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The identity and quantity of the light 1
matter on each side of the Earth’s inner 2
core boundary 3
A. AITTA 4
Institute of Theoretical Geophysics, Department of Applied 5
Mathematics and Theoretical Physics, University of Cambridge, 6
Wilberforce Road, Cambridge CB3 0WA, UK 7
Tel: 44 1223 337031 8
Fax: 44 1223 765900 9
e-mail: A.Aitta@damtp.cam.ac.uk 10
11
Light elements in the iron-rich core of the Earth are important 12
indicators for the evolution and dynamics of our planet. However, 13
there is a longstanding controversy surrounding the identity and 14
quantity of the light elements. The theory of tricritical phenomena 15
has been recently employed as a precise theoretical framework to 16
study solidification at the high pressures and temperatures of the 17
Earth. When combined with the most reliable iron melting data 18
and the seismic data, the theory provides the solidification 19
temperature at the inner core boundary (ICB) pressure for both 20
pure iron and for the alloy of iron and light elements in the actual 21
core melt. Here the theory is used further to find a value of about 5 22
mol.% for the amount of light matter in the core melt at ICB and 23
to calculate the density of liquid pure iron at its melting 24
temperature. These give the density of the light matter consistent 25
with it being MgSiO3, and this identification is supported by sound 26
velocity against density analysis. In addition, using the pure iron 27
density and volume contraction obtained from the same theory, 28
one finds the density of pure solid iron at its melting temperature. 29
The sound velocity against density analysis for different iron 30
crystal structures and various light compounds possibly present in 31
the solid at the ICB shows that the most likely structure here is 32
iron in its bcc
δ
phase with 4 wt.% Mg2SiO4 as light impurity. This 33
finding is close to the conclusion of recent shear velocity 34
calculations for Fe with Mg. The light matter identified here 35
challenges the commonly assumed exclusion of mantle material in 36
the core and this issue is discussed. 37
Keywords: light element, Earth’s core, tricritical point, high pressure, iron melting 38
temperature, iron melting density 39
1. Introduction 40
Although the Earth’s core is impossible to reach directly, seismic wave scattering can 41
provide information on the material properties in the deep layers of the Earth. By 42
comparing these to the experimental high pressure studies of various materials, Birch 43
(1952) was the first to be able to deduce the composition of the Earth’s core: “The 44
inner core is most simply interpreted as crystalline iron, the outer part as liquid iron, 45
perhaps alloyed with a small fraction of lighter elements.” He also found (1961a; b) 46
that sound velocities depend linearly on densities for materials relevant to the Earth’s 47
interior, a behaviour known later as Birch’s law. In addition, the change in 48
composition in isostructural materials shows a linear relation between sound velocity 49
and density. The linear behaviour of compositional change versus density has been 50
demonstrated, for instance, by Chung (1970). Birch’s law has been commonly used 51
as a tool for studies at high pressures. However, the experimental study by Lin et al. 52
(2005) demonstrated some sound velocity decrease with increasing temperature for Fe 53
at pressures up to 73 GPa and temperatures up to 1700 K, but also expected the 54
temperature effects to become suppressed at higher pressures due to the smaller 55
thermal expansion there. The validity of Birch’s law has been considered at some 56
length by Karato (2008), who concludes that Birch’s law is satisfied if the intrinsic 57
temperature derivative of the elastic constant is negligible, and if not, Birch’s law 58
applies only approximately. Experimentally, Birch’s law holds up in a recent study of 59
cobalt by Antonangeli et al. (2008), who discusses in detail the high pressure iron, its 60
compounds and the supporting numerical study by Vočadlo (2007) together with the 61
work by Lin et al. (2005), and concludes that “in the absence of simultaneous high- 62
pressure and high-temperature data, … (Birch’s law) is the most suitable 63
representation for extrapolations to very high densities”. Numerical calculations by 64
Tsuchiya and Fujibuchi (2009) show that both P and S wave velocities depend 65
linearly on the density for three iron phases and their Fe-Si alloy. Also, the 66
calculations by Kádas et al. (2009) at high temperatures (5000 K and 7000 K) for 67
sound velocities of bcc Fe-Mg alloys show a regular trend for each temperature, with 68
a small increase with increased temperature and pressure, as presented against 69
pressure, but their results are not shown against density. Interestingly, Lin et al. 70
(2003) found experimentally that Fe-Ni and Fe-Si alloys follow Birch’s law, but 71
Dubrovinsky et al. (2009) say their results indicate that Fe alloyed with 10 % Ni does 72
not. Unfortunately the latter data are on a narrower interval, only for pressures up to 73
50 GPa with three temperatures up to 850 K. Some error bars at the lowest 74
temperature are comparable to the range of the scatter. Again, the large scatter present 75
in the data of Lin et al. (2005) excluded it from the linearity determination by Badro 76
et al. (2007), whose results are employed in the later analysis in this paper. 77
Several light elements such as H, C, N, O, Mg, Si, P, S, K and some compounds 78
of these have since been considered as present in the core, but the identity and 79
quantity of the light matter has remained unresolved (Stevenson, 1981; Jeanloz, 1990; 80
Poirier, 1994; Li and Fei, 2003; McDonough, 2003; Wood et al., 2006; Georg et al., 81
2007). The seismic PREM data (Dziewonski and Anderson, 1981), known to an 82
accuracy of about 1 %, demonstrates that the density difference between solid and 83
melt at the Earth’s ICB is greater than that expected from contraction of matter in 84
solidification, indicating that the core melt has some light elements alloyed with iron, 85
but the densities at high pressure for iron and for light elements are much less well 86
established. Direct experimental study of material densities at high pressures and 87
temperatures is difficult and the estimates are based on either shock compression data, 88
whose temperatures are not well-known, or extrapolations using thermal expansion 89
measurements at much lower temperatures (see short summaries in Chen et al., 2007 90
and Dewaele et al., 2006). Theoretical simulations give approximate densities at these 91
extreme pressures, but the results seem to be changing with improving techniques 92
(Alfè et al., 2002; Koči et al., 2006). The uncertainty of these results is increased by 93
the unknown crystal structure which needs to be predetermined for these calculations, 94
and the uncertainty of the iron melting temperature, which, as found by the same 95
technique, Alfè et al. (2007) quotes to be about ± 600 K. Reliable and accurate 96
estimates of the densities and sound velocities of iron and light elements and 97
compounds up to the core pressures and temperatures are essential for success in 98
identifying the core’s light matter. 99
Here a theoretical approach is employed together with the best experimental 100
data. The theory of tricritical phenomena is used again, extending previous work 101
(Aitta, 2006; Aitta, 2009) which provides an estimate of the melting temperature of 102
pure iron over the whole range of core pressures, and also the temperature of iron-rich 103
melt as occurs at the Earth’s ICB. Here this theory is applied to calculate the 104
corresponding mole fraction of light matter in the melt, with no input from the iron 105
density. Next, again using the theory of tricritical phenomena, the densities of pure 106
iron, for both solid and melt at the melting temperature, with their uncertainties, are 107
found at core pressures, using the constraint given by PREM on the density at the 108
ICB. The calculated quantitity of the light matter in the melt allows one to identify 109
this light material to be MgSiO3. Next, the graph of sound velocity Vp against density 110
ρ
is employed for both melt and solid. The PREM data point (
ρ
, Vp) is considered to 111
be, following Birch’s statement of the composition change in isostructural materials 112
(1961a), on the same line and between the pure iron data point and the pure light 113
component data point, all at ICB conditions. From this one finds for the melt the same 114
light matter, MgSiO3, as was identified before. For the solid, the identity and quantity 115
of the light matter corresponding to each of the three iron crystal structures are found, 116
the most likely combination being the
δ
phase with Mg2SiO4. This phase compares 117
well with the shear velocity calculations of Fe-Mg by Kádas et al. (2009). 118
The identification of the light matter in the melt, MgSiO3, calls for further 119
investigations of the solubility and stability of about 1-9 wt.% of MgSiO3 melt in 120
molten iron at high pressures (around or above 140 GPa) and high temperatures 121
(around or above 4000 K). We have some experimental evidence that solid MgSiO3 122
dissolves in molten iron at high pressures. Knittle and Jeanloz (1991) have measured 123
the amounts of O, Si and Mg in molten iron foil in contact with crystalline magnesium 124
perovskite at high pressures. Their maximum values reached about 16, 10.5 and 7.5 125
wt.%, respectively, all found at 1 µm from the interface, thus all these three 126
components have some solubility in liquid Fe at high pressure (> 70 GPa). Takafuji et 127
al. (2005) studied the solvability of (Mg,Fe)SiO3 perovskite in liquid iron and found 128
at 97 GPa and 3150 K: 5.3 wt.% for O, 2.8 wt.% for Si and a “measurable amount” 129
for Mg. Sakai et al. (2006) found up to 6.3, 4.0 and 0.36 wt.% dissolved, respectively, 130
when metallic iron was coexisting with magnesium postperovskite at 139 GPa and 131
3000 K. Ozawa et al. (2009) investigated (Mg,Fe)SiO3 perovskite and molten iron up 132
to 146 GPa and 3500 K and found 3.9 wt.% for O and 3.5 wt.% for Si in quenched 133
liquid iron but they also say: “It is known that Mg is selectively removed during the 134
electron beam irradiation when the electron beam density is higher than the critical 135
value”. However, even though we do not yet have experiments on the solvability and 136
possible ionization of MgSiO3 with iron when both are molten at high pressures, the 137
solvability is presumably increased. The potential ionization would still maintain the 138
ratio of the light elements unchanged in the whole melt. Numerical calculations have 139
demonstrated that the likely Moon-forming collision melted most if not all of the 140
early Earth and thus provided contact for iron and the main mantle material MgSiO3, 141
both in a vigorously mixed molten state (Canup, 2008). 142
2. The theory of tricritical phenomena for solidification 143
The density of matter under pressure increases until the distance between atoms 144
becomes less than the atomic diameter; then atoms lose their individuality and the 145
matter changes phase from liquid or solid to plasma where nuclei and electrons are 146
free (see p. 317 in Landau and Lifshitz, 2001). This transition from condensed matter 147
to warm dense plasma has been suggested (Aitta, 2006) to have a tricritical point 148
where liquid, plasma and solid phases meet. The melting temperature of material 149
whose melting curve as a function of pressure has a horizontal tangent, was 150
investigated using Landau’s theory of tricritical phenomena (Landau, 1937; Landau 151
and Lifshitz, 2001; Aitta, 2006; 2009). The theory of tricritical phenomena is 152
applicable to a great variety of physical phenomena, including phase transitions 153
between liquid and crystals (Landau, 1937), and it is known to be very accurate in 154
problems in three spatial dimensions (Lawrie and Sarbach, 1984). It is also shown to 155
be quantitatively applicable along the phase transition line in a wide neighbourhood 156
around a tricritical point (Aitta, 1986). In the case of a melting curve, the branching 157
out of the vaporization curve and its end point are likely to have their own 158
neighbourhoods outside the validity regime of the tricritical point and its critical line 159
neighbourhood. However, this leaves a large pressure range from room pressure along 160
the melting curve to beyond the tricritical pressure where the theory should be valid, 161
excluding possibly the limited neighbourhoods of a few points where a different 162
curvature of the melting curve occurs locally in the presence of some triple points 163
related to the boundaries between different crystal structures in the solid phase. 164
One uses an order parameter x to describe first order phase transitions which 165
change to be second order at a tricritical point. x = 0 for the more ordered solid phase 166
which occurs at lower temperature, and in the less ordered phases, the liquid or 167
plasma,
!
x"0
. The Gibbs free energy density is proportional to the Landau potential, 168
which needs to be a sixth order polynomial in x: 169
!
"(P,T;x)=1
6x6+1
4g(P)x4+1
2
#
(P,T)x2+"0(P,T)
. (1) 170
No higher order terms in x appear in since they can be eliminated using 171
coordinate transformations as in bifurcation theory (Golubitsky and Schaeffer, 1985). 172
This method also scales out any dependence on physical parameters of the coefficient 173
of the highest order term. When the melting curve has a horizontal tangent at high 174
pressures, the coefficient g can be taken to depend linearly on pressure P while 175
allowing
ε
to depend on both P and T. g and
ε
are both zero at the tricritical point 176
(Ptc, Ttc). Using bifurcation theory one finds (Aitta, 2009) 177
!
g(P)=4T0(P/Ptc "1)
(2) 178
and 179
!
"
(T,P)=Ttc #T#(Ttc #4T0)(P/P
tc #1)2
(3) 180
where T0 is the melting temperature at P = 0. The thermodynamically stable state 181
occurs at the value(s) of x that globally minimizes
!
"
. For different (P, T) values, 182
!
"(x)
can have one, two or three local minima. These occur either at x = 0 and/or 183
!
x2="g/2 +g2/ 4 "
#
. (4) 184
The temperature TM of the solid-liquid melting transition is determined by the 185
condition that there are three minima of
!
"
, all equally deep. This occurs when g < 0 186
and 187
!
"
=3g2/16
(5) 188
giving from (3) 189
!
TM(P)=Ttc "(Ttc "T0)(P/P
tc "1)2, for P<P
tc
, (6) 190
!
and from (4) 191
!
x2="3g/ 4
. (7) 192
Below TM, the solid phase with x = 0 is the favoured stable state, but the liquid 193
phase (which corresponds to a local minimum with
!
x"0
) can exist as an unfavoured 194
stable state down to 195
!
"
=g2/ 4
with g < 0 (8) 196
where
!
"
changes from having three minima to having only a single minimum at x = 197
0. There the temperature (Aitta, 2009) is 198
!
T
S(P)=Ttc "Ttc P/P
tc "1
( )
2, for P<P
tc
. (9) 199
At P = 0, note that TS = 0, as liquids will always solidify at or above absolute zero. 200
One can make stable liquid to be favoured at temperatures lower than TM by using 201
impurities: for instance, in the familiar eutectic binary phase diagram the melting 202
temperature of the mixture at any concentration is lower than the melting temperature 203
of at least one of the pure end members. 204
Above TM, the liquid state is the favoured stable state, but the solid phase 205
(which corresponds to a local minimum with x = 0) can exist as an unfavoured stable 206
state up to
ε
= 0. The temperature corresponding to
ε
= 0 is 207
!
TL(P)=Ttc "(Ttc "4T0)(P/P
tc "1)2, for P<P
tc .
(10) 208
One can find expressions for the change of volume ΔV and entropy ΔS along the 209
melting line. Using a constant
!
"
as the coefficient of proportionality between the 210
Gibbs free energy and the Landau potential (1) one obtains 211
!
S="
#
G/
#
T="$
#
%/
#
T="$
#
%0/
#
T"1
2$x2
#&
/
#
T
. Since x = 0 for solid, one finds 212
using (7), (2) and (3) that 213
!
"S=3
2#T0(1 $P/P
tc )
. (11) 214
To solve
!
"
one can use the latent heat
!
L=T"S
whose value at P = 0 is denoted by 215
L0. Thus 216
!
"=2
3L0/(T0T0)
(12) 217
giving 218
!
"S=L0(1 #P/P
tc ) /T0
. (13) 219
The volume contraction can now be written as 220
!
"V="S dT/dP =2L0(Ttc #T0)(1#P/P
tc )2/(T0P
tc )
(14) 221
using (13) and (6). 222
The melt densities at the critical temperatures of (6), (9) and (10) are 223
correspondingly for P < Ptc 224
!
"
M(P)=
"
tc #(
"
tc #
"
0)(P/P
tc #1)2
, (15) 225
!
"
S(P)=
"
tc #(
"
tc #
"
S0 )(P/P
tc #1)2
(16) 226
and 227
!
"
L(P)=
"
tc #(
"
tc #
"
L0 )( P/P
tc #1)2
, (17) 228
where
ρ
tc is the density at the tricritical point,
ρ
0 the density at the melting point at P = 229
0 and
ρ
S0 the lowest possible melt density at P = 0, T = 0. The melt density 230
corresponding to the highest possible temperature ( = 4T0) for solid at P = 0 is
ρ
L0 231
which can be expressed as 232
!
"
L0 =4(
"
0#
"
S0 )+
"
S0 =4
"
0#3
"
S0
. (18) 233
Of these, only
ρ
0 is known. However, one can see from (9) that
!
(P/P
tc "1)2=Ttc "T
S
Ttc
. 234
Using the pressure where TS = 4T0 one knows from (18) that the density there is 235
!
"
S=4
"
0#3
"
S0
. Thus one can obtain from (16) 236
!
"
S0 =
"
0Ttc #
"
tcT0
Ttc #T0
. (19) 237
The values of Ttc and Ptc have been found for iron previously (Aitta, 2006; 238
2009) using a fit to good experimental data. To find
ρ
tc one also needs some 239
experimental input. The known PREM melt density at the ICB, denoted here as 240
!
"
melt
PREM,
is expected to take the value of
ρ
S(PICB) since convection in the core fluid has 241
lasted for a very long time and thus it has been able to adjust the temperature and 242
density differences inside the fluid to a minimum. This allows one to find using (16) 243
and (19) 244
!
"
tc =
"
melt
PREM #
"
0Ttc /(Ttc #T0)2(P
ICB /P
tc #1)2
1#(P
ICB /P
tc #1)2#T0/(Ttc #T0)(P
ICB /P
tc #1)2
. (20) 245
3. Results for pure iron and the melt in the Earth’s core
3.1. Temperatures
Figure 1. Iron melting temperature TM as a function of pressure P as calculated from equation (6)
(upper solid line) with its maximum uncertainty (dashed lines) and the consistent experimental data
whose selection details are discussed in Aitta (2006). Open symbols are for the most accurate static
measurements (squares Shen et al., 1998; diamond Shen et al., 2004; triangle Ma et al., 2004). Solid
symbols show all the supporting shock wave results (downward triangles Sun et al., 2005; triangles Tan
et al., 2005; diamond Nguyen and Holmes, 2004; square Brown and McQueen, 1986). Error bars are
from the original sources when given. The limit for liquid temperature, TS, as a function of pressure P
as calculated from equation (9) (lower solid line) with its maximum uncertainty (dashed lines). The
core boundary pressures are shown by vertical dotted lines.
For pure iron T0 =1811 K. The value for (Ptc, Ttc) can be estimated using the most
reliable high-pressure data for TM including only the most accurate static data with all
supporting shock data (the same data as selected earlier by Aitta, 2006, see the caption
2000
3000
4000
5000
6000
0 50 100 150 200 250 300 350
Temperature (K)
Pressure (GPa)
CMB ICB
T
M
T
S
of Figure 1 for references). This gives Ptc = 800 ± 100 GPa, Ttc = 8600 ± 800 K, with
the signs of the errors correlating. Fig. 1 shows the best fit (solid line) with its
uncertainty (dashed lines) and the data with their error bars. Also, TS is shown in Fig.
1 (the lower solid line), with its uncertainty (dashed lines). From (6) and (9), one finds
that at P = 329 GPa, the ICB pressure,
!
TM=6290"100
+80 K
and
!
T
S=5670"40
+30 K
, whose
uncertainties follow from the uncertainty in Ptc and Ttc. This pure iron melting
temperature TM is between the cluster of estimates around 6000 K obtained by various
methods (Anderson et al., 2003) and 6350 ± 600 K found in ab initio calculations
(Alfè et al., 2007). TS determines the absolute lower limit down to which any
impurities in the iron-rich core melt can lower the melting temperature. Its value is
very close to 5700 K, the value inferred for the temperature at the ICB from ab initio
calculations on the elasticity properties of the inner core (Steinle-Neumann et al.,
2001), and is in the range 5400 K – 5700 K reported by Alfè et al. (2007). The
temperature difference
!
TM"T
S=620"130
+110 K
at ICB agrees with the estimate 600 K to
700 K by Alfè et al. (2007) but is more than twice the 300 K used in the rather recent
energy budget calculations of the core (Lister, 2003).
3.2. Critical concentration at ICB
For sufficiently dilute solutions, perhaps up to about 10 mol.%, an iron-dominated
alloy can be approximated by an ideal solution. Then the iron concentration
!
cFe
can
be related to the alloy’s chemical potential and that of pure iron by
!
µ
alloy =
µ
Fe +RT lncFe
(see, for instance, Janssens et al., 2007). On the other hand, the
chemical potential is the Gibbs free energy per mole and thus proportional to the
Landau potential (1):
!
µ
="# =2
3L0#/(T0T0)
, where the proportionality coefficient
is taken from (12). When the liquid temperature equals TS,
!
"alloy ="0
(corresponding
to having three equally deep minima in the potential) and
!
"Fe ="0#g(P
ICB)3/ 48
which follows from (8) and (1) with x2 = – g(PICB)/2 given by (4) and (8). The
concentration for the impurities is therefore
!
ci=1"cFe =1"exp 8L0P
ICB /P
tc "1
( )
3/(9RT
S)
[ ]
giving
!
ci=5.1"1.6
+1.4 mol.%
, where the
uncertainties correspond to the maximum uncertainty in Ptc and TS. Thus, by this
method, one can find the concentration of light elements at the ICB without using the
iron density, which is not known very accurately. The small fraction of Ni expected
also in the core is here assumed not to change the iron behaviour due to the
remarkable similarities of Fe and Ni atoms.
3.3. Critical densities at ICB
Using the PREM values PICB=328.85 GPa and
!
"
melt
PREM(P
ICB)=12.16634 g/cm3
in (20)
one finds that the iron density at its tricritical point is
ρ
tc = 16.1± 1.2 g/cm3, where the
uncertainty corresponds to the uncertainty in Ttc and Ptc. This density is consistent,
within the uncertainties, with the similar densities at high pressure for iron’s principal
Hugoniot by Brown (2001), Koenig et al. (2005) and the older data tabulated by
Kogan et al. (2003) even though the temperatures they correspond to are not known
and need not be the melting temperature.
Figure 2. Densities as a function of pressure. Density of solid iron
ρ
M,solid (dotted line) and liquid iron
ρ
M (thin solid line) with its maximum uncertainty (dashed lines) at the melting temperature, the limit
for liquid density
ρ
S (dash-dotted line) and PREM (Dziewonski and Anderson, 1981) density (thick
solid line). Square shows the density found here for the light matter MgSiO3 in the melt at the inner
core boundary. To demonstrate its reasonability it is connected by a quadratic fit to PREM lower
mantle densities (long-dashed line).
Figure 2 presents
ρ
M from (15) as a thin solid line and its maximum
uncertainty as dashed lines, together with
ρ
S from (16) with (19) as a dot-dashed line,
and the PREM density data as a thicker solid line. In particular, at ICB,
ρ
M= 13.00±
0.17 g/cm3, about 7 % higher than
!
"
melt
PREM
and close to 2 % higher than the PREM
density of the solid
!
"
solid
PREM
.
One also finds that at the core mantle boundary (CMB) pressure,
ρ
M is less
than 0.5 % below the PREM melt density there, indicating that either the core melt
4
6
8
10
12
0 50 100 150 200 250 300 350
Density (g/cm3)
Pressure (GPa)
!
M
!
S
!
M,solid
!
PREM
hardly has light impurities there or some Ni compensates them in the melt. This is
about 8 % less than obtained from ab initio calculations (see, for instance, fig. 3 in
Koči et al., 2006) which at that pressure also give about 10 % (or 6 % with free
energy correction) greater temperatures for the melting (Alfè et al., 2002; 2007). But
the values of
ρ
M near CMB are in agreement with the similarity of compressional
wave velocity measurements against density in pure Fe and the seismic velocity
profile against density from PREM at the pressures of the upper outer core (see fig. 2
in Badro et al., 2007). This means that light matter brought to CMB by convection is
removed from the melt by solidification accompanied by some iron. From (8) the
melting temperature TM at the CMB pressure is 3945 ± 12 K, very similar to its
seismic estimate 3950 ± 200 K (van der Hilst et al., 2007). This is further evidence
that at the CMB the light components of the core melt are solidifying, as has been
suggested by Schloessin and Jacobs (1980) and Buffett et al. (2000; 2001). Even
though there is presently no experimental evidence concerning the solidification
temperature of relevant matter from iron-rich melt at the CMB pressure, the
temperature must be close to the pure iron melting temperature owing to the very
small amount of impurities.
The density
ρ
M,solid of pure solid iron is obtained from the density of the melt
ρ
M using
!
"
M,solid =A/(A/
"
M# $V)
, where the atomic weight A is taken to be the
standard value for iron on the Earth and ΔV is the volume contraction in solidification
with value of 0.045 ± 0.008 cm3/mol for iron at ICB pressure from (14). This is
approximately in the middle of the range of the following estimates (sources from
Anderson, 2003): 0.0318 cm3/mol (from the liquid and solid density difference
Δρ
m = − 100 kg/m3 in Gubbins et al., 1979 and Masters and Shearer, 1990), 0.0476
cm3/mol (from
Δρ
m = − 150 kg/m3 in Buffett et al., 1996) and 0.04 - 0.06 cm3/mol
(Poirier, 1986). It is slightly less than 0.0548 cm3/mol, based only on the two latter
papers, which is used for
ε
iron by Anderson (2003) and Anderson and Isaak (2002).
ρ
Μ
,
solid is shown in Figure 2 as a dotted line. In particular, at ICB,
ρ
Μ
,
solid = 13.14 ± 0.20
g/cm3 whose uncertainty only takes account of the uncertainties in
ρ
M and ΔV. These
results give density deficits relative to pure iron of 2.8 % in solid and 7.4 % in melt,
both being in their traditionally expected range, namely for inner core 2 ± 1 % and
outer core 6-10 % as presented by Li and Fei (2003).
3.4. Light matter in the melt at the ICB
Figure 3. Density (solid line, dotted lines showing its maximum uncertainty) and weight fraction
(dashed line) of the melt impurities as a function of molar mass for the obtained mole fraction of the
light impurities. To guide the eye, the molar masses for O, Si, S, SiO2 and the main mantle components
MgSiO3 and Mg2SiO4 are shown by vertical lines up to the density curve. Their likely densities are
indicated at ICB pressure. The densities for solid Si, S and SiO2 as well as for liquid O are estimated
from Kogan et al. (2003) (long grey bar for O, open circle for Si, filled triangle for S, short black bar
0
2
4
6
8
10
0
2
4
6
8
10
12
14
16
0 50 100 150 200
fraction (wt.%)
SSi MgSiO3Mg2SiO4
OSiO2
!
i (g/cm3)
Molar mass of the impurity (g/mol)
for SiO2). The filled diamond and circle express the extrapolation to ICB of the melt density along the
liquidus for MgSiO3 and Mg2SiO4, respectively, from data given by Mosenfelder et al. (2009).
The average density of the impurities
ρ
i in the fluid at ICB can be estimated using the
equation
!
1
"
melt
PREM =y
"
i
+1#y
"
M
, where y is the weight fraction of the impurities, valid
strictly for homogeneous solutions but probably also valid for the fluid in the core due
to the very small amount of impurities present and the very long time the core melt
has been mixed by convection. This gives
ρ
i
since y can be calculated for any molar
mass Mi of the light impurity with a known mole fraction ci as
!
y=100ciMi/(ciMi+(1"ci)MFe),
where for MFe the standard molar mass of iron is
used. In Figure 3, y and
ρ
i are presented as a function of molar mass using ci =5.1
mol.% as was obtained earlier in section 3.2. The solid line shows the densities for all
elements and their compounds as function of their molar mass (and also for
combinations of them; however, they remain useless for comparison while their high
pressure densities remain unknown). For compounds of Fe all iron atoms have already
been counted in the total iron. For clarity, only six light matter candidates are shown
explicitly by symbols in Fig. 3 together with approximations for their density at the
ICB pressure. As an example for others, one can find that C’s molar mass 12.01 g/mol
corresponds to
ρ
i = 1.85 g/cm3 which is far too low compared to its estimated density
at ICB pressure (Kogan et al., 2003). Generally single light elements have estimated
densities higher than the calculated
ρ
i corresponding to their Mi even though Si and S
have estimated densities at
ρ
i’s upper uncertainty range. This is in agreement with
previous research (Poirier, 1994; Wood et al., 2006; Alfè et al., 2007) which suggests
that the light matter needs to have at least two elements. The previously suggested
two likely elements are O and Si (Alfè et al., 2007; Knittle and Jeanloz, 1991;
Takafuji et al., 2005; Sakai et al., 2006). Only their compound SiO2 can be compared
to experiments giving its estimated solid density only slightly above
ρ
i; however, SiO2
as the dominant light matter is unlikely on the grounds of its sound velocity against
density as discussed below. In addition, by this method one can consider also the
common three-element compounds as possible light impurities, of which the most
relevant are the most abundant mantle compounds: forsterite, Mg2SiO4, and
magnesium silicate, MgSiO3. For MgSiO3,
!
"
i=7.3#1.2
+0.7 g/cm3
with
!
y=8.8"2.7
+2.3 wt.%
, and
this density range perfectly includes 6.7 g/cm3 obtained by a quadratic extrapolation
to ICB pressure of P(
ρ
) in Mosenfelder et al. (2009) for the density of liquid MgSiO3
at its melting temperature, also shown in Fig. 3. For Mg2SiO4 the calculated
!
"
i=8.3#2.4
+2.0 g/cm3
still includes, but less centrally, the value 6.9 g/cm3 obtained using a
quadratic extrapolation to ICB pressure of P(
ρ
) in Mosenfelder et al. (2009), the
pressure versus density curve for liquid Mg2SiO4 at its melting temperature. The
lower temperature at ICB has an effect to narrow the differences but not enough for
Mg2SiO4. For MgSiO3, the value of y implies that at ICB the melt has about 2.1±0.6
wt.% Mg,
!
2.5"0.8
+0.6
wt.% Si and
!
4.2"1.3
+1.1
wt.% O. The amount of the light matter found by
this method is rather similar to the recent results by Alfè et al. (2007) where one can
infer having 4.6 wt.% Si with 4.3 wt.% O, and Badro et al. (2007) which quotes 2.7
wt.% Si with 5.3 wt.% O; however, they only considered two light elements at a time
in their studies.
Figure 4. Compressional (P wave) sound velocity Vp against density
ρ
, for melt. Data for lower mantle
(LM, open squares), outer core (OC, open diamonds) and inner core (IC, dots) as given by PREM,
together with lines for
ε
Fe by Badro et al. (2007) (dotted line) and experimental results for solid
MgSiO3 (dash-dotted line via filled upward (Flesch et al., 1998) and downward triangles (Li and
Zhang, 2005)) and numerical result for liquid MgSiO3 at 4000 K for higher densities (solid line via plus
symbols whose coordinates are from figs. 3 and 4 in Stixrude et al. (2009) for five pressures). The line
for liquid Fe (dashed line) is estimated by shifting the solid Fe line by a constant amount (see the text).
A bold line is drawn through the inner boundary of the outer core (OC) and the liquid Fe density found
at ICB pressure (circle). On this line is marked by a filled square the value of
ρ
i found here for MgSiO3
as the light matter in the melt at ICB, and the thick gray line indicates its uncertainty. In addition, the
Vp(
ρ
) data for solid SiO2 in Kogan et al. (2003) is also shown (dashed line via open triangles), and
ρ
i
for SiO2 melt at ICB (crossed square), to support the discussion in the text.
The final considerations are based on the materials’ sound velocity against
density behaviour, Vp(
ρ
), whose linearity is called Birch’s law. First, Figure 4 shows
PREM data for the lower mantle, outer core and inner core, together with a previous
8
10
12
14
16
4 6 8 10 12
V
p (km/s)
Density (g/cm3)
solid
SiO2
LM
OC
IC
liquid Fe
solid
MgSiO3
liquid
MgSiO3
at 4000 K
!
Fe
"
i for MgSiO3
melt at ICB
"
i for SiO2
melt at ICB
linear estimate for
ε
iron by Badro et al. (2007). An approximation for the sound
velocities in iron’s liquid phase is made by subtracting a constant, since the velocity
difference in solid and liquid, if considered to be constant as a function of pressure in
fig. 2 of Nguyen and Holmes (2004), is 1.2 km/s between P=220 GPa and P=260
GPa. (However, Nguyen and Holmes’ calculated values of the velocities do not
correspond to their quoted pressure if compared to the equations for the iron
ε
phase
and its melt. They do not quote the densities used.) The line for liquid iron sound
velocity against density goes through the same value, 3.82 km/s, as Nasch et al.
(1994) obtained for the melting point density, both at room pressure. This also allows
the estimation of the liquid iron sound velocity at ICB pressure for the density
ρ
M
found earlier. This point is marked by a circle in Fig. 4. A bold line is drawn through
this circle and the PREM ICB melt data point, the extreme point of the outer core
data, in order to utilize the fact that the change in composition follows a line in the
(
ρ
,Vp) plane for isostructural materials as suggested first by Birch (1961a). This bold
line would then give the sound velocity for the light matter whose density was found
earlier at ICB melt. The molten light matter, MgSiO3, with density of 7.3 g/cm3 is
marked as a filled square on the bold line; a thicker gray line shows its uncertainty.
This density can be compared to the available experimental and numerical results.
Experimental data (Flesh et al., 1998; Li and Zhang, 2005) is only for solid MgSiO3,
and their linear fit almost aligns with the lower mantle PREM data, being on its
lighter side, and indicating an increased amount of iron with depth. For the MgSiO3
melt the numerical results by Stixrude et al. (2009) are at 4000 K, and in the pressure
range 60 - 140 GPa exhibit a linear behaviour. This temperature is about two thirds of
the way between room temperature, where the solid studies were made, and the ICB
temperature. The numerical result is almost parallel to the experimental estimate for
the solid, and the liquid line is lower as it was for iron. This liquid line goes
convincingly through the uncertainty range of
ρ
i for MgSiO3, being above the marked
point, in accordance to their temperature difference. However, if the light matter were
SiO2, its
ρ
i would be 5.75 g/cm3. Relevant data is only for solid SiO2 and its velocity
against density line crosses the bold line at about 6.17 g/cm3. One can expect its melt
velocity to be reduced which means a shift even further away from the value of
ρ
i for
SiO2, thus indicating SiO2 is not as convincing a candidate for the light matter as
MgSiO3. Nevertheless, it would be best to have true experimental sound velocity
against density data for these liquids.
The light matter found here, MgSiO3, suggests that in the partially or fully
molten early Earth, a small quantity of the molten main mantle material accompanied
the iron when it descended as a melt to the core. The compound MgSiO3 has in the
past (Alder, 1966; Knittle and Jeanloz, 1991) been considered as the core’s light
compound, and its elements, Si and O, alone or together, have been often suggested as
light elements in the core. Magnesium has been suggested more rarely: it was
suggested by Alder in 1966 but later discounted on account of the weak solubility of
MgO in iron melt even at high pressures (see Poirier, 1994). However, recently Mg
has been found adequately soluble in iron at high pressures and thus suggested as an
important light element in the core (Dubrovinskaia et al., 2005). The result found here
calls for further investigations of the liquid density and sound velocity of Fe, Ni and
MgSiO3, both pure and as alloys, in core conditions. It will also be worth
investigating the crystal structure and the Fe-concentration of the solid, and especially
the solidification temperature, as a small amount of molten MgSiO3 in Fe, or in Fe-Ni,
is cooled down at the CMB pressure.
3.5. Light matter in the solid inner core
Figure 5. Compressional (P wave) sound velocity Vp against density, for solid. Data for inner core (IC,
dots) from PREM are shown together with thin lines for
ε
Fe (dotted), FeO (dashed) and FeSi (tightly
dotted) all from Badro et al. (2007), Fe3C (Fiquet et al. 2009) (tightly dashed), SiO2 (Kogan et al.,
2003) (long dashes), Mg2SiO4 (Li, 2003) (solid) and MgSiO3 (Flesh et al., 1998; Li and Zhang, 2005)
(dash-dotted). Estimates of Vp for the
ε
(open triangle),
γ
(open square) and
δ
(open circle) Fe phases
are shown at the density
ρ
M,solid(ICB), all from this work. Isostructural lines (bold dashed for
ε
, bold
dash-dotted for
γ
and bold solid for
δ
) connecting these to the data point corresponding to the ICB are
drawn and extrapolated to lower densities. The crossing point densities are discussed in the text.
The small fraction of the light matter in the solid at the ICB can be approximated
using
!
1
"
solid
PREM =z
"
i,solid
+1#z
"
M,solid
(21)
10
15
20
25
4 6 8 10 12
V
p (km/s)
Density (g/cm3)
SiO2
IC
!
Fe
MgSiO3
Mg2SiO4
!
"
#
FeOFeSi Fe3C
where z is the weight fraction of the impurities in the solid. All the light matter
components in the fluid at ICB need not be solidifying into the inner core even though
all the material in the inner core must have come from the outer core. To find the
density of the light impurity compound,
ρ
i,solid, a diagram, Figure 5, similar to Fig. 4
is employed. Fig. 5 shows first the sound velocity against density, Vp(
ρ
), for the inner
core (PREM),
ε
iron (Badro et al., 2007) and the selected possible light components
SiO2 (from Kogan et al., 2003), Mg2SiO4 (Li, 2003) and MgSiO3 (Flesch et al. 1998;
Li and Zhang, 2005), all for solid phases. Similar lines for FeSi, FeO and Fe3C
(Fiquet et al., 2009) are also drawn.
Next the isostructural lines for different iron phases are obtained: If the ICB
solid is mainly
ε
iron, on the line
!
Vp
Fe(
"
)
the density
ρ
Fe
= ρ
M,solid = 13.14 g/cm3 is
chosen for solid iron at ICB pressure, as found above (open triangle in Fig. 5). A line
(bold dashed line in Fig. 5) through this point and
!
Vp
PREM(
"
ICB,solid )
gives, for
ρ
<
ρ
PREM,
!
Vp(
"
)=
Vp
Fe #Vp
PREM
"
Fe #
"
PREM (
"
i,solid #
"
PREM)+Vp
PREM
(22)
thus determining the sound velocity and density properties of the light matter which
can be present with
ε
iron in the ICB solid. The other iron phases do not have good
experimental Vp(
ρ
) data, so the Vp(
ρ
) values are approximated using their numerically
found difference from
ε
iron (Tsuchiya and Fujibuchi, 2009). Their isostructural lines
are given for
γ
phase by Vp = 10.613 km/s (open square) it being 0.27 km/s less than
!
VP
Fe(
"
ICB)
for
ε
phase, and for
δ
phase by Vp = 9.973 km/s (open circle) it being 0.91
km/s less than
!
VP
Fe(
"
ICB)
for
ε
phase in Tsuchiya and Fujibuchi (2009). A bold
dashed-dotted line (
γ
phase) and a bold solid line (
δ
phase) in Fig. 5 are drawn
through these reduced Vp with the same
ρ
M,solid , and the ICB data point. The point
where any of the impurity lines cross any of the isostructural lines gives the density
ρ
i,solid for the impurity with that particular phase structure. Using (21) one finds the
weight percentage for the impurity compound. From the known percentage of the
light element(s) in each compound one finds the weight percentages of each light
element. Using (21) for the light element, one finds its density.
The identity of the light matter and the phase of the iron are determined as
follows. The density value is required to agree with an estimate for the density of the
light material at ICB pressure based on experimental data. Its weight fraction must be
realistic. Finally, the shear wave velocity Vs for the light matter with its obtained
concentration, mixed with iron in its particular phase, should agree with PREM at the
inner core conditions.
The light matter found above in the liquid at ICB, MgSiO3, gives for
γ
phase
the density 6.59 g/cm3 with z = 2.96 wt.%. This is rather less than 7.1 g/cm3 as found
by extrapolating linearly the density data of postperovskite phase at its melting
temperature to ICB pressure in Mosenfelder et al. (2009). The density for
ε
phase is
even lower and for
δ
phase too high, so these phases can definitely be ruled out. In
order to decide whether MgSiO3 in this concentration with
γ
phase iron is the light
impurity of the inner core, one would need to find whether the corresponding shear
velocity Vs agrees with PREM results in the inner core conditions. This data is
presently not available. However, the numerical calculations by Tsuchiya and
Fujibuchi (2009) indicate that neither
ε
nor
γ
phase can satisfy the constraints given
by PREM for Vs since the impurities increase further the already too high pure iron
values for Vs. Thus for the light matter in the inner core MgSiO3 is unlikely, and other
possibilities are investigated next.
For Mg2SiO4 the high pressure data exist only for its
γ
phase, and this is
shown. Its density with the iron
δ
phase is 7.82 g/cm3, which agrees with an
extrapolation to ICB pressure of the data in Kogan et al. (2003) for forsterite (only its
ceramic phase has adequately data, but none above 104 GPa, thus leaving uncertainty
in the extrapolation) but more than 7.4 g/cm3, a linear extrapolation to ICB pressure of
the density data of postperovskite at its melting temperature found by Mosenfelder et
al. (2009); the short pressure range in their fig. 11(b) does not allow any better
extrapolation. This difference can also be a consequence of the use of data for
different Mg2SiO4 phases: the isostructural line gives an estimate for a hypothetical
δ
phase of Mg2SiO4 (see Birch, 1961a). For this density, z = 4.33 wt.% corresponding
to 0.9 wt.% of Si, 1.5 wt.% of Mg and 2.0 wt.% of O. For the
γ
phase the density is
5.99 g/cm3, less likely in the light of the data shown in Kogan et al. or Mosenfelder et
al. The corresponding z = 2.46 wt.%. Furthermore, Vs
for
γ
phase with any light
impurity seems to be higher than the PREM result as discussed earlier. However, the
calculation by Kádas et al. (2009) for
δ
phase of Fe-Mg alloy show that both Vp and
Vs agree with PREM results for the solid inner core if the temperature is 5000 or 7000
K with Mg concentrations of 9 and 5 at.%, respectively. Linear interpolation to the
temperature found here (5670 K) corresponds to 3.5 wt.% of Mg. This is rather close
to the overall light matter concentration (4.3 wt.% of Mg2SiO4) found here, bearing in
mind it includes also some O and Si. Overall, Mg2SiO4 with the iron
δ
phase is a very
reasonable candidate for the light matter in the solid inner core. One can note that 3.5
wt.% of Mg cannot be the sole light element in the inner core since from (21) one
finds the corresponding density to be much too high for its extrapolated density at
ICB pressure (Kogan et al., 2003).
The density of SiO2 with
ε
phase, 6.11 g/cm3, is only slightly higher than its
density estimate 6.01 g/cm3 in Kogan et al. (2003) making it close to be a possible
light matter in the inner core. Correspondingly z = 2.56 wt.%. It may be worthwhile to
find whether its Vs with
ε
phase and this concentration is in agreement with PREM
even though the numerical results suggest not due to the
ε
phase. The concentration is
slightly more than the experimental estimate of 2.3 wt.% for Si and 0.1 wt.% for O
found by Badro et al. (2007) but less than the ab initio estimate of 3.7 wt.% for Si and
almost no O which may be inferred from Alfè et al. (2007). Its other phases give too
high densities.
The fraction of Si with
ε
phase (here 2.24 wt.%) favourably compares to the
result 2.3 wt.% by Badro et al. (2007). However, its density is 5.68 g/cm3, too high for
its estimate 4.72 g/cm3 at ICB pressure (Kogan et al., 2003). Its density for the other
phases is even higher. So this method indicates Si cannot be the main light element in
the inner core.
The fraction of O with
ε
phase (here 1.62 wt.%) is also similar to the result 1.6
wt.% by Badro et al. (2007). However, its density is 4.7 g/cm3, which is rather
unlikely even though data for solid O at high pressures are not available. O cannot be
the main light element in the inner core with
ε
phase. Nor can it be with the
γ
phase,
but for
δ
phase its density is 7.4 g/cm3, with z = 3.80 wt.%, and this could be a
reasonable amount higher than the estimate for liquid O’s density at ICB pressure
(Kogan et al., 2003) so that O with iron in the
δ
phase remains among the possibilities
but needs better experimental data to be confirmed.
Carbon with
δ
phase corresponds to density 4.868 g/cm3 and z = 1.73 wt.%,
and this density agrees with the density 4.8642 g/cm3 at 326.6 GPa (Kogan et al.,
2003). The densities for the other phases are not reasonable, but the fraction of C with
ε
phase (here 0.90 wt.%) is similar to the value 1 wt.% by Fiquet et al. (2009). In
order to confirm that C in this concentration with
δ
phase is the light impurity of the
inner core, one would still need to verify that the corresponding Vs agrees with PREM
results in the inner core conditions. The data in Gao et al. (2008) suggests otherwise.
The fraction and density of S can be calculated from the equations for FeS and
FeS2 given by Badro et al. (2007) even though they are not drawn in Fig.5. The
weight percentages of S are for
ε
phase 10.45 and 1.68 wt.%, compared to 9.7 and 1.6
wt.% (Badro et al.), respectively. The results are rather similar, but the bigger
concentration shows the difference in our calculations. Even more noticeable is that
the weight percentages for S from FeS2 and FeS also disagree with each other in both
studies, indicating S cannot be the relevant light element even though from FeS2 the
density with the
ε
phase for S is 4.76 g/cm3, rather close to 4.87 g/cm3, which is the
ICB pressure approximation for the density for rhombic sulphur (Kogan et al., 2003).
The other phases give too high densities.
There is also the following supporting evidence for the light matter being
Mg2SiO4 with iron: Iron and olivine, (Mg,Fe)2SiO4, occur together typically in
pallasite meteorites thus making Mg2SiO4 as light impurity in iron the most common
in nature although meteorites are believed to originate at lesser pressures and can have
much more light material than present in the Earth’s core. The calculations by Côté et
al. (2008), based on the combined effect of temperature and light elements such as Si
and O, imply that the inner core is more likely to be in the
δ
phase than
ε
.
These arguments suggest that the solid iron at the ICB is most likely to be in
the
δ
phase, and the main light impurity in iron is Mg2SiO4 with concentration of 4.3
wt.%. This choice is the best candidate to fulfil all three constraints given by PREM
and high pressure experiments for density, Vp and Vs at the inner core boundary
pressure and temperature. Mg2SiO4 is different from the light impurity MgSiO3 found
in the melt at ICB, which is perfectly possible.
However, the estimated values for Vp are likely to be subject to change in
future with more extensive experimental and numerical data. Thus MgSiO3 could
after all be the light compound in the solid inner core, if iron Vp were about 10.5 km/s
at ICB pressure, which is only about 0.5 km/s higher than Vp as estimated above for
the
δ
phase.
4. Conclusions
The theory of tricritical phenomena provides a concise framework to calculate
precisely the quantities essential to understand the Earth’s core. Quantitative
predictions are obtained for the iron melting temperature and melting density at high
pressures, and for the temperature and the amount and identity of the light matter in
the real Earth at the inner core boundary in both melt and solid. The light matter in the
solid inner core depends on the iron phase there: most convincingly ICB solid is made
of
δ
iron with 4.3 wt.% of Mg2SiO4. The density of light matter in the melt
corresponds to MgSiO3 and its amount is about 8.8 wt.%. Thus the core consists of
the four most abundant elements of the Earth in addition to the commonly expected
Ni not considered here separately. It is likely that MgSiO3 originates from the molten
early mantle. Conceptually, it is more likely that the light matter in the molten part of
the core is mainly the most common light compound of the whole Earth than that it is
an arbitrary mixture of many individual light elements.
These results agree with the understanding that O is the principal light element
in the core, and the expectation that the core has some Si. Additionally this work
identifies the third main light element to be Mg. This challenges the present
geochemical calculations where no O or Mg are assumed in the core. Furthermore,
this study indicates that the light matter solidifies out at CMB with some iron, in
agreement with previous suggestions.
Acknowledgement 1
I thank an anonymous referee for suggestions for improvements, Paul Asimow for 2
useful comments on an earlier, shorter version of this manuscript (arXiv:0807.0187), 3
and also Don Isaak for providing the background information on ΔV values in the 4
literature and a copy of the Anderson paper (2003). 5
References
Aitta, A., 2009. Tricritical points and liquid-solid critical lines. In: C. Hobbs and S.
Paycha (Editors), European Women in Mathematics: Proceedings of the 13th General
Meeting, World Scientific, pp. 93–102. Preprint: arXiv:0902.4578.
Aitta, A., 2006. Iron melting curve with a tricritical point. J. Stat. Mech., 2006:
P12015.
Aitta, A., 1986. Quantitative Landau model for bifurcations near a tricritical point in
Couette-Taylor flow. Phys. Rev. A, 34: 2086–2092.
Alder, B.J., 1966. Is the mantle soluble in the core? J. Geophys. Res., 71: 4973–4979.
Alfè, D., Price, G.D., and Gillan, M.J., 2002. Iron under Earth's core conditions:
Liquid-state thermodynamics and high-pressure melting curve from ab initio
calculations. Phys. Rev. B, 65: 165118.
Alfè, D., Gillan, M.J., and Price, G.D., 2007. Temperature and composition of the
Earth's core. Contemp. Phys., 48: 63–80.
Anderson, O.L., 2003. The three-dimensional phase diagram of iron. In V.M. Dehant,
K.C. Creager, S. Zatman and S. Karato (Editors), Earth's Core: Dynamics, Structure
and Rotation (Geodynamics Series 31), American Geophysical Union, Washington,
DC, p. 89.
Anderson, O.L., and Isaak, D.G., 2002. Another look at the core density deficit of
Earth's outer core. Phys. Earth Planet. Inter., 131: 19–27.
Anderson, O.L., Isaak, D.G., and Nelson, V.E., 2003. The high-pressure melting
temperature of hexagonal close-packed iron determined from thermal physics. J.
Phys. Chem. Solids, 64: 2125–2131.
Antonangeli, D., Krisch, M., Farber, D.L., Ruddle, D.G. and Fiquet, G., 2008.
Elasticity of hexagonal-closed-packed cobalt at high pressure and temperature: A
quasiharmonic case. Phys. Rev. Lett., 100: 085501.
Badro, J., Fiquet, G., Guyot, F., Gregoryanz, E., Occelli, F., Antonangeli, D., and
d'Astuto, M., 2007. Effect of light elements on the sound velocities in solid iron:
Implications for the composition of Earth's core. Earth Planet. Sci. Lett., 254: 233–
238.
Birch, F., 1952. Elasticity and constitution of the Earth’s interior. J. Geophys. Res.,
57: 227–286.
Birch, F., 1961a. The velocity of compressional waves in rocks to 10 kilobars, Part II.
J. Geophys. Res., 66: 2199–2224.
Birch, F., 1961b. Composition of the Earth's mantle. Geophys. J. R. Astron. Soc., 4:
295–311.
Brown, J.M., 2001. The equation of state of iron to 450 GPa: another high pressure
solid phase? Geophys. Res. Lett., 28: 4339–4342.
Brown, J.M., and McQueen, R.G., 1986. Phase transitions, Grüneisen parameter, and
elasticity for shocked iron between 77 GPa and 400 GPa. J. Geophys. Res., 91: 7485–
7494.
Buffett, B.A., Huppert, H.E., Lister, J.R., and Woods, A.W., 1996. On the thermal
evolution of the Earth's core. J. Geophys. Res. 101: 7989–8006.
Buffett, B.A., Garnero, E.J., and Jeanloz, R., 2000. Sediments at the top of Earth's
core. Science, 290: 1338–1342.
Buffett, B., Garnero, E., and Jeanloz, R., 2001. Porous sediments at the top of Earth's
core? Response. Science, 291: 2091–2093.
Canup, R.M., 2008. Accretion of the Earth, Philos. Trans. R. Soc. Lond., A 366:
4061–4075.
Chung, D.H., 1970. Effects of iron/magnesium ratio on P- and S-wave velocities in
olivine. J. Geophys. Res., 75: 7353–7361.
Chen, B., Gao, L., Funakoshi, K.-i., and Li, J., 2007. Thermal expansion of iron-rich
alloys and implications for the Earth's core. Proc. Natl Acad. Sci. USA, 104: 9162–
9167.
Côté, A.S., Vočadlo, L., and Brodholt, J.P., 2008. Light elements in the core: Effects
of impurities on the phase diagram of iron. Geophys. Res. Lett., 35: L05306.
Dewaele, A., Loubeyre, P., Occelli, F., Mezouar, M., Dorogokupets, P.I., and Torrent,
M., 2006. Quasihydrostatic equation of state of iron above 2 Mbar. Phys. Rev. Lett.,
97: 215504.
Dubrovinskaia, N., Dubrovinsky, L., Kantor, I., Crichton, W.A., Dmitriev, V.,
Prakapenka, V., Shen, G., Vitos, L., Ahuja, R., Johansson, B., and Abrikosov, I.A.,
2005. Beating the miscibility barrier between iron group elements and magnesium by
high-pressure alloying. Phys. Rev. Lett., 95: 245502.
Dubrovinsky, L., Glazyrin, K., McCammon, C., Narygina, O., Greenberg, E.,
Übelhack, S., Chumakov, A.I., Pascarelli, S., Prakapenka, V., Bock, J., and
Dubrovinskaia, N., 2009. Portable laser-heating system for diamond anvil cells. J.
Synchrotron Rad., 16: 737–741.
Dziewonski, A.M., and Anderson, D.L., 1981. Preliminary reference Earth model.
Phys. Earth Planet. Inter., 25: 297–356.
Flesch, L.M., Li, B., and Libermann, R.C., 1998. Sound velocities of MgSiO3 -
orthopyroxene to 10 GPa at room temperature. Am. Mineral. 83: 444–450.
Fiquet, G., Badro, J., Gregoryanz, E., Fei, Y., and Occelli, F., 2009. Sound velocity in
iron carbide (Fe3C) at high pressure: Implications for the carbon content of the Earth's
inner core. Phys. Earth Planet. Inter., 172: 125–129.
Gao, L., Chen, B., Wang, J., Alp, E.E., Zhao, J., Lerche, M., Sturhahn, W., Scott,
H.P., Huang, F., Ding, Y., Sinogeikin, S.V., Lundstrom, C.C., Bass, J.D., and Li, J.,
2008. Pressure-induced magnetic transition and sound velocities of Fe3C: implications
for carbon in the Earth’s inner core. Geophys. Res. Lett., 35: L17306.
Georg, R.B., Halliday, A.N., Schauble, E.A., and Reynolds, B.C., 2007. Silicon in the
Earth's core. Nature, 447: 1102–1106.
Golubitsky, M., and Schaeffer, D.G., 1985. Singularities and Groups in Bifurcation
Theory, vol. 1. Springer-Verlag, New York, pp. 257–288.
Gubbins, D., Masters, T.G., and Jacobs, J.A., 1979. Thermal evolution of the Earth’s
core. Geophys. J. R. Astron. Soc., 59: 57–99.
Janssens, K.G.F., Raabe, D., Nestler, B., Miodownik, M.A., and Kozeschnik, E.,
2007. Computational materials engineering: an introduction to microstructure
evolution. Academic Press, Amsterdam, pp. 21–28.
Jeanloz, R., 1990. The nature of the earth’s core. Annu. Rev. Earth Planet. Sci., 18:
357–386.
Kádas K., Vitos, L., Johansson, B., and Ahuja, R., 2009. Stability of body-centered
cubic iron-magnesium alloys in the Earth's inner core. Proc. Natl Acad. Sci. USA,
106: 15560–15562.
Karato, S.-i., 2008. Deformation of Earth Materials. Cambridge University Press,
Cambridge, pp. 59–74.
Knittle, E., and Jeanloz, R., 1991. Earth's core-mantle boundary: Results of
experiments at high pressures and temperatures. Science, 251: 1438–1443.
Koči, L., Belonoshko, A.B., and Ahuja, R., 2006. Molecular dynamics study of liquid
iron under high pressure and high temperature. Phys. Rev. B, 73: 224113.
Koenig, M., Benuzzi-Mounaix, A., Ravasio, A., Vinci, T., Ozaki, N., Lepape, S.,
Batani, D., Huser, G., Hall, T., Hicks, D., MacKinnon, A., Patel, P., Park, H.S.,
Boehly, T., Borghesi, M., Kar, S., and Romagnani, L., 2005. Progress in the study of
warm dense matter. Plasma Phys. Control. Fusion, 47: B441–B449.
Kogan, V.E., Levashov, P.R., and Lomov, I.N., 2003. Shock wave database.
http://www.ihed.ras.ru/rusbank/.
Landau, L.D., 1937. Zur Theorie der phasenumwandlungen I. Phys. Z. Sowjetunion,
11: 26–35. English translation in: D. ter Haar (Editor), 1965. On the theory of phase
transitions. Collected Papers of L. D. Landau, Pergamon, London, pp. 193–216.
Landau, L.D., and Lifshitz, E.M., 2001. Statistical Physics, Part 1, third ed.
Butterworth-Heinemann, Oxford, p. 317; pp. 493–498.
Lawrie, I.D., and Sarbach, S., 1984. Theory of tricritical points. In: C. Domb and J.L.
Lebowitz (Editors), Phase Transitions and Critical Phenomena, Vol. 9. Academic
Press, London, pp. 1–161.
Li, B., 2003. Compressional and shear wave velocities of ringwoodite γ-Mg2SiO4 to
12 GPa, Am. Mineral., 88: 1312–1317.
Li, B., and Zhang, J., 2005. Pressure and temperature dependence of elastic wave
velocity of MgSiO3 perovskite and the composition of the lower mantle. Phys. Earth
Planet. Inter., 151: 143–154.
Li, J., and Fei, Y., 2003. Experimental constraints on core composition. In: R.W.
Carlson (Editor), The Mantle and Core, Vol. 2. Treatise on Geochemistry (eds. H.D.
Holland and K.K. Turekian). Elsevier-Pergamon, Oxford, pp. 521–546.
Lin, J.F., Struzhkin, V.V., Sturhahn, W., Huang, E., Zhao, J.Y., Hu, M.Y., Alp, E.E.,
Mao, H.K., Boctor, N., and Hemley, R.J., 2003. Sound velocities of iron-nickel and
iron-silicon alloys at high pressures. Geophys. Res. Lett., 30: 2112.
Lin, J.F., Sturhahn, W., Zhao, J.Y., Shen, G.Y., Mao, H.K., and Hemley, R.J., 2005.
Sound velocities of hot dense iron: Birch’s law revisited. Science, 308: 1892–1894.
Lister, J.R., 2003. Expressions for the dissipation driven by convection in the Earth's
core. Phys. Earth Planet. Inter., 140: 145–158.
Ma, Y., Somayazulu, M., Shen, G., Mao, H.-k., Shu, J., and Hemley, R.J., 2004. In
situ X-ray diffraction studies of iron to Earth-core conditions. Phys. Earth Planet.
Inter., 143-144: 455–467.
Masters, T.G., and Shearer, P.M., 1990. Summary of seismological constraints on the
structure of the Earth's core, J. Geophys. Res., 95: 21691–21695.
McDonough, W.F., 2003. Compositional model for the Earth's core, In: R.W. Carlson
(Editor), The Mantle and Core, Vol. 2. Treatise on Geochemistry (eds. H.D. Holland
and K.K. Turekian). Elsevier-Pergamon, Oxford, pp. 547–568.
Mosenfelder, J.L., Asimow, P.D., Frost, D.J., Rubie, D.C., and Ahrens, T.J., 2009.
The MgSiO3 system at high pressure: Thermodynamic properties of perovskite,
postperovskite, and melt from global inversion of shock and static compression data.
J. Geophys. Res., 114: B01203.
Nasch, P.M., Manghnani, M.H., and Secco, R.A., 1994. Sound velocity measurements
in liquid iron by ultrasonic interferometry. J. Geophys. Res., 99: 4285–4291.
Nguyen, J.H., and Holmes, N.C., 2004. Melting of iron at the physical conditions of
the Earth's core. Nature, 427: 339–342.
Ozawa, H., Hirose, K., Mitome, M., Bando, Y., Sata, N., and Ohishi, Y., 2009.
Experimental study of reaction between perovskite and molten iron to 146 GPa and
implications for chemically-distinct buoyant layer at the top of the core. Phys. Chem.
Miner., 36: 355–363.
Poirier, J.-P., 1986. Dislocation-mediated melting of iron and the temperature of the
earth's core. Geophys. J. Roy. Astron. Soc., 83: 313–328.
Poirier, J.-P., 1994. Light elements in the Earth’s outer core: A critical review. Phys.
Earth Planet. Inter., 85: 319–337.
Sakai, T., Kondo, T., Ohtani, E., Terasaki, H., Endo, N., Kuba, T., Suzuki, T., and
Kikegawa, T., 2006. Interaction between iron and post-perovskite at core-mantle
boundary and core signature in plume source region. Geophys. Res. Lett., 33: L15317.
Schloessin, H.H., and Jacobs, J.A., 1980. Dynamics of a fluid core with inward
growing boundaries. Can. J. Earth Sci., 17: 72–89.
Shen, G., Mao, H.-k., Hemley R.J, Duffy, T.S., and Rivers, M.L., 1998. Melting and
crystal structure of iron at high pressures and temperatures. Geophys. Res. Lett., 25:
373–376.
Shen, G., Prakapenka, V.B., Rivers, M.L., and Sutton, S.R., 2004. Structure of liquid
iron at pressures up to 58 GPa. Phys. Rev. Lett., 92: 185701.
Steinle-Neumann, G., Stixrude, L., Cohen, R.E., and Gülseren, O., 2001. Elasticity of
iron at the temperature of the Earth's inner core. Nature, 413: 57-60.
Stevenson, D.J., 1981. Models of the Earth's core. Science, 214: 611-619.
Stixrude, L., de Koker, N., Sun, N., Mookherjee, M., and Karki, B.B., 2009.
Thermodynamics of silicate liquids in the deep Earth. Earth Planet. Sci. Lett., 278:
226–232.
Sun, Y.-H., Huang, H.-J., Liu, F.-S., Yang, M.-X., and Jing, F.-Q., 2005. A direct
comparison between static and dynamic melting temperature determinations below
100 GPa. Chin. Phys. Lett., 22: 2002–2004.
Takafuji, N., Hirose, K., Mitome, M., and Bando, Y., 2005. Solubilities of O and Si in
liquid iron in equilibrium with (Mg,Fe)SiO3 perovskite and the light elements in the
core. Geophys. Res. Lett., 32: L06313.
Tan, H., Dai, C.D., Zhang, L.Y., and Xu, C.H., 2005. Method to determine the
melting temperatures of metals under megabar shock pressures, Appl. Phys. Lett., 87:
221905.
Tsuchiya, T., and Fujibuchi, M., 2009. Effects of Si on the elastic property of Fe at
Earth's inner core pressures: First principles study. Phys. Earth Planet. Inter., 174:
212–219.
van der Hilst, R.D., de Hoop, M.V., Wang, P., Shim, S.-H., Ma, P., and Tenorio, L.,
2007. Seismostratigraphy and thermal structure of Earth's core-mantle boundary
region. Science, 315: 1813–1817.
Vočadlo, L., 2007. Ab initio calculations of the elasticity of iron and iron alloys at
inner core conditions: evidence for a partially molten inner core? Earth Plan. Sci.
Lett., 254: 227–232.
Wood, B.J., Walter, M.J., and Wade, J., 2006. Accretion of the Earth and segregation
of its core. Nature, 441: 825–833.
