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Temperatures Within the Earth The present temperature distribution in the earth depends on (1) the original temperature distribution shortly after formation; (2) the dis- tribution and intensity of heat sources, all of which are time dependent; and (3) the mechanism of internal heat transfer, whether by conduction or convection or both. Conversely, if the present temperature were known, it should be possible, at least in theory, to extract from it some information on, and possibly set upper or lower limits to, the distribu- tion of internal heat sources. Much effort has been devoted in the past to calculating the tempera- ture distribution in the-mantle, particularly the upper mantle, from the heat conduction equation. The main datum is the surface heat flow; a steady state is commonly assumed. Examples of such calculations are the often quoted "shield" and "oceanic" geotherms of Clark and Ring- wood (1964) and the calculations by MacDonald (1965), which are now only of historical interest, if only because it is now certain that heat is also carried in the upper mantle by convection; not all heat transfer takes place by conduction in the radial direction. It is clear that, for instance, much of the heat flowing across the oceanic floor has been brought there by horizontal motion of hot material rising under, and moving away from, oceanic ridges. Curiously, it escaped everyone's (or almost everyone's) notice that these calculations were self- contradictory, insofar as the large horizontal temperature variations between subcontinental and suboceanic mantle that emerged from 29

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30 ENERGETICS OF THE EARTH them (e.g., MacDonald, 1965) were sufficient to cause the very convec- tion that had been assumed not to take place. But when convection is included, there is no simple way of calculat- ing the temperature, short of solving the full set of time-dependent equations expressing conservation of mass, momentum, and energy, which contain all the parameters of the conduction equation plus some poorly known parameters describing theological properties and their dependence on pressure and temperature. Only partial solutions for idealized models, generally two-dimensional ones, are available at this time; there is still some debate as to whether convection embraces the whole mantle or is restricted to its upper few hundred kilometers. Furthermore, to get a solution one must first postulate what and where the heat sources are; thus we obviously cannot use these calculations to determine the heat sources, which is our main objective at this stage. We must thus turn to other, more direct methods for estimating what the temperature may be at a given place. Broadly speaking, there are two methods for doing this. In the first method, one attempts to determine the temperature at a given depth from a known property of material at that depth. The property may be physical (e.g., velocity of elastic waves, electrical conductivity) or chemical (e.g., phase equilibrium between solid and liquid, or between polymorphs). In the second method, one tries to determine the tem- perature gradient in a homogeneous layer from an observed variation with depth of a physical property of the layer. In both methods, one must first allow for the effect of pressure, which, of course, also in- creases with depth. Results, as we shall see, are on the whole rather disappointing. The main difficulty generally lies in estimating the effects of pressure. Pres- sure in the earth is tolerably well constrained, to better than 1 percent, but pressure coefficients of physical properties are hard to measure at the very high pressures of the earth's interior. Physical properties in the earth are generally not known very precisely. Density at a given depth, for instance, is not known to better than a few percent, which is also the magnitude of any likely temperature effect. The exact composition and nature of the mineral phases present in the lower mantle are still debated, and so is the composition of the core. Some physical proper- ties (e.g., electrical conductivity) may be sensitive to variables (oxidation state, grain size, lattice defects, impurities) that, in the ab- sence of representative samples, cannot be estimated at all. We now review, in descending order from the top of the mantle to the inner core, some of the results.

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Temperatures Within the Earth 31 THE UPPER MANTLE THE LOW-VELOCITY ZONE The low-velocity layer, as the name implies, is a layer in which the seismic velocities vp and vs are slightly lower than they are in the layers immediately above or below. The decrease in velocity with increasing depth is more pronounced for vat (shear wave) than for vp (compres- sional wave), and more pronounced under oceanic areas than under continents, where the low-velocity layer may either be absent or occur at greater depth. In oceanic areas the top of the low-velocity layer almost reaches the surface under ridges. Its depth increases progres- sively with distance from the ridge to a maximum of about 100 km; nowhere does it coincide with a sharp discontinuity. There is some evidence that the low-velocity layer coincides with the asthenosphere or "weak" zone, so named for its enhanced ability to flow as compared with the more rigid lithosphere above it. Lithospheric thickness is generally determined from seismic surface-wave dispersion; it appears to be well correlated with surface heat flow (Chapman and Pollack, 1977). There are several possible interpretations for the low-velocity layer. Thomsen (1971) has shown that it could be due to a steep temperature gradient. The effects on seismic velocities of increasing temperature and increasing pressure being generally of opposite signs, the normal increase in velocity with increasing depth (pressure) could be reversed in a zone where the temperature gradient is sufficiently steep to over- come the pressure effect. The required temperature gradient is of the order of 5-10/km for vs and 10-20/km for vp. At the moment, the preferred interpretation is that the reduced seis- mic velocities in the low-velocity layer are caused by the presence of a very small amount, of the order of 1 percent or so, of melt that forms a thin fluid film separating the grains of an otherwise solid rock. The evidence in favor of this interpretation is summarized by Solomon (1976~. The temperature at the top of the low-velocity layer must then be the temperature at which melting begins at the corresponding pressure. The temperature of incipient melting depends of course on the composition of the rock but does not depart much from 1100C (at P = 1 bar) for the several types of peridotite whose density and elastic properties make them likely constituents of the upper mantle. The melting point for dry peridotites rises with pressure at a rate of about 10/kbar, bringing it to about 1400-1500C at a depth of 100 km, where

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32 ENERGETICS OF THE EARTH the pressure P -30 kbar. In the presence of water, however, the incipient melting point first decreases with increasing pressure, reach- ing a minimum of about 1000C at 30 kbar for a water content of only 0.1 percent (Wyllie, 1971, Figure ~181. The amount of water present in the mantle (presumably mainly in amphiboles and micas) is not exactly known. Carbon dioxide may also be present. Mysen and Boettcher (1975) have reported experiments on the melting of four different perid- otites (two spinet and two garnet lherzolites) in the presence of water and CO2 in variable proportions, and in the pressure range from 7.5 kbar to 30 kbar. When present alone, CO2 does not lower much the solidus temperature from its value for the dry system, in contrast to water, which may lower the solidus to less than 900C at 15-20 kbar. Mysen and Boettcher's experimental conditions are such that there is enough water present to saturate the melt at all pressures, a condition that may not be met in the mantle. Furthermore, the lherzolite speci- mens used in these experiments (inclusions in Hawaiian lavas and/or kimberlite pipes) may represent the fraction of the original mantle rock that is left after partial melting and removal of the most easily fusible components, and are therefore not necessarily typical of undepleted mantle or pyrolite. What may be concluded from this is that if we accept the hypothesis that the top of the low-velocity zone or bottom of the lithosphere is indeed the depth of incipient melting, the temperature there is probably in the range 950-1300C where the lithosphere is 50 km thick, and in the range 900-1400C where its thickness is lOO km. In continental areas where the lithospheric thickness is 200 km, the temperature of incipient melting is probably not greater than 1200C. The lower limit of the ranges quoted here applies when there is enough water present to saturate the melt, while the upper limit is for the "dry" case of no water at all. If the bottom of the low-velocity zone is defined as the depth at which the shear velocity resumes the value it has at the top of the low-velocity zone, the thickness of the low-velocity zone comes out at about 10~150 km. The important point is that the thickness is finite, implying that at depths greater than 200 250 km the mantle is again entirely solid, or that the temperature gradient has dropped sufficiently for the effect of pressure on velocity to once more become preponderant, as it is above the low-velocity zone. Return to subsolidus temperatures could be caused either by a sharp drop in the gradient temperature, as in Figure 3-la, by a sharp rise in the solidus temperatures, as in Figure 3-lb, or by a combination of the two. Melting in the presence of water entails a minimum solidus temperature.

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Temperatures Within the Earth 33 (a) SOLI DUS ACTUAL TEMP. (b) 1 l B DEPTH- t ACTUAL TEN SOLIDUS// I CL ,'_ , ! A B DEPTH- FIGURE ~1 Possible temperature profiles in the upper mantle. The low-velocity layer extends from A to B. In (a), the temperature of incipient melting (solidus) rises linearly with increasing pressure or depth ("dry mantle"); the finite thickness of the low-velocity layer is due to a drastic reduction in slope of the temperature curve. In (b), temperature increases linearly while the solidus exhibits a min~rnum ("wet mantle"). The situation in the mantle is probably somewhere between cases (a) and (b). GEOTHERMS FROM NODULES IN KIMBERLITE Geotherms (i.e., temperature-depth curves) for subcontinental areas in the depth range 10~250 km have been estimated from the inferred temperature and pressure of equilibration of the several minerals pres- ent in lherzolite inclusions in kimberlites. The equilibrium distribution of the major elements calcium and magnesium between coexisting clinopyroxenes and orthopyroxenes in lherzolites is a function of tem- perature but is relatively insensitive to pressure. On the other hand, the distribution of aluminum between coexisting garnet and orthopyroxene is sensitive to pressure. The distribution coefficients are then compared with distribution coefficients measured on natural samples to determine the temperature and pressure at which the samples last equilibrated. Experiments are difficult because of the great length of time needed to reach equilibrium; the influence of other ubiquitous elements (e.g., iron) on distribution coefficients must also be ascertained. Investigating in this manner a number of nodules from kimberlite pipes in Lesotho, Boyd and Nixon (1973) reported temperatures in the range 900-1400C at depths ranging from 140 km to a little over 200 km. The individual points fall nicely on two curves ("inflected" geotherms) indicating a sharp steepening of the temperature gradient at about 180 km. Specimens from below that depth are generally sheared, whereas

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34 ENERGETICS OF THE EARTH those from above are granular. Boyd and Nixon point out that in the absence of marked differences in thermal properties above and below the inflection, continuity of steady-state heat flow across the inflection requires a continuous temperature gradient. They see in the inflected geotherm evidence for an event that caused major heating in the depth range 15~200 km. (The eruption of kimberlite that carried the nodules to the surface is in itself evidence for a perturbation of some sort, since kimberlite is not erupted continuously everywhere.) Figure 3-2 shows results of many investigators, as plotted by Harte (19781. Although there is much regional variation, the plot suggests possible tempera- tures of about 1000C at 150 km, and of 1200-1400C at 200 km (65 kbar) and that the Mg/Fe ratio is 9; from Akimoto's data we would estimates for a given temperature, from 65 kbar to 40 kbar in one instance; corresponding depth estimates are reduced by 6(}65 km. Harte (1978) suggests lowering the estimates shown in Figure 3-2 by perhaps 30~5 km. The matter can apparently not be resolved until we get more reliable experimental data on the distribution coefficients of major elements in coexisting phases. Perhaps all that can be said at the moment is that nodules in kimberlite suggest temperatures of about 1000 + 200C at 100-km depth, and 1200 + 200C at 150 km. 1400 1200 C' (a) ~i// / '1 - (b) i: L 8001 100 150 200 100 depth/km 1000 ~ / of o o o )~` oo 2/ o Ye . . . . . . . . 150 200 FIGURE ~2 Estimated temperatures and depths of equilibration of garnet-lherzolite xenoliths in kimberlite pipes. Plot (a) is for pipes in northern Lesotho; (b) represents pipes in southern and southwest Africa and in Udachnaya (USSR). Symbols are as follows: triangles, Premier; squares, Kimberly; dots, Louwrencia; diamonds, Udachnaya. The depth scale may need reduction, in some instances by perhaps as much as 60 km (see text). Reproduced with permission from Harte (1978).

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Temperatures Within the Earth 35 THE MANTLE'S TRANSITION ZONES Ever since Bullen's early work (Bullen, 1936, 1940), seismologists have noted that the rate of increase with depth of the body-wave velocities vp and as is particularly rapid in what Bullen called zone C, roughly between 350 and 900 km. Details of the seismic velocity distribution have long been, and to some degree still are, obscure. After muc h debate, it is now generally recognized that there is no discontinuity of the first order (discontinuous jump in velocity), or even of the second order (discontinuous change in velocity gradient), in this depth range. There do seem to exist, however, two narrow zones, at around 400 and 650 km, respectively, in which the velocity of both vp and vs increases with depth more rapidly than it does either immediately below or above these transition zones; the thickness and depth of these zones are still rather uncertain. Bullen had already recognized that velocities in zone C are incompatible with the assumption of a homogeneous mantle. Birch (1952) confirmed that heterogeneity might be due to phase changes rather than to changes in gross chemical composition. Most of the minerals likely to be present in an upper mantle peridotite (olivine, pyroxenes, garnet) have now been found experimentally to undergo phase transformations at pressures such as exist in the mantle's C zone. For instance, at 10(10C olivine undergoes a major transformation at a pressure in the neighborhood of 11~120 kbar for olivines with a molar fraction of Mg2SiO4 (forsterite) greater than 0.8. The olivine transformation is now generally believed to account for the velocity transition zone near 400 km. Common olivine being a two- component system (Mg2SiO4-Fe2SiO4), any phase transition (e.g., melting) will take place, at fixed temperature, over a finite range of pressure. This spreading out of the transformation over a finite depth range accounts for the absence, near 400 km, of any first-order discon- tinuity in either density, elastic properties, or seismic velocities. The transformation has been studied experimentally by Ringwood and Major (1970) and by Akimoto (19721. Olivines with less than about 70 mol percent Mg2SiO4 invert to a so-called ~ phase with the spinet structure; but magnesium-rich olivines invert to a phase ~) with a much distorted spinet structure that, according to Ringwood and Major, remains stable up to very high pressures. Akimoto, on the contrary, thinks that the ,B phase transforms to a By phase at a pressure not greatly in excess of that of the olivine-,6 transition. At 1000C, and for an olivine with 90 percent Mg2SiO4, the olivine-,`3 transition begins at about 119 kbar, and the beginning of the ,l3-~y transition is at about 135 kbar. With 80 percent Mg2SiO4, the olivine-^y transition begins at about

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36 ENERGETICS OF THE EARTH 100 kbar; at 110 kbar the iron-rich By phase inverts to ad phase that disappears again at 135 kbar. Pure Mg2SiO4 inverts to the ,6 phase at 125 kbar and 1000C; it is not known at what pressure it transforms to the phase, if indeed it ever does. From Akimoto's phase diagrams at 800 and 1000C, rough values of the temperature coefficient of the transfor- mation pressure can be determined. It appears to be about 45 bar/de" (dT/dP = 20.8/kbar) for pure Mg2SiO4, somewhat less (30 bar/de", dT/dP = 33.3/kbar) for 80 percent Mg2SiO4. The depths in the mantle at which the transformation begins and ends are not precisely known, and are likely to vary from place to place. Suppose that the transformation starts somewhere at 400 km (P = 132 kbar) and that the Mg/Fe ratio is 9; from Akimoto's data we would infer a temperature of about 1350C. Ringwood and Major find a some- what higher temperature (1600C) at 400 km if that is the mean depth of the transformation zone and the olivine has 11 percent Fe2SiO4. Considering the high uncertainty (50 percent) affecting the tempera- ture coefficient of the transformation pressure, and our ignorance of the exact iron content of mantle olivines, of the precise depths at which the transformation begins and ends, and of whether the olivine-spinel phase transformation is indeed responsible for the seismic anomaly near 400-km depth, all we can say is that the temperature at that depth is probably in the range 1300-1600C. This is not a very precise estimate, but it is so far the best we have. It agrees with an earlier estimate by Graham (1970) of 1450 + 120C at 370 km. Near 650-km depth (P-250 kbar) there is another thin zone in the mantle in which the seismic velocities increase downward abnormally quickly. This has been variously attributed to a breakdown of olivine (spinet form) to MgSiO3 + MgO (Liu, 1975b), to a transformation of pyroxenes to phases with ilmenite or perovskite structure, or to a breakdown of all silicates to their constitutent oxides, with SiO2 in the form of stishovite. The situation is summarized in Figure 3-3 taken from Liu (19771. None of the relevant phase equilibria is sufficiently well known to allow even an approximate determination of the tem- perature at which the phase change occurs. THE LOWER MANTLE LAYER D From differences in the gradient of seismic velocities vp and vs. Bullen (1950) divided the lower mantle into two zones, namely D' and D". D' extends from 984 km to 2700 km and is characterized by a monotonic

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Temperatures Within the Earth 37 300 2SO . 200 Periclase + Perovski e ~ ~- ll l 1 ! c ! Spinel + Stishovite I ~ + Stishovite c 150 1 100~ TO ~1 1 o A I ~I , 1 ~I 05 Mole fraction SiO2 FIGURE ~3 Schematic phase diagram for the system MgO-SiO2 at about 1000 C. The dashed line represents a mixture 60 mol per- cent olivine plus 40 mol percent pyroxene. Reproduced with permis- sion from Liu (1977). downward increase in velocity. D" extends from 2700 km to the core boundary (2885 km). It is now generally recognized that the velocity gradients are much smaller in D" than in D' and may in fact be negative (Bolt, 1972; Cleary, 19741. The lower mantle, from 700 km to the top of D", is commonly con- sidered to be mineralogically homogeneous even though some seismic observations have suggested to Johnson (1969) the possible presence of anomalous zones corresponding perhaps to phase transitions with small changes in density and elastic constants. For the time being, we shall

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38 ENERGETICS OF THE EARTH ignore them and take D' to be homogeneous. We will return to the matter later. In a homogeneous layer, in hydrostatic equilibrium, the variation in density p with depth is due to the effects of pressure P and temperature T. The pressure varies as dP = -g p dr, where r is distance from the center of the earth and g = gory is the acceleration of gravity. The variation of density is then given by the relation dp gp2 dT = - -ap dr KT dr (3.1) where KT = p~dP/8p)T is the isothermal incompressibility and cY = - (l/p)~6p/8T)p is the coefficient of thermal expansion. If ~ and KT were known from the equation of state p = p(P, T) for the material forming the layer, a knowledge of the density variation with depth dp/dr would be sufficient to determine the temperature gradient dT/dr. The difficulty is that the equation of state for the lower mantle is not known a priori, since we aren't quite sure what the lower mantle is made of. A major step was taken by Birch in his classical paper of 1952. An ingenious and rigorous transformation of Equation (3.1) leads to Birch's celebrated generalization of the Adams-Williamson equation: 1 _ ~ d!+ = (8KT) + Ta{-YA + (TCY)12B + g , (3.2) where ~ is the seismic parameter vp2 -~413)vs2. The left-hand side of this equation is thus derivable from seismic observations. ~ is Grueneisen's ratio EKT y= = = pcv pep cp (3.3) KS is the adiabatic (isentropic) incompressibility, and cv and cp are, respectively, the specific heat at constant volume and constant pres- sure; ~ is the superadiabatic gradient defined by dT _ Tag _ = - -7, dr cp (3.4) where the first term on the right-hand side of Equation (3.4) is the

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Temperatures Within the Earth 39 "adiabatic" gradient derived from the thermodynamic relation for the change in temperature with pressure at constant entropy S. Tc' pep (3.5) In equation (3.2), A, B. and C are dimensionless functions of param- eters such as (dKT/8P)T, (8KT/8T)P, (~~/6T)P' (dcp/dT)p; all these quantities are, in principle, derivable from the equation of state, pro- vided that the equation of state be known in the pressure-temperature range of the lower mantle; then Equation (3.2) can, again in principle, be used to find T and r. The difficulty is that the temperature terms, for instance ATya, are small compared to the first term, (6KT/dP)T, which must therefore be known very accurately. As things stood in 1952, Birch concluded that all one could say was that a temperature of 5000C produces no "serious discrepancy." Since Birch's early work, some progress has been made in determin- ~ng an equation of state applicable to the lower mantle. An interesting empirical discovery was made of a linear relationship between density and compressional wave velocity Up, or bulk sound velocity c = (Ks /p) I/2, for materials of the same mean molecular weight, regardless of crystal structure. Shock wave experimental data on a variety of minerals and rocks have also provided much information on density at very high pressures. Graham and Dobrzykowski (1976), and more recently Watt and O'Connell (1978), have calculated the temperature distribution in the mantle between 700 and 1200 km that best fits density and seismic velocities, using third-order finite strain theory and assuming adiaba- ticity. Results depend on assumed composition (e.g., the ratio of (Mg,Fe)O to SiO2), on mineralogy (e.g., mixed oxides versus phases with the perovskite structure), and, of course, on which of the many proposed density distributions is used. Graham and Dobrzykowski (1976) exclude pyroxene composition ((Mg,Fe)O/SiO2 = 1), as it leads to "a temperature profile which is clearly outside the range of reason- able solutions" (~2500C at 600 km). Peridotite and dunite composi- tions lead to a temperature of 1600 + 400C at the 671-km seismic discontinuity. Watt and O'Connell suggest temperatures about 1300-1700C at 700 km and an adiabatic gradient of about 0.3/km. In what is probably the best study of this kind to date, Wang (1972) used shock wave data to calculate temperatures in the lower mantle between 1300 and 2800 km. He starts from an acceptable density distri

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56 ENERGETICS OF THE EARTH however, be considered to be much more than a coincidence, consider- ing the amount of guessing that goes into the choice of y. THE TEMPERATURE AT THE INNER CORE-OUTER CORE B OUNDARY Since the inner core is solid and presumably consists of iron (plus, possibly, some nickel and small amounts of a few other less abundant elements), while the outer core, consisting mostly but not entirely of iron, is liquid, it is rather natural to assume that the temperature at the boundary must be such that solid iron is in equilibrium with its melt, at the prevailing pressure. The temperature at which a pure substance is in equilibrium with its melt is called the melting point Tm. It depends only on pressure, as dTm AV dP (3.15) where AV = Vl - Vet is the difference in volume between liquid and solid, and AS is the corresponding difference in entropy. In a multi- component system, the temperature at which a solid phase is in equilibrium with liquid depends not only on pressure but also on the composition of the liquid. For instance, in a system consisting of iron end x percent of sulfur atP = 1 bar, solid iron can be in equilibrium with liquid at any temperature between 1809K (for x = 0, pure iron) to 1261K (for x = 31.4~. Since addition to a pure substance of any com- ponent that is soluble in the melt necessarily lowers the melting point of the pure substance, the melting point of pure iron at the pressure of the inner core boundary sets an upper limit to the temperature there. THE MELTING POINT OF IRON The pressure dependence of the melting point of iron has been the subject of much debate in recent years (for good summaries see Jacobs, 1975, and Boschi, 1975~. Some measurements of doubtful accuracy have been carried to 200 kbar. The most frequently cited data are those of Sterrett e' al. (1965) up to 40 kbar. The problem is to extrapolate those results to the pressure (3.3 Mbar) of the inner core. It is curious that we are still unable to explain why solids melt, or, for that matter, why the liquid state exists at all. All theories of melting are essentially empirical. A theory that has had much success is that of Lindemann. Lindemann supposes that melting occurs when the ampli

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Temperatures Within the Earth 57 tude of thermal vibrations reaches a fraction ha of the lattice spacing a . In that state, at temperature Tm, the kinetic energy of vibration 3/2 kTm = i/2 m (6a)2~2, assuming that atoms of mass m vibrate harmonically with frequency co; k is Boltzmann's constant. The eject of pressure on Tm will arise from its effect on a, which is proportional to Vi/3 (V iS volume), and on w. It is convenient here to introduce the Debye tem- perature ~ = h cu/k (h is Planck's constant), and recall that from Debye's theory ~ _ Vl/3 v, where v is the mean elastic velocity defined in Equa- tion (3.91. Thus Tm is directly proportional to v2, and the ratio of the square of the elastic velocity in the inner core to that of iron at P = 1 bar gives the ratio of the melting temperatures. Alder (1966) obtained in this manner a melting temperature of 7720K. Attempts have been made to refine the Lindemann law by reformu- lating it in less empirical terms. Boschi (1974), using a melting criterion by Ross, derives the relation Tm = lo ~ l 3 Vo 6 (3 ) ( VO ) . . . } , (3 16) where Tm is the melting point at a pressure at which the volume of the solid is V, To is the melting point in some reference state (e.g., P = 0) at which the volume is VO, and 1` V = VO - V. Here n is the exponent in the expression for the potential ~ of the repulsive interatomic force, assumed to be of the form ~ r-n, where r is the interatomic distance. For iron, n _ 8.4, as determined from isothermal compression tests. The validity of (3 .16) depends on how well the power law r-n represents the interatomic potential; different expressions for that potential lead to widely different values of Tm when /`V/V is large. Kraut and Kennedy (1966) suggested that melting points can very generally be represented as Tm = To ~ 1 + C-~ (3.17) where TV and VO have the same significance as above, and C is a constant that can be determined from the initial slope of the melting point curve. Higgins and Kennedy (1971) extrapolated the data of Sterrett et al. (1965) to 3.3 Mbar to obtain a melting point of 4250C (~4500K).* How the volume V of the solid at its melting point at 3.3 *But the initial slope of the melting curve of Sterrett et al. (2. 85C/kbar) does not agree with the slope (3.8/kbar) calculated from measurements of ~ V and AS and the Clapeyron equation (3.15).

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58 ENERGETICS OF THE EARTH Mbar is determined is not said, though the authors do state that the density at 1.5 Mbar is assumed to be 11.4 g/cm3. Although Equation (3.17) represents fairly well the melting point behavior of some substances at small compression, it is unlikely that it will apply to all substances, as Kraut and Kennedy (1966) had originally claimed. Validity of the simple linear relation is now claimed only for metals (Higgins and Kennedy, 1971~. If (3.17) is valid it follows that dips = To Cp-, where ,B is compressibility. Since,l3 and V are necessarily positive quantities, dTm IdP cannot change sign and Tm can go through neither a maximum nor a minimum. Yet there are many metallic elements (cesium, barium, gallium, silicon, antimony, bismuth, selenium, tel- lurium, cerium, uranium) that do just that; for lithium and potassium the slope of the melting point almost goes to zero, even though ,8 does not. (For a compilation of phase diagrams at high pressures, see Cannon, 1974.) It is true that many of the observed changes in slope of the melting curve (Tm, P) occur in conjunction with a change of phase of the solid, yet the maximum melting point for barium occurs between 10 kbar and 20 kbar, whereas the nearest phase change occurs along the melting curve only above 60 kbar. The melting point curve of selenium has a maximum near 50 kbar, but no reported phase change (note that selenium is mentioned by Higgins and Kennedy as an example of an element satisfying the linear law). Attributing changes in slope to phase changes does not help much for iron, since between the pressure range of the observations by Sterrett et al. and of the inner core, iron under- goes at least two phase changes: ~ ~ ~ and ~ ~ ~ (Liu, 1975a). On the basis of the "y-e transition, Birch (1972) estimated that even if the linear extrapolation were correct, the melting point at core pressures might have to be raised by some 700. A comparison of (3.17) and (3.16) makes it clear that the linear rela- tion (3 .17) is an approximation that can be valid only in the limit of small 1\ V/VO or if n is small, as it would be for soft metals with high compres- sibility. Quite apart from the effect of phase changes, the linear law for iron almost certainly breaks down, perhaps seriously, at pressures in the megabar range. Boschi (1974), using Equation (3; 16), finds a melting point at 3.2 Mbar of 6600C, as against Higgins and Kennedy's 4250C. Note, however, that the figure arrived at by Boschi strongly depends on his choice for the repulsive potential. The volume dependence of the

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Temperatures Within the Earth 59 melting point, as with A, to which the melting point is related in Lindemann's theory, is a sensitive function of the interatomic potential. Most theories of melting suffer from the defect that they attempt to predict the melting point by considering only properties of the solid phase. Since the melting point is, by definition, the temperature at which the free energy of the solid equals that of the liquid, it should reflect properties of both the solid and liquid phases. Leppaluoto (1972a,b) has attempted to do this, using Eyring's significant-structure theory to predict properties of liquid iron. It will be recalled that this theory represents a liquid as consisting on the one hand of "solidlike" atoms with vibrational degrees of freedom as in the solid, and on the other hand, of "gaslike" atoms that have acquired translational degrees of freedom that allow them to jump into unoccupied sites or "holes," the number of which determines the difference in volume between liquid and solid. A partition function is then written to represent the "solidlike" and "gaslike" structures and any other structures (magnetic, diatomic, etc.) that may be significant. Leppaluoto's work has been dismissed by Boschi (1974) as "lack- ing credibility" because an early attempt by Tuerpe and Keeler (1967) to apply significant-structure theory had led to anomalous results. Leppaluoto's work was designed, however, to circumvent these dif- ficulties. He first writes down a partition function for liquid iron at P = 1 bar in which the parameters are chosen so as to give the correct TV and AS of melting and the correct Gibbs free energy change 2\G = 0 at the melting point (1809K). The lattice or "solidlike" proper- ties of solid iron at high temperature are calculated in the Einstein approximation. The partition function so designed predicts within 5 percent the thermal expansion and compressibility of liquid iron, a slightly high (20 percent) specific heat, and a good temperature depen- dence of viscosity. The melting point at high pressure is then determined as the tempera- ture at which the free energies of solid and liquid are equal. Properties of the solid are determined from shock wave data. The calculations involve a quantity AV*, an activation volume, which measures the pressure dependence of the free energy of activation a gaslike particle must have to move into a hole; to occupy a hole a molecule of liquid must do work P /` V* against the external pressureP. lo V* is similar, but not exactly equal, to the activation volume for self-diffusion, and is not well determined. Leppaluoto finds, however, that agreement between his calculated melting point and the melting point calculated from the Clapeyron equation requires AV* to be constrained between 0.07 and

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60 ENERGETICS OF THE EARTH 0.18 cm3/mol at pressures up to 3.3 Mbar. The lower limit for ~ V* gives Tm = 7000K at 3.3 Mbar; the upper limit gives 9500K. For AV* = 0, which leads to a melting temperature inconsistent with the Clapeyron equation, the melting point at 3.3 Mbar is close to the value predicted by Higgins and Kennedy (1971~. There is also some uncertainty in the melting point arising from the fact that shock wave data presumably refer to the ~ (hop) phase of iron, not to the ~ (bcc) phase present at the melting point at P = 1 bar. Taking this into account, Leppaluoto sug- gests Tm = 7500 + 2000K for iron at 3.3 Mbar. Alder's (1966) and Boschi's (1974) results fall within these limits. MELTING IN THE FE- S SYSTEM As mentioned earlier, the melting point of pure iron at 3.3 Mbar (~7500K) is an upper bound on the temperature at the inner core-outer core boundary, since the outer core presumably does not consist of pure iron. The temperature at which solid iron is in equilibrium with a multicomponent melt will always be less than the melting point of pure iron by an amount that depends on the nature and proportion of the other components. Oxygen, for instance, has very little effect on the melting point of iron until its proportion reaches about 23 percent by weight; but even then the lowering of the melting point is less than about 150. On the other hand, addition of sulfur may lower the melting point of iron by more than 500. The lowest temperature at which solid iron can be in equilibrium with an Fe-S melt is 988C at P = 1 bar (eutectic temperature); the melt then contains 31.4 percent of sulfur. At higher sulfur content, the solid phase in equilibrium with the melt (above 988C) is FeS. The phase diagram for the Fe-S system at 1 bar is shown in Figure 3~. Just as the melting point of pure iron is an upper bound to the temperature at the inner core boundary, so the eutectic temperature is a lower bound. The effect of pressure on the eutectic temperature Te has been measured at 30 kbar by Brett and Bell (1969) and at from 30 to 100 kbar by Usselman (1975a). From an initial value of 988C at P = 1 bar, Te first rises very slowly, to 998 + 5C at 55 kbar, then much more rapidly, to 1190C at 100 kbar. The break in slope, which occurs near 52 kbar, is presumably connected with a phase change in solid FeS, or with the substitution of FeS2 for FeS as the stable form of iron sulf~de; this is possible because the reaction 2FeS = FeS2 + Fe

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Temperatures Within the Earth 61 1600 1 539 1400 1200 1 000 800 600 400 J I l L i q u i d / Two Liquids 083 7430 a> 11 1 ~ 1 . 1 1_ He 20 40 60 80 S FIGURE 3 6 Phase diagram for the Fe-S system at ordinary pressure. has a large negative volume change is V = -5.31 cm3 at ordinary pres- sure. As the pressure rises, the sulfur content of the eutectic mixture first decreases from 31.4 percent at P = 1 bar to 24 percent at 55 kbar, and remains essentially constant from there on. Attempts to extrapolate the eutectic temperature to core pressures have been made by Usselman (197Sb) and Stacey (1977~. Usselman uses the Kraut-Kennedy linear extrapolation, but since the compres- sibility of solid mixtures of Fe and FeS is very poorly known, the volume of the solid phases at 3.3 Mbar has to be guessed. Stacey uses a form of Lindemann's theory that requires knowledge of A, the Grueneisen ratio for the solid phase; since this is not known, Stacey uses the value he derived for the liquid outer core. Both calculations ignore the ~y-e transition in iron, which is likely to change the slope

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62 ENERGETICS OF THE EARTH dTe/dP just as the phase transition in FeS steepens it at 52 kbar. At 3.3 Mbar, Te is between 3750 and 4050K according to Usselman, and 4168K according to Stacey. The effect of pressure on the eutectic is formally described by the relations (Prigogine and Defay, 1954, p. 365) ~ xv +x2Av2 ~ v'-us dP xiAhi +X2Ah2 (3.18) dx2= _x ---1- ~ _-- 1 , (3 .19) dP RT (xi ~ h ~ + x2 Ah2 ~ (d On x2~21/dx2 ~ lkh~Av2-Ah2Av~ where x~ and x2 are, respectively, the mole fractions of components 1 and 2 at eutectic composition, GYM is the activity coefficient of com- ponent 2 in the melt, and Ahi = hi'-his, Avi = vi'-vis, (i = 1, 2), where hi and vi are, respectively, the partial molar enthalpy and volume of component i, and superscripts l and s refer to liquid phase and solid phase, respectively. It is important to note that although viS at any pressure could be calculated from an adequate equation of state, the partial molar volume ~ cannot because it depends on volume changes that occur, at constant P and T. when Fe and FeS mix. If the FeS melt were a perfect solution, with no volume change upon mixing, vi' = vie, the molar volume of pure liquid i. The same remark applies to enthalpy, which would be calculable only if there were not heat of mixing, as in a perfect solution. That FeS liquids are not perfect solutions is shown experimentally by the very fact that the slope of the eutectic tempera- ture curve is essentially zero up to 55 kbar. This requires the numerator of (3.18) to be zero, and since x~ and x2 are both positive quantities, either Ivy or Av2 must be smaller than 0. The volume of the eutectic liquid is less than the sum of the volumes of its two pure liquid com- ponents, so contraction occurs upon mixing. Because of the markedly nonideal nature of the FeS system, it is in fact impossible to predict what its phase diagram will look like at high pressure. The solid phase FeS may become unstable with respect to FeS2 or a high-pressure phase of it. Pyrite (FeS2) melts incongruently at 743C at P = 1 bar. The very existence of a eutectic between Fe and FeS (or FeS2) may be in doubt. As noted by Kullerud (1970), most sulfur-metal systems exhibit liquid immiscibility, i.e., the existence of not just one but of two coexisting liquid phases. The systems Cu-S,

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Temperatures Within the Earth 63 Pb-S, and Hg-S have in fact two immiscibility ranges, one of which occurs at low sulfur concentration. In the Cu-S system at P = 1 bar, for instance, there is a eutectic with only 0.77 percent sulfur at 1067C, barely below the melting point of pure copper (1083C). At 1105C, a liquid with 1.5 percent sulfur is in equilibrium with a liquid containing 19.8 percent sulfur, the composition of which is close to that of Cu2S, which melts congruently at 1129C. The second immiscibility gap occurs at much higher concentrations of sulfur. This led Verhoogen (1973) to suggest that immiscibility in the Fe-S system could perhaps account for the properties of the lower few hundred kilometers of the outer core, which were once thought to form a separate layer (Bullen's layer F) with properties (e.g., density) different from those of the rest of the outer core (Bolt, 19721. It now seems that the properties of layer F are not sufficiently different to warrant its recognition as a separate entity, even though the decrease in slope of the seismic velocity versus depth curve has not been satisfactorily explained. Quite apart from the highly questionable propriety of applying, as Usselman and Stacey do, to a eutectic questionable theories of melting in one-component systems, it would seem that, from the very nature of the Fe-S system (or of the Fe-O system, for that matter), the determi- nation of its eutectic temperatures at very high pressure is even more uncertain than determining the melting point of pure iron. For all it is worth, one could perhaps venture to guess that since the slope (4.2/kbar) of the eutectic temperature curve above 55 kbar is slightly steeper than the initial slope of the melting curve of iron (3.8/kbar), pressure would tend to reduce the difference between the melting point of iron and the eutectic temperature. But this extrapolation ignores possible differences in compressibility between liquid iron and Fe-S melts that could reduce the slope of the melting curve more quickly for the eutectic liquid. In spite of all this uncertainty, Stacey (1977) has recently proposed a temperature distribution for the whole earth that is anchored to a temperature of 4168K at the boundary of the inner core. That tempera- ture of 4168K is, it will be recalled, Stacey's estimate of the eutectic temperature at inner core pressure, the basic assumption being that the liquid outer core and solid inner core have the same (eutectic) compo- sition. This assumption is almost certainly false, for two reasons. The first reason is that the density of the inner core is compatible with it consisting of iron (or iron plus a small amount of nickel), but there is no evidence to support the contention that an Fe-FeS solid eutectic mix- ture (zero pressure density ~ 5.5 g/cm3) could attain a density of about 13 g/cm3 at 3.3 Mbar (see Table 3-1~. The second reason is that, if the

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64 ENERGETICS OF THE EARTH 7000 6000 . 5000 A o `t 4000 ~ 3000 llJ 2000 1000 II 1 ~ MANTLE .,-OUTER CORE_,_lCo ER_ O1000 2000 3000 4000 5000 6000 6371 DEPTH, KM FIGURE ~7 Temperatures in the earth, estimated by methods explained in the text. Points shown refer to, from left to right: (1) xenoliths in kimberlite; (2) the olivine-spinel transition at 400 km; (3) the lower mantle at 1300 km, from Wang (1972); (4) the lower mantle at 2800 km, also from Wang; (5) the core-mantle boundary; (6) the inner core boundary; and (7) the center of the earth, from Bukowinski (1977). Estimates of uncer- tainties are somewhat arbitrary, as they include many diverse factors, such as regional variations at point (1). The error bar at point (7) (center of the earth) reflects only the uncertainty inherent in Bukowinski's calculations, which assume that the inner core density is exactly that given by the PEM model of Dziewonski et al. (1975). Bukowinski estimates that an error of 1 percent in the density adds about 500 to the uncertainty on the temperature. inner and outer cores have the same eutectic composition, the density jump of 0.6 0.8 g/cm3 at their interface (Table 3-1) necessarily repre- sents solely the effect of melting at constant composition. When intro- duced in Equation (3.18) for the pressure dependence of the eutectic

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Temperatures Within the Earth 65 temperature, it leads to a very high value, of the order of several degrees per kilobar, for the slope of the eutectic line, which is incon- sistent with Stacey's value of the eutectic temperature itself. It seems safe to conclude that the sulfur content of the outer core is less than that of the eutectic and that, accordingly, the temperature Ti at the bound- ary of the inner core is higher than the eutectic temperature, whatever that temperature may be. In summary, then, all that can be said about Ti is that it is less than the melting point of pure iron at 3.3 Mbar (7500 + 2000C), but how much less is not known. Bukowinski's estimate of 5450K is not ex- cluded; but recall that it is calculated from the properties of ~ iron, which may be irrelevant to the inner core, and for a particular density model that assigns to the inner core a somewhat lower density than other models do (Table 3-1~. All in all, it seems likely that temperatures in the core are in the broad range between 4500 + 800K at the core-mantle boundary and 6000 + 500K at the center. Those temperatures are somewhat higher than the estimated initial temperatures calculated from accretion theory (Hanks and Anderson, 1969), which are particularly low for the core because the rate of release of gravitational energy is necessarily low at the beginning of accretion, when the gravitational pull of the accreting body is still quite small. Our temperatures suggest that the core has heated up in the course of time. Several sources immediately suggest them- selves (Chapter 2~. If the core now contains 0.1 percent of potassium, the total heat generated by it through 4.5 billion years amounts to 103 J. enough to heat up the core by some 700; if one half of the gravita- tional energy of separation of the core (1 x 103} J. according to Flasar and Birch, 1973) were transformed into heat in the core, it would raise the core's temperature by some 3500. SUMMARY In Figure 3-7 we have plotted as a function of depth the possible temperatures, or temperature ranges, deduced from the considerations outlined above. These temperatures can be summarized as follows: 1. From temperatures of incipient melting at base of lithosphere: oceanic lithosphere 50 km thick: 1075 + 225C oceanic lithosphere 100 km thick: 1150 + 250C continental lithosphere 200 km thick: <1200C

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66 ENERGETICS OF THE EARTH 2. From kimberlite nodules: at 100 km: 1000 + 200C at 150 km: 1200 + 200C Clearly it is not possible to draw a single solution through these points. Regional variations are considerable. A (possibly meaningless) average temperature gradient in the first 100 km might be 1~12/km. 3. From olivine-spinel transition: at 400 km: 1450 + 150C 4. Mean adiabatic (isentropic) gradient between 100 700 km: 1.5 + 0.5/km (including entropy change due to phase transitions) err ~ nls gives: at 700 km: 2000 + 350C = 2300 + 350K 5. Below 700 km, the isentropic gradient falls to 0.3~.4/km. The following temperatures are those calculated by Wang (1972), assuming adiabaticity: at 1300 km: 2800 + 800K at 2800 km: 3300 + 800K 6. In boundary layer D", the gradient is estimated at 10-12/km. The temperature at the core boundary is then 4500 + 800~. The melting point of pure iron at the pressure of the core-mantle boundary is 5000 + 500K (Leppaluoto, 1972a,b). 7. Guessing that an average value of ~ suitable to the outer core is about 1.3, the temperature at the ICB comes out as 5740K, with an uncertainty possibly as large as 1000. Bukowinski's (1977) estimate is 5450 + 60K. Bukowinski's temperature at the center of the earth, plotted in Figure 3-7, is 5684 + 65K, a value predicated on the as- sumptions (1) that the inner core consists of iron in its ~ (fcc) form, and (2) that the density of the inner core is that given by the so-called PEM model. Bukowinski estimates that a change in density of 1 percent from the PEM value entails a change in temperature of about 500.