density distribution within the Earth

density distribution within the Earth Our estimates of the total mass of the Earth, M, and therefore its mean density, rely on the measurement of gravity. Outside the Earth, at a distance r from the centre, this is, by Newton's law,g = GM/r²  (1)

with small corrections for ellipticity and, if measurements are made on the surface itself, also for rotation. G is known as the gravitational constant. Observations of the orbital accelerations of satellites have yielded values of the product GM with 9-figure accuracy. G is, however, by far the least well known of the fundamental constants of nature, and this limits our knowledge of M to about 1 part in 8000. Using the present best estimate of G, obtained at the National Bureau of Standards in the USA, the mean density of the Earth is 5515 kg m⁻³ with an uncertainty of one in the last figure.

The mass of the Earth, obtained in this way, constrains the estimates of density at all levels. If we imagine the value of G to be revised downwards by 1 per cent (a hypothesis that seemed just possible in the 1980s), then M and the densities at all levels would be revised upwards by the same factor. A similar upward revision of the elasticities of the deep Earth materials would leave gravity at all depths, the observed seismic velocities, and the frequencies of the Earth's free oscillation all unaffected. The mean density is thus a scaling factor for density throughout the Earth.

Another whole-Earth property that any model of the Earth's density structure must match is the moment of inertia (see moment of inertia and precession). It was first used in Earth model studies in the 1930s by K. E. Bullen, whose models became the reference standards for several decades. The axial moment of inertia, C, is related to the mass, M, and the equatorial radius, a, by a numerical coefficient which is known much more precisely than the mass or mean density:C = 0.330695 Ma².  (2)

If a model of the Earth's density profile were to be rescaled to match a new value of M (perhaps arising from a new and better measurement of G) in Eqn (2), then the numerical coefficient would be unaffected. It is therefore a scaling factor with a role that is rather different from that of M. The mass controls the absolute densities in a model; C/Ma² gives a control on relative densities at different levels.

Seismological observations

Although they are not entirely distinct, there are three kinds of information derived from seismology that are used to obtain earth models: body waves, surface waves, and free oscillations give complementary data. The earliest Earth model studies used the travel times of body waves through the interior to obtain the variations in velocity of compressional (P) and shear (S) waves. Major boundaries show up as discontinuities or as abnormally rapid increases in velocity with depth. These boundaries separate the regions within which the velocity increases can be explained as due to increasing pressure on homogeneous material. Body-wave speeds give the ratios of elastic moduli to density. Density is not obtained independently but must be deduced from additional data. Before free oscillations were observed this meant total mass, moment of inertia, and high-pressure (shock wave) observations on properties of likely minerals.

Surface waves are guided by the layered structure near the surface of the Earth. The amplitude of wave motion is virtually restricted to a depth of a half or a third of a wavelength and falls off exponentially at greater depths. The longer wavelengths (lower frequencies) thus sample greater depth ranges. Since the (body wave) speeds generally increase with depth, the surface waves of longer wavelength sample more of the high-velocity material and so travel faster. They show strong dispersion (velocity variation with wavelength or frequency), unlike the body waves which are almost non-dispersive. But like the body waves, surface waves give ratios of elastic moduli to density. They give information about near-surface layering and are important in indicating lateral variations in structure, but give no independent density information.

Free oscillations are literally the ringing of the Earth like a large bell after it has been struck (by an earthquake). There are two basic types of free oscillation: toroidal, or torsional, and spheroidal. The simplest torsional oscillation is a twisting motion between two hemispheres. No radial motion or change in shape occurs and no independent information about density is conveyed. The spheroidal oscillations entail radial motion, causing small changes in the Earth's shape, and so gravity contributes to the restoring forces and hence the mode frequencies. This means that density enters the equations of motion in two ways, both as a ratio with elasticity (as in the body and surface waves) and coupled with gravity. The spheroidal mode frequencies thus give an independent measure of the density structure, but one that is still subject to the total mass of the Earth as a scaling factor.

Variation of density with depth

The most widely used Earth model is known by its acronym, PREM (Preliminary Reference Earth Model). It was published in 1981 by A. M. Dziewonski of Harvard and D. L. Anderson of the California Institute of Technology at the request of an international committee that was set up in response to a perceived need for a generally accepted reference model. A very large number of data points were used in preparing the model, the free oscillation frequencies being particularly important. The word ‘preliminary’ in the name implies that replacement by improved models was expected, but this is certainly not happening quickly, essentially because the changes would be small and, for most purposes, would not justify the effort of producing a new model.

PREM specifies density and the P- and S-wave velocities as polynomial functions of radius in each of a number of radius ranges. It is a spherically averaged model, except in the outermost part, where different continental and oceanic structures are recognized.

Calculation of zero pressure densities requires a finite strain theory, so called because conventional elasticity theory applies only to infinitesimal strains. For pressure ranges that are not very small compared with elastic moduli, the moduli cannot be treated as constant but increase with pressure. In fact they increase much more strongly than does density, which is why the seismic velocities increase with pressure. The value for the bulk modulus (usually called incompressibility in geophysics) in the inner core is about eight times the zero pressure value for iron. Compression becomes progressively more difficult with increasing pressure. Numerous empirical formulae have been fitted to Earth model tabulations to account for this. Although there is no really satisfactory theory of finite strain, the differences between alternative calculations are slight and the extrapolated zero pressure densities are well constrained.

Temperature as well as pressure increases with depth in the Earth. The deep interior minerals are therefore thermally expanded and the temperature profile of the Earth acts to oppose the increase in pressure with depth. For much of the Earth a temperature gradient of about 7 °C per kilometre would sufficie to cancel the effect of pressure on density. This gradient is exceeded by a factor of three or so in the upper crust, but in the rest of the Earth, except for a thin layer at the core–mantle boundary, the gradient is smaller by a factor of about 20 and the density variation by self compression is completely dominant. Nevertheless, when we interpret zero pressure densities the temperature effect is important.

One of the targets of fundamental Earth science is to determine the deep interior composition. The zero pressure densities give a basis for initial guesses. In the shallowest layer of the mantle, where minerals have familiar low-pressure forms, the matching of possible compositions to the observed density is straightforward. At greater depths high-pressure phases of closer atomic packing appear, and these may be observable in laboratory measurements only in very high-pressure apparatus. High-pressure experiments on likely deep-Earth compositions are therefore vital to the interpretation of the density data.

The core

The abundance of iron in meteorites and in the solar atmosphere made it the obvious candidate material for the Earth's dense core, as soon as the core was identified by seismologists. For this reason the high-pressure properties of iron have received considerable attention. F. Birch, who pioneered finite strain theory, first pointed out that the core density was less than that of pure iron at core pressures. How much less depends on our still insecure knowledge of the high-pressure phase diagram of iron and on temperature, but it may be as much as 10 per cent. There is little doubt that the core composition is dominated by iron, but there are divergent views on the light ingredients that reduce the density to the observed value. Oxygen, sulphur, carbon, silicon, and hydrogen all have their advocates. It is probable that these elements and others are all present to some extent. There is almost certainly some nickel and other transition metals, but they do not reduce the density.

The boundary between the liquid outer core and the solid inner core is marked by a density jump of about 650 kg m⁻³. It is not very precisely observed, but certainly exceeds the density increment due to solidification of a homogeneous alloy at core pressure, which is estimated to be about 200 kg m⁻³. The difference must be accounted for by a difference in composition. The inner core contains less of the light ingredients (but they are not entirely absent). Since the Earth is cooling slowly, the inner core must be growing by progressive solidification, and there is therefore a continuing process of rejection of the light ingredients by the solid. This leaves an excess of the light ingredients at the bottom of the outer core. The process of upward mixing into the whole outer core releases gravitational energy, which is believed to be the dominant (or sole) source of power for the geomagnetic dynamo.

The outer core is liquid and so lacks a defined crystal structure, but the inner core is solid with a crystal structure that is still in doubt. Under ordinary laboratory conditions iron has a body-centred cubic structure (the alpha phase), but this is stable only at low pressures and is not relevant to the core. Even at zero pressure a face-centred cubic structure (the gamma phase) appears at high temperatures, and this is more strongly favoured at higher pressures. However, at low temperatures increasing pressure causes alpha iron to transform to the denser hexagonal close-packed (epsilon) form. Until recently it was believed that this replaced also the higher-temperature gamma form when core pressures were approached. There is now evidence of further phase transitions, so that the solid phase of iron at core pressures is unclear. In any case the lighter core ingredients probably modify the phase diagram. However, the observation that the inner core is seismically anisotropic, with P-waves travelling faster from pole to pole than across the Equator, is easier to explain if the inner core is composed of a phase with an anisotropic crystal structure, such as epsilon iron. It is on the basis of a comparison of the core with epsilon-iron that the density decrement of the core relative to pure iron is believed to be close to 10 per cent.

The lower mantle

By convention the lower mantle extends from a depth of about 670 km to the outer boundary of the core at 2890 km depth. In Fig. 1 it is seen as a region in which density progressively increases with depth but without discontinuities and over most of the depth range appears to be homogeneous in composition and phase. There are heterogeneities in a layer just above the core, known in the jargon of geophysics as D (dee-double-primed), and probably also in the 100 km or so just below 670 km, but these do not show in the figure. The significance of the selection of 670 km as the upper boundary of the lower mantle is that this marks the deepest of the series of phase transitions by which upper mantle minerals are converted by high pressure to the close-packed crystal structures characteristic of deep mantle mineralogy.

In 1976 an experiment at the Australian National University by L. G. Liu, using the then new technique of compressing very small specimens between diamond anvils, showed that when important upper mantle minerals are heated at lower mantle pressures the predominant product is a ferromagnesian silicate, (Mg, Fe) SiO₃, with a perovskite structure. This is a crystal structure first found for the relatively rare mineral perovskite, CaTiO₃. A second mineral, (Mg, Fe)O, also appeared. This is usually referred to as magnesio-wustite, wustite being FeO, but ferro-periclase might have been a better name as the magnesium is dominant. In a subsequent experiment in the same laboratory, S. E. Kesson and J. D. Fitzgerald examined very small inclusions in diamonds that were believed to have originated in the lower mantle. They inferred that the Fe/Mg ratios in lower mantle minerals are about 1:20 in the silicate perovskite and about 1:7 in the magnesio-wustite; that is, iron favours the magnesio-wustite.

Table 1 Zero pressure densities and elasticities for lower-mantle minerals

	Silicate	Magnesio-
perovskite	wustite
(Mg0.95Fe0.05) SiO3	(Mg0.88Fe0.12) O
Data from L.G. Lin, S. E. Kesson and J. D. Fitzgerald
Density (P = 0, T = 0)	4163 kg m−3	3880 kg m−3
Incompressibility
(P = 0, T = 0)	263 Gpa	162 Gpa

With these compositions of silicate perovskite and magnesio-wustite we know the zero pressure densities and elasticities of what are believed to be the two dominant. minerals in the lower mantle (Table 1). These values can be compared with the extrapolated zero pressure properties of the lower mantle, but remembering that the lower mantle values still refer to a high temperature, although at zero pressure.

Lower mantle (P = 0)	Density	3977 kg m−3
	Incompressibility	205 Gpa

Knowing how density and incompressibility vary with temperature, we can find a mix of the two minerals and a temperature that match the extrapolated properties of the lower mantle. We find about 75 per cent perovskite, 25 per cent magnesio-wustite, and a temperature of about 1800 °C. This temperature estimate allows us to calculate the range of actual lower mantle temperatures, 2300 to 2700 °C, that would be obtained by compressing the mixture to lower mantle densities starting from 1800 °C. These estimates must of course be treated with caution: the mineralogy of the lower mantle is probably not quite as simple as is assumed here.

Densities of the crust and upper mantle

The upper part of the crust has been sampled by numerous drill cores and is found to be very variable in composition and density. Sedimentary rocks are generally much less dense than the deeper igneous material, but they represent only a thin veneer on the Earth and are not very significant for the total density structure of the planet. The two representative types of igneous rock in the crust are granite (with a density of about 2700 kg m⁻³) and basalt (density about 2900 kg m⁻³). These are both much less dense than the uppermost mantle (3370 kg m⁻³), in which the lighter continental rocks are effectively floating, as revealed by gravity observations. Between the uppermost mantle and the 670-km boundary the mantle minerals undergo a succession of phase transitions to progressively denser structures, as shown in Fig. 1. High-pressure experiments have now identified these intermediate phases as well as the lower mantle minerals.

Lateral density variations

Lateral heterogeneity of the crust is very obvious, especially in the division of the Earth's surface into continents and oceans. Differences in structure between continental and oceanic areas are seen to persist to depths of at least 100 to 200km, but the variations diminish with depth. In the deep mantle the lateral variations appear to be very small, but they exist at all depths down to the core. At the core–mantle boundary itself there is a layer of quite strong heterogeneity. Seismologists are studying the lateral structure of the mantle by techniques known collectively as tomography, and are obtaining a three-dimensional picture of the mantle. Tomography indicates that there are variations in the velocities of seismic waves within the mantle. The interpretation of these variations in terms of densities is still a matter of contention, but a particularly interesting aspect of this work is that lateral density variations mean buoyancy variations; they must therefore be related to the convective pattern that causes the tectonic processes that make the surface of the Earth such an interesting place.

Frank D. Stacey

Bibliography

Bullen, K. E. (1975) The Earth's density. Chapman and Hall, London.
Stacey, F. D. (1992) Physics of the Earth (3rd edn). Brookfield Press, Brisbane.