ELASTICITY AND CONSTITUTION OF THE EARTH'S INTERIOR*

Transcription

1 JOURNAL OF GEOPHYSICAL RESEARCH VOLUME 57, NO. 2 JUNE, 1952 ELASTICITY AND CONSTITUTION OF THE EARTH'S INTERIOR* BY FRANCIS BIRCtt Harvard University, Cambridge, Massachusetts (Received January 18, 1952) ABSTRACT

2 The one-dimensional Earth according to Jeffreys (1939a) and Gutenberg (1948, 1951) 14 VELOCITY, KM/$EC 12 1 Wave Velocities Iooo 2ooo ooo 4ooo sooo DEPTH, KM Depth (km) 4 5 6

3 Quick aside: radial vs. equal area vs. equal volume 11.6 Equal Volume Equal Area 11.6 lower mantle (% area corresponds to % volume)

4 14 Vp 12 1 Wave Velocities 8 6 Vs Williamson-Adams V - 4/3 V -- K g/p - - (OP/Op dpf p g(r) dr/ h(r appropriate for: 4 hydrostatic homogeneous 2 adiabatic layer Depth (km)

5 Build-a-Planet Part II PRESSURE looo IO G CM/SEO 2 GRAVITY... '"'"" 5OO 5 DENSITY I DEPTH, IN 13 KM I F'IG. 2.--DENSITY, PRESSURE, AND ACCELERATION WITHIN THE EARTH (AFTER BULLEN)

6 V bulk (m/s) Outer Core Inner Core density (g/cm 3 ) 4 2 Lower Mantle Pressure (GPa)

7 Constant parameters K=45 GPa ρ=5 g/cc 1 1 V bulk (m/s) 8 6 Lower Mantle 8 6 density (g/cm 3 ) Pressure (GPa)

8 Linear Increase of density, constant K K=45 GPa 1 1 ρ=4.2 g/cc dρ/dp=.1 V bulk (m/s) 8 6 Lower Mantle 8 6 density (g/cm 3 ) Pressure (GPa)

9 K=-V dp/dv V~1/ρ ρ=4.2 g/cc Hooke s law Earth 1 1 dρ/dp= V bulk (m/s) 8 6 Lower Mantle 8 6 density (g/cm 3 ) K: Bulk Modulus (GPa) Pressure (GPa) Pressure (GPa)

10 K=-V dp/dv Definition of K= -VdP/dV V~1/ρ dρ/dp-polynomial 1 1 K: Bulk Modulus (GPa) fit (4th order) V bulk (m/s) Lower Mantle density (g/cm 3 ) Pressure (GPa) Pressure (GPa)

11 Introduction to Elasticity 1. Deep Earth context 2. Stress, strain and elastic tensors 3. Elasticity and symmetry/anisotropy 4. Thermodynamics of elasticity 5. Beyond Hooke: stress-strain equations of state on board 6. Introduction to lattice dynamics 7. Thermoelasticity 8. Experimental technique 9. Mysteries of Elasticity a. Composite behavior b. Softening: -phase transformations -with iron and volatiles c. Frequency dependence

12 Why mineral physicists like salt 2P 1.5 : 5 1. LITHIUM SODIUM P()TASSIUM RUBIDIUM CESIUM ?/?. 2. FIG. 3--COMPRESSION OF THE ALKALI METALS

13 Birch-Murnaghan Equation of state [ (V ) 7/3 ( V ) 5/3 ] P(V, T ) = 3 2 K T { V V [ 1 3 (V 4 (4 K T ) V ) 2/3 1]}

14 Mie-Gruneisen-Debye Equation of state P(V, T ) = P(V, T ) + P th (V, T ) P th (V, T ) = γ(v ){E th(θ, T ) E th (θ, T )} V ( ) 3 T θ/t t 3 E th (θ, T ) = 9nRT θ (e t 1) dt γ(v ) = γ ( V V ) q θ(v ) = θ exp { [γ γ(v )] q }

15 14 Adiabatic temp gradient 14 dt/dp = (ot/op)s = Ta/pC V bulk (m/s) Outer Core Birch s thermal Williamson-Adams Inner Core density (g/cm 3 ) Lower Mantle ( -.o /g) = - g2 ( - c /g) dr combination o y = Ks/pC and τ ~ deviation from adiabaticity 15 2 Pressure (GPa)

16 Elasticity of a mechanical composite I

17 Elasticity of a mechanical composite II Reuss Bounds: constant stress boundary condition Voigt Bounds: constant strain boundary condition Polycrystalline elasticity constraints = rotations

18 Introduction to Elasticity 1. Deep Earth context 2. Stress, strain and elastic tensors 3. Elasticity and symmetry/anisotropy 4. Thermodynamics of elasticity 5. Beyond Hooke: stress-strain equations of state (Introduction to lattice dynamics) (Thermoelasticity) 8. Experimental Techniques 9. Mysteries of Elasticity a. Composite behavior b. Softening: -phase transformations -with iron and volatiles c. Frequency dependence

19 Introduction to Elasticity 1. Deep Earth context 2. Stress, strain and elastic tensors 3. Elasticity and symmetry/anisotropy 4. Thermodynamics of elasticity 5. Beyond Hooke: stress-strain equations of state (Introduction to lattice dynamics) (Thermoelasticity) 8. Experimental techniques 9. Mysteries of Elasticity a. Composite behavior b. Softening: -phase transformations (Landau) -with iron/volatiles? c. Frequency dependence

20 Determining Elastic Properties 1. Directly measure density as a function of pressure, temperature 2. Measure wavespeeds directly (ultrasonic techniques) 3. Probe lattice dynamics (Brillouin spectroscopy and Nuclear Inelastic X-ray spectroscopy) 3b. Calculate lattice dynamics 4. Shockwave techniques: Determine internal energy directly at high P,T

21 High Pressure Diamond Anvil Cell

22 X-ray beam experimental procedure I 2θ 2θ Laser heating Unknown Phase stability Density Elastic properties Standard as a function of temperature

23 High Pressure before about 1 years ago, no in situ high P,T. All was quenched from high Temperature. High Temperature Characterize Sample Synchrotron added in situ It has added two major discoveries, and a host of minor embarassments Synchrotron X-ray diffraction

24 Results for FeO (Crowhurst et al., 28) Fig. 1. Acoustic wave velocities as a function of pressure for propagation in the (1) plane of singlecrystal (Mg.94,Fe.6 )O in an argon pressure-transmitting medium. Circles indicate present data acquired by impulsive stimulated scattering. Uncertainties are given by ± 2s, where s is the formal SE. Squares indicate data obtained by Jackson et al. via Brillouin scattering(19). Lines are calculated velocities based on linear extrapolations of the elastic moduli obtained by Jackson et al.(19). (Top) Body wave velocities. Solid circles and open circles are data for propagation along[11]and[1],respectively.(bottom) Velocities of the wave that propagates at the interface between the sample and the pressure-transmitting medium. The interfacial wave has no dependence on direction under these conditions. ncemag.org SCIENCE VOL JANUARY

25 Fig. 3. (A)Bulkmodulusversuspressure.Circlesindicatepresentdatacalculatedonthebasisofthec ij shown in Fig. 2. Squares are values calculated on the basis of the data of Jackson et al.(19). The solid line is based on a thermodynamic description of the HS-to-LS transition with parameters scaled from those that fit the compression data of Lin et al. (11). The dashed line is the average of separate equations of state for the HS and LS phases [see (B)]. Error bars indicate two SEs obtained from fits to the measured velocities. (B) Experimental data of Lin et al. (11) showingdensityof(mg.83,fe.17 )O versus pressure. The dashed lines represent equations-of-state (fourth-order Eulerian finite-strain) fits (22) to the HS (low pressure) and LS phases. The thick solid line is a fit to the data based on Eq. 1. The thin solid line is the first-principles theoretical prediction of Tsuchiya et al.(15). Results for FeO (Crowhurst et Fig. 2. Measured pressure dependence of the elastic moduli c ij and anisotropy factor A =(c 11 c 12 )/2 c 44.Circlesindicatethepresentdata,andsquaresaredataofJacksonet al.(19). Lines are linear fits to the latter data. Error bars indicate two SEs obtained from fits to the measured velocities. al., 28) A B

26 Elasticity of hydrous phases Structure and elasticity of serpentine at high-pressure Mainak Mookherjee a,, Lars Stixrude b

27 Single crystal elastic anisotropy (Inner core anisotropy?)

28 Waves propagating through anisotropic media Slide borrowed from Ronald D. Kriz Engineering Science and Mechanics Virginia Polytechnic Institute and State University Blacksburg, Virginia 2461