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.Physics of the Earth and Planetary Interiors 109 1998 179197

The Gruneisen parameter for iron at outer core conditions and theresulting conductive heat and power in the core

Orson L. Anderson )

Department of Earth and Space Sciences and Institute of Geophysics and Planetary Physics, Uniersity of California, Los Angeles, Los

Angeles, CA 90095-1567, USA

Received 6 April 1998; accepted 11 August 1998

Abstract

The aim of this paper is to find the conductive power in the outer core. Before the heat conduction equation can be

usefully applied, however, a careful examination of the Gruneisen parameter, g, and the thermal conductivity, k , of the ccore is required. The focus of this paper is on these two parameters at outer core conditions, using primarily experimental

data and one theoretical evaluation of g. The melting g of the core is found to be 1.3, the adiabatic g , slightly less, andm ad

k is found to be 44 W my1 Ky1 in the upper limit and 28 W my1 Ky1 in the lower limit. To obtain the coresc

.temperature, T , from T of pure iron at core conditions , the freezing point depression is found. This requires assumptionsm m

about the impurities in the iron core. Recent experimental data greatly restrict the type and concentrations of these

impurities. Several allowable combinations are found, however; all of these lead to a DT of ; 1000 K. At the core side of . y1the core-mantle boundary CMB , the adiabatic temperature is 3900 K and dTrd r is 0.9 K m , resulting in a conductive

12

.power of 4.4 " 1 TW 1 TWs10 Watt flowing out of the core along the adiabatic gradient. New experimental results onthe thermal conductivity of mantle minerals and new theoretical insights into the heat flow in mantle plumes support the casethat the conductive power along the cores adiabatic gradient passing to the upper mantle can be transported away by both

conduction and convection in the upper mantle. q1998 Elsevier Science B.V. All rights reserved.

Keywords: Gruneisen parameter; Iron; Core conduction; Mantle conduction

1. Brief description of the model

This paper examines the conductive power flow-

ing from the core and especially the parameters

found in the thermal gradient at the core-mantle

.boundary CMB and the thermal conductivity. At-tention is focused on finding the values of the pa-

rameters involved in calculating the conductive heat,

)

Corresponding author. Fax.: q1-310-206-3051; E-mail:[email protected]

especially the Gruneisen parameter and the thermalconductivity. The values of these parameters are

found to be similar to those by Braginsky and Roberts .1995 , but different from those found in most recent

reports.

I disallow the cores thermal gradient from beingsubadiabatic because I find the conductive power

12from the core to be 3.4 to 5.4 TW 1 TWs10.Watt , which is within the range of power leaving

the CMB on the mantle side by conduction and

convection. According to this model, any excess

power, which will be convective, will be returned to

0031-9201r98r$ - see front matter q 1998 Elsevier Science B.V. All rights reserved. .P I I : S 0 0 3 1 - 9 2 0 1 9 8 0 0 1 2 3 - X

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( )O.L. AndersonrPhysics of the Earth and Planetary Interiors 109 1998 179197180

the core by the mechanism of compositional convec- .tion of Loper 1978a,b .

In this model, there is no attempt to explore the

physics of the cores convective power or to find all

the sources that comprise the cores conductive

power. I do, however, estimate the power generated

by the crystallization of the inner core boundary

since the parameters for this power source have been

determined.

2. Parameters needed for the heat flow analysis

The heat conducted from the core to the mantle is

given by:

ETq syk , 1 .c c /Er Swhere k is the thermal conductivity of the core andcr is its radiusthe radial distance from the center of

Earth to the point considered. The outer core is

known to be liquid and is ordinarily assumed to be

under adiabatic compression, so one can obtain the

slope of the temperature profile by means of the

adiabatic equation:

ET ygg T1s , 2 . /Er fS

where g is the Gruneisen parameter of the liquid1state, g is gravity, and f is the seismic parameter.

The departure from adiabaticity is considered to be

small. Although the values of g and f at any radius

can be readily obtained from seismological tables,

evaluation of the other parameters requires special . .effort. Values of k and ETrEr in Eq. 1 must bec

. .found. Before ETrEr can be found, g and T r inl . .Eq. 2 must be evaluated. T r in turn depends on

g , the gamma of melting. Thus, to begin an evalua-mtion of the power, Q , one must first understand thecGruneisen parameter in the various states and phasesof iron.

3. g in the solid, liquid, and melting states of iron

Fig. 1 shows the theoretical and experimentalvalues of g for pure iron in its various phases see

Fig. 1. Gruneisen parameter, g, vs. density for iron. The top scale,for P in GPa, shows various starting values of P s0 for the bcc

.phase, the hcp phase, and liquid. The main focus is the g e vs. rs .curve starting at G, as deduced from Mao et al. 1990 EoS

.measurements, extending along the Hugoniot F from Brown and .McQueen 1986 and terminating at melting at the edge of the

. .solid phase D calculated by Stacey 1995 . Other lines represent . .g vs. r at low pressure and g a E at low pressure measuredl s

.by Boehler and Ramakrishnan 1980 . Note that g for the core ismgiven as 1.3"0.3, which coincides with the solid phase D by

Stacey. The curve g for the core, ---, is coincident with DX

andmD

Y. The liquid for the isentrope, shown by P , is slightly lower.

.phase diagram of iron, Fig. 2 . We now review the

literature data on g as plotted in Fig. 1. The line forvalues of g has Points A, B, and C Stevenson,l

1981; Verhoogen, 1980; Chen and Ahrens, 1997,.respectively marked as triangles. This line descends

rapidly with pressure, P. Point E, showing g in the . .solid state g , represents the a- bcc iron datum ats

.P s0. The line extending from Point E, called g a ,s

represents measured values at higher densities see.figure caption for data references . The line emanat-

ing from E represents the measurements for bcc iron:

g s 1.66 and q s0.6 in the equation:0

g r0s , /g r0

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( )O.L. AndersonrPhysics of the Earth and Planetary Interiors 109 1998 179197 181

Fig. 2. The phase diagram of iron in V, T space. The shaded areas .represent DV at phase boundaries. The beta phase b is accepted

as part of the phase diagram, making b the dominant phase at core

conditions. The Hugoniot passes through the e and b phases well

below melting, until it emerges into the liquid phase at 240 GPa.

where the subscript 0 means at P s0. The solid X Y .squares in Fig. 1 D , D, and D represent g of thes

.core at the core-mantle boundary CMB , the inner .core boundary ICB and the middle of the outer

.core but in the solid state , as calculated by Stacey .1995 . To these literature data on g is now added

.the value of g e , Point G, corresponding to the essolid phase.

.We find g e at P s 0, Point G in Fig. 1, fromsthe relationship between K

Xand g for hcp iron, as0 0

. .derived by Stacey 1995 see his Eqs. 34 and ..35 , where:

g s 1r2 KX y0.95, 3 . .0 0X . Xand where K s EK rET . To obtain K for e0 T Ps0 0

.iron listed in Table 1 , one uses P, V data on hcp .iron measured to 330 GPa by Mao et al. 1990 at

Ts 300 K. KX is found from these data by solving0 .the BirchMurnaghan EoS, Eq. 4 . The values of

K and V used in the calculation of KX

are listedT 0 00in Table 1:

22 V 3X ~K s4 q P0 / /3 K VT 0 0

y1y15 2 V 4 V3 3 y y1 y 1 . / / /V 3 V0 0 4 .

Table 1 shows that the average value of KX

for hcp0 .iron is 5.31, as found from Eq. 4 . Using the Mao et

Table 1

Values of KX

for e iron from measured P, V dataa using the0

BirchMurnaghan EoS whereK

s165 GPa andV

s6.73 cm3

T 00moly1

X3 y1 y5 3 y1 . . .P GPa V cm mol V =10 m kg K0

34.8 5.8110 10.408 4.915

35.6 5.8120 10.410 5.257

56.1 5.4740 9.805 4.795

80.0 5.2760 9.450 5.687

101.6 5.0800 9.099 5.587

112.4 4.9300 8.831 4.997

125.4 4.9030 8.872 5.538

132.1 4.8090 8.614 5.135

141.2 4.7140 8.443 4.878

142.2 4.7010 8.420 4.835

156.7 4.6690 8.362 5.193170.8 4.6130 8.263 5.324

197.3 4.4750 8.015 5.244

208.7 4.4580 7.985 5.448

210.1 4.4680 8.383 5.553

212.1 4.4170 7.912 5.278

227.2 4.3850 7.854 5.452

235.7 4.3260 7.748 5.287

244.8 4.3210 7.740 5.456

247.3 4.3100 7.720 5.443

256.2 4.2540 7.620 5.294

258.2 4.2870 7.679 5.531

264.0 4.2890 7.682 5.663

270.1 4.2430 7.600 5.500

270.4 4.2130 7.546 5.330277.8 4.1700 7.469 5.223

292.8 4.1450 7.424 5.338

293.5 4.1370 7.410 5.306

294.7 4.1440 7.423 5.364

304.1 4.0990 7.342 5.268

a .P and V data from Mao et al. 1990 .

Average value of KX

is 5.31.0

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al. P, V data with the Vinet EoS, one finds KX s5.310

and, using same data with the BornMie EoS, KX s0

5.11. For e iron, KX s5.3 is selected from the Mao0

.et al. 1990 data, yielding g s1.7 for e iron from0 .Eq. 3 . The value of g for hcp iron is represented0

.by Point G in Fig. 1. We now seek g hcp at highspressure.

.Stacey 1995 found theoretically the values of gsfor close packed crystalline iron at core pressures

X Y .shown by the squares D, D , and D in Fig. 1,

covering the pressure range of the outer core. They X . . Y .are g D s1.27, g D s1.30, and g D s1.33.

.Gilvarry 1956 argued that the melting of a solid

occurs at the limit where TT , e.g., at the meltmboundary of the solid and liquid, but on the solid

side. Therefore the melting line shown as a dashed.line in Fig. 1 is drawn adjacent to Staceys points

.solid squares . Noting Staceys value of g at mid-

core pressure, e.g., near 240 GPa, we now connectthe Hugoniot to Staceys Point D. I assume that for

the hcp phase, g is independent of T at constant V.s .This makes the Hugoniot value, g P , identical toH

.the hcp value, g P . I now use the experimentalsobservation that the curve representing the P, V data

on the Hugoniot egresses from the solid hcp phase at

P s 240 GPa, because at that pressure, the shearvelocity of pure iron approaches zero, as seen in Fig.

3. Thus, the onset of melting for the Hugoniot, .T 240 , is at a pressure of 240 GPa. The g of them

Hugoniot, g , is connected to Staceys midpointHvalue, which is at 240 GPa, the same pressure at

which the Hugoniot enters the liquid state. This gives . .g 240 sg s1.30. Since g e is a function of PH m s

but insensitive to T at constant r, the Hugoniots gH . is identical to g e or g of b iron above Ts 1500s

. .K . Thus the terminus of the g e curve, as well assthat of g , is Staceys Point D. A line connectsH

. y3 . .g e at P s0 rs8280 kg m and g e ats s y3 . .P s240 GPa rs12200 kg m . From Eq. 3 it

.is found that qs 0.7, so that g r is:e

0. 7r0g r s1.7 . 5 . .

e /rThe gr diagram, Fig. 1, is now complete. What is

significant for core physics from the diagram is that .g 240 s 1.3, which is identical to the solid statem

.value of Stacey 1995 and also connects the equa-

Fig. 3. The Hugoniot experimental P, V data, plotting r vs. P.The sound velocity of iron measured along the Hugoniot path. The

melting of the Hugoniot, placed at 240 GPa, occurs when ysplunges to zero. Dr of melting is found from the separation

between the two Hugoniot curves in the liquid state. Data repre- .sented by solid circles are from Brown and McQueen 1986 ; data

.represented by diamonds are from Altshuler et al. 1962 .

tion of state of hcp iron and the Hugoniot of iron to

the melt at midcore pressure.

We are now in a position to evaluate the meltingtemperature at 240 and 330 GPa. We follow the

.procedure shown by Anderson and Duba 1997 and .also by Stixrude et al. 1997 evaluating T at itsH

. .melting pressure 240 GPa and then finding T 330m .from T 240 by the Lindemann law. The tempera-m

ture profile of the Hugoniot must now be evaluated.

4. Evaluation of the Hugoniot temperature

The Hugoniot temperature, T , at any PV pointH .can be calculated McQueen, 1964 from the integra-

tion of:

T V yV d P q P yP dV . .S 0 0dT syg dVq ,H s

V 2CV

6 .

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Table 2 .Calculation of T and DT the isentropic contribution to DTS S H

y3 y3 y3 y6 3 y1 . . . . .P J m r =10 kg m V =10 m kg g Drrr g DT K T Kave S S

0 1.7 a40 9.49 10.5 1.55 650

60 9.99 10.0 1.49 0.0513 1.52 51 701

80 10.42 9.6 1.45 0.0490 1.47 50.8 752

100 10.72 9.31 1.42 0.0280 1.435 30.7 783120 11.04 9.06 1.39 0.0215 1.405 30.46 814

140 11.3 8.85 1.34 0.0232 1.38 26.15 840

160 11.54 8.69 1.35 0.0210 1.36 24.09 864

180 11.75 8.51 1.33 0.018 1.34 20.94 885

200 11.94 8.38 1.32 0.016 1.325 18.82 904

220 12.12 8.25 1.31 0.0150 1.315 17.84 922

240 12.20 8.14 1.30 0.0130 1.305 15.80 937

. 8 y3 . y5 3 y1P sP 40 s40 =10 J m ; V sV 40 s10.5 = 10 m kg .0 0a .Andrews 1973 .

or:

d SdT sdT q , 7 .H S

CV

where g represents g of the solid state for hcp iron,sdT is the temperature increment due to isentropicS

compression, and d SrC is the temperature incre-V

.ment arising from P yP under compression dV.HThus d S can be considered as a type of work term,

.dW see also Anderson, 1995, p. 321 . The well-known P and r data along the Hugoniot see Table

.2 of Brown and McQueen, 1986 are reproduced asthe first two columns in Table 2. The first term on

el .Fig. 4. The electronic specific heat factor at pressures along the Hugoniot. The inset shows C calculated by Stixrude et al. 1997 forVisochores vs. T. The final CelrR curve was found by assuming a temperature distribution T vs. r so as to calculate Cel vs. pressure andV V

.then recalculate from Eq. 6 the resulting T vs. P distribution and recycle until convergence has been achieved.

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Table 3

Work contribution to T , DTH Hel 2 aP C in terms C Total =10 Work DT : W DT DT : DT Hugoniot T,V V W S H W

9 y3 y1 y1 2 y1 . . . . . . .=10 J m of R J kg K =10 J kg %C K K qDT K T KV S H

0 4.461 0b40 0 650

60 0.3 4.907 1054.8 215.0 51.0 266.0 916

80 0.45 5.130 1766.6 344.4 50.8 395.2 1311100 0.60 5.353 2084.2 389.4 30.7 420.1 1731

120 0.77 5.606 2491.5 444.3 30.5 474.3 2205

140 0.94 5.859 2729.9 465.9 26.2 492.1 2698

160 1.10 6.097 2976.2 488.1 24.1 512.1 3210

180 1.28 6.364 3110.9 488.8 20.9 509.7 3720

200 1.45 6.617 3245.6 490.5 18.9 509.4 4229

220 1.55 6.766 3406.0 503.4 17.8 521.2 4750

240 1.62 6.870 3469.1 505.0 15.8 520.8 5271

a . . 4P yP dVq V yV d P r2; P, P , V, V from Table 2.0 0 0b .Boundary condition: Andrews 1973 .

Also listed: DT and T .S H

.the right of Eq. 7 arises from the adiabatic tempera-ture profile; step-by-step details of its evaluation

along the Hugoniot are listed in Table 2. This term .may be readily evaluated since g V for e iron is

.now known from Eq. 5 . The numerator of the .second term in Eq. 7 can be considered to be the

work necessary to displace the isentrope to the . . 4Hugoniot: Ws 1r2 V y V d P q P y P dV .0 0

.The chief problem in evaluating Eq. 6 consists of

finding values of C , composed of the classicalVlattice term 3R, but with an additional term arising

from the free electrons, Cel

, which is linear in T andValso depends on density. The calculation of the

fraction of dT arising from Cel must be doneH Vcautiously because Cel depends on T, and T dependsV

el .on C . Stixrude et al. 1997 found and plotted CV Ve .of iron vs. T and r their Fig. 2 ; their plot is shown

as an inset in Fig. 4. A solution was found for CelVvs. P; it is the prominent curve shown in Fig. 4. The

solution for T vs. P for the Hugoniot is listed inHthe last column of Table 3, the previous columns of

which give the step-by-step evaluation of d S. The

integrating constant, T , is the value of T at the0

phase boundary between the a and e phases, given .as 650 K by Andrews 1973 , Fig. 9. The solution

. .for T P is given in Table 3, which lists T 240H H . sT 240 s5271 K 5300 K to two significantm

. . .figures . Boness and Brown 1990 found T 240 sH .5625 K and Brown and McQueen 1986 found

.T 240 s5510 K. Results found here are differentH

from theirs because the Cel results of Stixrude et al.V .1997 and a different and more exact solution of

.g V for e iron were used.

It is significant that the temperature rise due to

isentropic compression is less than 10% of the total

temperature rise. Therefore it would take an unrea-

sonable value of g to shift the Hugoniot temperaturesby a significant amount. We find that g is exactlys

.fixed by Eq. 5 .

5. The value of T of iron at the ICB pressurem

.Taking T 240 s5300 K for iron, the value ofm .T 330 for iron can be found from the Lindemannm

equation for melting. The Lindemann formulation in .T, r space is Anderson, 1995, p. 281 :

E ln Tms2 g y 1r3 . 8 . .m /E ln r

From Fig. 1, we see that g changes very littlembetween P s135 and 330 GPa. Thus the above

.equation can be approximated by taking 2 g y1r3mas a number independent of density, e.g., 2 g ym

. .1r3 s1.9 for g s1.3 across the outer coresm .density range. By integrating Eq. 8 for this special

case, it is found that:

1. 9T rm 22 s . 9 . / /T rm 11

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Table 4

Values of parameters for pure iron and the core

bRadius and r, pure iron:core r, core A surface M mass of DV pure T purec c m ma c . .pressure km conditions area interior iron iron K

3 y3 3 y3 13 2 24 3 y1 . . . . .10 kg m 10 kg m =10 m =10 kg m kg

d .r 135 GPa , CMB 11.28 9.9035 15.26 1.85 4380 .r 200 GPa 12.12 5020

y6

.r 240 GPa 12.47 11.29 2 = 10 5300y6 .r 330 GPa , ICB 13.38 12.166 1.9 0.097 0.7 = 10 6100

a . .For e-iron solid Mao et al. 1990 for room temperature data followed by T correction from aT.b

PREM.c .Stacey 1992 .d .Melchior 1986 .

.Values of r P needed for the evaluation of Eq. .9 are listed in Table 4, from which we have

. . .r 330 rr 240 s 13.38r12.47. Thus, T 330 rm . . .T 240 s1.15 by Eq. 9 , and T 330 s 6095 Km m

for pure iron, 6100 K to two significant figures. .Using Eq. 9 and the densities listed in Table 4,

we find the melting point of pure iron at mantle-core .boundary MCB pressure to be about 4380 K. At

200 GPa, T is 5020 K, considerably larger than them .experimental value reported by Boehler 1993 ,

. .T 200 f3950 K. Boehler 1993 extrapolated hism .experimental T data and obtained T 330 s 4800m m

K, whereas 6100 K was obtained here. .The value of T 240 obtained here involves nom

estimates. The calculations used in finding g for eiron were based on P, V measurements on hcp iron

.in the diamond cell Mao et al., 1990 . The determi-

nation that the Hugoniot crosses the melting curve at .240 GPa has been made experimentally see Fig. 3 .

The Lindemann theory is used to find the value of . .T 330 from T 240 ; g is determined from Fig. 1.m m m

For the calculation of T along the Hugoniot, CelH Vvalues at all pressures along the Hugoniot see Fig.

.4 are needed. These were extracted from the paper . elof Stixrude et al. 1997 , who calculated C vs.V

volume and temperature. .My calculated value of T 330 for pure iron,m

6100 K, agrees reasonably well with the value calcu- .lated by Poirier and Shankland 1994 using disloca-

.tion theory, T 330 s6150 K, and the value foundm .by Stevenson 1981 . Many others have found

. .T 330 to be near 6000 K, including Gilvarry 1956 ,m . . .Zharkov 1962 , Birch 1972 , Boschi 1975 , Liu

. .1975 , Brown and McQueen 1986 , Anderson and . .Young 1988 , Boness and Brown 1990 , Kerley

. .1994 , Wasserman et al. 1996 and Anderson and

.Duba 1997 .

6. Models of core impurity composition

.Our value for T 330 is for pure iron. To findm .T 330 for the core, we must calculate the effect ofm

the cores impurities on the melting temperature of

pure iron. In looking for a plausible set of impurities

for the core, one observes wide division of opinion

on this matter among cosmochemists and geophysi-

cists. Table 5 is a summary of various past modelsfor core impurity composition.

The uncertainty in the choice of light element

impurities has been greatly reduced by several recent

advances. .First, Poirier 1994 provided equations and fig-

ures that help restrict the concentration of light ele-

ments. The concentration of all impurities must pro-

duce a total impurity density equal to 10% of the

density of pure iron at core conditions. This is called

the core mass deficit, e.g., the difference between the

density of iron and PREM density at core conditions. .The equations of Poirier 1994 allow one to derive

the contribution of each impurity in the core to the

10% core mass deficit. The relation between the

weight percent of an impurity and the core mass

deficit is shown in Fig. 5. The three lower lines, for

oxygen, sulfur, and silicon, are taken from the analy- . .sis of Poirier 1994 see his Fig. A1 .

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Table 5

Models of core composition impuritiesa

Previous chemical models

Sulfur only

.Murthy and Hall 1970 .Ahrens 1982

Small amount of sulfur, large amount of silicon

.Allegre et al. 1995` .Dreibus and Palme 1996No sulfur, silicon only

.Saxena and Benimoff 1977 .McDonough and Sun 1995

Silicon and sulfur

.MacDonald and Knopoff 1958

Sulfur and oxygen

.Boehler 1992

Features added

Oxygen and silicon almost mutually exclusive .Sulfur with large DV of mixing Sherman, 1997

a

Impurities considered are the abundant light elements Si, S, andO.

.Second, the paper by ONeill et al. 1998 pro-vides experimental evidence on the solubility of Si,

S, and O in an Fe-rich metal. Their results restrict

the amount of S in an Fe-rich metal to less than 6%

by weight because of high volatility, and they found

Fig. 5. Curves showing the contributions of a given weight

percent of an impurity element towards the density deficit of the

outer core. S represents sulfur assuming an FeS ideal solution.

S) results from the volume of mixing in the FeS solution . according to Sherman 1997 . The curves for O, S , and Si are

. )taken from Fig. A1 of Poirier 1994 . The curve for S is found .from data presented by Sherman 1997 .

O to be even more cosmochemically volatile than S.

The abundance of Si is tied to the abundance of

SiO , and there is no evidence that there was a large2enough loss of O from the primitive Earth to free

sufficient Si for the core. ONeill et al. concluded

that . . . from the chemical view, none of the

cosmochemically abundant light elements appears

able to account for the mystery light component in

the core, at least when considered individually. .Third, Sherman 1995 pointed out that the pres-

ence of silicon eliminates oxygen as a core impurity

candidate because oxygen demands oxidizing condi-

tions and silicon demands reducing conditions. .ONeill et al. 1998 came to the same conclusion

about the incompatibility of silicon and oxygen.

They noted, however, that a small amount of silicon

and oxygen can coexist in iron, according to results

known and used in the steelmaking industry, and

they presented a curve taken from the steelmakingliterature shown in Fig. 6. By this figure, 2-1r2wt.% silicon could coexist with 2-1r2 wt.% oxygen.

.Fourth, Sherman 1997 discovered that there is a .high excess volume of mixing density deficit in

FeFeS alloys at 250 GPa in the FeFeS phase .diagram see Fig. 7 . This density deficit is suffi-

ciently large that it shifts the sulfur curve towards )smaller levels of weight impurity the S curve in

Fig. 6. The mutual dependence of the solubilities of Si and O in

liquid Fe at 1 bar, showing that Si and O are mutually exclusive

except along the two axes. From the steelmaking literature and . .modified from ONeill et al. 1998 their Fig. 10 .

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Fig. 7. Density vs. composition for the FeFeS phases at 250 GPa .after Sherman, 1997 . The calculated nonideal mixing curve

relative to Fe and FeS is compared with the ideal mixing curve.

The first up arrow on the left corresponds to 2.3 wt.% sulfur

projected to the calculated curve. The first down arrow on the left

represents 4.3 wt.% from the ideal mixing curve. The second up

arrow on the left represents 5.9 wt.% projected to the calculated

curve. The second down arrow on the left represents 10 wt.%

from the ideal mixing curve. A concentration of 5.9 wt.% sulfur

satisfies the 10% density deficit of the core.

.Fig. 5 , so that only 5.9 wt.% of sulfur would offset

all 10% of the outer cores density deficit. ONeill et .al. 1998 conclude that the amount of S in the core

should be less than 6 wt.% or about half the

amount required for the density deficit, according .to Fig. A1 of Poirier 1994 .

However, that conclusion was based on the S

curve in Fig. 5. Taking into account the discovery of .Sherman 1997 of the high volume of mixing of

sulfur in FeS, the S) curve in Fig. 5 was con-

structed; it shows that sulfur could completely ac-

count for the core deficit and be within the limits .stated by ONeill et al. 1998 .

My approach is to find combinations of S, O, and

Si that are within the limits described above: sulfur ) .identified in this paper as S to be held to less than

6 wt.%, and O and Si in combination to be taken

from the 2750 K isotherm in Fig. 6.

Of the six proposed impurity compositional mod-

els listed in Table 6, only the first four satisfy the

limits stated above. The last two are for the cases of

S alone and Si alone. For all six cases considered,

the value of the weight percent sum is constrained to

account for the 10% core density deficit using the

curves in Fig. 5.

( )7. Depression of T ICB by impuritiesm

Finding the temperature depression, DT, is

straightforward if known impurities with known con-

centrations in iron form as a solid solution, but the

problem is complex if the solute forms a eutectic

with the solvent. The concentrations of the impurities

in the first four cases listed in Table 6 are suffi-

ciently small that I assume that the impurities doindeed form a solid solution. Case f probably does

not form a solid solution, but it is listed in order to

show, as I do below, that the melting point depres-

sion, DT, is unreasonably large.

The equation used for calculating DT for outer

core conditions is:

DTsyT) ln 1y x , 10 . .i i ,l

where T) is the pure iron melting point at the ICB

pressure and x is the molar fraction of the ithi,l .impurity in the liquid outer core. Eq. 10 is derived

in Appendix A. .Evaluating Eq. 10 for case a in Table 6, where

x)s 0.048, x s 0.074, and x s0.032:S Si o) 4DTsyT ln 0.952 q ln 0.926 q ln 0.968

sy0.159T) .

Using T)s 6100 K, DTsy970 K for case a.Results from the first four cases listed in Table 6

vary from y700 to y1070 K. It is interesting that .Poirier 1994 stated that many authors assume that

the depression of the melting point is 7001000 K,

but they do not generally explain how this result

was obtained. Following the comment of Poirier .1994 that the iron core would likely have more

than one impurity, I average the results from cases a,

b, and c, and choose DTsy1000 K. Accordingly,6100 K y1000 K s5100 K is assigned as the value

of T at the ICB Braginsky and Roberts, 1995,m

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Table 6

Calculation of the freezing temperature depression, DT, resulting from various models of the composition of the core

)Model Comment Impurities DTrT DT )S S Si O

Case a Si-rich

Wt.% 3 4 1

Molar frac. 0.048 0.074 0.032 y0.159 y970

Case b O-richWt.% 2 0.5 4

Molar frac. 0.031 0.009 0.125 y0.1755 1070)Case c S -rich

Wt.% 4 1.2 1.2

Molar frac. .077 0.022 0.038 y0.143 y871)Case d S alone

Wt.% 5.9

Molar frac. 0.109 y0.115 y700Case e S alone

Wt.% 11

Molar frac. 0.177 y0.194 y1183Case f Si alone

Wt.% 18.5

Molar frac. 0.311 y0.372 y2271

T) represents melting at the ICB; S represents the case of sulfur with volume of mixing in FeS solution; S) represents sulfur in an ideal

solution of FeS. .Only light impurities considered. Cases a, b, c, and d constrained by limits set by ONeill et al. 1998 . Cases e and f presented to illustrate

.extreme cases, though they exceeded limits set by ONeill et al. 1998 .

estimated 5300 K, and Stacey and Stacey, 1998,.assumed 5000 K .

The ratio of temperatures at the ICB core to pure

. .iron , T rT , is therefore 0.936. Eq. 10 is basedm mc ion the assumption of an ideal dilute solution. Since

both silicon and sulfur have a volume of mixing,

neither silicon nor sulfur forms an ideal solution with

iron at high pressure. Thus there may be some error

in the above DT calculations. Nevertheless, the re- .sults from Eq. 10 listed in Table 6 give a good idea

of the DT effect since the magnitude of the errors

arising from lack of ideality is probably small com-

pared to the variability in DT values arising from the

choice of impurities themselves. Except for case f,

concentrations of impurities described in the six

models satisfy the criteria for a dilute solution. I

conclude that an FeSi core is improbable because if

DT due to crystallization were y2270 K, then theadiabatic T of the core at the CMB would be about

2750 K, close to the adiabatic T of the mantle at the

CMB. This would result in a negligible thermal

gradient across DY

, preventing DY

from being a

useful thermal boundary as required by convection .modelling. ONeill et al. 1998 also found an FeSi

core to be improbable. They stated, . . . neither O

nor a combination of O with Si can dissolve in liquidFe-rich metal in sufficient amounts to account for the

presumed density deficit in the Earths outer core.

8. The melting temperature and the thermal gra-

dient

Values of T and g are needed for calculationsm mof the thermal gradient. The Lindemann law in the

.form given by Eq. 8 is useful because it is ex-

pressed in terms of the density, values of which are . .listed in Table 4. Using Eq. 9 to find T CMBm

.from T ICB s5100 K requires the density ratiom . . 9.904r12.166 . Thus, T CMB s 3450 K seem

..Table 7 for other values of T P . The equation formfinding the value of T along the isentrope anchoredad

.at 5100 K is found from Eq. 2 . Rewriting this

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Table 7

Details of calculating the thermal gradient and values of thermal conductivity

a b y2 b y1 2 c . . .Radius T , g g m s T , f km s dTrd r, Q TWm l ad cy1 . . .core K core K core K km

d e .r 135 GPa , CMB 3450 1.33 10.68 3900 67.33 y0.82 3.4 , 5.4 .r 240 GPa 4400 1.30 7.94 4630 89.95 y0.53 .r 330 GPa , ICB 5100 1.27 4.40 5100 105.38 y0.27

a .From Stacey 1995 .b

From PREM.c

Dimensions, W my1 Ky1 .d y1 y1 .ks28.6 W m km from Stacey 1972 .eks43 W my1 kmy1 from this work.

equation in densitytemperature coordinates, we

have:

E ln Tads g . 11 .l

E ln r

.Since the volume decreases at melting Fig. 2 , it isexpected that g will be smaller than g . The vol-l mume decrease at melting is on the order of 1%. Thus,

we take g s1.29. Since g changes very little atl l .outer core conditions, Eq. 11 becomes:

1.29T r .ad 22s . 12 . /T r .ad 11

Values of T for three radii in the outer core foundad . .from Eq. 12 , including T ICB , are shown inad

Table 7. The core temperature at the CMB is 3900

K. These values are close to those presented by .Braginsky and Roberts 1995 in their Table E2. .With these data, dTrd r can be found using Eq. 1 ;

.values of dTrd r for the three selected radii arealso shown in Table 7.

Now that g and T for the core at the ICB havem mbeen found, there is sufficient information to deter-

mine the heat of crystallization, DH . The formulam .for DH , derived from Anderson and Duba 1997 ,m

is:

DV Km S mDH

s,

m 2 g y1r3 1 qag T . .m m m

where DV is the volume change at melting, K atm Sm .330 GPa is 1370 GPa, according to PREM, a ICB

y5 y1 .s1.7 = 10 K Stacey, 1995 , and, from thediscussion above, g s 1.3 and T s5100 K.m m

. y6Anderson and Duba 1997 found DVs 2 = 10m3 kgy1 from Hugoniot data, so that DH s 1.38m

= 106 J kgy1 from the above formula. To getpower, we must know the age of the inner core t in

. .billion years . Labrosse et al. 1997 use ts1.7, and .Stacey and Stacey 1998 use ts2.3. For ts 2, the

power, Q , is 1.5 TW. Glatzmaier and Robertsm .1996 estimated Q

s3 TW. Stacey and Stacey

m . 1998 estimate a much smaller value of DV they.find DVrVf1.1% , which would depress the value

of DH to 0.5 = 106 J kgy1 and that of Q to 0.4m mTW.

9. The thermal conductivity in the core

We have determined g and dTrd r across thecore, but we also need the value of thermal conduc-

tivity to find the gradient power, Q , across the corec

side of the CMB. According to the Fourier law, .power is equal to Eq. 1 multiplied by area:

dTQ sA k , 13 .c c c

dz

where A is the surface area of the core, and k isc cits thermal conductivity. There are two recent esti-

mates of the value of k . Braginsky and Robertsc . y1 y11995 proposed 40 W m K ; Labrosse et al. . y1 y11997 proposed 60 W m K . Other estimates of

y1 y1 .k are 35 W m K Buffett et al., 1996 , 40 Wcy1 y1 . y1 y1m K Stevenson, 1981 ; and 28.6 W m K

.Stacey, 1972; Stacey, 1992, p. 331 .

In order to justify the magnitude of these values

of k for the core, we need to explore the theoretical

and experimental background by which k is ob-ctained. The chief problem for core physics is that

values of k are needed at temperatures very muchclarger than most measurements can be taken. Data on

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.k at P s0 or at small pressure even at high T arebeset by several transitions, as can be seen by exam-

ining the P s0 isobar in Fig. 2. In order to find kof the core, we need data at high P and T near the

path shown by the Hugoniot in Fig. 2. A physical

principle called the WiedemannFranz ratio, by

which the value of thermal conductivity in metals is

found from measurements of the electrical conduc-

tivity, is very helpful.

Electrical conductivity, s, can be measured more

easily at high pressure than thermal conductivity, k,

because the latter is a transport property requiring

accurate measurements of gradients, whereas the for-

mer requires the simpler measurements of current

and field. The WiedemannFranz ratio, by which k

is calculated from s, is founded in the classical free

electron model of atoms. A summary of that theory .following the derivation of Joos 1958 is presented

in Appendix B, and the result is given by:22k p ke

s . 15 . /s 3 ek is the electronic contribution to the thermal con-eductivity, s is the electrical conductivity, k is the

Boltzmann constant, and e is the electrons charge. .The quantity on the right of Eq. 15 is often

called the Lorenz number, KL : note that it is com-

posed only of fundamental constants and numbers.

KL can be simplified by including in the thermal

conductivity equation the magnetic diffusivity . 5 y1Braginsky and Roberts, 1995 , hs8 = 10 s ,where h is in units of m2 sy1 and s is in units of S

my1. For this case, k sKL = Trh and KL s 0.02eW m sy1 Ky1. The WiedemannFranz ratio is

remarkable because, though it is derived from free

electron theory, none of the parameters of that theory

are retained. Many physicists early in the century did

experiments to see whether the WiedemannFranz

ratio is independent of T and of the metal tested. An .example of such a test is given by Sprackling 1991 ,

p. 305, who showed that sr

kT is the same for the

metals copper, silver, lead, and zinc over a broad

range of temperature. The WiedemannFranz ratio is

also very useful for core physics because k iseproportional to sT at all pressures and temperatures.

Thus if s is measured at a high T, then k is

measured at the same high T. The WiedemannFranz

ratio provides a very useful method for extrapolation

of measurements made at accessible temperatures to

core temperatures. Further, there is a suggestion that

the measurement of the effect of an impurity on

electrical conductivity may be independent of the

kind of impurity, depending only on concentration.

The electrical conductivity at pressures along the .Hugoniot was measured by Matassov 1977 on sam-

ples of iron with various levels of silicon impurities.

His shock wave results are reproduced in Fig. 8. The

plot shows s vs. P along the Hugoniot up to 140

GPa, barely within outer core pressure. On the

Hugoniot the temperature increases with P, but at .maximum 140 GPa , T ; 2100 K for an ironsili-H

con alloy, while the core temperature is about 3900 .K see Table 7 . I assume that the effect of one mole

of Si on electrical conductivity is the same as that of

one mole of S or one mole of oxygen. Thus a mole .fraction impurity level of roughly 0.150 Table 6 is

Fig. 8. Electrical conductivity of Fe with various levels of silicon

impurities at high-pressure shock conditions measured by Matassov . .1977 up to 140 GPa . The FeSi curve for x s0.181 is usedSito obtain the electrical conductivity, ss8.5=103 mho cmy1 s105 S my1 , at maximum P. Since T of the Hugoniot is about

.2200 K, the value of s must be corrected downward to obtain .core temperatures ; 3900 K , and must be further corrected

downward to go from the solid to the liquid state.

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needed to achieve the 10% core deficit level. The

FeSi curve in Fig. 8 for x s 0.181 is used to findSiss 8.5 = 10 5 mho my1 s 8.5 = 10 5 S my1.Matassov reports that in the high pressure range s

scales as 1rT, so s at 140 GPa and the solidus T is5 . 5 y1 8.5 = 10 2100r3900 s4.6 = 10 S m follow-

.ing Braginsky and Roberts, 1995 , but this is still in

the solid state. Another reduction in s is required

because melting always further reduces the conduc-

tivity. Assume a further 10% drop due to melting,

obtaining ss4 = 10 5 S my1. Thus, for Ts3900 Kand hs 2 m2 sy1 , k s0.02 = 3.9 = 10 3r2 s39 Wemy1 Ky1.

.In his Table 18.8, Matassov 1977 lists g for the . 5 y1Earths core as 7.30 4.87 = 10 S m , so appar-

ently he did not reduce s due to melting as was .done by Braginsky and Roberts 1995 . Conse-

quently he finds k for the core to be higher, 54 We

my1 Ky1. .Gardiner and Stacey 1971 , using data then avail-

able on ironsilicon alloys, extrapolated the resistiv-

ity of pure liquid iron at low pressure to core temper-

atures and found, after correcting for the expected

impurity concentrations, that ss3 = 10 5 S my1 ,y1 .corresponding to k s28.6 W m K Stacey, 1992 .e

.Secco and Schloessin 1989 made measurements

of electrical conductivity on solid and molten pure

iron up to 7 GPa in a large volume press and found

ss7.8 = 10 5 S my1 for pure iron at this pressure.

They reasoned that this value could be assumed to bethe same as the conductivity of the iron diluted by

impurities at outer core P and T. Taken at face

value, this leads to k f77 W my1 Ky1, which iseprobably too large. References to the value of s at

.core conditions listed in the table of Matassov 1977 ,

may not be corrected for temperature between Hugo-

niot T and core T and for the solidliquid state

transition. This could lead to values on the order of

k s 6070 W my1 Ky1.eThe value of k s3 9 W my1 Ky1 seems ae

reasonable upper limit for the core. To this must be

added the lattice contribution, k , so that:l

k s k q k . 16 .c e l

Taking k to have the same value as the lattice k inly1 y1 the deep lower mantle, k f4 W m K Kieffer,l

. y1 y11976 . Therefore, I take k s43 W m K .c

10. Conductive heat from the core to DY

and the

mantle

Choosing k s28.6 W my1 Ky1 as the lowestclimit and k s43 W my1 Ky1 as the highest limitcfor the outer core, one calculates the limits in con-

.ductive heat flow from the core by Eq. 13 .

A is listed in Table 4. The two limiting values ofcthe present-day power, Q , leaving the core alongcthe gradient are 3.4 and 5.4 TW, or Q s4.4 " 1cTW. According to the curve in Fig. 2 of Buffett et al. .1992 , the power leaving the core is Q s 5 TWc .corresponding to their ts2 billion year lifetime

.curve .

The power from the surface of the Earth is 44 =12 . .10 W 44 TW , according to Pollack et al. 1993 .

Therefore the conductive power flowing into the

mantle is about 10% of the surface power leaving

Earth.c .On the core side of the CMB, T 2970 km s 3900

K, as given by T in Table 7. Stacey and Loperad . Y . y11983 found dTrdZ D s9.6 K km . The changein temperature across D

Yis thus f1440 K for a

thickness of DY s 150 km. Therefore, on the mantle

m .side of the CMB, T 2820 km f 2460 K, which .agrees with the value Brown and Shankland 1981

found from their isentrope of the lower mantle. .The Stacey and Loper 1983 thermal gradient in

DY

is 12 times the thermal gradient of the core at the

CMB, and therefore DY

would conduct the samepower as the core if k Y is 1r12 that of k , e.g.,D ck Y f3.7 W my1 Ky1. This suggests that DY isDcomposed primarily of mantle material with little

core material contained in it. I therefore conclude

that DY

is a thermal boundary layer, not a composi-

tional boundary layer.

11. Thermal conductivity and the power flowing

into the lower mantle

It is commonly accepted that the thermal conduc-

tivity in the deep lower mantle is about 45 W my1

y1 .K . Kieffer 1976 estimated that k s 4.2 Wlmmy1 Ky1 by calculating the heat transfer due to

phonons in a dielectric solid. If the deep mantle has

this value of k , then the lower mantle couldlmconduct less than 4.2r43s11% of the cores con-

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ductive power, leaving one to find some other mech-

anism of heat transfer for the large amount of resid- .ual power 3.9 TW . However, in the assumption

that k is the same as the lattice thermal conductiv-lmity, there is the implicit assumption that conductivity

is determined by phonon transfer alone. This implicit

assumption will probably result in an underestima-

tion of the capacity of the lower mantle to conduct

power towards the surface. .In addition to lattice thermal conductivity, k ,L

there is also likely radiative transfer connected with

infrared electromagnetic waves arising from the opti-

cal properties of a dielectric solid. The radiative

thermal conductivity associated with infrared electro-

magnetic waves, k , is given by Zharkov and Tru-R .bitsyn 1978 , p. 57, according to the equation:

2 316 s)n T

k s " ,R 3 a

where s) is the StefanBoltzmann constant, a is

the absorption coefficient, and n is the refractive

index. Note the T3 factor, which has a pronounced

effect at lower mantle temperatures. In a recent .report, Hofmeister 1998 showed that the optic

modes are the primary mode of heat transport deter-

mining thermal conductivity at lower mantle condi-

tions, contrary to the assumption made in the calcula-

tion of k where phonons are considered as thelm

only transport mechanism. Taking into account re-cent infrared reflectivity measurements and correct-

ing for the pressure dependence of the constants, she

found that k for the deep lower mantle is suffi-Rciently large that k q k is about 2.3 times theL Rvalue of k .L

Thus we have sufficient evidence to suppose that

k is roughly 10 W my1 Ky1 rather than about 4.2lmW my1 Ky1. Taking the thermal gradient of the

y1 lower mantle to be 0.5 K km Brown and Shank-.land, 1981 , 0.6 TW and perhaps more is transported

conductively towards the lithosphere. Could the

residual of the cores conductive power, 3.8 TW, be

transported upwards in the lower mantle by convec-

tion? . Tackley et al. 1993, 1994 and Tackley 1995,

.1996 have shown that the convective heat in the

mantle is transported to the surface by a number of

mantle plumes originating at the base of DY

. When

these plumes interact with the lithosphere, they cre-

ate hotspot intrusions that can be studied by geo-

physical methods. The number of plumes that are

currently interacting with the Earths surface equals

the number of known hotspot intrusions. .G. Schubert private communication reported that

he, Don Turcotte and Peter Olson, who are writing a

book on mantle geodynamics, have identified a total

of 38 current hotspot intrusions. They have calcu-

lated that these intrusions collectively transport 2.3

TW of power to the Earths surface. Thus the lower

mantles conductive heat, 0.6 TW, and the convec-

tive heat from the known hotspot intrusions, 2.3 TW,

yield 2.9 TW, close to the lower limit of our sug-

gested core conductive power, 3.4 TW. In addition

to the plumes evidenced by hotspot intrusions, there

are, however, plumes that have not yet been made

evident by surface activity, as revealed by the snap-

shots of plume activity given by numerical simula-tion done by Tackley and his colleagues. Plumes

grow from DY

, slowly ascend to the surface, contact

the surface, detach from DY

, and eventually disap-

pear, but, at any given time there are more active

plumes than the number of plumes in contact with .the surface giving hotspot intrusions . Thus, a rea-

sonable assumption is that the number of hot plumes

that have not yet made contact with the surface is

about half of the number that have been identified

through surface activity. I assume that the total

convective power arising from DY

via the plumes is1.5 times the power for known hotspot intrusions

calculated by Schubert and his colleagues that is, a.total of about 3.5 TW . Under this assumption, the

total estimated power coming from the CMB both.convective and conductive would be about 4.1 TW,

close to the mean of the upper and lower limits of .conductive core power 4.4 TW . I therefore conjec-

ture that a rough balance exists between the conduc-

tive power transmitted from the core to the mantle

and the power transferred by conduction and convec-

tion from the upper mantle towards the lithosphere.

12. The convective heat in the outer core

The outer core must be in a state of convection so

that the Earths magnetic field can be maintained.

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Maintenance of the magnetic field requires a smallfraction of a TW 0.2 TW, according to Stacey and

. .Loper, 1983 in mechanical energy. Loper 1984 .and Loper and Roberts 1983 have proposed that the

mechanical energy requirements may be largely sup-

plied by vigorous convective motion driven bycompositional buoyancy of the ejecta enriched

.with light elements from the inner core to the outer

core. The convective power arising from the core

may be small compared to its conductive adiabatic .power. Buffett et al. 1996 emphasized that a

modest heat flux in excess of that conducted down

the adiabatic gradient is sufficient to power the

geodynamo, even in the absence of compositional .convection and latent heat release. Gubbins 1977

stated that the gravitational energy released by

rearrangement of matter in the core is completely

converted to magnetic dissipation enabling a large

magnetic field to be generated with a low heat flowfrom the core. Although convective heat may be

transferred from the core to DY

and then on to the

lower mantle, there appears to be no convincing

evidence that the cores convective power is large

compared to that conducted down the adiabatic gra-

dient. Any excess power may be returned to the core .by compositional convection Loper, 1978a or con-

tained by a subadiabatic thermal gradient near the

boundary of DY

, as proposed by Labrosse et al. .1997 . In my proposed model, the mantle success-

fully transports away the conductive power arisingfrom the cores adiabatic gradient. Therefore, I as-

sume that a large subadiabatic gradient is not neces-

sary and that any power from core convection alone

in excess of the power conducted down the adiabatic

gradient will be returned to the core by composi-

tional convection.

Acknowledgements

I acknowledge helpful comments from Frank

Stacey, Stanislav Braginsky, Paul Roberts, Gary

Glatzmaier, Jerry Schubert, and Paul Tackley on the

geophysical aspects of an early version of this paper.

I acknowledge helpful comments from Dave Sher-

man, Giulio Ottonello, Surendra Saxena, Francois

Robert, and John Wasson on its geochemical aspects.

The critiques of two unknown reviewers were also

quite helpful. Support from NSF grant EAR-96-

14654 acknowledged. Support by ONR acknowl-

edged. IGPP contribution no. 5073.

Appendix A

Calculation of the temperature depression requires

the assumption of thermodynamic equilibrium be-

tween the solid and liquid phases. As a beginning,

assume that all impurities reside in the liquid. Be-

cause the condition of thermal equilibrium requires

that the Gibbs free energy be equal for both phases .DG s0 , we must deal with the chemical potentialof the pure liquid and the pure solid. Stevenson .1981 showed that if all the impurities reside in the

liquid, the thermal equilibrium arising from DG s0

is given by:

ml P ,T qRT ln 1y x s ms P ,T , A1 . . . .0 i ,l 0

where m is the chemical potential, and x is the0 i,lmolar fraction of the ith species in the liquid and

where the super and subscripts l and s refer to the

liquid and solid, respectively.

Note that the weight percent of an impurity ele-

ment was used in calculating the fraction of the

cores density deficit that it can account for. But

now, using thermodynamic functions to obtain freez-

ing depression, we need to express concentrations inmolar fraction because molar fraction identifies the

number of atoms on a particular thermodynamic site.

The molar fraction of the impurities in the six cases

is listed in Table 6. .Eq. A1 is oversimplified for use in core theory

because we now are sure that light impurities exist in

the solid inner core. As the core solidifies, a large .fraction less than unity of the impurities passes into

the liquid, leaving a small fraction behind in theinner core see, for example, Jephcoat and Olson,

.1987 . Impurities in the solid inner core are ac-counted for by adding a term to the right of Eq. .A1 , giving:

ml P ,T q TR ln 1 yx . .0 i ,l

sms P ,T q TR ln 1 yx , A2 . . .0 i ,s

where x - x .i,s i,l

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Assuming that v is the fraction of all impurities .existing in the outer core, then 1 yv is the fraction

residing in the inner core. We make the approxima- .tion that the second term on the left of Eq. A2 and

.the second term on the right of Eq. A2 are multi- . plied by v and 1y v , respectively it would be

more accurate to multiply all x by v and all xi,l i,s .by 1 yv , but for small concentrations the approxi-

.mation used here is sufficiently accurate . Thus, Eq. .A2 can be replaced by the approximation:

ml P ,T q vRT ln 1y x . .0 i ,l

sms P ,T q 1y v RT ln 1y x . . . .0 i ,s

Using the general relationship Dm sDHy TDS,where H is enthalpy and S is entropy, and referenc-

.ing Chapter 9 of Landau and Lifshitz 1958 as the

. .authority, Stevenson 1981 replaced Eq. A1 elimi-nating ms, leaving the following for the freezing

depression, DT, of the outer core:

RT)

DTsyv ln 1 yx .i ,l /DSvT)

sy ln 1 yx , A3 . .i ,lln 2

where DS is the entropy of melting and v has been

incorporated. For the core, T) is 6100 K. The

parameter v takes into account the fact that the inner

core contains some impurities. Thus:

T)vDTsy ln 1 yx . A4 . .i ,l

0.693 .

.Anderson and Duba 1997 calculated that at the . y3ICB, Dr freezing s200 kg m . Masters and

.Shearer 1990 found from seismic data on the core . y3 that Dr ICB

s550 kg m , leaving Dr chemical

. y3differentiation s350 kg m . The outer core has adensity 1270 kg my3 lower than that of pure iron so

that the concentration of impurities in the outer core

accounts for 1270y350 or 920 kg my3. Thus vs920r1270s 0.72. Within the approximations cre-ated by the uncertainty in exact chemical composi-

tion, the value of v is close enough to the value in

.the denominator of Eq. A4 that vrln 2 can be .replaced by unity. Thus, Eq. A4 is simplified to:

DTsyT) ln 1 yx . A5 . .i ,l

Appendix B

The free electron model of metals, which pre-

ceded wave mechanics, was very successful in the

formulation of a number of properties of metals,

including electrical conductivity. In this classical

model, the valence electrons are able to move about

the lattice freely having intermittent reactions with

the lattice. Five parameters are of significance in the

free electron theory of metals: the drift velocity of

the electrons, y; the relaxation time, t, measuring

the time between collisions with the lattice; thenumber of electrons per unit volume, n; and the

mass and electrical charge of the electron, m and e.

The electrical conductivity, s, of a metal is the

parameter relating the current j to the electrical field

E, where:

j ssE. B1 .

Separate solutions of y with E giving ys . .errm E and y with J Jsney combined with Eq. .B1 eliminate the parameters J, E, and . We then

find the electrical conductivity in terms of the pa-

rameters of the free electron model:

ssne2trm. B2 .

We proceed to find the thermal conductivity of

the free electron model. In solids, k is found from .the Fourier law, Eq. 13 , so that:

Q dzk solid s . . /A dTAnalogous to this is the thermal conductivity of

gases, which, according to the kinetic theory of

gases, is:

k gases s 1r3 Cul, . .

where u is the average particle velocity in the direc-

tion of heat flow, l is the mean free path, and C is

the contribution of each gas molecule to the specific .heat per unit volume. In the equation for k solid ,

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we note that k is inversely proportional to the

temperature gradient, dTrdz. The thermal gradientis also found in k of gases, located in C, the specific

. . .heat: Cs dErdT s dErdz dzrdT . Thus k ofgases is also inversely proportional to the thermal

.gradient. The 1r3 comes from integration of thecosine angle, because the particles will be travelling

in all directions with the velocity . The need to find

the vector component in one selected direction leads,

by integration, to the factor 1r3. The specific heat,C, is the product of n and k, where n is the number

of molecules per unit volume, and k is the energy of

one particle with one degree of freedom, the Boltz-

mann constant.

The conduction electrons are in a state of chaotic

thermal agitation, somewhat like the atoms in an

ordinary gas, except that l of free electrons is the

measure of the distance between collisions of the

electron with the lattice. The equation for k for freeeelectrons is the same as for gases:

1k free electrons s nkul, B3 . .e /3where k is the electronic contribution to the totalethermal conductivity, the subscript emphasizing that

the lattice conductivity, k , also has yet to be takenl . .into account. Dividing Eq. B3 by Eq. B2 :

k 1 nkulmes .2s 3 ne t

Using tslru:

k 1 ke 2s u m. B4 .2 /s 3 eUsing the theorem of equal partition of thermal

and kinetic energy:

12mu skT.

2

.Eq. B4 is transformed into the WiedemannFranz

ratio, giving:

2k 2 kes . B5 . /sT 3 e

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