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Physics of the Earth and Planetary Interiors 174 (2009) 24–32

Contents lists available at ScienceDirect

Physics of the Earth and Planetary Interiors

journa l homepage: www.e lsev ier .com/ locate /pepi

hermal conductivity of lower-mantle minerals

lexander F. Goncharova,∗, Pierre Becka, Viktor V. Struzhkina,enjamin D. Haugena,b, Steven D. Jacobsenc

Geophysical Laboratory, Carnegie Institution of Washington, 5251 Broad Branch Road N.W., Washington, DC 20015-1305, USADepartment of Geological Sciences, University of Colorado, Boulder, CO 80309, USADepartment of Earth and Planetary Sciences, Northwestern University, Evanston, IL 60208, USA

r t i c l e i n f o

rticle history:eceived 17 October 2007eceived in revised form 15 April 2008ccepted 23 July 2008

eywords:arth’s mantle dynamicshermal conductivityower mantleron containing mineralslectronic spin transitionigh pressureptical propertiesiamond anvil cell

a b s t r a c t

Geodynamic models of heat transport and the thermal evolution of Earth’s interior require knowledge ofthermal conductivity for high-pressure phases at relevant temperatures and pressures. Here we presentnew data on radiative and lattice heat transfer in mantle materials determined from optical spectroscopyand time-resolved optical radiometry. The pressure dependence of optical absorption in ferropericlase(Mg,Fe)O, and silicate perovskite (Mg,Fe)SiO3, has been determined in the IR through UV regions up to133 GPa. Whereas (Mg,Fe)O exhibits a strong pressure dependence of absorption and spectral changesassociated with the high-spin (HS) to low-spin (LS) transition of Fe2+ [Goncharov, A.F., Struzhkin, V.V.,Jacobsen, S.D. 2006. Reduced radiative conductivity of low-spin (Mg,Fe)O in the lower mantle. Science 312,1205–1208], the pressure dependence of optical absorption in (Mg,Fe)SiO3 is relatively weak. We observea moderate increase in absorption with pressure for (Mg,Fe)SiO3 in the visible and infrared spectral rangedue to a red-shift of absorption in ultraviolet, however the crystal-field transitions of Fe2+ become weakerwith pressure and disappear above 50 GPa as a result of the HS–LS transition in (Mg,Fe)SiO3. Intervalencecharge-transfer transitions in silicate perovskite shift to higher energies with pressure. The tempera-ture dependence of the optical absorption of (Mg,Fe)O measured up to 65 GPa and 800 K is moderatebelow 30 GPa and weak above 30 GPa. Thus, the temperature correction of the radiative conductivity isinsignificant. The estimated total pressure-dependent radiative conductivity (in approximation of a largegrain size) is lower than expected from the pressure extrapolation of the ambient and low-pressure data

[Hofmeister, A.M., 1999. Mantle values of thermal conductivity and the geotherm from phonon lifetimes.Science 283, 1699–1706; Hofmeister, A.M., 2005. Dependence of diffusive radiative transfer on grain-size,temperature, and Fe-content: implications for mantle processes. J. Geodyn. 40, 51–72]. A new methodhas been developed to measure thermal diffusivity of mantle materials at high P–T using time-resolvedradiometry combined with a pulsed-IR source. Here, the technique is tested on MgO to 32 GPa and usedto obtain a functional pressure dependence of thermal diffusivity and calculated thermal conductivity of

aotami

the lower mantle.

. Introduction

The lower mantle is thought to consist mainly of silicate per-vskite (Mg,Fe)SiO3 (PV), and ferropericlase (Mg,Fe)O (FP), whichre electrically insulating materials that can transport heat by bothattice vibrations (phonons) and radiation (photons). However, val-

es for lattice (klatt) and radiative conductivity (krad) for theseaterials have not been determined at lower-mantle pressures

nd temperatures. The importance of temperature and pressure-ependent thermal conductivity (Hofmeister, 1999; Hofmeister

∗ Corresponding author. Tel.: +1 202 478 8947.E-mail address: goncharov@gl.ciw.edu (A.F. Goncharov).

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031-9201/$ – see front matter © 2008 Elsevier B.V. All rights reserved.oi:10.1016/j.pepi.2008.07.033

© 2008 Elsevier B.V. All rights reserved.

nd Yuen, 2007) has been recognized in recent geodynamic modelsf mantle convection, which incorporate variable thermal conduc-ivity (Dubuffet et al., 2002) and variable viscosity (Yanagawa etl., 2005; van den Berg et al., 2005). Likewise, optical properties ofinerals under relevant conditions are also necessary for model-

ng the heat transport (Hofmeister, 1999, 2005, 2007) because theyirectly affect radiative heat transfer.

Electronic spin-pairing transitions of iron at lower-mantle pres-ures (60–120 GPa) in ferropericlase (Badro et al., 2003; Lin et al.,

005; Speziale et al., 2005; Persson et al., 2006; Tsuchiya et al.,006) and silicate perovskite (Badro et al., 2004; Li et al., 2004,005; Jackson et al., 2005) may also affect the crystal-field dipolective transitions (Goncharov et al., 2006; Keppler et al., 2007).hanges in the optical properties of iron-bearing phases associated

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A.F. Goncharov et al. / Physics of the Ear

ith spin transitions may therefore influence the radiative part ofhermal conductivity (Burns, 1993; Sherman, 1991). The need forxperimental constraints on phonon and photon transport proper-ies of lower-mantle phases undergoing electronic transitions hasrompted the present development of new methods for studyingigh-pressure phases at lower-mantle conditions.

Previous measurements of thermal diffusivity (Dlatt = klatt/�Cp,

here � is density and Cp is heat capacity) in mantle mineralst high pressures have been carried out in the 5–10 GPa pressureange (i.e. 150–300 km depth) at room temperature (Ross et al.,984; Abramson et al., 1999; Chai et al., 1996; Gibert et al., 2003),ith a few optical absorption studies of the direct radiative compo-ent of conductivity in olivine or silicate spinels at 20–30 GPa (Maond Bell, 1972; Keppler and Smyth, 2005). At high temperatures,hermal diffusivity in olivine has been measured up to 1500 ◦C atmbient pressure (Pertermann and Hofmeister, 2006). Thus, geo-ynamic modeling has relied heavily on extrapolation of lower P–Tata to deep-Earth conditions using the existing theoretical mod-ls, or by indirect measurements (Ross et al., 1984; Hofmeister,999, 2007; Pertermann and Hofmeister, 2006). Simple theoriesescribing the P–T dependence of thermal diffusivity (see, e.g., Rosst al., 1984; Hofmeister, 1999) are often satisfactory, but predic-ions related to changes in crystal structure (Roufosse and Jeanloz,983), spin transition, or disorder require further development.irst principle calculations (e.g., Cohen, 1998) are extremely rare.irect measurements of the thermophysical properties on synthe-

ized deep-mantle materials will offer better confidence for futureodels and guide theoretical development.Recently, optical absorption spectra of ferropericlase have been

eported to about 80 GPa at room temperature (Goncharov et al.,006; Keppler et al., 2007) showing a strong dependence of absorp-ivity with iron concentration, pressure, and possibly spin state. Inerropericlase, interpretation of optical spectra are complicated byhe defect-structure of (Mg,Fe)O, resulting from variable amountsf Fe3+ in samples annealed at different conditions of oxygen fugac-ty and pressure (Keppler et al., 2007). Absorption spectra fore-bearing silicate perovskite have been reported (Keppler et al.,994; Shen et al., 1994) but only at ambient P–T conditions.

Here we present an overview of thermal transport in lower-antle phases. We describe a new optical spectrometer and

ime-resolved radiometry systems operating under conditions ofigh pressure and temperature using the diamond anvil cell (DAC).rom measurements of optical absorption spectra in silicate per-vskite and ferropericlase to 133 GPa at room temperature, andptical absorption spectra in ferropericlase to 65 GPa up to 800 K,e infer the optical conductivity of the lower mantle. Finally, new

esults on lattice conductivity of MgO (periclase) are used to con-train the total heat conductivity in the lower mantle.

. Experimental techniques

.1. Thermal conductivity

Measurements of the thermal conductivity usually involveetermination of the temperature gradient across the sample which

s subjected to a steady heat flow (Bridgman, 1922; Lussanna,918; Andersson and Bäckström, 1976). These techniques are veryime consuming and delicate, and moreover they require a ther-

ocouple to be placed in the high-pressure zone. More recently

pulsed plane-wave technique (Pierrus and Sigulas, 1985) has

een designed, which improves the accuracy. Nevertheless, therere strong limitations because of geometrical constraints and theresence of materials with P–T-dependent thermophysical prop-rties, which reduce the accuracy of the measurements. These

adrpo

Planetary Interiors 174 (2009) 24–32 25

onstraints become even more severe when working at higherressures, because of increased spatial limitations and problemsith temperature determination (e.g., thermocouple calibration).oreover, contact resistance between thermocouple and sample

auses large systematic errors of ca. 25% (Hofmeister, 2007). Cur-ently, only measurements of the thermophysical properties to aew GPa have been reported (Ross et al., 1984; Pierrus and Sigulas,985; Osako et al., 2004) using these conventional methods. Sub-tantial efforts were made by the planetary science community todapt these techniques to multi-anvil apparatus, and pressures upo 20 GPa were reached (see for instance Osako et al., 2004; Katsura,993, 1995,1997; Xu et al., 2004).

Static experiments at substantially higher pressure require usef DACs. Implementation of an all-optical technique would benefithis kind of operation in the future to avoid cumbersome electricalonnections leading to the sample cavity. To our knowledge, pre-ious studies in the gemstone anvil cell (Pangilinan et al., 2000)ere limited to 2 GPa and near room temperature where tempera-

ure changes due to pulsed-laser heating was measured with a rubyensor, positioned away from the sample.

Picosecond transient grating spectroscopy or impulsive stimu-ated light scattering (ISLS) has been used successfully (Abramsont al., 1999; Chai et al., 1996) to determine the thermal diffusivityn 10 GPa pressure range using the DAC technology. This power-ul technique involves a use of three precisely synchronized laserulses, which should be aligned spatially and temporally. Thisechnique imposes sample dimension limitations related to thetanding thermal grating at the sample, so pressures on the orderf 40 GPa are considered the practical limit. Also, the thermophysi-al properties of the pressure transmitting medium (if present) willffect the measurements. Measurements under simultaneous highressure at high temperature are rare (Chai et al., 1996).

Measurements of the thermophysical properties at ambientressure are more straightforward (Pertermann and Hofmeister,006). A widely used technique includes a laser-flash arrangementParker et al., 1961) using a thin, disk-shaped sample heated withlaser pulse from one side and temperature monitoring from thepposite side with a sensitive infrared (IR) detector. This techniquellows determining thermal diffusivity, thermal conductivity andeat capacity at variable temperatures, but high-pressure studiesave not been reported yet.

Our method of thermal conductivity measurements under con-itions of high pressure and temperature combines a pulsed-lasereating technique in the DAC with a time-resolved radiometricemperature determination (Beck et al., 2007). A portion of the laseradiation is absorbed by a thin metallic coupler sandwiched by theample material (Fig. 1). To provide such sample arrangement, twodentical polished discs of material and the coupler between themre positioned into a cavity of matching dimensions formed in ahenium gasket and then loaded in a DAC to the desired pressure.

The all-optical system (Fig. 2) uses an IR (1.06 �m wavelength)5 ns pulsed laser with 1–20 kHz repetition rate for periodic heatingf the sample in the DAC. An optically gated (5 ns gate width) multi-hannel detector is used to monitor the instantaneous temperatures it decreases from the peak value. The laser is focused into theample by an aberration-corrected long-focal length lens (120 mm)o a Gaussian spot of approximately 30 �m diameter. The incan-escent light is collected by an all-reflective microscope and relayptics (two off-axis paraboloids) into the entrance slit of a single-rating imaging spectrograph. The delay time between the laser

rrival and probe window is controlled electronically by using aelay generator triggering the detector, and the measurements cor-espond to averaging over approximately 20,000 cycles. The laserower and the repetition rate is chosen to allow the temperaturef the metallic coupling material to return to its initial tempera-

26 A.F. Goncharov et al. / Physics of the Earth and Planetary Interiors 174 (2009) 24–32

Ft

ttlsacdiutapwdThd

m

Fetr

aiitdbhd1

∇

wiat

ig. 1. Sample arrangement in the DAC for in situ high P–T measurements of thehermal diffusivity.

ure (or sufficiently low steady temperature) after each shot. Theemperature profile across the heating spot is determined by ana-yzing the variation of spectra along the vertical (nondispersive)pectrograph direction, where the temperature distribution is flatcross a ∼20-�m area. In the present study, only the temperatureorresponding to hottest part of the coupler is analyzed. Varyingetector delay enables one to study the foil surface temperature

n the time domain. An example of the temperature measurementsing temporally resolved spectra of incandescent radiation fromhe laser-heated coupler is shown in Fig. 3. The accuracy of temper-ture measurements was tested by measuring platinum meltingoint and found to be within 50 K (a current was applied to a Ptire located on the focal point of the microscope, and melting wasiagnosed as a discontinuity in the current/temperature curve).he accuracy of the temperature measurements (±100 K) in the

igh-pressure experiments (see below) was estimated as a largesteviation from the averaged set over three to four consecutive runs.

Our calculations show that if the coupler is sufficiently thin (sub-icron), then it becomes thermally equilibrated within a few ns

p

�

Fig. 2. A schematic diagram of the optical setup for in situ thermal diffusivity

ig. 3. Examples of temperatures measurements by instantaneous optical radiom-try. The detector gate was 10 ns. Red line: experimental data; black curve: Plank fito the data. (For interpretation of the references to color in this figure legend, theeader is referred to the web version of the article.)

fter the laser pulse arrival, and its temperature variation with times determined by the thermophysical properties of the surround-ng material. Because the dimension of the hot spot is much largerhan the coupler thickness, the heat transfer is essentially one-imensional, as has been verified by our model calculations (seeelow). Generally, the temperature distribution within the bothalves of the sample will be changing with time, t in a mannerescribed by the heat conduction equation (e.g., Carslaw and Jaeger,959):

(k(T, P))∇T = �Cp(T, P)∂T

∂t, (1)

here �, Cp and k are temperature independent density, heat capac-ty and thermal conductivity, respectively. In the simplest case ofn infinite sample, the temperature change at the distance x fromhe coupler can be determined as a function of time after the laser

ulse arrival as follows:

T(x, t) =(

A

2k

)(D

�

)1/2t−1/2 exp

(−x2

4Dt

),

/optical properties/Raman/fluorescence spectroscopy measurements.

A.F. Goncharov et al. / Physics of the Earth and

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ig. 4. Time-resolved surface temperature measurements of the iridium foil embed-ed in NaCl medium following the arrival of the laser heating pulse. The small blackots and gray shaded area of the lower panel show the IR pulse temporal shape.

here D = k/�Cp is the thermal diffusivity and A is the laser powerensity. By measuring T as a function of time at the coupler–sample

nterface (x = 0) and at a given x = x0, one can determine both Dnd k (if � and Cp can be estimated separately). In case of exper-ments in the DAC, numerical modeling of the heat transport of theaser-heated sample is needed to take into account the real exper-mental arrangement (see Section 3.1). Our data on pressure- andemperature-dependent thermal diffusivity of NaCl (see time seriesn Fig. 4) and KCl (not shown) which have been fitted accordingly,ielded the results comparable to those published before (Beck etl., 2007).

.2. Optical spectroscopy

The optical properties of minerals, determined by light absorp-ion experiments, provide information on the radiative componentf thermal conductivity (�rad). The radiative part of the thermalonductivity, �rad can be calculated as follows (Hofmeister, 2005):

rad(T) = 43

∫ ∞

0

�

[1 + ln 10 × � × ˛(�, T)]1�2

∂[n(�, T)2(1 − exp(−ln

here I(�, T) is the Planck function, n(�, T) is the refraction index, �s the grain size (scattering length), ˛ is the measured absorptionoefficient ˛(, T) (cm−1) = ln(10)/l, where l is the path length inhe material corresponding to the transmitted signal attenuationn 10 times. Assuming the grain size is much larger than the typi-al absorption length (∼1/˛), the radiative contribution to thermalonductivity in the lower mantle can be obtained using experimen-al absorption spectra as will be shown below. Here we describe ourptical micro-spectroscopy system designed for the DAC operation.

The infrared microscope described in the previous section haseen also used for near-IR, visible and ultraviolet (UV) measure-ents (2000–40,000 cm−1) of the optical properties of mineral

n the DAC (Fig. 2). A custom made (Syassen and Sonnenschein,982) all-reflective infrared microscope uses a double-confocal-perture configuration (one field aperture for the illumination andne for the transmitted light) in order to reduce the amount ofstray light, which may otherwise bypass the sample by scatter-

ng on optical imperfections. This double-aperture configuration

nsures that the optical absorption spectra of highly absorbingamples can be recorded. No refractive optics are used to avoidV absorption and chromatic aberrations. The light sources forV–vis include a fiber-coupled deuterium and halogen lamp andflash Xe lamp. The spectra are analyzed using a UV–vis grating

croai

Planetary Interiors 174 (2009) 24–32 27

� × ˛(�, T)))I(�, T)]d� (2)

onochromator-spectrograph with CCD detector in UV–vis andy FT-IR spectrometer (Nicolet 750 Magna) equipped by a halo-en lamp and a CaF2 beamsplitter for near IR spectral range. Noample adjustment is necessary when changing the spectral range.he same setup can be used for the pulsed heating/gated tempera-ure measurements (see above) and also (pulsed) ruby fluorescenceoptical pressure sensor) and (pulsed) Raman spectra, which makest very convenient for high-pressure high-temperature applica-ions.

High-pressure measurements were carried out in a symmet-ic DAC with various pressure media (Ar, Ne, silicon oil) to 87 GPar without pressure medium (to 133 GPa). The diameter of theight spot at the sample was about 20 �m in UV–vis and 30 �mn near IR. At each pressure, a spectrum through the sample, I(�)nd a spectrum through the pressure medium, I0(�) (or thoughhe reference cavity in case of the experiment to 133 GPa) werebtained. Absorbance spectra were calculated according to A(�) =og I0((I0�)/I(�)), thus neglecting the sample reflection, which ismall for the absorption coefficient range studied. Indeed the modelalculations, which use Lorentz oscillators to approximate thebserved optical absorption, show that reflectivity does not exceed.5% throughout the whole spectral range studied. The reflectivityorrection is smaller than other experimental uncertainties relatedo the sample thickness determination and reproducibility of mea-urements. The absorption coefficient ˛, which is necessary foralculations of the radiative conductivity krad(T) (1) is determineds ˛ = A ln 10/d, where d is the sample thickness. The initial samplehickness was measured from the white-light interferometry, andhe pressure-dependent one was calculated using an isothermalquation of state.

.3. Sample preparation

Synthesis of the single-crystal (Mg0.85Fe0.15)O sample isescribed elsewhere in detail (Jacobsen et al., 2002). Briefly,he sample was prepared by Mg–Fe interdiffusion betweeningle-crystal MgO [1 0 0]-plates (∼10 mm × 10 mm × 0.5 mm) andre-reacted (Mg,Fe)O powders in a 1-atm gas-mixing furnace.

nterdiffusion was carried out for 200 h at 1450 C at fO2 = 10−2 Pa,et by the ratio of CO/CO2 introduced into the furnace. After excava-ion of the (Mg,Fe)O single crystal from the sintered pellet, electron

icroprobe analysis (EMPA) revealed a homogeneous composi-ion of (Mg0.85Fe0.15)O at the level of the microprobe samplingvery 10–20 �m across the sample. Mössbauer spectroscopy indi-ates that 1 ± 1% of the iron is ferric (Fe3+). Single-crystal X-rayiffraction of the sample with a rotating-anode Mo source didot detect magnetite or spinel impurities (Jacobsen et al., 2002).iamond-cell samples of the (Mg0.85Fe0.15)O were prepared by pol-

shing parallel [1 0 0]-cleavage fragments of about 20 �m × 20 �mcross and about 8–10 �m thickness with diamond lappinglm.

Single-crystals of silicate perovskite were synthesized from syn-hetic 57Fe-enriched enstatite starting material (En90) at 23 GPa and800 ◦C in the 5000-ton press at Bayerishes Geoinstitut, Bayreuth,ermany. EMPA of the recovered samples confirmed approximatehemical formula (Mg0.9Fe0.1)SiO3. Mössbauer spectroscopy indi-

ates 11 ± 3% of the iron is ferric (Fe3+). Single-crystal structureefinements indicate iron occupies the dodecahedral (A) site, with-ut detectable iron in the octahedral (B) site. For the opticalbsorption measurements at high pressures, samples were pol-shed to 15–25 �m thickness and selected based on their visible

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8 A.F. Goncharov et al. / Physics of the Ear

larity. Sharp optical extinction in cross-polarized light confirmedheir crystalline state after polishing.

MgO for lattice thermal conductivity measurements wasressed in thin pellets of approximately 15 �m thickness, depositednto the opposite diamond anvils, and a thin Pt coupler was posi-ioned between them.

. Results and discussion

.1. Thermal diffusivity

We applied the transient heating technique for thermal diffu-ivity measurements (Section 2.1) for MgO (periclase). We chosehis material because it is one of the most abundant in the Earth’s

antle (doped by Fe) and because of a vast number of experimentalnd theoretical data available. Typical experimental time-resolvedemperature measurements for MgO are presented in Fig. 5. Wettempted to probe the temperature range, which is relevant forhe Earth’s lower mantle—2000–2500 K. Our radiometry systemas sensitive enough for measuring temperatures of approximately

500 K and above. The effect of pressure on the thermal diffusivitys substantially smaller in MgO compared to NaCl (Fig. 4) and KCl.rom the temperature relaxation curves, thermal diffusivity wasnferred by modeling the measured temperature relaxation profilesFigs. 4 and 5) using a finite difference method. These calculationsolve the one-dimensional diffusion Eq. (1) (Carslaw and Jaeger,959; Crank, 1967) by applying an explicit scheme. It models lasernergy deposition in the foil and subsequent heat diffusion withinhe sample. Boundary conditions are constant temperature (300 K)t the media–diamond interface, and diffusive flux continuity athe two foil–media interfaces. The time profile of the laser sourcentensity is approximated by a 8-ns width (experimental value)aussian pulse. The thermal diffusivity of the coupler Dfoil is fixed

o 10 mm2/s. This value does not affect the model result as longs the foil temperature equilibration time is below the pulse dura-

ion, and the media is thermally insulating with regard to the foilDfoil > Dmedia), which was the case in our experiments. The thinnessf the foil with regard to our heating spot size (1/200 ratio) enablesse of the planar approximation in solving the diffusion equation

ig. 5. Time-resolved surface temperature measurements of the iridium foil embed-ed in MgO medium following the arrival of the laser heating pulse. The circles ofifferent colors correspond to experimental data at different pressures. The smalllack dots and gray shaded area of the lower panel show the IR pulse temporalhape. (For interpretation of the references to color in this figure legend, the readers referred to the web version of the article.)

wtCppa

Planetary Interiors 174 (2009) 24–32

Chen and Grigoropoulos, 1997). The validity of this approach haseen previously tested by comparing with a set of 3D calculationsBeck et al., 2007). In our model calculations we use a temperature-ndependent thermal diffusivity D, so our results should be treateds averaged over the studied temperature range. In the correspond-ng temperature range, the variation of the thermal diffusivity isimited (see, e.g., Hofmeister, 1999), and our data do not allowetermining the temperature dependence of D with confidenceecause of insufficient accuracy of the radiative temperature mea-urements. Because the typical diffusion length scale is L ∼ 1 �mL ∝ √

Dt, with D ≈ 10−6 m2 s−1 and t ≈ 1000 ns), our measurementsre unlikely to be affected by the foil position inside the DAC.

As mentioned in Section 2.1, the errors in Figs. 4 and 5 (±100 Kn temperature) are estimated from the reproducibility of the mea-urement, i.e. if the same sample at the same power at the sameelay is measured many times in a row and standard deviation isalculated. The uncertainties of individual temperature measure-ents are smaller (they depend on temperature, but usually do

ot exceed 10 K). We believe that the major source of uncertaintys unequal coupling to the laser radiation of different parts of theoupler, which may also be time dependent.

The uncertainty of the D-values determined by fitting thexperimental T–t profiles is propagating from the uncertainties inemperature and time measurements. The latter one is due to anite width of our time gate (5 ns). Because of these uncertain-ies, there is a range of D-values, which are consistent with thebserved temperature decrease of the foil after the laser pulserrival. This range of values is the error reported in Fig. 6. Addi-ional sources of uncertainties of Dlatt include contact losses at theample–media interface (see Hofmeister, 2007 for a comprehen-ive discussion) and a heterogeneous spatial distribution of theaser light density. When converting thermal diffusivity to ther-

al conductivity (dividing by �Cp), additional sources of error arentroduced. Although the equation of state of MgO is well known,n error can appear the pressure distribution is heterogeneousithin the sample chamber (due to a thermal pressure contribu-

ions), but which is most likely below 10%. Uncertainties exist on

p but those decrease with increasing temperature. Since the tem-erature measurement is the major source of uncertainty, we arelanning to improve the sample preparation procedure by usingdvanced “clean room” microfabrication procedures. One should

Fig. 6. Pressure dependence of MgO thermal diffusivity (T = 2000 K).

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nally note that our modeling involves a temperature-independenthermal diffusion coefficient which represents the averaged overhe experimentally accessible temperature range value of D.

For the thermal diffusivity of MgO, we observe fair agreementetween our measured value at 6 GPa and 2500 K, and a previousmbient pressure extrapolation to high temperature (Hofmeister,999). Our data are also in fair agreement with high-pressurexperimental data at room temperature and 100 ◦C (Yukatake andhimada, 1978; MacPherson and Schloessin, 1982; Katsura, 1997).he pressure dependence of the thermal diffusivity (Fig. 6) deter-ined in this work allows us to examine the validity of differentodels that have been proposed to express variations of klatt with

ompression.Details on the models described below can be found in the

iterature (i.e., Ross et al., 1984; Hofmeister, 2007; Hofmeister etl., 2007). A convenient way is to define the density dependencehrough the g parameter:

=(

∂ ln klatt

∂ ln �

)T

(3)

According to Ross et al. (1984) g for non-metallic solids can beritten as

= 3 + 2q − 13

(4)

can be approximated as the thermodynamic Grüneisen parame-er and q is its density dependence q = (∂ ln /∂ ln �)T which may beonsidered a measure of bulk anharmonic effects.

Although the validity of Eq. (4) has been questioned (Hofmeister,007), it enables one to quantify the density dependence of k.or materials we studied (KCl, NaCl, MgO), the photon meanree path is much larger than the sample size. We can thus rea-onably neglect the radiative contribution to thermal diffusivitykrad klatt) (Hofmeister, 2005). Then:(

∂ ln klatt

) (∂ ln kTot

)

=

∂ ln �T

≈∂ ln �

T

Our values of g in KCl, NaCl, and MgO determined from the exper-ments performed under pressures up to 34 GPa (Beck et al., 2007)re significantly lower than those obtained in the lower pressure

Fpata

Fig. 7. Optical absorption spectra of (a) ferropericlase (15 mol% Fe) and (b) an

Planetary Interiors 174 (2009) 24–32 29

tudies (up to 2 GPa) for a variety of crystal structures, includingur materials of study (g = 6–12) (Ross et al., 1984). Perhaps, thisifference can be explained by a decrease in anharmonic effectsith pressure (q → 0), in which case g should decrease to 3 − 1/3

or solids in a limit of high pressures.An alternative model has been proposed by Hofmeister (2007)

ased on description of phonons in the damped harmonic oscillatorodel. The most simplified of them predicts the thermal diffusivity

o follow the pressure dependence of the bulk modulus:

d ln(klatt)dP

= d ln K

dP, (5)

here K is the bulk modulus.The Ross and Hofmeister models were applied to MgO to

ompare with the experimentally determined thermal diffusivityFig. 6). We calculated the expected density dependence of thermaliffusivity using Hofmeister Eq. (5), using volume-dependent val-es of g and q deduced from the thermal equation of state (Spezialet al., 2001) and also using Ross Eq. (4) (with q = 0). Our results aren a reasonable agreement with both models given relatively largexperimental uncertainties.

.2. Radiative conductivity

The lower mantle constitutes roughly half of the planet’sass and is thought to consist of about 80% silicate perovskite

Mg,Fe)SiO3 and 20% ferropericlase (Mg,Fe)O each with 10–20% ironepending on temperature and depth. Iron-free materials such asnstatite (MgSiO3) and periclase are insulators with a wide bandap, so they are transparent in this spectral range. In Fe-bearinginerals, the major absorption band is believed to be caused bycrystal-field transition of high-spin Fe2+ ion. This transition is

plit into several components by the Jahn–Teller effect; the num-er of components and the symmetry of the ground and excitedtates are determined by a local symmetry of the iron ion envi-onment. In ferropericlase the 5T2g → 5Eg crystal-field transition of

e2+ ion in an octahedral site is split into two components; in silicateerovskite 5Eg → 5T2g crystal-field transition of Fe2+ ion in a dodec-hedral site is split into three components (Burns, 1993). Theseransitions are in near infrared spectral range (7000–12,500 cm−1)nd they are expected to increase gradually in frequency with

d silicate perovskite (10 mol% Fe) up to 133 GPa at room temperature.

3 th and Planetary Interiors 174 (2009) 24–32

ptvSoatio

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Fig. 8. Radiative part of the Earth’s lower mantle thermal conductivity as a func-tion of depth. Three different geotherms were used following (Hofmeister and Yuen,2007): a hot lower mantle adiabat (blue dots), a whole mantle adiabat (black dots)and the extrapolated transition zone adiabat (blue dots). Previous estimate is pre-sented as dashed line (Hofmeister and Yuen, 2007). The absorption coefficient wasassumed to be temperature independent. We also suppose that the product ˛�is large (� > 0.5 mm). The radiative conductivity was calculated using a formula(1) in two limiting cases: pure perovskite mantle (ferropericlase has much largerahcr

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0 A.F. Goncharov et al. / Physics of the Ear

ressure. An abrupt change of absorption is expected at the spin-ransition; the crystal-field transitions were predicted to move toisible and ultraviolet, leaving the near-IR transparent (Burns, 1993;herman, 1991; Badro et al., 2004). Crystal-field bands of Fe3+ aref the low intensity in the spectral range of interest because theyre spin-forbidden. Other than crystal-field transitions contribu-ions include Fe2+–Fe3+ intervalence transitions (near 15,000 cm−1

n near infrared-visible range) and Fe–O charge transfer (in ultravi-let) (Rossman, 1988; Keppler et al., 1994).

Pressure-dependent optical absorption in ferropericlaseGoncharov et al., 2006) and silicate perovskite (Goncharov et al.,ubmitted for publication) reveals a variety of unexpected phe-omena which include substantial changes in absorption in thepectral range of interest (Fig. 7). Here we present a brief accountf our observations. Contrary to the predictions, the pressurehanges are mainly related to an increase of absorption of Fe–Oin perovskite and ferropericlase) and Fe–Fe (in ferropericlase)harge-transfer bands—these effects were largely overlooked inhe past. The changes in absorption related to the spin transi-ions are notable, but the effects are much smaller. Moreover,hese changes become even less pronounced at high temperatureecause of a smearing out of the spin transition pressure interval.he experiments have been conducted using various transmittingedia (Ar, Ne, oil) and without transmitting medium, but noeasurable differences were detected.The results suggest that the absorption of mantle rocks is mainly

overned by the iron concentration: Fe2+ in ferropericlase ande3+ in silicate perovskite (Goncharov et al., 2006, submitted forublication). In the case of ferropericlase, absorption is criticallyependent on Fe concentration: the absorption increases dras-ically near the percolation limit (∼12% Fe). In case of silicateerovskite, the presence of Fe3+ increases intervalence Fe2+–Fe3+

nd Fe3+–O charge-transfer bands (Fig. 7).The current results from optical absorption of silicate perovskite

nd ferropericlase (Fig. 7) with compositions relevant to Earth’sower mantle provide a basis for determining the direct radiativeeat transfer of minerals over a wide depth range in the mantle.lthough the effect of temperature on the absorption spectra maye substantial (Shankland et al., 1979), we do not expect this to behe case for silicate perovskite and ferropericlase because the mainbsorption mechanisms are different compared to that in olivinetudied in Ref. Shankland et al. (1979). Our absorption measure-ents on 15% Fe (Mg,Fe)O samples to 60 GPa and 800 K did not

how any substantial temperature dependence, which would affecthe calculations of the radiative conductivity presented below.

For these calculations the mantle is a modeled as a mixture of0% (Mg0.85Fe0.15)O and 80% (Mg0.9,Fe0.1)SiO3-perovskite and wessume that the radiative contribution of both phases can be lin-arly added to obtain the total mantle krad. Our estimation (Fig. 8)s substantially smaller than the previous estimation (Hofmeister,999) based on extrapolations of the optical properties of miner-ls to higher pressures. Substantial contribution from the UV–visbsorption tails to the radiative conductivity, which could only benferred in previous models (Hofmeister, 1999), is responsible forhe discrepancy. Because the absorption in the UV–vis tails is mostlyue to the O–Fe+3 charge transfer (Fig. 7), and the contribution fromhe Fe2+–Fe3+ intervalence charge transfer is also substantial, themount of the ferric iron in the mantle (i.e., redox state McCammon,997; Frost et al., 2004) may be controlling the radiative heat con-uctivity (Fig. 8).

.3. Thermal conductivity of the lower mantle

The total thermal conductivity across the mantle was calcu-ated by adding phonon (�latt) and radiative contributions (�rad).

sbtFc

bsorption coefficient and therefore will not contribute to the diffusive radiativeeat transfer) and a mixture of 80% of PV10 and 20% of ferropericlase with 15% Feomposition. (For interpretation of the references to color in this figure legend, theeader is referred to the web version of the article.)

he latter was obtained from our measured absorption spectras described above, whereas the pressure dependence of latticeontribution was calculated according to Ross et al. model (with= 0) or Hofmeister (2007) model. Those two models were chosenecause they were found to agree with our high-pressure measure-ents of MgO thermal conductivity. Equation of state for MgO was

aken according to Speziale et al. (2001) whereas measurementsf Shim and Duffy (2000) were used for perovskite. Since little isnown about the lattice conductivity of (Mg,Fe)O we used data forure MgO, but the presence of iron might not significantly affecthe lattice contribution to thermal conductivity.

In both cases, the temperature dependence of �latt was taken asescribed in Hofmeister (1999), and extrapolating room pressurend high-temperature measurements of Andersson and Bäckström1976) (MgO) and Osako and Ito (1991) (perovskite).

The mantle is modeled as a mixture of 20% ferropericlase and0% silicate perovskite (k = 0.8kPv + 0.2kFp). Results show that forhe two different geotherms used in the calculations, the total ther-

al conductivity is in the 6–11 W/mK range on top of the D′′ layerFig. 9).

Insights on the power output of the convecting outer core can beained by solving the temperature profile across the D′′ layer. The Pnd T dependence of k introduces non-linearity in the heat diffusionquation, which was then solved numerically using a finite differ-nce method, assuming the absence of radioactive heat source. Theteady-state solution was obtained by relaxing an arbitrary ini-ial temperature profile, and using the previously described P–Tependence of thermal conductivity. The two required boundaryonditions were chosen to be constant temperatures on top and athe bottom of D′′. These boundary conditions also imply fixed val-es for k at these positions. The temperature at the top of D′′ was

et to agree with the chosen geotherm, and the temperature at theottom of the layer was taken as T = 3900 K (Boehler, 2000). Oncehe initial profile is finally relaxed, T and k are known across D′′.or T = 3900 K at the bottom of D′′, the obtained values for � at theore mantle boundary are between 4.3 and 5.2 W/mK. For the dif-

A.F. Goncharov et al. / Physics of the Earth and

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4

msraontdp

vr(sd(lu(arlera

A

NwSceN

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A

A

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B

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B

B

C

C

CC

C

D

F

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G

G

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H

H

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H

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ig. 9. Thermal conductivity of the lower mantle calculated using three differenteotherms from Hofmeister (1999) (red and black curves) and Ito and Katsura (1989).For interpretation of the references to color in this figure legend, the reader iseferred to the web version of the article.)

erent geotherms used (and, therefore, the temperature gradientscross D′′), the integrated heat flux over the CMB are between 2.8nd 6.3 TW.

. Conclusions and outlook

We have presented measurements of thermal diffusivity inantle materials at high pressure–temperature conditions with

ufficient accuracy for testing current theoretical models. Ouresults are in agreement with the lower P–T experimental datavailable from the literature. Future improvement includes the usef a shorter laser pulse (∼1 ns), and detector gate (∼1 ns). This tech-ique can in principle be applied at megabar pressures, allowinghermal diffusivity measurements in previously unreachable P–Tomains and enabling unique insights into the thermal state oflanets.

Optical properties of lower-mantle minerals, synthesized atarious conditions, have now been studied over a wide spectralange including mid- and near-infrared, visible and ultraviolet2000–35,000 cm−1) and at high-pressures and temperatures initu. The estimated radiative contribution to the thermal con-uctivity does not exceed 0.54 W/mK on the top of the D′′ layercf., Hofmeister, 1999). The value of k in the thermal boundaryayer strongly influences its thinness and stability, where low val-es of k typically imply a less stable and thinner boundary layerTurcotte and Schubert, 2002). Moreover a decrease in k also impliesdecrease in plume temperature (Montague et al., 1998). These

esults should be taken into account when exercising model calcu-ations of the mantle dynamics (Dubuffet et al., 2002; van den Bergt al., 2005; Naliboff and Kellogg, 2007). Substantial efforts are stillequired to estimate the lattice contribution to thermal diffusivityt relevant pressures and temperatures.

cknowledgements

We acknowledge support from DOE/BES, DOE/NNSA (CDAC),SF (EAR-0711358), CIW, and the W.M. Keck Foundation. S.D.J.

ishes to thank D.J. Frost, C.A. McCammon, D.P. Dobson, J.R. Smyth,

.J. Mackwell, and H. Watson for help with sample synthesis andharacterization and support from NSF (EAR-0721449) and the Bay-rishes Geoinstitut Visitor’s Program. B.D.H. was supported by theSF Research Experience for Undergraduates (REU) Program at

K

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Planetary Interiors 174 (2009) 24–32 31

he Carnegie Institution of Washington. We are very grateful to A.ofmeister and an anonymous reviewer for very useful commentsnd suggestions, which improved the paper.

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