Chapter 18

Elasticity and solid-state geophysics

Mark well the various kinds of constants. TI1ese are usually the bulk modulus K minerals, note their properties and and the rigidity G or fL Young's modulus E and Poisson's ratio cr are also commonly used. TI1ere their mode of origin. are, correspondingly, two types of elastic waves; Petrus Severinus (I 57 I) the compressional, or primary (P), and the shear, or secondary (S) , having velocities derivable from The seismic properties of a material depend on pVi = K +4G/3 = K +4J.L/3 composition, crystal structure, temperature, pres­ pVl = G = fL sure and in some cases defect and impurity con­ centrations. Most of the Earth is made up of The interrelations between the elastic constants crystals. The elastic properties of crystals depend and wave velocities are given in Table 18.1. In an on orientation and frequency. TI1us, the inter­ anisotropic solid there are two shear waves and pretation of seismic data, or the extrapolation one compressional wave in any given direction. of laboratory data, requires knowledge of crystal Only gases, fluids, well-annealed glasses and or mineral physics, elasticity and thermodynam­ similar noncrystalline materials are strictly ics. But one cannot directly infer composition isotropic. Crystalline material with random ori­ or temperature from mantle tomography and a entations of grains can approach isotropy, but table of elastic constants derived from laboratory rocks are generally anisotropic. experiments. Seismic velocities are not unique Laboratory measurements of mineral elas­ functions of temperature and pressure alone nor tic properties, and their temperature and pres­ are they linear functions of them. Tomographic sure derivatives, are an essential complement cross-sections are not maps of temperature. The to seismic data. The high-frequency elastic-wave mantle is not an ideal linearly elastic body or 'an velocities are known for hundreds of crystals. ideal harmonic solid '. Elastic 'constants' Compilations of elastic properties are frequency dependent and have intrinsic, of rocks and minerals complement those extrinsic, anharmonic and anelastic contribu­ tabulated here. In addition, ab initio cal­ tions to the pressure and temperature dependen­ culations of elastic properties of cies. high-pressure minerals can be used to supplement the measurements. Elastic properties depend on both crystal Elastic constants of isotropic solids structure and composition, and the understand­ ing of these effects, including the role of The elastic behavior of an isotropic, ideally elas­ temperature and pressure, is a responsibility of tic, solid - at infinite frequency - is completely a discipline called mineral physics or solid-state characterized by the density p and two elastic geophysics. Most measu rem ents are made under 234 EL A STIC ITY AND SOLID - ST ATE G EOPHYSICS

Table 18.1 I Connecting identities for elastic constants of isotropic bodies

K G =i-L A. a A. A.+ 21-L/3 3(K - A.)/2 K - 2~-L/3 2(A. + !-L) 2(1 +a) I- 2a 2a A. A.--- ~-L -- 1-L 3( I - 2a) 2a I -2a 3K -A. l+a I -2a 3K- 2!-L A.-- 3K-- 3K-a- 3a 2+2a l+a 2(3K + 1-L) p(Vp2 - 4V l /3) pVs2 p(V p2 -2Vs2 ) I (Vp/Vs)2 - 2

l(Vp/V5 )2- I

A., 1-L =Lame constants 2 G =Rigidity or shear modulus= pV5 = 1-L K = Bulk modulus a = Poisson's ratio E =Young's modulus p =density Vp. Vs =compressional and shear velocities 4!-L- E pVi =A.+ 2!-L = 3K - 2A. = K + 4~-L /3 = ~-L--- 3t.L- E 3k + E 1 - a 2 - 2a 1 - a =3K =A.--=t.L--=3K-- 9K - E a 1 - 2a 1 +a

conditions far from the pressure and temper­ pressure is more complex. It is often not justi­ ature conditions in the deep crust or mantle. fied to assume that all derivatives are linear and The frequency of laboratory waves is usually far independent of temperature and pressure; it is from the frequency content of seismic waves. The necessary to use physically based equations of measurements themselves, therefore, are just the state. Unfortunately, many discussions of upper­ first step in any program to predict or interpret mantle mineralogy and interpretations of tomog­ seismic velocities. raphy ignore the most elementary considerations Some information is now available on the of solid-state and atomic physics. high-frequency elastic properties of all major The functional form of a(T, P), the coeffi­ rock-forming minerals in the mantle. On the cient of thermal expansion, is closely related other hand, there are insufficient data on any to the specific-heat function, and the neces­ mineral to make assumption-free comparisons sary theory was developed long ago by Debye, with seismic data below some 100 km depth. It is Griineisen and Einstein. Yet a(T, P) is some­ essential to have a good theoretical understand­ times assumed to be independent of pressure ing of the effects of frequency, temperature, com­ and temperature, or linearly dependent on tem­ position and pressure on the elastic and thermal perature. Likewise, interatomic-potential theory properties of minerals so that laboratory mea­ shows that the pressure derivative of the bulk surements can be extrapolated to mantle con­ modulus dK/dP must decrease with compression, ditions. Laboratory results are generally given yet the moduli are often assumed to increase in terms of a linear dependence of the elastic linearly with pressure. There are also various moduli on temperature and pressure. The actual thermodynamic relationships that must be sat­ variation of the moduli with temperature and isfied by any self-consistent equation of state, ELAST IC CON STANTS OF ISOTROPIC SOL I DS 235 and certain inequalities regarding the strain Anharmonicity dependence of anharmonic properties. Processes There are various routes whereby temperature within the Earth are not expected to give random affects the elastic moduli and seismic velocity. orientations of the constituent anisotropic min­ The main ones are anelasticity and anharmonic­ erals. On the other hand the full elastic tensor is ity. The first one does not depend, to first order, difficult to determine from seismic data. Seismic on volu me or density and therefore many geo­ data usually provide some sort of average of the dynamic scaling relations are invalid if anelas­ velocities in a given region and, in some cases, ticity is importan t. Elastic moduli also depend estimates of the anisotropy. The best-quality labo­ on parameters other than temperature, such as ratory data are obtained from high-quality single composition. The visual interpretations of tomo­ crystals. The full elastic tensor can be obtained in graphic color cross-sections assume a one-to-one these cases, and methods are available for com­ correspondence between seismic velocity, density puting average properties from these data. and temperatu re. It is simpler to tabulate and discuss average The thermal oscillation of atoms in their properties, as I do in this section. It should be (asymmetric) potential well is anharmonic or kept in mind, however, that mantle minerals are nonsinusoidal. Thermal oscillation of an atom anisotropic and they tend to be readily oriented causes the mean position to be displaced, and by mantle processes. Certain seismic observations thermal expansion results. (In a symmetric, or in subducting slabs, for example, are best inter­ parabolic, potential well the mean positions are preted in terms of oriented crystals and a result­ unchanged, atomic vibrations are harmonic, and ing seismic anisotropy. If all seismic observations no thermal expansion results.) Anharmonici ty are interpreted in terms of isotropy, it is possible causes atoms to take up new average positions of to arrive at erroneous conclusions. The debates equ ilibrium, dependent on the amplitude of the about the thickness of the lithosphere, the deep vibrations and hence on the temperature, but the structure of continents, the depth of slab pene­ new positions of dynamic equilibrium remain tration, and the scale of mantle convection are. nearly harmonic. At any given volume the har­ to some extent, debates about the anisotropy and monic approximation can be made so that the mineral physics of the mantle and the interpreta­ characteristic temperature and frequency are not tion of seismic data. Although it is important to explicit functions of temperature. This is called understand the effects of temperature and pres­ thequasi-harmonic approximation.ffit~ sure on physical properties. it is also important assumed that the frequency of each normal mode to realize that changes in crystal structure (solid­ of vibration is changed in simple proportion as solid phase changes) and orientation have large the volume is changed. There is a close relation­ effects on the seismic velocities. Tomographic ship between lattice thermal conductivity, ther­ images are often interpreted in terms of a single mal expansion and other properties that depend variable, temperature. Thus blue regions on tomo­ intrinsically on anharmonici ty of the interatomic graphic cross-sections are often called cold slabs potential. The atoms in a crystal vibrate about and red regions are often called hot plumes. Many equilibrium positions, but the normal modes are of the current controversies in mantle dynam­ not independent except in the idealized case of a ics and geochemistry, such as deep slab penetra­ harmonic solid. The vibrations of a crystal lattice tion, whole mantle convection and the presence can be resolved into interacting traveling waves of plumes can be traced to over-simplified or erro­ that interchange energy due to anharmonic, non­ neous scaling relations between seismic veloci­ linear coupling. ties and density. temperature and physical state. In a harmonic solid: Table 18.2 is a compilation of the elastic prop­ erties, measured or estimated, of most of the there is no thermal expansion; important mantle minerals, plus pressure and adiabatic and isothermal elastic constants are temperatu re derivatives. equal; Table 18.2 I Elastic properties of mantle minerals (Duffy and Anderson, 1988) formula Density Ks G -Ks -G 3 (structure) (g/cm ) (GPo) (GPa) K's G' (GPa/K) (GPa/K)

(Mg,Fe)2Si04 3.222+ 129 82- 5.1 1.8 0.016 0.013 (olivine) 1.182XFe 31 XFe (Mg,Fe)2Si04 3.472+ 174 114- 4.9 1.8 0.018 0.0 14 (.8 -spinel) I .24XFe 41XFe (Mg. Fe)2Si04 3.548+ 184 119- 4.8 1.8 0.017 0.014 (y -spinel) 1.30XFe 41XFe (Mg,Fe)Si03 3.204+ 104 77- 5.0 2.0 0.012 0.011 (orthopyroxene) 0.799XFe 24XFe CaMgSi206 3.277 113 67 4.5 1.7 0.013 0.010 (clinopyroxene) NaAISi20 6 3.32 143 84 4.5 1.7 0.016 0.013 (clinopyroxene) (Mg.Fe)O 3.583+ 163 - 131 4.2 2.5 0.016 0.024 (magnesiowustite) 2.28XFe 8XFe 77XFe AI203 3.988 251 162 4.3 1.8 0.014 0.019 (corundum) Si02 4.289 316 220 4.0 1.8 0.027 0.018 (stishovite) (Mg.Fe)3AI2Si3012 3.562+ 175+ 90+ 4.9 1.4 0.021 0.010 (garnet) 0.758XFe IXFe 8XFe Ca3(AI,Fe)2Si3012 3.595+ 169- 140- 4.9 1. 6 0.016 0.015 (garnet) 0.265X Fe I IXFe 14XFe (Mg. Fe)Si03 3.810+ 212 132- 4.3 1.7 0.017 0.017 (ilmenite) I.IOXFe 41 XFe (Mg.Fe)Si03 4. 104+ 266 153 3.9 2.0 0.031 0.028 (perovskite) 1.07XFe CaSi03 4. 13 227 125 3.9 1.9 0.027 0.023 (perovskite) (Mg,Fe)4Si4012 3.518+ 175+ 90+ 4.9 1.4 0.021 0.010 (majorite) 0.973XFe I XFe 8XFe Ca2Mg2Si4012 3.53 165 104 4.9 1.6 0.0 16 0.015 (majorite) Na2AI 2Si401 2 4.00 200 127 4.9 1.6 0.016 0.015 (majorite)

XFe is molar fraction of Fe end member. Alternative values for parameters in Table 18.2 and data for other minerals are given in the followi ng references. Anderson, 0. L. and Isaak, D. G. (1995) Elastic constants of mantle minerals at high temperature, in Mineral Physics and Crystallography: A Handbook of Pl1ysical Constants, pp. 64-97, ed. T. J. A11rens, American Geophysical Union, Washington, DC. Bass, J. D. (1995) Elasticity of minerals, glasses, and melts, in Mineral Physics and Crystallography: A Handbook of Physical Constants, pp. 45-63, ed. T. J. A11rens. American Geophysical Union, Washington, DC. Duffy, T. and Anderson, D. L. (1989) Seismic velocities in mantle minerals and the mineralogy of the upper mantle. ]. Geophys. Res., 94(B2), 1895-912. Li, B. and Zhang, J. (2005) Pressure and temperatu re dependence of elastic wave velocity of MgSi03 perovskite and the composition of the lower mantle, Pl1ys. Eart/1 Planet. Inter, 151, 143-54. Mattern, E., Matas, J., Ricard, Y. and Bass, J. D. (2005) Lower mantle composition and temperature fTom mineral physics and thermodynamic modelling, Geophys.]. Int. , 160 , 973-90. CORRECTING ELASTIC PROPERT IES FOR TEM PERATURE 237

the elastic constants are independent of pres­ approximation y is independent of temperature sure and temperature; at constant volume, and a has approximately the the heat capacity is constant at high temper­ same temperature dependence as molar specific ature (T > 0); and heat. the lattice thermal conductivity is infinite. TI1ese are the resu lt of the neglect of anharmon ic­ Correcting elastic properties ity (higher than quadratic terms in the inter­ for temperature atomic displacements in the potential energy). In a real crystal the presence of lattice vibra­ The elastic properties of solids depend primarily tion causes a periodic elastic strain that, through on static lattice forces. but vibrational or ther­ anharmonic interaction, modulates the elastic mal motions become increasingly important at constants of a crystal. Other phonons are scat­ high temperature. The resistance of a crystal to tered by these modulations. TI1is is a nonlinear deformation is partially due to interionic forces process that does not occur in the absence of and partially due to the radiation pressure of anharmonic terms. high-frequency acoustic waves, which increase The concept of a strictly harmonic crystal in intensity as the temperature is raised. If the is highly artificial. It implies that neigh boring increase in volume associated with this radiation atoms attract one another with forces propor­ pressure is compensated by the application of tional to the distance between them, but such a suitable external pressure, there still remains a crystal would collapse. We must distingu ish an intrinsic temperature effect. Thus, these equa­ between a harmonic solid in which each atom tions provide a convenient way to estimate the executes harmonic motions about its equilibrium properties of the static lattice. that is, K(V, 0) and position and a solid in which the forces between G(V. 0). and to correct measured values to differ­ individual atoms obey Hooke's law. In the for­ ent temperatures at constant volume. The static mer case, as a solid is heated up, the atomic lattice values should be used when searching for vibrations increase in amplitude but the mean velocity-density or modulus-volume systematics position of each atom is unchanged. In a two­ or when attempting to estimate the properties or three-dimensional lattice, the net restoring of unmeasured phases. force on an individual atom, when all the near­ The first step in forward modeling of the seis­ est neighbors are considered, is not Hookean. An mic properties of the mantle is to compile a atom oscillating on a line between two adjacent table of the ambient or zero-temperature proper­ atoms will attract the atoms on perpendicular ties, including temperature and pressure deriva­ lines, thereby contracting the lattice. Such a solid tives, of all relevant minerals. The fully normal­ is not harmonic; in fact it has negative thermal ized extrinsic and intrinsic derivatives are then expansion. formed and, in the absence of contrary infor­ TI1e quasi-harmonic approximation takes into mation. are assumed to be independent of tem­ account that the equilibrium positions of atoms perature. The coefficient of thermal expansion depend on the amplitude of vibrations, and can be used to correct the density to the tem­ hence temperature, bu t that the vibrations perature of interest at zero pressure. It is impor­ about the new positions of dynamic equilib­ tant to take the temperature dependence of a(T) rium remain closely harmonic. One can then into account properly since it increases rapidly assume that at any given volume V the harmonic from room temperature but levels out at high approximation is adequate. In the simplest quasi­ T. TI1e use of the ambient a will underestimate harmonic theories it is assumed that the frequen­ the effect of temperature; the use of a plus cies of vibration of each normal mode of lattice the initial slope will overestimate the volume vibration and, hence, the vibrational spectra, the change at high temperature. Fortunately. the maximum frequency and the characteristic tem­ shape of a(T) is well known theoretically (a De bye peratures are functions of volume alone. In this fu nction) and has been measu red for many 238 ELASTICITY A ND SOLI D -STA TE GE OPH YS ICS

mantle minerals (see Table 18.5). The moduli can {Ksh and { G}T respectively, and this notation then be corrected for the volume change using will be used later. equations given. The normalized parameters can For many minerals, and the lower mantle then be used in an equation of state to calculate M(P, T), for example, from finite strain. The nor­ (a lnKs/a lnG)T = (K s/G) malized form of the pressure derivatives can be is a very good approximation. assumed to be either independent of temperature or functions of V(T). Temperature is less effective in causing variations in density and elastic prop­ Intrinsic and extrinsic temperature erties in the deep mantle than in the shallow mantle and the relationships between these vari­ effects ations are different from those observed in the laboratory. Anharmonic effects also are predicted Temperature has several effects on elastic moduli; to decrease rapidly with compression. Anhar­ intrinsic, extrinsic, anharmonic and anelastic. monic contributions are particularly important The effects of temperature on the properties at high temperature and low pressure. Intrinsic of the mantle must be known for various geo­ or anharmonic effects probably remain impor­ physical calculations. Because the lower man­ tant in the lower mantle, even if they decrease tle is under simultaneous high pressure and with pressure. Anelastic thermal effects on mod­ high temperature, it is not clear that the sim­ uli are also probably more important in the plifications that can be made in the classical upper mantle than at depth. Large velocity vari­ high-temperature limit are necessarily valid. For ations in the deep mantle are probably due to example, the coefficient of thermal expansion, phase changes, composition and melting, or the which controls many of the anharmonic prop­ presence of a liquid phase. erties, increases with temperature but decreases with pressure. At high temperature, the elas­ tic properties depend mainly on the volume Temperature and pressure through the thermal expansion. At high pres­ derivatives of elastic moduli sure, on the other hand, the intrinsic effects of temperature may become relatively more important. The pressure derivatives of the adiabatic bulk The temperature derivatives of the elastic modulus for halides and oxides generally fall in moduli can be decomposed into extrinsic and the range 4.0 to 5.5. Rutiles are generally some­ intrinsic components: what higher, 5 to 7. Oxides and silicates having ions most pertinent to major mantle minerals (a InKs/ a In p )T = (a InKs/a ln p }r have a much smaller range, usually between 4.3 -a- '(a lnKs/ aT)v and 5.4. MgO has an unusually low value, 3.85. or IK s}P = IK sh - IK s}v The density derivative of the bulk modulus, for the adiabatic bulk modulus, K s , and a similar expression for the rigidity, G. Extrinsic and intrin­ sic effects are sometimes called 'volumetric' or for mantle oxides and silicates that have been 'quasi-harmonic' and 'anharmonic.' The anelastic measured usually fall between 4.3 and 5.4 with effect on moduli, treated later, is also important. MgO, 3.8, again being low. The intrinsic contribution is The rigidity, G, has a much weaker volume or density dependence. The parameter (dIn M / dT )v = a[(8 ln M / 8 1n p }r - (8 lnM/ 8 ln p)pj (a In G ;a In p)T = (KT / G )G ' where M is K , or G. There would be no intrinsic generally falls between about 2.5 and 2.7. The or anharmonic effect if a = 0 or if M(V, T) = above dimensionless derivatives can be written M(V). The various terms in this equation require INTRINSIC AND EXTRINSIC TEMPERATU RE EF FECTS I 239

Table 18.3 I Extrinsic and intrinsic of derivations Extrinsic Intrinsic

Substance clnK s) ~clnK s ) clnK s) a In p P (~)a lnp T a aT v aT v (I o-s jdeg)

MgO 3.0 38 0.8 2.3 CaO 3.9 4.8 0.9 2.6 SrO 4.7 5.1 0.4 1.6 Ti02 I 0.5 6.8 -3.7 -8.7 Ge02 10.2 6.1 -4.1 -5.6 AI 20 3 4.3 4.3 0.0 0.0 Mg2Si04 4.7 5.4 0.7 om MgAI 20 4 4.0 4.2 0.2 0.4 Garnet 5.2 5.5 0.3 0.5 SrTi03 7.9 5.7 -2.2 -6.2

Substance clnG) clnG) ~ clnG) a In p p a lnp T a aT v (~)aT v (I o-s fdeg) MgO 5.8 3.0 -2.8 -8.6 CaO 6.3 2.4 -3.9 -11.2 SrO 4.9 2.2 -2.7 -11.3 Ti02 8.4 1.2 -7.2 -16.9 Ge02 5.8 2.1 -3.7 -5.1 AI 20 3 7.5 2.6 -4.9 -16.5 Mg2Si04 6.5 2.8 -3.7 -9.2 MgAI20 4 5.9 1.3 -4.6 -7.4 Garnet 5.4 2.6 -2.8 -6.1 SrTi03 7.9 3.1 -4.8 -13.4

Sumino and Anderson (1984).

measurements as a function of both temperature create buoyancy at the base of the mantle. This and pressure. TI1ese are tabulated in Table 18.3 also means that all temperature derivatives of the for a variety of oxides and silicates. moduli and the seismic velocities are low in the Generally, the intrinsic temperature effect on deep mantle. TI1e inferred intrinsic temperature rigidity is greater than on the bulk modulus; effect on the bulk modulus is generally small G is a weaker function of volume at constant and can be either negative or positive (Table T than at constant P; and G is a weaker func­ 18.3). TI1e bulk modulus and its derivatives must tion of volume, at constant T, than is the bulk be computed from differences of the directly modulus. TI1e extrinsic terms are functions of measured moduli and therefore they have much pressure through the moduli and their pressure larger errors than the shear moduli. The intrin­ derivatives. sic component of the temperature derivative of The coefficient of thermal expansion, at con­ the rigidity is often larger than the extrinsic, stant temperature, decreases by about 80% from or volume-dependent, component and is invari­ P = 0 to the base of the mantle. It is hard to ably negative; an increase in ten1perature causes 240 ELASTIC ITY A N D SOLID-STA TE GEO PH YS ICS

Table 18.4 I Normalized pressure and temperature derivatives

Ks Substance cln G ) - clnG) (~)a In p T a In p T ( ~a In G ) T G ( ~a In p ) P a In p p ( ~aaT ) v ( ~aaT ) v

MgO 3.80 301 1.26 1.24 3.04 5.81 0.76 -2.81 AI203 4.34 271 1.61 1.54 4.31 7.45 O.Q3 -4.74 Olivine 5.09 2.90 1. 75 1.64 4.89 6.67 0.2 -3.76 Garnet 4.71 270 1.74 1.85 6.84 4.89 -2.1 -2. 19 MgAI204 4.85 0.92 5.26 1.82 3.84 4.19 1.0 1 -3.26 SrTi03 5.67 3.92 1.45 1.49 8.77 8.70 -3.1 - 4.78

G to decrease due to the decrease in density, can be used as a natural laboratory to extend con­ but a large part of this decrease would occur at ventional laboratory results. constant volume. Increasing pressure decreases It is convenient to treat thermodynamic the total temperature effect because of the parameters, including elastic moduli, in terms of decrease of the extrinsic component and the coef­ volume-dependent and temperature-dependent ficient of thermal expansion. The net effect is a parts, as in the Mie-Grtineisen equation of reduction of the temperature derivatives, and a state. This is facilitated by the introduction of larger role for rigidity in controlling the temper­ dimensionless anharmonic (DA) parameters. The ature variation of seismic velocities in the lower Gruneisen ratio is such a parameter. The pres­ mantle. This is consistent with seismic data for sure derivatives elastic moduli are also dimen­ the lower mantle. sionless anharmonic parameters. but it is use­ The total effects of temperature on the bulk ful to replace pressure, and temperature, by vol­ modulus and on the rigidity are comparable ume. This is done by forming logarithmic deriva­ under laboratory conditions (Table 18.3). There­ tives with respect to volume or density, giv­ fore the compressional and shear velocities have ing dimensionless logarithmic anhar­ similar temperature dependencies. On the other monic (DLA) parameters. They are formed as hand, the thermal effect on bulk modulus is follows: largely extrinsic, that is, it depends mainly on the change in volume due to thermal expansion. (alnM;a in p)T = MKT (aPaM ) = [Mh The shear modulus is affected both by the volume change and a purely thermal effect at constant (alnM;a ln p)p=(aM)- 1 (aM) =[M }p volume. aT T Although the data in Table 18.3 are not in (a lnM;aaT)v = [Mh - [M} = [M)v the classical high-temperature regime it is still possible to separate the temperature derivatives where we use braces {} to denote DLA parameters into volume-dependent and volume-independent and the subscripts T, P, V and S denote isother­ parts. Measurements must be made at much mal, isobaric, isovolume and adiabatic condi­ higher temperatures in order to test the various tions, respectively. The {}v terms are known as assumptions involved in quasi-harmonic approxi­ intrinsic derivatives, giving the effect of temper­ mations. One of the main results I have shown ature or pressure at constant volume. Deriva­ here is that, in general, the relative roles of tives for common mantle minerals are listed in intrinsic and extrinsic contributions and the rel­ Table 18.4. Elastic, thermal and anharmonic ative temperature variations in bulk and shear parameters are relatively independent of temper­ moduli will not mimic those found in the ature at constant volume, particularly at high restricted range of temperature and pressure temperature. This simplifies temperature correc­ presently available in the laboratory. The Earth tions for the elastic moduli. I use density rather SEISMIC CONSTRAINTS ON THERMODYNAMICS OF THE LOWER MANTLE 241

than volume in order to make most of the param­ particularly for the bulk modulus. and variations eters positive. of seismic velocities are due primarily to changes T11e DLA parameters relate the variation of in the rigidity. the moduli to volume, or density, rather than to Intrinsic effects are more important for the temperature and pressure. This is useful since the rigidity than for the bulk modulus. Geophys­ variations of density with temperature, pressure, ical results on the radial and lateral varia­ composition and phase are fairly well under­ tions of velocity and density provide constraints stood. Furthermore, anharmonic properties tend on high-pressure-high-temperature equations of to be independent of temperature and pressure state. Many of the thermodynamic properties at constant volume. The anharmonic parame­ of the lower mantle, required for equation-of­ ter known as the thermal Griineisen parame­ state modeling, can be determined directly from ter y is relatively constant from material to the seismic data. The effect of pressure on the material as well as relatively independent of coefficient of thermal expansion, the Griineisen temperature. parameters, the lattice conductivity and the tem­ perature derivatives of seismic-wave velocities should be taken into account in the interpreta­ Anelastic effects tion of seismic data and in convection and geoid calculations. Solids are not ideally elastic; the moduli depend T11e lateral variation of seismic velocity is on frequency. This is known as 'dispersion' and very large in the upper 200 km of the mantle it introduces an additional temperature depen­ but decreases rapidly below this depth. Velocity dency on the seismic-wave velocities. T11is can be itself generally increases with depth below about written 200 km. T11is suggests that temperature varia­ tions are more important in the shallow mantle d ln v l d In T = (1 12n) Q;;:,:U,[2E *I RT] than at greater depth. Most of the mantle is above where v is a seismic velocity, T is absolute tem­ the Debye temperature and therefore thermody­ perature, Q~,~x is the peak value of the absorp­ namic properties may approach their classical tion band and E* is the activation energy. T11e high-temperature limits. What is needed is pre­ temperature dependence arises from the relax­ cise data on variations of properties with temper­ ation time, which is an activated parameter that ature at high pressure and theoretical treatments depends exponentially on temperature. Many of properties of solids at simultaneous high pres­ other examples are given in the chapter on dis­ sure and temperature. sipation. The above example applies inside the I use the following relations and notation: seismic absorption band.

(i:lln KT ia In p}r = (aKTiaP )T = K -~ = {KTh Seismic constraints on (a ln Ksla ln p)T = (KT I K s )K ~ = {K sh thermodynamics of the lower mantle (a ln Ksli:l ln p)p = - (aKs)- '(aKslaT)r = 8s = {K s}p For most solids at normal conditions, the effect (alnG ia ln p}r = (KT I G)(aG iaP}r of temperature on the elastic properties is = (KTI G)G ' = G * = {G h controlled mainly by the variation of volume. (alnG ia ln p)r -(acr'(aGiaT)r Volume-dependent extrinsic effects dominate at = low pressure and high temperature. Under these = g = {G lr conditions one expects that the relative changes - (a In a I a In ph ~ 8s + y = - {a h in shear velocity, due to lateral temperature (alnK1-jaT)v = a[(a lnKTia ln p}r gradients in the mantle, should be similar to -(a In KTla In P)rl changes in compressional velocity. However. at high pressure, this contribution is suppressed, = a(K~- 8T) Table 18.5 I Dimensionlss logarithmic anharmonic parameters

a Ks G y y Substance (10-6/1<) (kbar) (kbar) (K sh (Gh (K T}P (K s}p (G} p (K Tlv {K s}v (G} v {K - G} v Ks/G thermal BR

LiF 98 723 485 4.90 3.97 4.69 2.42 6.35 0.5 2.5 -2.4 4.9 1.49 1.66 1.92 NaF 98 483 3 14 4.96 2.62 5.80 3.75 5.06 -0.6 I .2 -2.4 3.6 1.54 1.51 1.37 KF 99 318 164 4.81 1.97 5.05 3.18 6.17 0.0 1.6 -4.2 5.8 1.94 1.50 1.12 RbF 95 280 127 5.35 1.63 4.77 2.97 5.95 0.8 2.4 -4.3 6.7 2.20 1.43 1.05 LiCI 134 318 193 4.65 4.45 5.40 3.32 6.84 -0.4 1.3 -2.4 3.7 1.65 1.82 2.11 NaCI 118 252 148 5.10 3.00 5.45 3.74 5.07 -0.1 1.4 -2.1 3.4 1.71 1.51 1.55 Kcl 105 181 93 5.1 0 2.0 I 7.48 5.67 5.54 -2.0 -0.6 - 3.5 3.0 1.95 1.39 1.17 RbCI 119 163 78 5.09 1.79 5.81 4.1 I 5.30 -0.4 1.0 -3.5 4.5 2.10 1.44 1.09 AgCI 93 440 81 6.21 2.83 I 0.6 7.94 11.0 -3.8 -1.7 - 8.2 6.5 5.44 2.08 1.77 NaBr 135 207 I 14 4.63 3.15 7.34 5.28 4.24 -2.2 0.7 -1.1 0.4 1.81 1.72 1.59 Kbr 116 ISO 79 5.12 2.0 I 5.64 3.94 4.68 -0.2 1.2 - 2.7 3.8 1.90 1.45 1.16 RbBr 113 I 37 65 5.05 1.81 6.27 4.54 5.96 -0.9 0.5 -4.2 4.7 2.09 1.47 1.09 Nal 138 161 91 5.1 I 3.22 4.79 2.75 5.41 0.6 2.4 - 2.2 4.5 1.76 1.74 1.65 Kl 126 122 60 4.82 2.29 5.95 4.05 4.89 -0.8 0.8 -2.6 3.4 2.02 1.41 1.26 Rbl 119 I I I 50 5.14 1.97 6.17 4.49 6.05 -0.7 0.6 -4.1 4.7 2.21 1. 51 1.17 CsCI 140 182 I 0 I 5.20 4.75 6.28 3.82 6.0 I -0.6 1.4 -1.3 2.6 1.80 2.04 2.30 TICI 158 240 92 6.00 4.45 7.78 5.18 7.35 -1 .0 0.8 - 2.9 3.7 2.60 2.46 2.31 CsBr 138 156 88 4.97 4.53 6.33 3.86 5.90 -0.9 I .I - 1.4 2.5 1.76 1.98 2.18 TIBr 170 224 88 5.80 5.39 7.51 4.75 6.36 -0.8 I. I - I .0 2.0 2.55 2.76 2.66 Csl 138 126 72 5.06 4.51 6.40 3.86 6.09 -0.9 1.2 -1.6 2.8 1.73 1.94 2.18 MgO 31 1628 1308 3.80 3.0 I 5.48 3.04 5.81 -1.6 0.8 - 2.8 3.6 1.24 1.52 1.41 CaO 29 I 125 810 4.78 2.42 5.45 3.92 6.29 -0.6 0.9 -3.9 4.7 1.39 1.27 1.25 SrO 42 912 587 5.07 2.25 6.99 4.68 4.98 -1.8 0.4 -2.7 3.1 1.55 1.74 1.22 BaO 38 720 367 5.42 1.95 9.46 7.34 7.88 -3.9 -1.9 -5.9 4.0 1.96 1.56 1.17 AI 20 3 16 25 12 I 634 4.34 2.71 6.84 4.31 7.45 -2.5 0.0 -4.7 4.8 1.54 1.27 1.33 Ti20 3 17 2076 945 4.10 2.23 7.78 6.66 12.9 -3.6 -2.6 -II 8.2 2.20 1.13 1.15 Fe20 3 33 2066 91 0 4.44 1.63 5.70 3.68 3.34 -1.2 0.8 -1.7 2.5 2.27 1.99 0.95 Ti02 24 2140 I 120 6.83 1.09 12.7 I 0.5 8.37 -5.7 -3.7 -7.3 3.6 1.91 1.72 0.96 Ge02 14 2589 1509 6.12 2.1 0 I 1.9 10.2 5.83 -5.7 -4.1 -3.7 -0.3 1.72 1.17 1.27 Sn02 10 2123 I017 5.09 1.25 9.90 8.64 6.21 -4.8 -3.6 -5.0 1.4 2.09 0.88 0.85 MgF2 38 1019 547 5.01 1.34 5.36 4.17 3.83 -0.3 0.8 - 2.5 3.3 1.86 1.22 0.87 NiF2 23 1207 459 4.98 -1.4 8.80 7.47 1.96 -3.8 -2.5 - 3.4 0.9 2.63 0.88 -0.2 ZnF2 29 1052 394 4.51 0.13 9.49 8.23 4.73 -4.9 - 3.7 -4.6 0.9 2.67 0.97 0.39 CaF2 61 845 427 4.55 2.26 5.85 3.86 4.75 -1.1 0.7 -2.5 3.2 1.98 1.83 1.21 SrF2 47 714 350 4.67 1.66 6.02 4.75 4.94 - 1.2 - 0.1 - 3.3 3.2 2.04 1.30 0.98 Cd F2 66 1054 330 5.77 4.05 8.36 6.04 7.11 -2.2 - 0.3 -3.1 2.8 3. 19 2.45 2.10 BaF2 61 581 255 4.89 0.88 6.52 4.54 3.83 - 1.4 0.3 - 3.0 3.3 2.28 1.80 0.7 1 Opx 48 1035 747 9.26 3.18 7.26 5.43 3.34 2.2 3.8 - 0.2 4.0 1.39 1.87 1.95 Olivine 25 1294 791 5.09 2.90 6.70 4.89 6.67 - 1.6 0.2 - 3.8 4.0 1.64 1. 16 1.49 Olivine 27 1292 812 4.83 2.85 6.61 4.67 6.27 -1 .7 0.2 -3.4 3.6 1.59 1.25 1.44 Olivine 25 1286 811 5.32 2.83 6.55 4.73 6.50 -1.2 0.6 - 3.7 4.3 1.59 1.16 1.48 Garnets Fe l6 19 1713 927 4.71 2.70 7.89 6.84 4.89 - 3.1 - 2.1 - 2.2 0.1 1.85 1.05 1.38 Fe36 19 1682 922 4.71 2.67 7.42 5.98 5.05 - 2.7 - 1.3 -2.4 1.1 1.82 1.01 1.37 Fes4 24 1736 955 5.38 2.52 6.79 5.52 4.8 1 - 1.3 - 0.1 - 2.3 2.2 1.82 1.28 1.37 MgAI20 4 21 1969 1080 4.85 0.92 5.47 3.84 4.19 - 0.6 1.0 - 3.3 4.3 1.82 1.40 0.67 SrTi03 25 1741 1168 5.67 3.92 11.2 8.77 8.70 - 5.3 - 3.1 -4.8 1.7 1.49 1.63 1.06 KMgF3 60 751 488 4.87 2.98 5.02 3.46 4.72 - 0.0 1.4 - 1.7 3.1 1.54 1.60 1.50 RbMnF3 57 675 341 4.80 3.69 4.48 3.0 I 5.04 0.4 1.8 -1.4 3.1 1.98 1.49 1.80 RbCdF3 40 614 257 1.09 2.15 4.12 3.06 3.27 3.0 2.0 - 1.1 0.8 2.39 1.06 1.80 TICdF3 49 609 228 7.43 0.95 5.09 3.86 3.14 2.4 3.6 - 2.2 5.8 2.68 1.24 1.03 ZnO 15 1394 442 4.76 -2.2 9.60 6.22 3.02 - 4.8 - 1.5 - 5.2 3.7 3.15 0.81 - 0.4 BeO 18 2201 1618 5.48 1.19 5.17 3.08 4.19 0.3 2.4 - 3.0 5.4 1.36 1.27 0.79 Si02 35 378 455 6.37 0.35 3.28 2.43 - 0.3 3.1 3.9 0.7 3.3 0.85 0.67 0.40 CaC03 17 747 318 5.36 - 3.5 24.00 23.1 18.5 - 19 - 18 - 22 4.2 2.35 0.56 - 1.0 244 ELASTICITY AND SOLID-STATE GEOPHYSICS

Table 18.6 I Dimensionless logarithmic anharmonic derivatives

Substance {Ksh {Gh {Ks}P {G }p {Ks}v {G }v Averages Halides 5.1 2.6 4. 1 5.9 0.9 - 3.3 Perovskites* 4.8 2.7 4.4 5.0 0.3 - 2.2 Garnets* 4.9 2.6 6.1 4.9 - 1.2 - 2.3 Fluorites* 5.0 2.2 4.8 5.2 0.2 -2.9 Oxides 5.3 2.0 5.7 5.8 -0.4 -3.7 Silicates 5.6 2.8 5.4 5.4 0.2 -2.6 Grand average 5. 1 2.5 5.0 5.7 0. 1 - 3.2 (± 1.0) (± 1.3) (± 1.9) (± 1.9) (± 1.9) (± 1.9) Olivine 5.1 2.9 4.9 6.7 0.2 -3.8 O livine 4.8 2.9 4.7 6.3 0.2 -3.4 MgA120 4 -spi nel 4.9 0.9 3.8 4.2 -0.6 +1 .0 {M} = alnMj alnp * Structures.

Table 18.7 I Anharmonic parameters for oxides and silicates and predicated values for some high-pressure phases

Mineral {KTh {Ksh {Gh {KT}P {Ks}P {G }p a-olivine 5.0 4.8 2.9 6.6 4.7 6.5 ,B -spinel* 4.9 4.8 3.0 6.6 4.7 6.4 y -spinel* 5.1 5.0 3. 1 6.8 4.9 6.5 Garnet 4.8 4.7 2.7 7.9 6.8 4.9 MgSi03 (majorite)* 4.9 4.7 2.6 7.9 6.0 4.5 AI20 3-ilmenite 4.4 4.3 2.7 6.8 4.3 7.5 MgSi03-ilmenite* 4.7 4.5 2.7 7.0 4.3 6.0 MgSi03-perovskite* 4.5 4.5 3.5 7.5 5.5 6.5 * 4.2 4.1 2.8 6.2 4.0 5.7 Si02-(stishovite)* 4.5 4.4 2.4 7.5 6.4 5.0

* Predicted.

(a In G jaT )v = a [( a In G ;a In P)T for many halides, oxides and minerals. Average - (illn G ;a In P)r] values for chemical and structural classes are extracted in Table 18.6, and parameters for min­ Ks = KT(1 +ayT) eral phases of the lower mantle are presented in 8T "'=' 8s + y Table 18.7.

This notation stresses the volume, or den­ (a ln Vp ; a In p) sity, dependence of the thermodynamic vari­ = -1[ -3(a InKs/a ln p ) +-2(a In G ;a ln p) - 1 ] ables and is particularly useful in geophysical 2 5 5 discussions. Values of most of the dimensionless log­ where I have used Ks = 2G, a value appropriate arithmic parameters are listed in Table 18.5 for the lower mantle. SEISMIC C O NSTRAINTS ON T HE RM ODYNAMICS OF THE LOWER MANTLE 245

Theoretically, the effects of temperature (a ln G I a ln p )r 5.8 to 7.0 decrease with compression, and this is borne (a ln G I a ln p )T 2.6 to 2.9 out by the seismic data. In the lower mantle (alnK /a ln p)T 1.8 to 1.0 the lateral variation of seismic velocities is dom­ 5 inated by variations in the rigidity. This is sim­ (a ln K s!a ln p)T 2.8 to3.6 ilar to the situation in the upper mantle where (a In y ;a In p)T - 1 +e it is caused by nonelastic processes, such as par­ - (a lna;a ln p)T 3to2 tial melting and dislocation relaxation, phenom­ For the intrinsic temperature terms we obtain: ena accompanied by increased attenuation, and phase changes. In the lower mantle the effect -1 [--aIn G J ""-3.2to - 4.1 is caused by anharmonic phenomena and intrin­ a aT v sic temperature effects that are more important 2. [ a InKs] "" + 1 to +2.6 in shear than in compression. Iron-partitioning a aT v and phase changes, including spin-pairing and An interesting implication of the seismic data is melting, may be important in the deep mantle. that the bulk modulus and rigidity are similar One cannot assume that physical properties are a functions of volume at constant temperature. On function of volume alone, or that classical high­ the other hand, G is a stronger function of vol­ temperature behavior prevails, or that shear and ume and K s a much weaker function of volume compressional modes exhibit similar variations. at constant pressure than they are at constant On a much more basic level, one cannot simply temperature. adopt laboratory values of temperature deriva­ Extrapolation oflower mantle values to the sur- tives to estimate the effect of temperature on den­ face, ignoring chemical and phase changes, gives: sity and elastic-wave velocities in slabs and in the deep mantle. Po= 3.97-4.00gfcm3 If one ignores the possibility of phase changes K0 = 2.12 - 2.23kbar and chemical stratification, one can estimate G = 1.30-1.35kbar lower-mantle properties. From the seismic data 0 for the lower mantle, we can obtain the follow­ (K ~ ) s = 3.8 - 4.1 ing estimates of in-situ values: (G~) s = 1.5-1.8