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Physical Properties of Iron in the Inner Core Department of Geological Sciences, University of Michigan, Ann Arbor, Geophysical Laboratory, Carnegie Institution of Washington, Washington DC The Earth's inner core plays a vital role in the dynamics of our planet and is itself strongly exposed to dynamic processes as evidenced by a complex pattern of elastic structure To gain deeper insight into the nature of these processes we rely on a characterization of the physical properties of the inner core which are governed by the material physics of its main constituent, iron Here we review recent research on structure and dynamics of the inner core, focusing on advances in mineral physics We will discuss results on core composition, crystalline structure, temperature,and various aspects of elasticity Based on recent computational results, we will show that aggregate seismic properties of the inner core can be explained by temperature and compression effects on the elasticity of pure iron, and use single crystal anisotropy to develop a speculative textural model of the inner core that can explain major aspects of inner core anisotropy PACS numbers: INTRODUCTION The presence and slow growth of Earth's inner core is one of the most significant manifestations of the dynamics in the interior of our planet As it is inaccessible to direct observation, an understanding of the physical state of the inner core requires an integrative approach combining results from many fields in the geosciences Seismology, geo- and paleomagnetism, geo- and cosmo-chemistry, geodynamics, and mineral physics have advanced our knowledge of the structure and processes in the inner core, revealing many surprises Foremost among these have been the discoveries of anisotropy and heterogeneity in the inner core Long assumed to be a featureless spherically symmetric body, a higher number and higher quality of seismic data revealed that the inner core is strongly anisotropic to com-pressional wave propagation Morelli et al, 1986; Wood-house et al, 1986 Generally, seismic waves travel faster along paths parallel to the Earth's polar axis by 3-4% compared to equatorial ray paths Creager, 1992; Song and Helmberger, 1993 The presence of anisotropy is significant because it promises to reveal dynamical processes within the inner core Anisotropy is usually attributed to lattice preferred orientation, which may develop during inner core growth Karato, 1993; Bergman, 1997, or by solid state deformation Buffett, 2000 The source of stress that may be responsible for deformation of the inner core is unknown, although several mechanisms have been proposed Jean-loz and Wenk, 1988; Yoshida et al, 1996; Buffett 1996; 1997; Karato, 1999; Buffett and Bloxham 2000 An understanding of the origin of inner core anisotro-py will require further advances in our knowledge of the physical properties of iron at inner core conditions, and may rely critically on further observations of the detailed structure of the inner core For example, recent obser- vations indicate that the magnitude of the anisotropy may vary with position: heterogeneity has been observed on length scales from 1-1000 km Creager, 1997; Tanaka and Hamaguchi, 1997; Vidale and Earle, 2000 Inner core structure may change with time as well Song and Richards 1996 interpreted apparent changes in travel times of inner core sensitive phases in terms of super-rotation of the inner core with respect to the mantle Some recent studies have argued for a much slower rotation rate than that advocated originally, or questioned the interpretation of time dependent structure. |
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The inner core also plays an essential role in the dynamics of the overlying outer core The anisotropy and long magnetic diffusion time of the inner core may alter the frequency and nature of reversals, and influence the form of the time-averaged field Hollerbach and Jones, 1993; Clement and Stixrude, 1995 Moreover, important energy sources driving the geodynamo process are associated with solidification of the inner core: the density contrast across the inner core boundary is due to the phase transition from the liquid to the solid, and chemical differentiation during theincongruent freezing of the inner core Both of these processes provide energy for the dynamo through the release of latent heat Verhoogen, 1961 and the generation of chemical buoyancy Bragin-sky, 1963 Other energy sources for magnetic field generation are secular cooling of the Earth, gravitational energy from thermal contraction of the core, radioactive heat generation, and precession Verhoogen, 1980; Buffett et al, 1996 Both thermal and compositional contributions to the buoyancy depend on the thermal state of the core The more viscous mantle controls the cooling time scale of the Earth and facilitates the formation of a thermal boundary layer at the core mantle boundary The heat flux out of the core controls the rate of inner core growth and light 2 element partitioning during this process Buffett et al, 1996 Conversely, a reliable estimate on temperature in the Earth's core would advance our understanding of the current thermal state and evolution of the Earth Jean-loz and Morris, 1986; Yukutake, 2000 with important implications for the dynamics of the Earth Because the inner core is inaccessible, the study of model systems by theory and experiments is essential Here we consider the ways in which mineral physics may lend deeper insight into inner core processes and to the origin of its structure, extending previous reviews by Jeanloz 1990 and Stixrude and Brown 1998 We begin with geophysical background on the inner core including recent seismological advances, constraints on the composition, thermal state, and dynamics of the inner core As our subsequent discussions draw on various experimental and theoretical approaches in mineral physics we then give an overview of recent developments in methods in the following section, focusing on computational mineral physics (a recent review focusing on advances in experiments has been given by Hemley and Mao 2001) To the extent that the inner core is composed of nearly pure iron, physical properties of this element at high pressure and temperature govern the behavior of the inner core; we consequently review advances in our knowledge of the high pressure physical properties of iron, focusing on crystalline structure, equation of state, and elasticity at both static condition and high temperature In the final section we examine the implications of these results for inner core temperature, and integrate aspects of elasticity with considerations of the dynamics in the inner core to develop a simple speculative model of poly-crystalline structure that explains major aspects of its anisotropy. |
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GEOPHYSICAL BACKGROUND Aggregate Seismic Properties Lehmann 1936 discovered the inner core by recognizing weak arrivals of PKiKP within the P-wave shadow zone of the core The amplitudes of these arrivals were sufficient to invoke a discontinuous seismic boundary in the Earth's core The P-wave contrast across this boundary was soon established; Birch 1940 and Bullen 1946 argued that the inner core must be solid based on this estimate The best evidence for inner core solidity comes from studies of inner core sensitive normal modes Dziewonski and Gilbert, 1971: Earth models with finite shear modulus of the inner core provide a significantly better fit to eigenfrequency observations than those with a liquid inner core Recent observations of body wave phases involving a shear wave in the inner core (PKJKP, SKJKP, and pPKJKP) Okal and Cansi, 1998; Deuss et al, 2000 support solidity, but are still controversial The inferred shear wave velocity vS of the inner core is remarkable: it is low compared to the compressional wave velocity vP, a property which can also be expressed in terms of the Poisson's ratio aThe value of <r=044 for the inner core is nearly that of a liquid (05), leading to speculation that this region may be partially molten Singh et al, 2000 In principle density p, vP, and vS also depend on depth However, constraints on the depth dependence of p and vS are weak Seismic observations are consistent with an inner core in a state of adiabatic self-compression Anisotropy First evidence for deviations from a spherically symmetric structure came from the observation that eigen-frequencies of core sensitive normal modes are split much more strongly than predicted by ellipticity and rotation of the Earth alone Masters and Gilbert, 1981 Anomalies in PKIKP travel times were initially interpreted as topography on the inner core Poupinet et al, 1983 Morelli et al 1986 and Woodhouse et al 1986 interpreted similar observations of eigenfrequencies and travel times as inner core anisotropy Observation of differential travel times PKIKP-PKPBC Creager; 1992; Song and Helmberger, 1993 and a reanalysis of normal mode data Tromp, 1993 confirmed that the inner core displays a hexagonal (cylindrical) pattern of anisotropy with a magnitude of 3-4% and symmetry axis nearly parallel to Earth's rotation axis For example, PKIKP arrives 5-6 s earlier along polar paths than predicted from radially symmetric Earth models such as PREM Dziewonski and Anderson, 1981 It is worthwhile pointing out that some of the travel time differences could be due to mantle structure not accounted for in the reference model Breger et al, 1999; Ishii et al, 2002a; 2002b In particular, small scale heterogeneity in the lowermost mantle could be sampled preferentially for select body wave core paths Breger et al, 1999; Tromp, 2001; Ishii et al, 2002b, In further investigation deviations from first order anisotropy have been put forward, for example lateral variations in vP of the inner core on length scales ranging from hemispherical differences Tanaka and Hamaguchi, 1997; Creager, 1999; Niu and Wen, 2001, to hundreds of kilometers Creager, 1997, down to a few kilometers Vidale and Earle, 2000 Radial variations may also exist: weak anisotropy may be present in the uppermost inner core (to a depth of 50-100 km) Shearer, 1994; Song and Helmberger, 1995; Su and Dziewonski, 1995 and strong uniform anisotropy in its inner half Song and Helmberger, 1995; Creager, 1999 Seismological studies of the inner core are discussed in more detail elsewhere in this volume. |
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Composition Seismically determined properties of the core may be compared to laboratory measurements under high compression Measurements of the equation of state show that only elements with an atomic number close to that of iron satisfy the seismic constraints Birch, 1964 Additional arguments are necessary to uniquely implicate iron Jeanloz, 1990: iron is one of the most abundant elements in stars and meteorites, much more so than in the portions of the Earth that are directly observable Brown and Mussett, 1993; and a conducting liquid is necessary in the outer core to explain the existence of a long lived dynamo process that creates Earth's magnetic field Merrill et al, 1996 To the degree that we are certain about the main constituent of the core we are also sure that the core contains other lighter elements: pure iron can not satisfy the seismological constraints for both portions of the core Liquid iron is about 10% too dense to satisfy both the density and bulk modulus in the outer core Birch, 1964; Jeanloz, 1979; Brown and McQueen, 1986 and while solid iron can explain the bulk modulus of the inner core for reasonable temperatures it overestimates the density even for very high temperature (8000 K) Jeph-coat and Olson, 1987; Stixrude et al, 1997 The identity and amount of the light element is still uncertain, but based on cosmochemical arguments hydrogen, carbon, oxygen, magnesium, silicon, and sulfur have been proposed Poirier, 1994, with oxygen and sulfur being the most popular To infer information on the composition of the core from geochemistry, two questions are of central importance: did the core form in chemical equilibrium Karato and Murthy, 1997 and what are the physical conditions of the core forming event, as pressure and temperature critically determine the partition coefficient of various elements between silicate and metallic melt Ito et al, 1995; Li and Agee, 1996; Okuchi, 1997 Alternatively, one may use the available seismological information on the current physical state of the outer and inner core (p, vP, vS) and compare to the physical properties of candidate iron alloys at the appropriate pressure and temperature condition The compositional space of Fe-X with X any light element has been sparsely sampled in shock wave experiments at the conditions relevant for the core Only binary compounds in the Fe-S system (pyrrhotite Fe0gS Brown et al, 1984, troilite FeS Anderson and Ahrens, 1996, and pyrite Fe2S Ahrens and Jeanloz, 1987; Anderson and Ahrens, 1996) as well as wiistite FeO Jeanloz and Ahrens, 1980; Yagi et al, 1988 have been exposed to shock The data has been extrapolated to inner core conditions Stixrude et al, 1997 and compared to the required elastic parameters (Fig 1) This analysis indicates that small amounts of either S or O (few atomic percent)would be sufficient to match the Density (Mgm ) FIG 1: Properties of the alloy fraction that are required to match the seismically observed properties of the inner core Stixrude et al, 1997 For a given temperature (solid lines) the required effective bulk modulus is plotted as a function of required effective density The dashed lines connect points of common alloy fraction for any light element X (2, 5, 10, and 20% from left to right) Estimated uncertainties in the alloy fractions required are indicated with the error bars on the curve corresponding to 6000 K (1% in density and 5% in bulk modulus). |
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Superimposed are extrapolations of shock wave experimental estimates for FeO Jeanloz and Ahrens, 1980; Yagi et al, 1988, FeS Anderson and Ahrens, 1996, and Fe2S Ahrens and Jeanloz, 1987; Anderson and Ahrens, 1996 at 345 GPa and 6000 K (estimated uncertainties are 5% in density and 10% in bulk modulus) properties of the inner core Alfe et al 2000a; 2000b; 2002 combined the geophysical approach with a chemical argument They evaluated the liquid-solid partition coefficients of candidate light elements assuming thermodynamic equilibrium at the inner core boundary The results show that neither S, Si, nor O alone can satisfy the observed density contrast at the inner core boundary and that a ternary or higher mixture of small amounts of S or Si with O is required Thermal State Like the composition, the temperature of the inner core cannot be determined by direct observation Assuming that the inner core is growing in equilibrium from in-congruent freezing of the outer core liquid a knowledge of the melting behavior of iron-rich systems at the pressure of the inner core boundary (330 GPa) would yield an important fixed-point temperature for the construction of whole Earth geotherms Because the core is not a pure system, the temperature at the inner core boundary should differ from the melting point of pure iron Freez- 4 ing point depression in an eutectic system with no solid solution is given by the van Laar equation Brown and McQueen, 1982 which yields a value of 800 K for a melting point of iron of 6000 K and 10% mole fraction of the light element The value for the freezing point depression must be viewed as highly uncertain, however, since solid solution almost certainly exists at the high temperatures of the core An independent estimate of the temperature of the core may be obtained by comparing the elastic properties of iron with those seismologically determined We describe this approach as applied to the inner core below Seismic observations do provide constraints on some aspects of the thermal stateof the core In the outer core the compressional wave velocity vp equals the bulk sound velocity vb = yKsp In a homogeneous, con-vecting system, vB and p are related by adiabatic self-compression Deviations from this state are characterized by the Bullen 1963 inhomogeneity parameter n being different from one n is defined as вдр pg dr' (1) where g the gravitational acceleration, and r the radius n for the outer core is constrained by seismology to be 1005 Masters, 1979 This is consistent with (but does not uniquely require) a vigorously convecting outer core, and a resulting geotherm close to an adiabat, characterized by the gradient: dJL = _nT dr v% ' (2) where y is the Graneisen parameter Adopting the value measured for liquid iron at core conditions (y=15) Brown and McQueen, 1986 and a temperature at the inner core boundary of 6000 K, one finds a temperature contrast of 1500 K across the outer core The temperature contrast in the inner core is likely to be small We can place an upper bound on it by assuming that the inner core is a perfect thermal insulator If the inner core grows through freezing of the outer core its temperature profile will follow the solidus temperature Based on this assumption Stixrude et al 1997 estimated the total temperature difference across the inner core to be less than 400 K Conductive or convective heat loss will further reduce this temperature gradient relative to the insulating case Conduction is likely to be a very effective way to extract heat from the inner core, such that the temperature profile may fall below an adiabat. |
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Dynamics Song and Richards 1996 found that the differential travel time of PKIKP-PKPBC for earthquakes in the South Sandwich islands recorded in Alaska increased by 03 s over a period of three decades, and concluded that the inner core rotates relative to the mantle by 1year, a finding that was confirmed qualitatively using global data sets Su et al, 1996 Creager 1997 showed that part of the signal could be explained by lateral heterogeneity in the inner core, and reassessed the rotation rate to a lower value Recent years have seen body wave Souriau, 1998 and free oscillations studies Laske and Masters, 1999 that can not resolve inner core rotation, and put close bounds on rotation rate If it is present, differential inner core rotation would provide one of the few opportunities for direct observations of the dynamics in Earth's deep interior Moreover, differential rotation could have a significant effect on the angular momentum budget of the Earth yielding an explanation of decadal fluctuations in the length of day Buffett, 1996; Buffett and Creager, 1999 Its origin is not fully understood, but geodynamo simulations produce super-rotation by electro-magnetic coupling with the overlying outer core Glatzmaier and Roberts, 1996; Kuang and Bloxham, 1997; Aurnou et al, 1998 Gravitational stresses, arising from mass anomalies in the mantle, are also expected to act on the inner core Buffett, 1996; 1997 These tend to work against super-rotation by gravitationally locking the inner core into synchronous rotation with the mantle In detail, the interplay between forces driving and resisting super-rotation depend on the rheology of the inner core, which is currently unknown If the viscosity is sufficiently low, super-rotation may take place, with the consequence that the inner core undergoes continuous viscous deformation in response to the gravitational perturbations The interaction of super-rotation with gravitational stresses is just one of many proposed sources of internal deformation in the inner core The subject has received substantial attention because solid state flow in the inner core can result in lattice-preferred orientation, thought to be essential for producing seismically observed anisotropy Other proposed sources of stress in the inner core include: (a) Coupling with the magnetic field generated in the overlying outer core Karato, 1999; Buffett and Bloxham, 2000; Buffett and Wenk, 2001 Karato 1999 considered the radial component of the Lorentz force (Fr) at the inner core boundary which is caused by the zonal magnetic field (Вф) This is typically the strongest contribution to the Lorentz force in geodynamo models Glatzmaier and Roberts, 1995; Kuang and Bloxham, 1997 Buffett and Bloxham 2000 argue that the inner core adjusts to Fr in a way to minimize steady solid state flow: only weak flow in the inner core is induced that is largely confined to the outermost portion Considering additional terms to the Lorentz force F by including radial components of the magnetic field Br they conclude that the azimuthal 5 term Fф which is proportional to Br Вф induces a steady shear flow throughout the inner core (b) Thermal convection Jeanloz and Wenk, 1988; Wenk et al, 2000a As in any proposed model of inner core flow, the viscosity of the inner core remains an important uncertainty, as does the origin and magnitude of heat sources required to drive the convection. |
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Aspherical growth of the inner core Yoshida et al, 1996 Fundamental considerations, based on the expected cylindrical symmetry of flow in the outer core, and detailed geodynamo simulations Glatzmaier and Roberts, 1995 indicate that heat is transported more efficiently in the equatorial plane than along the poles, leading to an inhomogeneous growth rate of the inner core, and internal viscous relaxation A key question is whether the magnitude of the effect is sufficient to produce lattice preferred orientation In particular, the resulting strain rates are very small, and may not be sufficient to generate significant polycrystalline texture via recrystallization It has also been proposed that polycrystalline texture in the inner core may be acquired during solidification Karato, 1993; Bergman, 1997 However, if the inner core does experience solid state deformation, by one or more of the mechanisms described above, it is unclear to what extent the texture acquired during solidification would be preserved It is possible that texture in the outermost portions of the inner core is dominated by the solidification process, whereas lattice preferred orientation in the bulk of the inner core is produced by deformation Further progress in our understanding of the composition, temperature, dynamics, and origin of anisotropy in the inner core is currently limited by our lack of knowledge of the properties of iron and iron alloys at high pressures and temperatures A better understanding of elastic and other properties of iron at inner core conditions can provide a way to test hypotheses concerning the state and dynamics of the inner core MINERAL PHYSICS METHODS As elasticity plays a central role in deep Earth geophysics we will emphasize aspects of mineral physics that are directly related to the determination of elastic properties To gain deeper insight into complex elastic behavior, such as anisotropy, we need to know the full elastic constant tensor at the conditions in the Earth's center We will focus on methods based on first-principles quantum mechanical theory, but also briefly review experimental progress, as it relates to comparison and validation of theory A full review of experimental work has been given recently by Hemley and Mao 2001 Experimental Progress Determination of the elastic constants of metals in the diamond cell remains a particular challenge as now-standard techniques, such as Brillouin spectroscopy, cannot readily be used for opaque materials A variety of alternative methods have been developed and applied to study iron at high pressure and ambient temperature In the lattice strain technique Singh et al, 1998a; 1998b X-ray diffraction is used to study the strain induced in a polycrystal by uniaxial stress A full determination of the elastic constant requires the measurement of d-spacing for many (h, k, l) lattice planes, or additional assumptions such as homogeneity of the stress field in the sample Mao et al, 1998 Other efforts have exploited the relationship between phonon dispersion in the long-wavelength limit and elastic wave propagation: measurements of phonon dispersion have been used to estimate the average elastic wave velocity Lubbers et al, 2000; Mao et al, 2001, and the longitudinal wave velocity. |
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An approximate calibration has been explored in which the zone-center Raman active optical mode is related to the c44 shear elastic constant by a Brillouin-zone folding argument Olijnyk and Jephcoat, 2000 Finally, Anderson et al 2001 by analyzing pressure induced changes in the intensity of X-ray diffraction patterns from hcp iron, extracted a Debye temperature D, and thus average elastic wave velocity, which they equated with vS Whereas diamond anvil cell experiments most readily measure properties at ambient temperature, shock wave experiments achieve pressure and temperature conditions similar to those of the core through dynamic compres-sionBy varying the speed of the driver impacting the sample, a set of different thermodynamic conditions are accessed, along a curve in pressure-density space called the Hugoniot Temperature is not determined directly by the Rankine-Hugoniot equations, it must be measured using special techniques, such as optical pyrometry Yoo et al, 1993 or calculated on the basis of a thermodynamic model Brown and McQueen, 1986 Using temperature and Gruneisen paramter an adiabatic bulk modulus (KS) on the Hugoniot can be determined The impact of the driver plate on the sample not only sets up a shock in the sample but also in the plate itself When the shock wave reflects off the back of the impactor, pressure is released and a longitudinal (compressional) sound wave is set up traveling forward through the system of impactor and sample This has been exploited to determine vP; combining vP with KS the corresponding vS can be calculated Brown and McQueen, 1986 6 Computational Mineral Physics With the sparse probing of thermodynamic conditions relevant for Earth's inner core by the experimental methods discussed in the previous section, and the difficulty to obtain information on single crystal elasticity, first-principles material physics methods provide an ideal supplement to experimental study, with all of thermody-namic space accessible, and various approaches to determine elasticity at hand In the following sections we will introduce the basic principles of calculating such properties Total Energy Methods Density functional theory Hohenberg and Kohn, 1964; Kohn and Sham, 1965 provides a powerful and in principle exact way to obtain the energetics of a material with N nuclei and n interacting electrons in the ground-state (for a review see Lundqvist and March 1987), with the electronic charge density pe(r) being the fundamental variable It can be shown Hohenberg and Kohn, 1964 that ground state properties are a unique functional of pe(r) with the total (internal) energy Here T is the kinetic energy of a system of non-interacting electrons with the same charge density as the interacting system, and U is the electrostatic (Coulomb) energy containing terms for the electrostatic interaction between the nuclei, the electrons, and nuclei-electron interactions The final term Exc is the exchange-corre-lation energy accounting for many body interactions between the electrons Density functional theory allows one to calculate the exact charge density pe(r) and hence the many-body total energy from a set of n single-particle coupled differential equations Kohn and Sham, 1965 where ф^ is the wave function of a single electronic state, j the corresponding eigenvalue, and VKS the effective (Kohn-Sham) potential that includes the Coulomb and exchange-correlation terms from (3) The Kohn-Sham equations are solved self-consistently by iteration Density functional theory has been generalized to spin polarized (magnetic) systems. |
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While density functional theory is exact in principle the exact solution of the Kohn-Sham equations requires the knowledge of the universal form of the exchange-correlation potential which is yet unknown Approximations for Vxc however have been very successful The local density approximation (LDA) Lundqvist and March, 1983 replaces Vxc at every point in the crystal with the value of a homogeneous electron gas with the same local charge density This lowest order approximation yields excellent agreement with experiment for a wide variety of materials, but fails for some metals Most prominently for iron LDA wrongly predicts hcp as the ground state structure for iron at ambient pressure Stixrude et al, 1994 Generalized gradient approximations (GGA) include a dependence on local gradients of the charge density in addition to the charge density itself Perdew et al, 1996 GGA yields the correct ground state of iron at ambient pressure and predicts the phase transition from bcc to hcp iron at the experimentally determined pressure Asada and Terakura, 1992; Stixrude et al, 1994 In addition to total energy it is possible to calculate directly first derivatives of the total energy with first-principles methods This allows one to determine forces acting on the nuclei and stresses acting on the lattice Nielsen and Martins, 1985 All-electron, or full potential methods make no additional essential approximations to density functional theory Computational methods such as the Linearized Augmented Plane Wave (LAPW) method provide an important standard of comparison All-electron methods are very costly (slow), and are currently impractical for many problems of interest More approximate computational methods have been developed, which, when applied with care, can yield results that are nearly identical to the all-electron limit In the pseudopotential approximation the nucleus and core electrons are replaced inside a sphere of radius rc (cut-off radius) with a simpler object that has the same scattering properties (for a review see Pickett 1989) The pseudopotential is much smoother than the bare Coulomb potential of the nuclei, and the solution sought is only for the pseudo-wavefunctions of the valence electrons that show less rapid spatial fluctuations than the real wavefunction in the core region or those of the core electrons themselves The construction of the pseu-dopotential is non-unique and good agreement with all-electron calculations must be demonstrated Iron provides a particular challenge For example, all-electron results show that pressure-induced changes in the 3p band are important for the equation of state Stixrude et al, 1994, and so should be treated fully as valence electrons in a pseudopotential approach For our work on the high temperature elasticity of hcp iron Steinle-Neumann et al, 2001 we have constructed a Troullier-Martins Troullier and Martins, 1991 type pseudopotential for iron in which 3s, 3p, 3d, and higher electronic states are treated fully as valence electrons Agreement with all-electron calculations of the equation of state and elastic constants is excellent: for hcp iron the pressure at inner core densities is within 1 % of all-electron (LAPW) results (Fig 2), and the elastic constant tensor at inner core density is within 2% rms The predictions of density functional theory can be compared directly with experiment, or with geophysical Equation of state for hcp iron obtained from all electron results (dashed line) Steinle-Neumann et al, 1999 and the pseudopotential used in the calculation of high temperature thermoelasticity (solid line) observations. |
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For example, by computing the total energy as a function of volume, one obtains the equation of state This equation of state is static, that is the effects of thermal vibrations are absent This athermal state is one that is not attainable in the lab where zero-point motion cannot be eliminated Static properties are often directly comparable to experimental measurements taken at ambient temperature since the effect of 300 K is small for properties such as the density, or the elastic constants However, for comparison with the earth's core, thermal effects are essential Calculation of the effects of temperature from first-principles is more involved because one must calculate the energies associated with atomic displacements, including those that break the symmetry of the lattice High Temperature Methods Statistical mechanics provides the tools to deal with material properties at high temperature The thermody-namic behavior of any physical system is uniquely defined by the so-called fundamental relation, which, for a nonmagnetic or Pauli-paramagnetic solid in the canonical ensemble (particle number N, volume V, and temperature T held constant) takes the form where F is the Helmholtz free energy The total energy E(V, T) is now a function of T as well, because we explicitly have to account for thermal excitation of electrons according to Fermi-Dirac statistics Sel is the entropy associated with this excitation of the electrons McMahon and Ross, 1977, and Fvib is the vibrational part of the free energy Fvib is derived from the partition function for a system with N atoms, which in the classical limit, appropriate at high temperature conditions (significantly above D) is Zvib is a 3N dimensional integral over the coordinates of the nuclei located at i?i with the electronic free energy Fei = E — TSei uniquely defined by the coordinates of the atoms and Т A = к\2тгткТ is the de Broglie wavelength with h the Planck and k the Boltzmann constant, and m the nuclear mass A naive attempt to evaluate the integral (7) fails because of the large dimensionality, and because most configurations contribute little to the integral What is required is a search of configuration space that is directed towards those configurations that have relatively low energy In the particle-in-a-cell (PIC) method and the lattice dynamics method described next, atoms are restricted to vibrations about their ideal crystallographic sites, that is diffusion is neglected This is not a severe approximation to equilibrium thermodynamic properties at temperatures below the premelting region Molecular dynamics, described last, in principle permits diffusion, although in practice computationally feasible dynamical trajectories are sufficiently short that special techniques are often required to study non-equilibrium processes Particle-in-a-Cell Here the basic approximation motivates the division of the lattice into non-overlapping sub-volumes centered on the nuclei (Wigner-Seitz cell Д WS) with the coordinates of each atom restricted to its cell A second basic assumption in the PIC model is that the motions of the atoms are uncorrelated We can expect this approximation to become increasingly valid with rising temperature above D and below melting. |
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