Download Properties of iron at the Earth`s core conditions

Geophys. J . R. astr. Soc. (1986) 84,561-579
Properties of iron at the Earth’s core conditions
Orson L. Anderson
Institute ofGeophysics and Planetary Physics, and
Department ofEarth and Space Sciences, University of California, Los Angeles, CA 90024,
USA
Accepted 1985 July 29. Received 1985February 15
Summary. The phase diagram of iron up to 330 GPa is solved using the
experimental data of static high pressure (up to 11 GPa) and the experimental
data of shock wave data (up to 250 GPa). A solution for the highest triple
point is found (P=280 GPa and T = 5760 K) by imposing the thermodynamic constraints of triple points. This pressure of the triple point is less
than the pressure of the inner core-outer core boundary of the Earth. These
results indicate that the density of iron at the inner core-outer core
boundary pressure is close to 13 g ~ m - which
~ , lies close to the seismic
solutions of the Earth at that pressure. It is thus concluded that the Earth’s
inner core is very likely to be virtually pure iron in its hexagonal close packed
(hcp) phase.
It is shown that four properties of the Earth’s inner core determined from
seismology are close in value to the corresponding properties of hcp iron at
inner core conditions: density, bulk modulus, longitudinal velocity, and
Poisson’s ratio. The density-pressure profile of hcp iron at inner core
conditions matches the density-pressure profile of the inner core as
determined by seismic methods, within the spread of values given by recent
seismic models.
This indicates that the Earth is slowly cooling, the Earth’s inner core is
growing by crystallization, and the impurities of the core are concentrated in
the outer core. The calculated temperature at the Earth’s centre is 6450 K.
Key words: Earth’s core, equation of state, iron, phase diagram of iron
1 The composition of the core
Although the composition of the Earth’s core has long been debated, the consensus now
heavily favours an iron core where the inner core is very nearly pure iron, and the outer core
is a mixture of iron and some solute. Birch (1952) argued that iron is alloyed with some
lighter element, especially in the outer core, strictly on density grounds. Since that time,
562
0. L. Anderson
although the nature and distribution of the alloying element(s) has been a matter of
contention, there is good agreement that in the core the major element is iron.
The cosmochemical argument in favour of a predominantly iron core relies on the
assumption of approximate homogeneity in the Solar System in so far as element distribution
is concerned. If the element distribution is to be approximately maintained, the number of
iron atoms should be close to the number of silicon atoms in the Earth (Ross & Aller 1976;
Ganapathy & Anders 1974; Wasson 1984). Since iron is depleted relative to silicon in the
Earth’s mantle, it needs to be greatly enriched in the Earth’s core. The volume of the mantle
is much greater than the core. Consequently, to maintain the cosmic abundance ratio, the
core has to be composed essentially of iron.
Experimental geophysics has provided a great deal of data supporting the model of an
iron core. The most important experimental evidence is from shock-wave physics. This
evidence is reviewed elsewhere in a companion paper (Anderson 1985).
An alternative to the iron core model is to assume, as was done by Lyttleton (1963,
1965) and earlier by Ramsey (1949, 1950), that the core is a metallized form of the mantle
ionized under pressure. Experimental geophysics is now capable of observing any phase
change that would transform a silicate into a metallic conductor at core pressures. Phase
changes in silicate exist at high pressure, but the density change is small compared to that
required for the mantle-core boundary of the Earth. But shock wave results have shown
that a silicate most probably cannot duplicate the sound velocity of the core (Birch 1963;
Anderson 1985).
One report has been made (Pavlovskii et al. 1978) in which a large density change for
SiOz has been observed. An isentropic compression with a super strong magnetic field (at
about 150 GPa) apparently yielded a density corresponding to core conditions. This
anomalous result is inconsistent with all other shock wave experiments on SiOz, and needs
experimental confirmation. No such shock wave densification of a-quartz has been observed.
Trunin et al. (1971), for example, concluded that no discontinuities exist in quartz above
the stishovite transition even up to 650 GPa.
In summary, the experimental evidence for a silicate core consists of one questionable
result where a large density increase is found in a-quartz at 150 GPa. The experimental
evidence for an iron core, described elsewhere (Anderson 1985), consists of the agreement
of four independent physical properties of iron at inner core conditions with corresponding
inner core seismic properties: q5, the bulk sound velocity;p, the density; Vp, the longitudinal
sound velocity; and u, the Poisson’s ratio. The shock wave experimental evidence against a
silicate core is that a silicate with the requisite density cannot have the requisite q5.
2 The phase diagram of pure iron at low pressure
The iron we are most familiar with in everyday life is the a-phase, with a body-centred cubic
(bcc) structure. This is certainly not the phase of iron that is in the Earth’s inner core which
has been suggested to be the hcp phase (hexagonal closed packed) or the fcc (face-centred
cubic) phase. As will be shown, the evidence is in favour of the hcp phase. There is also a
high temperature-low pressure phase, 6 , which, like a, is body-centred cubic. The phase
diagram shown in Fig. l(a) is well measured up to 20 GPa and 2000°C. We see that at high
temperatures and at pressures well above 5.2 GPa, liquid iron is in equilibrium with the fcc
phase, and this boundary has been measured up to 20 GPa (Bundy 1965). There are four
solid phases and one liquid phase, so we should have three triple points (tp). The fcc-6liquid tp is at 1990 K and 5.2 GPa (Strong, Tuft & Hannemann 1973) and must be the
beginning point of any extrapolation of the iron solidus into core pressures and
Properties of iron at Earth's core
5.2 GPa
liquid
7000 -
.-.A^
I
563
^^
5760' \
Y
5000
-
4000
3OOo-,
(a)
0
lo
200
Pressure GPa
I
,
I
,
,
300
,
l
(b)
Figure 1. The pressure-temperature phase diagram of iron. (a) modified from Liu (1975), indicates the
tp's at 5 . 2 GPa as determined by Mao, Bassett & Takahash, (1 967) and at 11 GPa as determined by Strong
ef al. (1973). The details of the tp at 280GPa, shown in (b), will be discussed in Section 4.
temperatures. The bcc-hcp-fcc tp is at 750 K and 11 GPa (Bundy 1965). The slope
boundary between the hcp and fcc phases has been in dispute, but the extreme estimates fall
above and below Bundy's (1965) measured value of 28"GPa-'. The slope of a phase
boundary a t the tp must be in accordance with the boundary conditions at a tp (e.g. the
sums of the volume changes, and the sum of the entropy changes is zero).
Since ( A S / A V ) = AP/AT, then AP/AT at the fcc-hcp boundary can be found if two
other slopes, M I A T , are known and the three volume changes around the tp are known. On
this basis, Liu (1 975) estimated dT/dP for the e-y boundary to be 31°CPa-'. An earlier
estimate by Takahashi & Bassett (1964) gave dT/dP= 20"GPa-' . The total evidence suggests
that dT/dP for the fcc-hcp phase boundary is between 20 and 30"CPaC' (but probably
closer to the upper value) at 11 GPa.
The highest tp (hcp-fcc-l) occurs somewhere at high pressure, and the absence of direct
experimental information limits our knowledge of the structure of iron at inner core
pressures. If the t p occurs at pressure less than 320 GPa, then the inner core is in the hcpiron field. If the tp is greater than 350 GPa, then the inner core is in the fcc-iron field. Liu
(1 975) showed that a straight extrapolation of the slope 30"GPa-' would place the t p somewhere around 93.5 GPa, in which case the inner core is €-iron. As we shall see below, this
straight line extrapolation (30"GPa-') is not justified. Nevertheless, we shall conclude that
the inner core is in the ephase field, but see Spiliopolous & Stacey (1984) for a different
view .
There is no guarantee that dT/dP is invariant with pressure. If this were so, then the A S
would diminish at the same rate as AV. That Is not the case f o r the alkalI halides. where
AT/AP is proportional to V (Jeanloz 1982).
'I'he measurement of the volume change between the bcc and the fcc phases (750 K and
11 GPa P) shows a decrease in volume (-0.21 cm3 mole-') (see Mao et al. 1967). Birch
(1972) reported a smaller value at one atmosphere (-0.075 at 1184 K). The volume change
564
0. L. Anderson
between the bcc and the hcp phases (at 750 K and 11 GPa) also shows a decrease in volume
(-0.38 cm3 mole-‘) (Ma0 et al. 1967). We shall adopt the thermodynamic values of Mao
et al. for the 11 GPa t p and the thermodynamic values for the 5.2 GPa t p as given by Strong
et al. (1973). These properties are shown in Fig. 2 .
The properties of bcc iron can be considered secure because this phase is well measured.
For hcp iron, as shown in Table 1 , the value for K O found by Mao & Bell (1 979) is smaller
than KO for bcc iron. K O ordinarily increases between phases in proportion to density
increase, so KO for hcp iron should bc greater than 166 CPa. Gephcoat, Mao & Bell (1986)
showed that this low value was attributed t o non-hydrostatic pressure conditions. When the
experiment was redone with a hydrostatic media, the bulk modulus was found to be 178 GPa
and Kb was 4.4. Thus, Gephcoat’s values are probably the best choice for hcp iron, and will
be used here.
Because p o for fcc iron is between that for bcc iron and hcp iron, the other physical
properties of fcc iron should be between that o f the two other phases.
.
T
O K
A: 300
B: 1990
C: 750
D: 5760
E: 6210
V
P
P
cm3/m. g m k c GPa
7.873 7.091 0
7.40
7.54 5.2
8.66 il.
6.45
4.44
12.57 280
4.29 13.00 330
VOLUME cc/mole
Figure 2. The volume-temperature phase diagram of iron, showing A V across the phase boundaries for
the three triple points. The lower two tp’s are solutions by Strong et a l . (1973) (point B), and by M a o
et al. (1967) (point C). The upper t p solution is discussed in Section 4.
Properties of iron at Earth S core
565
3 The measurement of sound velocity, V p ,in highly compressed iron
Figs 1 and 2 show that the hcp phase ( E iron) is stable above P = 280 GPa, which indicates
that the inner core of the iron is composed of the hcp phase. Some authors have proposed
that fcc iron comprises the inner core (Bukowinski 1977; Spiliopoulos & Stacey 1984), but a
new set of shock wave experiments has indicated that the upper triple point lies at a pressure
below that of the inner core-outer core boundary pressure (Brown & McQueen 1980, 1982,
1986).
Most shock wave analyses arise from experiments which measure the shock velocity
versus particle velocity.
us = $0 -+ s u p
where Go is the bulk sound velocity
($0
=
4Gm
(1)
(2)
and KO is the uncompressed bulk modulus, p o is the uncompressed density, and s is the
inferred slope. From these data, the Hugoniot, which is the pressure density trace under
shock conditions, is calculated. The shock velocity is the primary measured variable and all
derived properties are in P- V-T space. Because the bulk velocity is sometimes insensitive to
small changes in volume at phase transitions, the Hugoniot sometimes does not give clues to
the crossing of phase boundaries. Such is the case for iron.
Although the bulk sound velocity is sometimes insensitive to a solid-liquid transition, the
sound velocities, Vp and VS,are quite sensitive to such a transition because VS vanishes in
the liquid state. A measurement of V p near the liquid-solid phase boundary has the
following properties: Vp is greater than the bulk sound velocity in the solid, but equal to the
bulk sound velocity in the liquid. At the transition point, Vp abruptly slows from the value
of a compressional wave to a lower value representing the bulk wave velocity.
The Hugoniot is known to rise within the hcp phase. Up until the Brown & McQueen
experiment, it was not known whether the Hugoniot crossed into the fcc phase before the
liquid state was reached. Brown & McQueen (1982) found the hcp-fcc boundary was
reached at 200 f 2 GPa, with an estimated temperature of 4400 f 300 K. They found that
the pressure where Vp converged to $ occurred at 250 f 10 GPa (later revised downward to
243 f 2 GPa; (Brown & McQueen 1986)). From the Hugoniot data, they first estimated the
liquid transition to be between 5000 and 6000 K, later revised to 5400 f 400 K (Brown &
McQueen 1986). A plot of their fundamental curve (V, versus P) is given in Fig. 3.
Thus hcp iron has the larger value of V p , requiring that hcp iron have the larger Poisson
ratio value. This is reasonable since fcc iron has the higher temperature phase (see Fig. 1).
Note from Fig. 3 that Vp for hcp iron extrapolates into the PREM seismic values of Vp for
the inner core (Dziewonski & Anderson 1981), whereas the Vp for fcc iron does not.
From Fig. 3, the PREM velocity profile of the outer core lies substantially below both
fcc iron and hcp iron, agreeing with the general notion that the outer core is not pure iron
but is diluted with lighter elements. The joining of the experimental points with the bulk
sound velocity indicates the pressure where the Hugoniot crosses the fcc-liquid boundary.
Additional analysis of the data described in Fig. 3 provides further evidence that the inner
core has the properties of pure hcp iron. From the bulk velocity and the compressional
velocity, Brown & McQueen (1986) estimated Poisson’s ratio, u, for iron at high pressure
(Fig. 4). The value, (I, for hcp iron extrapolates into the seismically determined values of u
for the inner core, but u for fcc iron is too high for the inner core.
In my view, the shock wave evidence, as shown in Figs 3 and 4, persuades us that the
physical properties of the inner core are identical to that of solid hcp iron at inner core
566
0. L. Anderson
-
IRON RAREFACTION VELOCITIES
I
I
I
I
I
I
200
400
PRESSURE (GPA)
Figure 3. Sound velocity, V p versus pressure for pure iron (Brown & McQueen 1982, 1986). The first
peak is at the hcp-fcc transition, and the second peak is at the fcc-liquid transition. PREM refers to the
velocity profile of the Dziewonski & Anderson (1 981) seismic model.
0.5
t
INNER-CORE
11
12
13
DENSITY (Mg/m’)
Figure4. Poisson’s ratio, u, versus density for shocked iron. Note that the hcp iron data for u extrapolates into the inner core data. PREM data is from Dziewonski & Anderson (1981); 1066B and 1066A
data are from Gilbert & Dziewonski (1975). (From Brown & McQueen 1986, with permission).
Properties of iron at Earth S core
567
pressures. Therefore, the tp (hcppfcc-liquid) must lie at a pressure below the inner coreouter core (IC-OC) boundary pressure.
4 Calculation of the high pressure tp properties
4.1
T H E FCC S O L I D U S
The density of the upper tp is important for understanding the physics of the inner core. In
Fig. 2, the density is shown to be 12.57 gm c - ~ .Before proceeding to discuss the
implications of this number, a review of how it was obtained will be made. The complete
analysis of the phase diagram, of which this section is a synopsis, is given by Anderson
(1 986).
The upper tp is the intersection of the fcc solidus, projected upwards from point B in Fig.
2, with the hcp-fcc boundary projected upwards from point C in Fig. 2. This intersection
gives point D in Fig. 2.
The fcc solidus is controlled by an appropriate equation of state. This equation of state
(EOS) must satisfy the known thermodynamic data of the fcc phase, and the shock wave
data (eg. the derived Hugoniot), and melting theory.
Use is made of the Lindeman (1910) law for melting, given in its differential form,
d In T ,
dlnp
= 2 (y
-
1/3)
(3)
where y is the Grfineisen parameter, T, is the melting temperature, and p is the density.
Equation ( 3 ) , as developed by Lindemann, does not have as a postulate the criterion that
there is equilibrium between the liquid and the solid state, but it relies upon the calculation
of the breakdown of a long-range order of the solid state.
However, there have been at least two recent theoretical papers which show that a more
fundamental approach, accounting for the liquid state, yields the Lindemann equation with
small variations.
Stevenson (1980) derived a melting theory from the liquid state theory, and Stacey &
Irvine (1977) derived a melting theory using a thermodynamic equilibrium approach starting
from the Clausius-Clapeyron equation. The results of these two theories differ slightly from
equations ( 3 ) , but not significantly, considering the uncertainty of other measurements of
iron at high pressure.
The parameter, y, is also a function of volume.
Using the general relationship between y and density
where 4 is an arbitrary constant (Anderson 1968), and using (1 1) in (lo), it is found that
Equation ( 2 ) of Spilioupolos & Stacey (1984) can be obtained from ( 5 ) by taking q = 1.
Equation (5) describes T(p) along the solidus. The complete EOS along the solidus is
obtained when p is determined from P. To obtain p(P), we use the bulk modulus-pressure
curve K(P), and integrate the resulting differential equation.
An empirical K(P) curve can be obtained from the shock wave Hugoniot. Since the value
of K(P) of the fcc solidus must be less than that of the K(P) of the hcp solid, an estimate can
0. L. Anderson
568
be made of K ( p ) . Using the K(P) data derived from the Hugoniot by Jeanloz (1979) (see his
fig. 8),
K = K o + K,'P= 1 6 0 + 4P.
(6)
This linearity in Ks is often sufficiently accurate when the temperature increases with
pressure. For example, Bullen (1968) found linearity between K and P for both the lower
mantle and the outer core of the Earth. However, such a linear relationship and the resulting
EOS is not very accurate along an isotherm.
Integrating (6) (since K s = p (aP/ap)), the P-p equation for the fcc solidus is
p/po = [l
+4
(7)
;]OfS
where po is the uncompressed density of the fcc solidus at 1990 K.
From the data in Table 1, it is found that po in (7) is given by po(O, 1990) = 7.313
g L3.Using (5) and (7), T(P) and p(P) can be found along the solidus, providing yo and q
are specified. The value of yo is found from (3), since Tm and p and dT/dP are known at
point B (Fig. 2), resulting in yo = 2.09. The value of the remaining parameter, q , is found by
requiring that T,(P) duplicate the results of Brown & McQueen (1986), where the Hugoniot
crosses the liquidus, e.g. T,,, = 5400 f 400 K at P = 243 f 2 GPa. The midpoint value
requires q = 1.7.
4.2
THE HCP-FCC
EQUILIBRIUM BOUNDARY
The T ( p ) part of the EOS is obtained from the slope, dT/dP, of the phase boundary which is
known at the lower tp (at point C) in Fig. 2. In many solids dT/dP is a constant independent
of pressure, but in others dT/dP decreases with pressure. In general, assume that
AS = (AV)Vr
where r is independent of pressure. Then
d T = C ( P o / p ) 'dP
(8)
where C i s equal to the slope of the hcp-fcc phase boundary evaluated at the 1 1 GPa tp.
Table 1. Equation of state parameters for phases of iron at room temperature pressures
KO
Iron phase
(P=O,T=300K)
Po
01
7.873
7.873
166.6
166.6
5.29
5.97
Guinan & Beshers (1 968)
Rotter & Smith (1966)
E
8.28
178.2
5.15
Reduced shockwave data (Brown & McQueen
1982)
E
8.28
156.2
5.4
Static compression. Murnaghan EOS (Ma0 &
Bell 1979)
4.29 c 0.36 Gephcoat et ~ l (1985)
.
KO
E
8.28
193.0
E
8.38
182.7
Y
8.0
?
?
(Liquid iron
1990 K )
(7.0)
(136.0)
(5.0)
at
Source
(GPa)
Andrews (1973)
Phase diagram and sound velocity (Birch 1972;
Filipov, Kazakov & Pronin 1966)
Properties of iron a t Earth's core
569
Table 2. Thermodynamic properties of the fcc-solidus between the t p a?
5.2 and 330 GPa.
P
(GPd
V
(cm3 mole-' )
P
(g cm-3 1
T (K)
(4 = 1.7)
Y
(4 = 1.7)
5.2
20
50
100
135
150
175
200
24 3
280
300
330
7.40
6.69
6.23
5.58
5.28
5.17
5.01
4.84
4.66
4.54
4.47
4.38
7 .54
8.34
8.96
10.03
10.58
10.80
11.14
11.45
11.93
12.30
12.49
12.75
1990
2724
342 1
421 1
4655
4816
5051
5255
5550
5758
5858
5990
2.09
1.76
1.56
1.29
1.14
1.11
1.08
1.02
0.95
0.91
0.89
0.86
A relationship between p and P is needed in (8) to obtain T(p). This is found by again
assuming a linear relationship between K and P. The constants K O and KA will be related to
those listed in Table 2 for hcp iron. It is found that for the hcp phase boundary
K
=
173 + 5.3 P = KOiKAP
(10)
and that the density for the integrating constant is
po(O, 750) = 8.20 gm ~ m - ~ .
The equation for density for the hcp phase at the phase boundary is found by integrating
(10)
A]
('/5.3)
p=
[1 + 5.3
8.20
.
Equation (1 1) can be used to eliminate P in (8).
Using the relationship
dP= - ( K / V ) d V
(12)
in (8), and integrating
T I = 750 K , p o = 8.20 and p I = 8.655.
Now C is estimated to be 30" GPa-' by Liu (1975), and 20" GPa-' by Takahashi &
Bassett (1 964), while Bundy's (1965) measurements indicate 28" GPa-' . Take
C = 25" GPa-' .
From (13), T(p) is defined once r is specified. For r = 0, the slope of dT/dP is a constant
or the change in entropy is proportional to AV. For r = 1, the slope of dT/dP diminishes as
V diminishes and the change in entropy is proportional to the product of AV and V.
The choice of q and r determines the upper tp, but there are thermodynamic constraints
t o maintain. At the tp point C A V = 0, C A S = 0 , and dT/dP= AS/AV. Further, AVshould
diminish as the pressure increases.
0. L. Anderson
570
Spiliopoulos & Stacey (1984) derived an equation for the volume change along the
melting curve in terms of the change in configurational entropy, ASc, along the melting
curve. The importance of their equation arises because ASc changes very little, if at all, as
the pressure increases. The change in volume passing to the liquid is controlled by ASc and
two other terms: (py),,,Cy and (dP/dT),, where Cv is the specific heat. The term ( P ~ ) ~ C ~
is small compared to (dP/dT),, so a good estimate of AVat the tp is obtainable. Using their
formulae, and their estimate of the configurational entropy change (0.86 eu), and the value
for hcp-fcc (dT/dP), obtained above (5.6" GPa-'), it is found that the hcp-fcc AVat 280
GPa is 0.04 cm3 mole-' . The associated entropy is AS = 1.07 eu.
At the tp, the two intersecting branches must have the same value of P and T , and they
must satisfy the measurements of Brown & McQueen (1986) at 200 and 243 GPa.
Combining all these constraints, a solution is found where the upper triple point is located
) . solution is
at P = 280GPa, T = 5760K, and V=4.44cm3mole-' (or p = 1 2 . 5 7 g ~ m - ~This
the intersection of the upper branch where 4 = 1.7 and the lower branch where r = 1.
The solution q = 1.7 for the fcc phase is analogous to the solution for 4 = 1.62 for the
hcp phase found by Jeanloz (1 979). The solution r = 1 indicates that dT/dP decreases as V ,
just as found for the NaC1-CsC1 transition of the alkali halides (Jeanloz 1982; Bassett et al.
1968; Stishov 1975).
The detaih of the phase diagram near the tp, as shown in Fig. l(b), are quite similar to
the phase diagram of iron proposed by Jeanloz (1985).
The relative change of volume at 280 GPa is 0.04/4.625, so that AV/Vis less than 1 per
cent, even at the lower pressure where the Hugoniot crosses the solidus (250 GPa). This
volume change is too small to be observed in P-V-T space, and thus the transition cannot
be detected in ordinary shock wave analysis. It is, however, detectable by the measurement
of the drop in Vp as demonstrated by Brown & McQueen (1982).
The properties of the calculated fcc solidus are shown in Table 2.
5 The temperature and density at the IC-OC pressure
The extension of the temperature curve above the tp involve (dT/dP) of the liquid-hcp
boundary. From Fig. l(b), dT/dP= 10.2" GPa-' at 280 GPa, which will decrease gradually
with pressure. Take an average value of dT/dP= 9" GPa-' from 280 to 330 GPa. Thus, AT
between 280 and 330 GPa is about 450", and T a t the IC-OC boundary pressure is close to
6210 400 K.
This solution is close t o that found by Spiliopoulos & Stacey (1984) (6120 ? 5 7 9 , but
that is a coincidence. The approaches are quite different. In the Spiliopoulos & Stacey
solution, inner core pressures are in the fcc phase of iron (e.g. there is no high tp at core
pressure) whereas in this solution the hcp phase is found at inner core pressure.
Previous estimates of the temperature at the IC-OC boundary are shown in Table 3.
Taking the density at the hcp side of the tp to be 12.57 gm c - (as
~ found in Section 4.2),
the value of A p , going from P = 280 to 330 GPa, is determined by the value of AT going
from 280 to 330 GPa and the Grheisen parameter of the hcp solidus. Equation (3) applies,
so that
*
P
1
Ap = AT)
T
2 (7- 1/31
From Young & Grover (1984), y for the hcp solidus is about 4/3 at V c 4 . 6 . Thus,
Properties of iron at Earth's core
571
Table 3. Predicted temperatures of the melting point
of pure iron at the pressure of the inner-outer core
boundary.
Tm(K)
Gilverry (1957)
Zharkov (1962)
Bundy & Strong (1962)*
Higgens & Kennedy (197 1)*
Birch (1972)*
Leppaluoto (1972)
Boschi (1975)t
Liu (1975)$
Stacey (1977)
Boschi, Mulargia & Bonafede (197911
Abelson (1981)$
Stevenson (1 981)
Brown & McQueen (1982)$
Anderson (1982)
Brown & McQueen (1984)s
Spiliopoulos & Stacey (1984)s
Young & Grover (1 984)
This reports
6200
6200
6100-81 00
4250
5100
7000-9000
6000
5125
3157
4500-7000
7 800
6300
6200 f 500
5900 i 700
5800 f 500
6140 f 575
6600
6210 i 400
Based upon an extrapolation from experiments at
6 GPa.
t A special case of the Ross (1969) theory.
$Based upon Monte Carlo theory.
§Based upon an extrapolation from experiments at
243 GPa.
Ap ~ 0 . 4 5gm cm-3 going from 280 to 330 GPa, making the density at the IC-OC
boundary pressure about 13 g ~ m - ~ .
This density, calculated from constructing the phase diagram which satisfies the experimental data on iron, is slightly larger than that reported in the PREM seismic model
.
it is less than the value of p
(Dziewonski & Anderson 1981), which is 12.86 g c - ~ However,
predicted by the CAL-8 seismic model (Bolt 1982), which is 13.4 gm c - ~at the solid side of
the IC-OC boundary.
One may conclude that the density of the pure iron at inner core conditions calculated by
a self-consistent detailed thermodynamic approach agrees with the seismically determined
density of the inner core, considering that the errors in the thermodynamic approach are
comparable to the seismic models.
The temperature at the Earth's centre is estimated by applying the adiabatic compression
law :
(a 1 nT/a 1 np)s
= y.
(14)
Using y = 1.7 for the inner core and fixing T at 6210 K at the inner-outer core
boundary, it is found that A T = 244", that is, the temperature at the centre is 6450 2 400 K.
6 The pressure-density profile of hcp iron at inner core conditions
It will be shown that the pressure of the hcp phase computed at inner core temperature and
density conditions correlates well with the pressure as determined from seismology studies.
572
0. L. Anderson
There is not general agreement about the density of the inner core among seismologists.
For example, the QM, model of Jordan & Anderson (1974) has virtually no jump in density
at the inner-outer core boundary, while the PREM model (Dziewonski & Anderson 1981)
shows a jump of 0.63 g ~ m - and
~ , as the extreme case, the CAL 8 model (Bolt 1982) shows
a jump of 1 .17 g cm-3 . In the following we shall use the CAL 8 model and the PREM model
as upper and lower limits. A table showing P versus p for these two models is found on pages
180 and 181 of Bolt's (1982) book.
No statement about the best seismic model for the inner core will be made here; that
decision is a seismological question. But we note that it is extremely difficult to choose
between models of the core based upon the geodetic and cosmological boundary conditions.
The mass and moment of inertia of the inner core make only a very small contribution to M
and 1 of the whole Earth. The difference between the jump in p from one model to another
quoted above is not easily determined from the Earth's boundary conditions.
To proceed, an equation of state accounting for both temperature and pressure of the
thermal pressure at outer core conditions must be calculated. In this calculation, the simple
linear relationship between K and P, represented by (6) and (10) is not sufficient, since the
calculation will first be along an isotherm at 0 K.
The EOS in its most general form is
where P o ( V ) is the pressure-volume relationship at absolute zero and PTH is the thermal
pressure. PTH( V, T ) will be hereafter designated PTH , which is given by
(Anderson 1984), and where M / p is the average atomic weight, here 55.83.
An estimate of y for the inner core is needed. Bukowinski (1977) found y = 1.87;
Jamieson, Demarest & Schiferl (1 978) found y = 1.5 ; and Anderson (1979) found y = 1.6
for the lattice contribution alone. For metals, an electronic contribution must be added. The
detailed calculation can be found in Stacey (1977) or in Jamieson et al. (1978), but as
shown by them, adding a small term to y suffices in place of the detailed calculation because
of the smallness of the electronic contribution relative t o the lattice term. Jamieson et al.
(1978) found that adding 0.1 to y suffices for the electronic correction. We shall use
y = 1.7
for the inner core.
In the previous section the temperature at the Earth's centre is found to be 6450 K f 400.
P T H= 61 f 5 GPa at the centre.
The temperature is about 240" less at the inner-outer core boundary than in the centre,
so the value of P,, at the midpoint of the inner core is about 60 f 5 GPa. P T Hmust be subtracted from the seismically determined P in order to obtain the zero degree estimate of
p, PO(V).
After correcting the values of pressure to absolute zero, the pressure of the PREM data,
for example, is found and is shown in Table 4.
Values of p o , K O , and K i of the hcp phase at absolute zero must be identified. From
Table 1, the values for hcp iron at 300 K are: po = 8.28,Ko = 173, and K i = 4.4. Correcting
these values to absolute zero, it is found that po = 8.29, K O = 175 GPa and K,' = 4.4. The
value of K i is weakly constrained from the experiments. This is seen by the three values of
K: for (Y iron in Table 1. However, it is futile to attempt t o constrain K: to better values
Properties of iron at Earth’s core
573
Table 4. P ~ e s s u ~ cins the inner core from PRILM (as found
in tables 4 and 5 0 1 Bolt 1982), but correctcd t o absolute
xro.
Radius
Depth
(km)
(km)
Pressure - PREM
Pressure
Corrected to 0 K
(CPa)
(GPa)
0
6371
6271
6171
6071
597:
5871
5771
5671
5571
5471
5371
5271
5171
5149
36 3.9
363.6
362.9
361.7
360.0
357.9
355.3
352.2
348.7
344.6
340.2
335.4
330.0
328.9
100
200
300
400
500
600
700
800
900
1000
1100
1200
1221.5
299-309
299-309
298-308
297-307
296-306
294-304
291 -301
288-298
284-294
2 80 -290
275-285
271 -281
265-275
264-274
because, as we shall see, there is a trade-off between Kd and KoK;, and the latter parameter
is completely unmeasured. (K; is the second derivative of K with respect to P at zero
pressure and K o K ; is dimensionless.)
To compute the density at the inner core pressure, an isothermal equation of state, that is
Po ( V ) in (1 5 ) , must be chosen. A more fundamental approach for finding the isothermal
EOS by the quantum mechanics method, such as done by Bukowinski (1977) for the fcc
phase, has not yet been made for the hcp phase of iron.
It is sufficient here to show that several well known semi-empirical isothermal EOS fit
the seismological data. We will use five EOS that have been used by various authors to
estimate properties of iron at high P,all of which have been discussed by Stacey, Brennan &
Irvine (1 981). These EOS’s, as applied below, are not entirely empirical, nor are they used
here as curve-fitting devices with floating values of the parameters. The three parameters p o ,
K O ,Kd , all of which are determined by experiment, fix the density-pressure trajectory once
the EOS has been selected.
Two important variations of one EOS are listed below (see Stacey e t a f . 1981):
Birch-Murnaghan (third order)
Birch-Murnaghan (forth order)
KoK{
+ (Kl
- 4) (KO’ - 3)
+
1’
9
[ ( P / P o ) ~ ’-~ 112
} + ..
In addition to the above, the Morse EOS is used (see equation (96) of Stacey et al. 1981).
The Born-Meyer will be used (see equation (90) of Stacey et al. 1981).
0. L. Anderson
574
Other EOS’s used to estimate iron properties are the Born-Mie EOS (see equation (75)
with m = I in Stacey et al. 1981) and the Stacey EOS (see equation (1 16) of Stacey et al.
( 1 981).
The use of the Morse equation for compressed iron has been recommended by a number
of authors (for example, Boschi et al. 1979) because of its appropriateness for metals.
The Born-Meyer (called the Method of Potentials by many Soviet writers) has been used
by scientists in the Institute of Physics of the Earth, with good results (Zharkov & Kalinin
1971).
The Born-Mie is perhaps the simplest third-order equation of state giving useful results at
very high prcssures (Anderson 1970).
The Stacey EOS has a rigorous thermodynamic basis (Brennan & Stacey 1979), and has
been applied to iron at high pressures (Spiliopoulos & Stacey 1984).
The Birch-Murnaghan EOS is widely used among geoscientists. It also has an additional
advantage: KOK: can be defined independently of Kd , by using equation (1 9).
The values of P versus p are computed for hcp iron for the five EOS’s listed above, where
the zero-temperature values of the experimental parameters used are po = 8.29;
Kd = 180 GPa, Kd = 5.2. The results are plotted in Fig. 5 . It is concluded that all five EOS’s
come reasonably close to the density profile of the inner core (as corrected to 0 K), within
the errors inherent in the seismic model approach.
Adjustments in the values of Kd taken well within the experimental determination of this
parameter would change the extrapolations, so that any one of the five EOS could be made
to intercept either the CAL-8 data or the PREM data. The uncertainty in the measured
density of the core is evidenced by the two data sets (PREM and CAL-8).
A fourth-order EOS is preferable at these high compressions, so the Birch-Murnaghan
0
v2
13.5
0
E
5
z
w
13.0
12.5
200
240
280
320
360
PRESSURE, GPa
Figure 5 . Trajectories of five third-order equations of state to inner core pressures, using the zerotemperature value of hcp iron for p o , K O and K,’ determined experimentally. These trajectories are compared with the seismic data of density and pressure (corrected t o absolute zero) of two earth models.
Properties of iron at Earth S core
5.a
rI-. .
Huncertaintv. A K. = 5
"
* uncertaintv. AD,= .05
5.6
Kd
5.4
575
-
5.2 -
-
-
I
I
5.0
0
KO K Z
Figure 6 . Values of KA and K , K i required to make the trajectory of the fourthurder Birch-Murnaghan
EOS intersect with the pressure density of two seismic models (corrected to absolute zero).
EOS is useful for demonstrating the sensitiveness of the extrapolations t o the physical
parameters o f the EOS.
In the third-order Birch-Murnaghan EOS, the choice of KA = 5 . 2 , implies a value of
KoK: = -6.53 (Stacey et al. 1981), but KoK: should be independent of K ; . Allowing
K o K l t o be arbitrary, one can map the pairs of values in KA and KoK: that make the
Birch-Murnaghan fourth-order EOS exactly fit either the PREM data or the CAL-8 data.
The resulting values are plotted in Fig. 6. Here it is seen that by choosing very reasonable
sets of values of KA and K,K:, the Birch-Murnaghan EOS trajectory can be made t o agree
exactly with either the PREM data or the CAL-6 data.
Since the required values of K o K i and KA are within the bounds of experimental
uncertainty, we conclude that pure hcp iron gives values of density at inner-core conditions
in harmony with seismic data of the inner core.
7 Geophysical implications
It is found that the choice of hcp iron for the inner core as indicated by the Brown &
McQueen shock wave data is consistent with our present knowledge of the seismic data for
the inner core. Within the errors of the experiments, and the seismic modelling method,
agreement is found for the pressure, density, bulk modulus, longitudinal velocity, and
Poisson's ratio.
The errors involved would allow the inner core to be either slightly more dense than pure
hcp iron by choosing the CAL-8 model, or slightly less dense than pure hcp iron, by
choosing the PREM model. If the former case can be proven, then a slight amount of nickel
could be added t o the iron in agreement with opinions on nickel in the core (Brett 1976;
McQueen & Marsh 1966; Al'tschuler, Sinakov & Trunin 1968). In the latter case, a slight
amount of a lighter element such as sulphur could be added to the iron. One could postulate
that the effects of a lighter element (sulphur) would exactly cancel the effects of the nickel
on the iron properties. But for the physicist, a convenient and sufficient model is that there
are n o impurities in the inner core. In any case, such impurity effects would have only a
slight effect on calculated physical properties of the Earth.
0.L . Anderson
576
This means that a process has been going on which sweeps most (if not all) of the
impurities from the inner to the outer core, and the inner core-outer core boundary is a
chemical boundary which separates essentially pure iron from quite impure iron. The
analogy t o ‘zone-refining’, or fractional crystallization by a moving solid-liquid interface
used in metallurgical processing of pure materials, is striking. The assumption by Verhoogen
(1961) that the Earth’s deep interior is slowly cooling, and is accompanied by a
corresponding growth of the inner core by crystallization, is confirmed by these results. The
release of light elements into the liquid outer core with its low viscosity creates gravitational
energy which can be directly converted into magnetic fields because the rising elements drive
the fluid motions directly (Verhoogen 1961, 1980; Gubbins 1977; Gubbins & Masters 1979;
Loper 1978). Thus, these results reinforce the idea that gravity may be important in driving
the Earth’s dynamo.
The amount of separation between the adiabat and the liquidus in the outer core affects
the physics of the dynamo. Higgens & Kennedy (1971) pointed out that if the adiabat
crosses the liquidus at a pressure less than that of the IC-OC boundary, stratification must
result. If the adiabat lies quite close t o the liquidus, then an iron slurry will be produced
which would affect the motion of the dynamo (Braginsky 1984).
The ratio of the slopes of the adiabat and liquidus in T-p space is given by the ratio of
(14) t o (3).
d In Tsld In p
d In T,/d
Y
-
In p
2 (y
-
1/3).
A value of unity for (20) would indicate the presence of a slurry, but the adiabat would
always have <he smaller slope if y > 2/3. Our calculations of y are for the iron solidus, but
the values of y for the liquidus cannot be very different. The change of volume and Cv
across the phase change is small, as is the change of thermal expansivity and bulk modulus at
high pressures. Taking y from Table 2, the value of (20) would be 0.63 at 135 GPa (the
core-mantle boundary) and 0.79 at the tp. Using y = 4/3 from Young & Grover’s (1984)
determination, we find 0.68 for (20) at the IC-OC boundary. This means that the adiabat
of the outer core should be several hundred degrees higher than the liquidus except
close to the IC-OC boundary. Even if there are subadiabatic regions in certain regions of
the flow regime, the adiabat is too far removed from the liquidus for consideration of a
crossing. The adiabatic gradient is less than the melting gradient, and affects the gravitationally driven dynamo accordingly (Loper 1978).
Acknowledgments
The author acknowledges financial support by the Department of Energy (DOE FG03-84ER 13203) and by the Lawrence Livermore National Laboratory. IGPP Contribution
No. 261 1 .
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