexions (not fit) 1 and 2 round-trips
20
-1
10 0 0
5
t0
10
15
20
25
30
35
Time (ns)
- for bulk waves the R(t) function is non-linear - for surface waves, R(t) is linear surface skimming bulk waves (SSBW) L & T
R(t ) v e 2 e0 R(t ) v(t t0 )
2
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surface skimming bulk waves (SSBW) also called CRL (critically refracted longitudinal wave) - generated at a critical angle c
Refraction law for acoustics, where n 1/v
diamond
sin 1 sin 2 VL1 VL 2
liquid
0
c
sin cL
VL1 VL 2
-2 critical angles : for vL and vT SSBWs : L and T they propagate in the diamond (vL=18 km/s, and vT=12 km/s)
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Taking account diffraction S.Ayrinhac et al, to be published (2015)
When e0 decreases, the wavefront is not strictly spherical
light beam
sound beam
Rayleigh length zR Gaussian acoustic transducer
The radius of curvature vary with z
zR 2 C ( z ) z 1 z
2 2 e z 2 2 2 R R (e e0 ) 1 e e0 e
corrective term zR as a free parameter when fitting R(t)
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Comparison imagery / temporal methods
Imagery method
Temporal method
gives velocity v and thickness e0
t0 is accurately determined 1 scan ~ 10 s (very fast)
1 scan ~ 4h (time consuming)
e0 is needed to obtain v
2 complementary methods to measure one isotherm
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Picosecond acoustics at HP : summary • the temporal domain of study of ps acoustics (~10 ns) is well adapted to the volume of the sample in DAC (vsound ≈ 5000 m. s-1 = 5 µm.ps-1) • study of opaque, transparent, single- or poly-crystalline solids, liquids, nanomaterials, etc transparents samples need a metallic coating • viscoelastic, thermal, thermodynamical properties • frequential domain complementary to other techniques [1GHz–1THz] • non-contact technique • possibility to reach extreme P and T > 300 GPa (DAC) > 1000 K (laser heating)
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Outline 1. Why studying matter at extreme conditions ? 2. Combining diamond anvil cell with ps acoustics 3. Some achievements
Son et lumière: from microphotonics to nanophononics Les Houches, France, February 16-27th, 2015
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Phase transitions
sound velocity (km/s)
sound velocity is a good probe to detect phase transitions Tin (Sn)
Gallium (Ga)
L.Xu et al, JAP 115 164 903 (2014)
S.Ayrinhac et al, to be published (2015)
(e0 supposed constant)
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another method to detect phase transitions is the modification of the pattern (example in Ga)
elliptic pattern (polycristalline solid)
circular pattern (liquid)
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Melting line New determination of the melting line of liquid Ga
S.Ayrinhac et al (2015)
Simon-Glatzel equation
P Pt T Tt 1 a
1/ c
Parameters (fit) : a= 24 GPa c=0.51 Triple point Ga-II/III/liq. Pt=3 GPa Tt=316 K
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Adiabatic sound velocities
in liquid Ga, up to 8 Gpa and 540 K
no abrupt or gradual transition in this range of P and T why the propagation of sound is adiabatic rather than isothermal ?
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Why adiabatic ? N.H.Fletcher, AJP, 42 487 (1974) A.B.Coppens et al, JASA 41 1443 (1966)
In a majority of cases, the propagation of sound can be considered adiabatic (isentropic dS=0)
The heat do not flow from high T regions to low T regions if fwave < flim f lim
v 2 0CV 2
In liquid metals flim ~ 0.1 THz 43
Density from acoustics L.A.Davis R.B.Gordon, J.Chem. Phys 46 2650 (1967)
From adiabatic sound velocity vS to isothermal equation of state (P,T)
T C P S CV
where is the adiabatic coefficient
T P2 T S CP
with S
1 v 2S
and T
1 P2 2 T CP P T v S
1 P T
P
P P2 1 ( P, T ) ( P0 , T ) 2 dP T dP v Cp P 0 S P 0 P
a smoothing function vS(P) is necessary
iterative procedure
1 T P
T Cp 2 P T
1 2 1 T P
This method gives density with an excellent accuracy E.Wilhelm, J. Solution Chem. 39 1777 (2010) 44
Density from acoustics From adiabatic sound velocity to isothermal equation of state
L.A.Davis R.B.Gordon, J.Chem. Phys 46 2650 (1967) Daridon et al, Int. J. Thermophys. 19 145 (1998)
input v S ( P, T )
(smoothed and interpolated)
( P0 , T ) P ( P0 , T ) C P ( P0 , T )
P0
P
T=1 K
T
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Density from acoustics From adiabatic sound velocity to isothermal equation of state input v S ( P, T )
(smoothed and interpolated)
( P0 , T ) P ( P0 , T ) C P ( P0 , T )
L.A.Davis R.B.Gordon, J.Chem. Phys 46 2650 (1967) Daridon et al, Int. J. Thermophys. 19 145 (1998)
T P2 1. T S CP 1 2.
T P
3. Cp T 2 P T
1 2 1 T P
P=0.01 GPa P0
...
P1
P2
P
T=1 K
T
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Density from acoustics From adiabatic sound velocity to isothermal equation of state input v S ( P, T )
(smoothed and interpolated)
( P0 , T ) P ( P0 , T ) C P ( P0 , T )
output
T P2 1. T S CP
( P, T ) P ( P, T )
1 2.
T P
3. Cp T 2 P T
Davis & Gordon JCP 46 2650 (1967) Daridon et al, International journal of thermophysics 19 145 (1998)
C P ( P, T )
1 2 1 T P
P=0.01 GPa P0
...
P1
P2
P
T=1 K
T
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Density from acoustics From adiabatic sound velocity to isothermal equation of state input v S ( P, T )
(smoothed and interpolated)
( P0 , T ) P ( P0 , T ) C P ( P0 , T )
output
T P2 1. T S CP
( P, T ) P ( P, T )
1 2.
T P
3. Cp T 2 P T
C P ( P, T )
1 2 1 T P
P=0.01 GPa P0
...
P1
P2
Davis & Gordon JCP 46 2650 (1967) Daridon et al, International journal of thermophysics 19 145 (1998)
P
- All the thermodynamical quantities are obtainable (bulk modulus, etc)
T=1 K
T
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Density of liquid Ga large discrepancies between data obtained by various techniques
20°C 50°C 100°C 150°C 200°C 250°C 300°C
Yu et al, JAP 111 112269 (2012) Lyapin et al, JETP 107 818 (2008) Li et al, APL 105 041906 (2014) Köster et al, BBPC 74 43 (1970) Tamura et al, JNCS 156 650 (1993)
our data from sound velocity are in excellent agreement with Köster Who have used a more direct technique to determine 49
Equation of state (EOS)
EOS is an analytical formula of (P,T) • based on statistical physics • based on intermolecular potentials • empirical (reasonable assumptions)
EOS permits • extrapolation to extreme P and T conditions not experimentally accessible (without the presence of a phase transition) • interpolation (to obtain accurately the derived quantities) The best known example : ideal gas law
PV nRT
However, in liquids (and solids), it is more complicated…
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Murnaghan EOS F.D.Murnaghan, Proc. Symp. Appl. Math. 1 167 (1949)
For a small increase in P, bulk modulus B(P) is constant
BT B0
and
P BT V V T
V V0e P / B0
However, at high P, this relation becomes invalid Because B increases with pressure due to the variation of elastic properties with P
BT B0 B P B P ... with ' 0
'' 0
2
2B B '' B B0 2 P P 0 P P 0 ' 0
At first order : Murnaghan equation
P BT B0 B P V V T ' 0
B V V0 1 P B0 ' 0
1/ B0'
valid up to ~ 10 GPa
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Many other EOS J.R.McDonald, Rev. Mod. Phys. 41 316 (1969)
1/ 3
V 1 X (1 X ) 3 ' Vinet P 3B0 e ; X ; ( B0 1) 2 X 2 come from a particular potential V0 P. Vinet et al, J. Phys. C: Solid State Phys. 19 L467 (1986)
Birch-Murnaghan F.Birch, J. Geophys.Res., 57 227 (1952)
7 5 2 3 3B0 V 0 3 V 0 3 3 V 0 ' P 1 (4 B0 ) 1 V 2 V V 4
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Which is the better EOS ? J.R.McDonald and D.R.Powell, J.Res.Nat.Bur.Stand 75 441 (1971)
2 methods are possible : • usually, ρ(P,T) is fitted by different EOS B0 and B0' are unknown the better fit determines the better values • Comparison between empirical EOS (without fit) knowing B0, B0' and ρ(P,T)
calc exp (%) exp If this quantity is zero, the EOS match perfectly ρ(P,T)
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Equations of state of l-Hg S.Ayrinhac et al, J.Chem.Phys. 105 041906 (2014)
Comparison between empirical EOS
l-Hg, T=240°C B0=18.5 GPa B0'=10.7
Birch Murnaghan EOS can be used in the case of liquid mercury instead of more complicated formulas (soft spheres, etc). 54
Elastic constants (stiffness tensor) in Si monocrystal J.P.Wolfe, Physics Today, pp.44-50, dec. 1980 J.P.Wolfe, Physics Today, pp.34-40, sept. 1995 S.Guilbaud, B.Audoin JASA 105 2226 (1999) F.Decremps et al PRB 82 104119 (2010)
Hooke's law (for a common spring)
F kl Generalized Hooke's law
ij Cijklkl
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Elastic constants (stiffness tensor) in Si monocrystal J.P.Wolfe, Physics Today, pp.44-50, dec. 1980 J.P.Wolfe, Physics Today, pp.34-40, sept. 1995 S.Guilbaud, B.Audoin JASA 105 2226 (1999) F.Decremps et al PRB 82 104119 (2010)
Hooke's law (for a common spring)
F kl Generalized Hooke's law
ij Cijklkl
The elastic information on an anisotropic solid is given by his stiffness tensor C (expressed as a matrix) with 21 components
For Si (and cubic crystals) : 3 independent constants C11, C12 and C44 0 0 C11 C12 C12 0 C C C 0 0 0 12 11 12 C C12 C11 0 0 0 12 C 0 0 C44 0 0 0 0 0 0 0 C 0 44 0 0 0 0 0 C 44 56
complex patterns appearing on the surface 3 polarizations waves
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Experiment
Simulation
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Equation of state : bulk modulus B
C11 2C12 B 3
P B V V 59
Lattice stability
Stability criterium (Born criterium) critical pressure which the ideal lattice become unstable, for example :
1 C11 C12 0 2 F.Mouhat, F.-X.Coudert PRB 90 224104 (2014)
Violation of the stability criterium around 120 GPa
→ precursor to the amorphization or melting 60
Sound velocity in polycristalline Fe F.Decremps et al Geophys.Lett. 41 1459 (2014)
Fe is the most abundant element in the Earth core
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up to 152 GPa (one order of magnitude higher than previously published ultrasonic data)
F.Decremps et al Geophys.Lett. 41 1459 (2014)
PREM
bcp-hcp structural transition
density is obtained with an EOS clearly out of the PREM model, confirming that light elements should be present in the Earth's core
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Thank you for your attention
Son et lumière: from microphotonics to nanophononics Les Houches, France, February 16-27th, 2015
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