Thermoelasticity of a mineral composite and a reconsideration of lower mantle properties

Physics of the Earth and Planetary Interiors 106 Ž1998. 219–236

Thermoelasticity of a mineral composite and a reconsideration of lower mantle properties Frank D. Stacey

)

CSIRO, DiÕision of Exploration and Mining, PO Box 883, Kenmore, Queensland 4069, Australia Received 18 August 1997; accepted 3 February 1998

Abstract A re-examination of thermal and elastic properties of a mineral composite, emphasising derivative properties, especially pressure and temperature dependences of incompressibility, resolves residual doubts about the application to the lower mantle of thermodynamic estimates of these properties. The only counter-intuitive conclusion is that ŽEK SrEP . S f 4.21 at P s 0, which is significantly higher than the values for component minerals. This leads to a revised lower mantle X equation-of-state and also to a new one which requires as input only K 0X s 4.21 and K`X s 1.425: K X s K 0X Ž K`X rK 0X . K` Pr K . These equations and three other finite strain theories fit the lower mantle PREM tabulation very well and agree on the extrapolation to P s 0: K 0 s 203.1 " 1.6 GPa, m 0 s 128.2 " 1.2 GPa, r 0 s 3972.5 " 4.0 kg my3. A further six finite strain equations are rejected as incompatible with homogeneity of the lower mantle. Application of the theory to thermal properties gives d S s yŽ1ra K S .ŽEK SrET . P decreasing with depth from 2.7 to 1.6 over the lower mantle depth range. Conversely ´ s yŽ1ram .ŽEmrET . P increases with depth from 7.1 to 8.7. Corresponding values of ŽElnVSrElnVP . P vary from 1.7 to 2.3. This conclusion is not consistent with a simple thermal interpretation of tomographic observations of lower mantle velocity anomalies. Compositional variations must be invoked. q 1998 Elsevier Science B.V. All rights reserved. Keywords: Thermoelasticity; Mineral composite; Lower mantle properties

1. Introduction In spite of considerable theoretical attention Žsee for example Watt et al., 1976., there is still no complete general theory that allows precise calculation of the elasticities of a rock from properties of its constituent minerals. This problem extends to other thermoelastic properties. Budiansky Ž1970. extended elasticity theory to thermal properties of a composite, but there appears to have been no discussion of higher derivative properties, pressure and tempera)

Corresponding author.

ture dependences of elasticity and expansion coefficient, that are now central to the mineral physics of the deep Earth. These higher derivative properties are the principal target of the analysis reported here. The results are applied to a simple mineral mix approximating the lower mantle. Two simple, extreme assumptions that put bounds on the elasticity of a rock have been used for many years. One, the Voigt assumption, is that, when a composite sample is stressed, individual grains are equally strained, in spite of their different elasticities. This implies stresses between grains that support part of the externally applied stress. It allows the mathe-

0031-9201r98r$19.00 q 1998 Elsevier Science B.V. All rights reserved. PII S 0 0 3 1 - 9 2 0 1 Ž 9 8 . 0 0 0 8 4 - 3

220

F.D. Staceyr Physics of the Earth and Planetary Interiors 106 (1998) 219–236

matical simplification that the volume fractions of the constituents are independent of stress Žor of temperature., but does not apply to any physically real situation. The other ŽReuss. limit assumes that intergranular stresses are completely relaxed so that all grains are equally stressed. This limit is applicable to very high stresses or to situations of very prolonged stress at high temperatures, although it is not then strictly an elastic deformation that is considered because inelastic deformation is implied. As Kumazawa Ž1969. and Thomsen Ž1972. pointed out, this applies to deformations accompanying pressure and temperature variations in the Earth, because these are slow enough to allow complete relaxation of internal stresses. However, the passage of seismic waves causes small, unrelaxed stresses that must be represented by higher moduli; Stacey Ž1997. pointed out that this essential difference between moduli representing steady and transient deformations compromised use of the Adams–Williamson equation and Bullen’s parameter h as measures of the homogeneity of seismologically observed layers in the earth. Hill Ž1952. argued that an average of the Voigt and Reuss limits was a satisfactory approximation to small rapid elastic deformations but this has sometimes been doubted and calculations imposing tighter limits have been presented, notably by Hashin and Shtrikman Ž1963. and Miller Ž1969.. However, use of the Voigt and Reuss limits continues because they are simpler to apply, and this is particularly true for the analysis reported here. The Voigt and Reuss assumptions can be applied readily to the derivative properties that are sought. For the case of incompressibility, the Voigt–Reuss–Hill ŽVRH. average value for the composition considered here falls between the more restrictive Hashin–Shtrikman bounds, giving some justification to the VRH averaging process for the other properties. Although the Voigt limit is useful only as component of the VRH average, it is of interest to note that a Voigt approach has, until now, been implicitly assumed in equation-of-state studies of the deep Earth. When a composite material is compressed the component with higher incompressibility contracts less than the average and so its volume fraction increases. There is therefore an increase in incompressibility arising from the increase in relative vol-

ume of the less compressible component, additional to the increase intrinsic to the individual constituents. Calculations that ignore this effect make the Voigt assumption that the volume fractions are constant. The calculations presented here require simultaneous consideration of both Reuss Žrelaxed. and unrelaxed responses to stress and to temperature variations. The purpose is to investigate the effects of mixed mineralogy on the pressure and temperature dependences of bulk modulus, K. The applications are to the effects of pressure and temperature variations within the earth which cause fully relaxed elastic responses, but are investigated by seismic waves which superimpose unrelaxed stresses. Investigation of the pressure dependence of the adiabatic modulus, K s , is motivated by a study of an equation-of-state for the lower mantle in which ŽEK SrEP . S is a vital parameter ŽStacey, 1995; Stacey, 1996.. The behaviour of such derivative properties can be counter-intuitive. In particular d Krd P for a mixture of silicate perovsite and magnesiowustite was found to be higher than the values for the independent minerals ŽStacey, 1996.. Interest in the temperature dependence of K S arises from the thermal interpretation of tomographically observed seismic velocity variations. This refers, in principle, to any depth in the Earth, but in the lower mantle Robertson and Woodhouse Ž1996a,b. observed that the ratio of S- to P-velocity variations implied that d S s Ž1ra K S .ŽEK SrET . P ™ 1 at 1900 km depth, where a is the volume expansion coefficient. Such a low value of d S appears incompatible with other thermodynamic properties but the complication of mixed mineralogy prompts the reinvestigation of this question. Su and Dziewonski Ž1997. reported that in the deepest part of the mantle the velocity anomalies required d S - 1. According to the analysis presented here, this cannot be explained as a thermal effect. It is of interest to consider partial relaxation of stress by thermal diffusion. The isothermal modulus, K T , must, by definition, be thermally relaxed, that is it causes no temperature changes, whereas the adiabatic modulus K S may be thermally relaxed or unrelaxed, even when there is no other elastic relaxation. For a compression that is sufficiently rapid to allow no time for thermal diffusion between grains, the individual adiabatic modulus is applicable to each grain. However, differences in thermodynamic

F.D. Staceyr Physics of the Earth and Planetary Interiors 106 (1998) 219–236

properties cause the grains to be unequally heated. If a compression is slow enough to allow temperature equalisation between grains but no heat transfer to or from the sample as a whole, then thermal relaxation gives a slightly lower value of K S , here denoted by K S ). In general there will still be stresses between grains of unequal elasticities and an unrelaxed modulus applies, but it is a reduced, thermally relaxed or partially relaxed value, rather than the completely unrelaxed modulus. Grain sizes with thermal relaxation times t between 1 s and 50 min are approximately Žht .1r2 s 1 mm to 6 cm, for thermal diffusivity h s 1.3 = 10y6 m2 sy1 . Thus, the range of frequencies used in seismology is such that both thermally relaxed and unrelaxed values of K S are relevant, although the difference must be small if the very slight anelastic losses generally assumed for pure compressions are believed Žas in PREM-Dziewonski and Anderson, 1981.. Equations showing the difference between thermally relaxed and unrelaxed adiabatic moduli are presented, but this difference is seen to be too small to affect the main themes of this paper. In comparing the lower mantle with calculated properties of a particular mineral composite, I have considered a simple mixture of ŽMg,Fe.SiO 3 perovskite and ŽMg,Fe.O magnesiowustite with MgrFe ratios derived by matching density and incompressibility to the lower mantle ŽStacey, 1996.. This omits the probable presence of Ca perovskite. The logic of this simplification is that the density and incompressibility data suffice to match the mixture to the mantle very well. If a more complicated mixture is admitted then we need additional data to constrain it and, in any case, the properties of Ca perovskite are not well known. But the property that matters most is d Krd P and we may reasonably suppose that this is similar for Ca and Mg perovskites. Thus, the equation-ofstate for the lower mantle and the extrapolation of properties to the foot of the adiabat do not appear to be seriously compromised by this simplification. 2. The Reuss approximation Consider a rock composed of two minerals, with properties subscripted 1 and 2, in volume fractions Ž1 y x . and x, respectively. Applying an isothermal compression, with equal pressure increments D P on

221

all grains by the Reuss assumption, the volume for each grain and changes are proportional to Ky1 T the standard Reuss equation applies 1

1yx s

KT

x

Ž 1.

q K T1

K T2

The same argument follows exactly for the thermally unrelaxed adiabatic modulus 1

1yx s

KS

x

Ž 2.

q K S1

KS2

Now allow thermal relaxation to a common temperature increment, DT, from the initial temperature rises DT1 and DT2 , so that the volume compressions become DV1

DP sy

V1

K S1

DV2

DP sy

V2

KS2

q a 1 Ž DT y DT1 .

Ž 3.

q a 2 Ž DT y DT2 .

Ž 4.

and therefore the total, thermally relaxed modulus, K S ), is given by DV

DP sy

V

s

DV1 q DV2

KS )

V1 q V2

s Ž1yx . y

qx y

DP K S1

DP KS2

q a 1 Ž DT y DT1 .

q a 2 Ž DT y DT2 .

Ž 5.

where a is volume expansion coefficient. Since the whole process is adiabatic the temperature changes are related by

Ž 1 y x . r 1CP1Ž DT y DT1 . q x r 2 CP 2Ž DT y DT2 . s 0

Ž 6.

where DT1 s

g 1T K S1

DP;

DT2 s

g 2T KS2

DP

Ž 7.

F.D. Staceyr Physics of the Earth and Planetary Interiors 106 (1998) 219–236

222

and r is density, CP the specific heat at constant pressure and g is the Gruneisen parameter ¨

g s a K Srr CP

Ž 8.

Using Eqs. Ž6. – Ž8. in Eq. Ž5. 1

1yx s

KS )

q

K S1 =

ž

x Ž 1 y x . r 1CP1 r 2 CP 2

x q

Ž 1 y x . r 1CP1 q x r 2 CP 2

KS2

g1

y

K S1

g2 KS2

Ex

2

/

T

Ž 9.

By Eq. Ž9., K S ) is smaller than K S ŽEq. Ž2.. because the third term in Eq. Ž9. must be positive. The magnitude of this term can be seen more clearly by rewriting the squared factor in an equivalent form

ž

g1 K S1

y

g2 KS2

2

/

1 s

g 1 a1

r 1CP1 K S 1

ž

= 1y

Now we can consider the variations of K T , K S and K S ) with ambient pressure. Within the Earth Žbut not for laboratory observations. the response to ambient pressure variations must be relaxed, that is the Reuss assumption applies, whether we consider a Reuss or Voigt response to superimposed transient stress. Thus x depends on pressure according to the difference between component moduli. For isothermal variation

a 2 r 1CP1 a t r 2 CP 2

ž / EP

/ž

g 2 K S1 g1 KS2

/

T

K T2

EKT

ž / EP

s T

K S ) s K T Ž 1 q ga T .

a s Ž 1 y x . a1 q x a2

Ž 12 .

and g is given by Eq. Ž8. with K S ) for K S , and

r CP s Ž 1 y x . r 1CP1 q x r 2 CP 2

ž /

K T21

Ex

ž / EP

sx Ž1yx . S

Ž 15 .

EP

ž

E K T2

x q T

K T2 2

1

ž / EP

1 y

K T1

K T2

T

2

/

Ž 16 .

=

1

ž

1 y

K S1

KS2

/

qx Ž1yx .

x r 2 CP 2 a 1 q Ž 1 y x . r 1CP1 a 2

=

r CP

ž

g1

g2

y

K S1

KS2

/

T

Ž 17 .

Thus the adiabatic pressure dependence of the thermally unrelaxed modulus is 1 K S2

E KS

ž / EP

s

Ž 1 y x . E K S1 K S21

S

Ž 13 . Ž 14 .

/

K T2

1 y x E K T1

q

r s Ž 1 y x . r1 q x r2

K T1

For an adiabatic variation of the ambient pressure an additional term is required in the variation of x, to account for the temperature equalization, as in Eqs. Ž3. and Ž4..

Ž 11 .

which applies to K S ), not K S , because a is, by definition, a property of the thermally relaxed state. With the Reuss assumption of equal stresses on both minerals

1 y

qx Ž1yx .

Ž 10 . This shows that the fractional difference between K S and K S ) is of order a T times the square of a fractional difference in component properties. It would make a small contribution to the dispersion of P-waves, but is not significant to the main theme of the present paper. It may be noted that we can also write K S ) in terms of a standard thermodynamic identity

1

ž

Note that no variation in composition is implied by Eq. Ž15.. It simply represents the fact that under increasing pressure the less compressible mineral occupies an increasing fraction of the total volume. This contribution to d Krd P has not been generally recognised. Using this result with Eq. Ž1. 1

y

sx Ž1yx .

ž

ž / EP

1

1 y

K S1

KS2

E KS2

x q S

K S22

Ex

/ž / EP

ž / EP

S

Ž 18 . S

F.D. Staceyr Physics of the Earth and Planetary Interiors 106 (1998) 219–236

with ŽE xrEP . S given by Eq. Ž17.. The first term in the x variation is equivalent to the corresponding term in Eq. Ž15. and is of a magnitude that makes it important to the present discussion, but the second term is smaller by a factor of order a T and so appears negligible in cases of interest. The pressure-dependence of K S ) differs from Eq. Ž18. by the pressure variation of the third term in Eq. Ž9., which was counted as too small to be significant in the present context. That this remains true of its pressure derivative is seen by considering the pressure-derivative of the product Ž a T .

ž

E Ž aT . EP

/

aT s KS )

S

= Ž d T y K XT .

Ž 19 .

w h e re q s Ž E ln g r E ln V . T a n d d T s Ž1ra K T .ŽEK T rET . P . The square-bracketed factor in Eq. Ž19. is typically 2 to 3 and is not sufficient to demand further consideration. Temperature dependences of the Reuss-limiting K T and K S are obtainable in the same way, noting that the variation in x with T depends on the difference between component expansion coefficients. 1 K T2

EKT

ž / ET

s P

1 y x E K T1

ž /

K T21

ET

x q P

K T2 2

E K T2

ž / ET

P

q x Ž 1 y x . Ž a1 y a2 . = 1 K S2

E KS

ž / ET

s P

ž

1

1 y

K T2

K T1

1 y x E K S1

ž /

K S21

ET

/

Ž 20 . x

q P

K S22

E KS2

ž / ET

P

=

ž

1 y

KS2

K S1

/

ž /

ž /

ž / EP

s Ž1yx . T

Ea 1

Ea 2

ž / ž / EP

qx

T

EP

q x Ž 1 y x . Ž a1 y a2 .

ž

Ž 21 .

As with the pressure-dependences, the third terms in these equations are of interesting magnitudes, but the difference between K S ) and K S is not. There is no term in Eq. Ž21. corresponding to the second term

T

1

1 y

K T2

K T1

/ Ž 23 .

3. The Voigt limit The Voigt assumption that all components are equally strained requires the volume fraction x to be independent of pressure and temperature. This means that component stresses are, in general, unequal. Then the total pressure, D P, on a composite sample is given by asserting that VD P s ÝVi D Pi , that is D P s Ž 1 y x . D P1 q xD P2 Ž 24 . with DV DV1 DV2 s s Ž 25 . V V1 V2 So that, for isothermal compression or for thermally unrelaxed adiabatic compression K T s Ž 1 y x . K T1 q xK T 2 Ž 26 . K S s Ž 1 y x . K S 1 q xK S 2

q x Ž 1 y x . Ž a1 y a2 . 1

in Eq. Ž17.. Noting that there is a general inverse relationship between a and K for many minerals Žgiving a sensibly constant product a K . we normally expect the third terms in Eqs. Ž20. and Ž21. to be negative. Since K decreases with T the negative temperature dependences are increased in magnitude by these terms. The pressure dependence of expansion coefficient can be written down directly from Eq. Ž20. by the identity Ea 1 EKT s 2 Ž 22 . EP T KT ET P so that Ea

g q 1 y K XT y q q ga T

223

Ž 27 .

As in the case of the Reuss limit, the Voigt limiting value of the thermally relaxed modulus, K S ), is given by Eq. Ž11., which does not apply to the unrelaxed K S . In this case a K T s Ž 1 y x . a 1 K T1 q x a 2 K T 2 Ž 28 . so that gr C V s Ž 1 y x . g 1 r 1C V1 q xg 2 r 2 C V 2

Ž 29 .

F.D. Staceyr Physics of the Earth and Planetary Interiors 106 (1998) 219–236

224

with

compared with

r C V s Ž 1 y x . r 1C V1 q x r 2 C V 2

Ž 30 .

Reuss:

by analogy with r CP for the Reuss case. However, it is difficult to estimate the small difference between K S and K S ) using Eq. Ž11.. A Voigt equation similar to Eq. Ž9. can be obtained by writing the component pressure increments for an adiabatically applied pressure as

1

1 y

KS )

1 s

KS

K S1 =

V

q a 1 K T1Ž DT y DT1 .

Ž 31 .

q a 2 K T 2Ž DT y DT2 .

Ž 32 .

DV D P2 s yK S 2

V

where DT1 , DT2 are given by Eq. Ž7. and DVrV is common to both minerals. The relaxed common temperature is DT s

Ž 1 y x . r 1C V1 DT1 q x r 2 C V 2 DT2 Ž 1 y x . r 1C V1 q x r 2 C V 2

Ž 33 .

ž

x Ž 1 y x . r 1C V1 r 2 C V 2

Ž 1 y x . r 1C V1 q x r 2 C V 2

EKT

ž / EP

s Ž1yx . T

E K T1 EP

qx Ž1yx .

2 Ž g1 y g2 . T

E KS

ž / EP

s Ž1yx . S

=

x Ž 1 y x . r 2 CV 2

Ž 1 y x . r 1C V1 q x r 2 C V 2

ž

g1

ž

2

/

EP

T

Ž KT

2

y K T1 .

T

2

Ž 38 .

K T1 K T 2

E K S1 EP

Ž 36 .

E KS ) EP

S

E KS2

S

Ž KS

2

EP

y K S1 .

K S1 K S 2

S

2

qx Ž1yx .

r CP

ž

g1

y

K S1

EP

g2 KS2

E KS

/ ž / s

qx

x r 2 CP 2 a 1 q Ž 1 y x . r 1CP1 a 2

=

Voigt: K S y K S )

g2

qx

ž / ž /

qx Ž1yx .

This is similar to Eq. Ž9. and Ž K S y K S ). is comparable in the two cases, being given by a product of a T and the square of a property difference. Subtracting K S ) from K S ŽEq. Ž27.. and rewriting to make this point more obvious

=g 1 a 1T 1 y

E K T2

ž / ž /

Ž 35 .

s K T1

/

Similarly the Žrelaxed. adiabatic variation of x is given by Eq. Ž17. so that this must be applied to differentiation of Eqs. Ž27. and Ž35. to obtain

K S ) s Ž 1 y x . K S 1 q xK S 2 y

g1 KS2

2

In calculating the pressure variations, since the aim is to describe the situation in the Earth, ambient conditions are assumed to be relaxed, so that the isothermal variation of x is given by Eq. Ž15.. Applying this to Eq. Ž26. we have, for the Voigt isothermal modulus

Ž 34 .

from which we obtain

g 2 K S1

Ž 37 .

and the common pressure increment is D P s Ž 1 y x . D P1 q xD P2 s yK S ) DVrV

Ž 1 y x . r 1CP1 q x r 2 CP 2

=g 1 a 1T 1 y

DV D P1 s yK S 1

x Ž 1 y x . r 2 CP 2

q´

/Ž

K S 2 y K S1 . T

Ž 39 . Ž 40 .

S

where ´ is a very small term due to the third term of Eq. Ž35..

F.D. Staceyr Physics of the Earth and Planetary Interiors 106 (1998) 219–236

225

Table 1 Properties of a composite of 72% perovskite and 28% magnesiowustite, calculated from component properties at P s 0, T s 290 K Property

Perovskite

Magnesiowustite

Voigt

Reuss

VRH Average

K T ŽGPa. K S ŽGPa. )ŽEK T rEP . T , ŽEK S rEP . S ŽEK T rET . P ŽGPa Ky1 . ŽEK S rET .. P ŽGPa Ky1 . a Ž10y6 Ky1 . ŽEarEP . T Ž10y6 Ky1 GPay1 . Ž a K T . Ž10y3 GPa Ky1 .

262.5 264.0 a 4.0 c y0.021c y0.011c 15.7 c y0.30 4.12

161.2 162.5 b 4.1b3 y0.027 e y0.0145d 31.6 f y1.04 5.09

234.1 235.6 4.08 y0.023 y0.0123 18.8 y0.41 4.40

223.2 224.7 4.34 y0.0273 y0.0139 20.1 y0.507 4.48

228.7 230.1g 4.21 y0.025 y0.0131 19.4 y0.46 4.44

a

Yeganeh-Haeri Ž1994.. Jackson and Niesler Ž1982.. c Jackson and Rigden Ž1996.. d Bass Ž1995.. e Anderson and Isaak Ž1995.. f Suzuki Ž1975.. )ŽEK T rEP . T f ŽEK S rEP . S q 0.029. The difference is less than uncertainties and is neglected. g Hashin–Shtrikman bounds 229.9 to 230.7 GPa. b

The temperature variations are calculated similarly

EKT

ž /

s Ž1yx .

ET

P

E K T1

E K T2

ž / ž / qx

ET

P

ET

P

q x Ž 1 y x . Ž a 2 y a 1 . Ž K T 2 y K T1 .

Ž 41 . E KS

ž / ET

s Ž1yx . P

E K S1

E KS2

ž / ž / qx

ET

P

ET

P

These equations are used in Table 1 to calculate properties of a mixture that is a plausible approximation to the composition of the lower mantle, using measured properties of the two constituents at ambient laboratory conditions Ž P s 0, T s 290 K.. The VRH averages are simply the arithmetic means of the Reuss limits by Section 2 and Voigt limits by Section 3. In the case of K S Žor K T . the VRH mean falls between the much more restrictive bounds of Hashin and Shtrikman Ž1963..

q x Ž 1 y x . Ž a 2 y a 1 . Ž K S 2 y K S1 .

Ž 42 .

4. Lower mantle equation-of-state

and, as before

ž

E KS ) ET

E KS

/ ž / s

P

ET

qh

Ž 43 .

P

where h may be derived from the third term of Eq. Ž35. but is too small to be of interest here. As in the Reuss case, we expect the third terms in Eqs. Ž41. and Ž42. to increase the magnitudes of the negative temperature variations. Also, Eq. Ž22. applies and so

Ea K T2

ž / EP

T

s Ž 1 y x . K T21

Ea 1

ž / EP

T

q xK T2 2

Ea 2

ž / EP

T

q x Ž 1 y x . Ž a 2 y a 1 . Ž K T 2 y K T1 .

Ž 44 .

The most significant conclusion of Table 1 is that the composite value of d Krd P, either for K S or K T , is above the component values. In the case of ŽEK SrEP . S , the temperature dependence is extremely small ŽJackson and Rigden, 1996; Stacey, 1996., so the higher value Ž4.2 1 . is appropriate for the foot of the lower mantle adiabat as well as room temperature. Therefore, to be realistic an equationof-state for the lower mantle must meet this value of K 0X . It is slightly higher than the value used by Stacey Ž1996., which neglected the pressure dependence of x ŽEq. Ž17.., so a revised equation-of-state is presented here. It is important to the comparison with the lower mantle that the calculated pressure

F.D. Staceyr Physics of the Earth and Planetary Interiors 106 (1998) 219–236

226

dependence is completely relaxed, as is the compression of the minerals in the lower mantle, although the incremental incompressibility is a response to very small superimposed pressure changes and is not relaxed. I have advocated the use of a linear relationship between rigidity, m , incompressibility, K, and pressure P for close-packed materials. The theoretical reason for such a relationship and a demonstration that it fitted the lower mantle extremely well were given by Stacey Ž1992. Žpp. 264–267. and implications for core and mantle equations-of-state were pursued later ŽStacey, 1995, 1996.. The most useful way of presenting the equation is

m

m

s K

ž /ž K

0

1 y K`X

P K

/

Ž 45 .

where subscript zero indicates a zero pressure value and K`X is the asymptotic limit of d Krd P as P ™ `. At this limit PrK ™ Ž PrK .` s 1rK`X and mrK vanishes. All finite strain equations have values of K`X , whether recognized or not, and it is of interest to compare them with the lower mantle value, K`X s 1.425, obtained by fitting Eq. Ž45. to the PREM tabulation. But the approach that I favour is the use of a finite strain equation that is matched to both extreme conditions, K 0X s 4.21 and K`X s 1.425. By presenting finite strain as a relationship between d Krd P s K X and PrK, we have two fixed end points that can be applied to the lower mantle fitting, Ž PrK s 0, K X s K 0X . and Ž PrK s 1rK`X , K X s K`X .. Moreover, plots of K X vs. PrK are smooth and not very far from linear. This was the justifica-

tion for presenting a finite strain equation of the form K

X

s K 0X q K 0 K 0Y

P

1

K 02 K 0Z

P

2

ž /

q Ž 46 . K 2 K where the coefficient of the third term is given by 1 2 Z K 0 K 0 s K`X 3 y K 0X K`X 2 y K 0 K 0Y K`X Ž 47 . 2 allowing a reliable estimate of its value for the lower mantle. I now refer to Eq. Ž46. as the quadratic K X y PrK finite strain equation. We may reasonably suppose that Eq. Ž46. is not complete but the beginning of a series. If this is so then examination of the values of the coefficients obtained by fitting lower mantle data suggests that the series looks like the expansion of an exponential and I have therefore examined a new equation K 0 K 0Y P X X K s K 0 exp Ž 48 . K 0X K

ž

/

where K 0 K 0Y s K 0X K`X ln Ž K`X rK 0X .

Ž 49 .

so that a more convenient form of Eq. Ž48. is K X s K 0X Ž K`X rK 0X .

X

K `PrK

Ž 50 . X

I refer to this as the exponential K y PrK equation. Eqs. Ž46. and Ž50. both give excellent fits to the PREM lower mantle tabulation, as in Table 2. For this purpose, Eq. Ž46. requires a fitted datum in addition to K 0X and K`X and I have used the volume average lower mantle values Ž PrK s 0.1457, K X s 3.369. to give K 0 K 0Y s y6.245, K 02 K 0Z r2 s 3.243. Eq. Ž50. does not require this additional datum,

Table 2 X Adiabatic extrapolation of lower mantle properties to zero pressure by two K y PrK equations Radius Žkm.

5600 5200 4800 4400 4000 3630

P ŽGPa.)

28.29 46.49 65.52 85.43 106.39 126.97

K ŽGPa.)

313.3 380.3 444.8 508.5 574.4 641.2

r Žkg my3 .)

4443.17 4678.44 4897.83 5105.90 5307.24 5491.45

)PREM tabulation at 1 s period ŽDziewonski and Anderson, 1981..

Eq. Ž46.

Eq. Ž50.

K0

r0

K0

r0

202.98 204.77 204.16 202.30 201.18 203.14

3977.33 3977.07 3973.96 3967.39 3963.21 3971.52

203.19 205.16 204.68 202.92 201.83 203.81

3977.55 3977.64 3974.91 3968.64 3964.59 3972.92

F.D. Staceyr Physics of the Earth and Planetary Interiors 106 (1998) 219–236

being completely specified by the values of K 0X and K`X only. The fitting procedure is to integrate the equations numerically to values of PrK required to match the listed PREM values. This relates rrr 0 , PrK 0 and KrK 0 to PrK s Ž PrK 0 .rŽ KrK 0 ., so that corresponding zero pressure values K 0 and r 0 are obtained by comparison with the tabulated K, r values. Eqs. Ž46. and Ž50. are compared with nine other finite strain equations in Table 3. The equations are summarized in an appendix. All were fitted to the same PREM values by the same procedure. The quoted uncertainties are standard deviations of individual values about the means. Formal Ž1 s . uncertainties of the mean values would be smaller by the factor 65. Categorization into goodrmarginalrunacceptable fits was based on the requirement that K 0 and r 0 should be independent of depth; the standard deviations were not used for this assessment but reflect the trends. The five equations classed as giving good fits showed no obvious trends and are therefore compatible with homogeneity of the lower mantle, as represented by PREM. Bearing in mind the slight departure of PREM from the Adams–Williamson equation, as seen in Bullen’s seismological homogeneity parameter ŽStacey, 1997., closer fits to any equation could hardly be expected. It appears remarkable that the five apparently very different

227

equations selected in this way should agree so closely on the values of K 0 and r 0 at the foot of the lower mantle adiabat. The conclusion must be that the details of the equations are not important and that these values are robust. Certainly, it is evident that extrapolation to K`X hardly affects the extrapolation to P s 0 because entries Žiii., Živ. and Žv. in Table 3, which give good fits, have more extreme values of K`X than do the rejected equations. However, successful equations are restricted to those with K 0 K 0Y close to y6. Uncertainties quoted in Table 3 are those apparent from ‘scatter’ of individual values about their mean and omit possible systematic errors. There are four sources of error that would not contribute to the quoted standard deviations and need to be allowed for in assessing the precision with which K 0 and r 0 of the lower mantle are estimated here. One, the uncertainty in K`X , can be dismissed as unimportant because the value of K`X has no obvious effect on quoted fits Žiii., Živ. and Žv.. It is essential for Ži. and Žii. but is more precise than K 0X , the uncertainty of which is more important. Second is the mean lower mantle data point assumed in fitting equations Ži. and Žiii.. If we consider only entries Žii., Živ. and Žv. in Table 3, which do not use this point, the average of the ‘good’ fits is hardly affected, so this also can be disallowed as a serious error. The value of K 0X has a

Table 3 X PREM lower mantle tabulation fitted to 11 finite strain equations, with the constraint K 0 s 4.21 X

Y

K 0 ŽGPa.

r 0 Žkg my3 .

K`

Good fits Ži. K X y PrK quadratic Žii. K X y PrK exponential Žiii. Birch, 4th order Živ. Morse potential Žv. Rydberg potential

203.1 " 1.3 203.6 " 1.2 202.4 " 1.2 202.1 " 1.5 203.9 " 1.2

3972 " 6 3973 " 5 3972 " 5 3970 " 6 3975 " 4

1.425 1.425 11r3 2r3 2r3

y6.245 y6.500 y5.432 y5.72 y6.00

Marginal fits Žvi. Brennan–Stacey equation Žvii. Davydov potential

207.6 " 3.1 200.7 " 2.2

3984 " 9 3966 " 9

4r3 4r3

y6.38 y5.57

Unacceptable Žviii. Birch, 3rd order Žix. Keane’s equation Žx. Born–Mie potential Žxi. Meyer potential

190.0 " 9.1 230.7 " 16.0 187.7 " 10.8 195.8 " 5.1

3937 " 32 4033 " 30 3931 " 38 3954 " 18

3 1.425 2.88 4r3

y4.14 y11.72 y3.84 y9.70

K0 K0

228

F.D. Staceyr Physics of the Earth and Planetary Interiors 106 (1998) 219–236

more obvious effect. If K 0X is reduced by 0.1, then the fitted values of K 0 and r 0 are raised by 1.6 GPa and 1.5 kg my3 , respectively. This appears to be the largest uncertainty for K 0 but not for r 0 . An error in K 0X as large as this is possible, not because of doubt about the calculations in Sections 2 and 3 but because the measured component values in Table 1 are uncertain. The remaining doubt concerns the model tabulation with which the theories are compared. This problem cannot be judged by means of equations that are themselves being tested. A comparison of PREM with HB1 ŽHaddon and Bullen, 1969., PEM ŽDziewonski et al., 1975. and ak135ŽKennett et al., 1995; Montagner and Kennett, 1996. shows differences in r and K through the lower mantle of order 1%, with PEM and PREM agreeing reasonably well but ak135 being nearer to HB1. A closer look at the most recent model Žak135. shows a variation in Bullen’s homogeneity parameter from 0.86 to 1.9 compared with an expected value close to 1.00 for a homogeneous region, with which PREM agrees quite well ŽStacey, 1997.. If ak135 is correct in this matter then lower mantle heterogeneity makes equation-ofstate fitting irrelevant, but ak135 also gives quite strong heterogeneity in the outer core, which is implausible. Thus, we are left with a doubt that is unresolved beyond indicating that earth model values of r and K could be in error by as much as 1% but that the values adopted from PREM are the most reliable that we have. The parameter fits in Table 3 are only slightly different from those obtained by Stacey Ž1996. using a lower value of K 0X . They do not justify a readjustment of the estimated lower mantle composition as that would lead to a completely insignificant change to the estimate of K 0X . The conclusion of the finite strain fitting is therefore that we have reliable values for r 0 , K 0 and Žby Eq. Ž45.. m 0 for the lower mantle as represented by PREM

r 0 s 3972.5 " 4.0 kg my3 K 0 s 203.1 " 1.6 GPa m 0 s 128.2 " 1.2 GPa

5. Effect of temperature and interpretation of lower mantle tomography The second principal target of this enquiry is the temperature-dependence of incompressibility, as expressed by the range of lower mantle values of the adiabatic Anderson–Gruneisen parameter. ¨

d S s Ž 1ra K S . Ž E K SrE T . P

Ž 52 .

This is calculable if we know the behaviour of the Gruneisen parameter ŽEq. Ž8.., because we have a ¨ thermodynamic identity

d S s K SX y 1 y g q q y Ž E lnC VrE lnV . S where q s y Ž E lngrE ln r . T

Ž 54 .

The last term in Eq. Ž53. can certainly be neglected at lower mantle temperatures and, to a good approximation, also at low temperatures because T remains an almost constant fraction of the Debye temperature during adiabatic compression. Since g is linked to K XT , which is readily calculated from K SX , it follows that d S is directly calculable from an equation-of-state. We may therefore estimate d S for the lower mantle from the equations in Section 4. For the purpose of this calculation it is necessary to know that the modified free volume formula for g , which Stacey Ž1996. applied to perovskite, can be used directly for a mineral composite with magnesiowustite. This is given by 1

gsh

2

ž

K XT y

r 1y

1

f y

6

2 P f 3 K

3

ž

1y

1 P 3 K

/

/

with an additional uncertainty of perhaps 1% arising from doubt about details of the model.

Ž 55 .

where f s 2.27, K XT is the low temperature value and h s 1 q 1.45 Ž K XT y 1 . Ž yD rrr . P

Ž 51 .

Ž 53 .

Ž 56 .

is a higher anharmonic correction factor expressed in terms of the total thermal dilation ŽyD rrr . P at the prevailing pressure, P. K XT is related to the observed K SX by another identity K SX s K XT q ga T Ž K XT y d S y d T y q .

Ž 57 .

F.D. Staceyr Physics of the Earth and Planetary Interiors 106 (1998) 219–236

229

Table 4 Ža. Thermoelastic properties of the lower mantle by the quadratic K X y PrK equation ŽEq. Ž46.. X Radius Žkm. K g q dS

T ŽK.

a Ž10y6 Ky1 .

Zero pressure reference 5600 5200 4800 4400 4000 3630

1725.0 1999.6 2127.2 2243.3 2352.6 2454.4 2537.7

33.5 21.8 17.9 15.3 13.3 11.8 10.8

1725.0 1996.8 2122.7 2237.4 2345.5 2446.5 2529.3

33.5 21.1 17.4 14.8 13.0 11.6 10.6

4.210 3.673 3.495 3.361 3.252 3.165 3.101

1.366 1.231 1.166 1.118 1.081 1.052 1.031

1.544 1.212 0.985 0.842 0.741 0.667 0.617

3.388 2.653 2.315 2.084 1.912 1.779 1.686

Žb. Thermoelastic properties of the lower mantle by the exponential K X y PrK equation ŽEq. Ž50.. Zero pressure reference 4.210 1.366 1.669 3.513 5600 3.662 1.219 1.194 2.637 5200 3.486 1.155 0.951 2.282 4800 3.354 1.110 0.798 2.042 4400 3.248 1.076 0.691 1.863 4000 3.163 1.049 0.613 1.726 3630 3.101 1.030 0.560 1.631

in which the negative second term is small and decreases with pressure, but K XT and K SX converge at low temperatures and since ŽEK SX rET . P s 0 within present uncertainties ŽJackson and Rigden, 1996; Stacey, 1996. it follows that K SX from the lower mantle equation-of-state can be used directly for K XT Žlow T . in Eqs. Ž55. and Ž56.. Eq. Ž55. is obtained by identifying g as the ratio of thermal pressure, PTh , to thermal energy. Where PTh s

T

H0 Ž a K

T

. dT

Ž 58 .

integrated at constant volume. In a composite material thermal pressure is shared in precisely the same manner considered in calculating a and K T separately in Sections 2 and 3. So Voigt and Reuss limits and the ŽHill. average of them are also appropriate for the product Ž a K T .. This is corroborated by the VRH average value of Ž a K T . in Table 1 which may be obtained either from the value of this product for each mineral or as the product of the VRH averages of a and K T separately, although, in this case the Voigt and Reuss limits are quite close and there is little scope for ambiguity anyway. Table 4a and b give values of K X calculated by Eqs. Ž46. and Ž50., with corresponding values of g

by Eq. Ž55., and q obtained by differentiating Eq. Ž55.: P

KX K qsy 2 P 1y f 3 K

ž ž

1y

/ /

d KX

1

2g d Ž PrK .

E ln h

y

ž / E ln r

f q

9g

2 q 3

f

Ž 59 . T

Then d S is obtained from Eq. Ž53. with neglect of ŽElnC vrElnV . S . The first term in Eq. Ž59. is written in a form that makes it readily calculable from Eq. Ž46. or Eq. Ž50.. To calculate the second term we need the variation of ŽyD rrr . P with r at high temperatures. By assuming thermal expansion to follow a Debye curve, we can write the total thermal expansion at T ) u as Dr

ž / ž y

r

fa Ty

P

3 8

u

/

Ž 60 .

where u is the Debye temperature, and use the relationship d ln urd ln r f g s Ž E lnTrE ln r . S

Ž 61 .

F.D. Staceyr Physics of the Earth and Planetary Interiors 106 (1998) 219–236

230

Then with

Ž E ln arE ln r .

X T s1yKT yq

Ž 62 .

´ s Ž 1ram . Ž EmrE T . P

for constant C V we find d ln h

1.45 s

d ln r

3 a Ty u h 8

q

ž

ž

1 y K XT

P

/

/

Ž K XT y 1 . Ž 1 y K XT y q .

3 8

Em

ET

s dP

KS

P

P

EK

K

0

s

ET

Ž 64 .

Ž 65 .

to each of the pre-set values of PrK. This gave g ,q, a and T at each depth and then d S was obtained from Eq. Ž53.. For this purpose C V s 1162.3 J kgy1 Ky1 was assumed to be constant, being slightly less than the classical, Dulong–Petit, value for the assumed mean atomic weight, 20.97, according to the higher anharmonic correction calculated at zero pressure ŽStacey, 1996, Eq. 39.. This, and the assumed values of T0 and u 0 have only slight effects on the other parameters. The reference value T0 s 1725 K gives T s 1960 K by adiabatic compression to 650 km depth and is consistent with the temperature estimate at that depth derived from the P–T conditions required for the conversion of transition zone minerals to perovskite plus magnesiowustite ŽBoehler and Chopelas, 1991.. Now that we have values of d S calculated directly from lower mantle equations of state, we can

=

E Ž mrK . 0 ET

P

m

g s

dP

K y K`X P .

/Ž ž /ž / ž ET

Ž 63 .

The method of solution, for each of the equations of state ŽEqs. Ž46. and Ž50.. was first to solve iteratively for the parameters g ,q,h, Žd lnhrd ln r . and a at zero pressure, using Eqs. Ž55., Ž56., Ž59., Ž63. and Ž64., with K X and Žd K XrdŽ PrK .. given by Eq. Ž46. or Eq. Ž50. and zero pressure values T0 s 1725K and u s 900 K. Then the equations were integrated numerically with T and u varying according to d ln u

E Ž mrK . 0 m

The expansion coefficient a also enters these equations and is solved simultaneously using the identity

a s gr C V Ž 1 y ga T . rK S

s

q

gau

Ž 66 .

Multiplying Eq. Ž45. by K and differentiating with respect to T at constant pressure, P

ž / ž

d K XT

K d Ž PrK .

y 1.45 Ž K XT y 1 .

d lnT

use Eq. Ž45. to calculate also the equivalent quantity for rigidity

EK

m

q

Ž mrK . 0

ž / žE / K

0

T

/

P

Ž 67 . P

Here we must note that m and K refer to an arbitrary high pressure P but Ž mrK . 0 is the intercept at P s 0 of the graph of mrK vs. PrK. Since the lower mantle is assumed here to be adiabatic, Eq. Ž45. represents an adiabat. The temperature T0 at the foot of the adiabat is therefore adiabatically related to T at pressure P and so dT0

T0

s

dT

Ž 68 .

T

Thus

Em

ž / ž ET

s

E Ž mrK . 0

P

dT0

T Ž mrK . 0

ž / žE / K

m

EK

m

q

/

T0

0

T

Ž 69 . P

Rewriting in terms of ´ by Eq. Ž66. and d S by Eq. Ž52.

´sy

T0 d ln Ž mrK . 0

aT

dT0

q

Ž mrK . 0 d Ž mrK . S

Ž 70 .

Here d lnŽ mrK . 0rdT0 is a constant of the equation, being the temperature dependence of mrK at P s 0. We can estimate its value by combining the intercept value, Ž mrK . 0 s 0.631 at T s 1725 K, with a value at 290 K from laboratory data. For this purpose it is most satisfying to average the limits on m and K obtained from the theory of Hashin and Shtrikman Ž1963., which gives, for the 72% perovskite y28% magnesiowustite mixture, m 0 s 162.8 GPa. Thus, Ž mrK . 0 Ž290 K. s 0.7076 . The variation of Ž mrK . 0 is not uniform over the temperature

F.D. Staceyr Physics of the Earth and Planetary Interiors 106 (1998) 219–236

231

Table 5 Temperature variations of seismic velocities in the lower mantle Radius Žkm.

a Ž10y6 Ky1 .

dS

´

ŽElnVP rET . P Ž10y6 Ky1 .

ŽElnVS rET . P Ž10y6 Ky1 .

ŽElnVS rElnV P . P

Zero pressure reference 5600 5200 4800 4400 4000 3630

33.5 21.5 17.7 15.1 13.2 11.7 10.7

3.46 2.65 2.30 2.06 1.89 1.75 1.66

6.51 7.07 7.41 7.76 8.09 8.43 8.72

y64.5 y37.7 y30.0 y25.1 y21.8 y19.4 y17.7

y92.3 y65.1 y56.6 y50.9 y46.6 y43.4 y41.3

1.43 1.72 1.89 2.03 2.14 2.24 2.33

range 290 to 1725 K, but increases with temperature. It is more nearly correct to assume that d S and ´ are independent of temperature. In the case of d S we can compare the value from Table 1 Ž d S 0Ž290 K. s 2.94. with the high temperature value from Table 4a,b Ž d S 0Ž1725 K. s 3.45.. Since ŽEK SrET . P for perovskite is poorly constrained by observations these values of d S agree within present uncertainties. Therefore, the best way of estimating dŽ mrK . 0rdT0 at 1725 K is to assume that over the temperature range 290 to 1725 K it is proportional to expansion coefficient, a , and that this follows a Debye curve, with a higher anharmonic correction as in Eq. Ž56.. Then we have d ln Ž mrK . 0 dT0

Ž 1725 K . s y1.00 6 = 10y4 Ky1 Ž 71 .

Table 5 lists values of d S obtained by averaging the values in Table 4a,b, with corresponding values of ´ by Eq. Ž70., using the result ŽEq. Ž71.., and averages also of a and T from Table 4a and b. The combination of d S , ´ and a allows calculation of the temperature dependences of seismic velocities, as also listed, using the relationships

ž

E lnVP ET

/

a sy 2

P

=



K S Ž d S y 1. q KS q

4 3 4 3

m

E lnVS ET

/

a sy P

2

Ž ´ y 1.

ž / E lnVP

Ž 73 .

s P

4 m 3 KS

Ž d S y 1. r Ž ´ y 1. q

4 m

Ž 74 .

3 KS

As seen in the final column of Table 5, this quantity increases quite strongly with depth. This is qualitatively similar to the observations of Robertson and Woodhouse Ž1996a,b. of an increasing ratio of S to P velocity anomalies with depth in the lower mantle. However, their observations suggest that at about 1900 km depth ŽElnV PrElnVS . P approaches a value coinciding with 3r4 Ž VPrVS . 2 , requiring d S ™ 1 at this depth. The theory presented here gives d S s 1.9 at this depth and I see no prospect of finding an error of 0.9. The condition d S ™ 1 corresponds to temperature independence of Ž K Srr ., or equivalently of the ‘bulk sound velocity’, Ž K Srr .1r2 . Su and Dziewonski Ž1997. considered this quantity more explicitly and reported that in the deepest mantle it was negatively correlated with shear wave speed, implying d S - 1. They appealed to the fact that Ž1ra .ŽEln K SrET . V is positive. Using the identity

a

0

1q

E lnVS

1

m Ž ´ y 1.

Ž 72 .

ž

In tomographic studies some emphasis has been given to the ratio of Eqs. Ž72. and Ž73.

ž

E ln K S ET

/

s Ž yd S q K SX . r Ž 1 q ga T .

Ž 75 .

V

the results in Tables 4 and 5 give values of this quantity varying from q0.97 to q1.40, that is, it is positive throughout the lower mantle. However, this constant volume derivative is not relevant; the constant pressure derivative, yd S , is certainly negative everywhere. The observations of Robertson and

232

F.D. Staceyr Physics of the Earth and Planetary Interiors 106 (1998) 219–236

Woodhouse Ž1996b. and Su and Dziewonski Ž1997. reinforce the conclusion ŽStacey, 1992,Stacey, 1995. that we should seek a compositional interpretation of tomographically observed velocity anomalies in the lower mantle.

6. Conclusions The lower mantle parameter-fitting considered here is restricted to the range between radii of 3630 and 5600 km, which is modeled as a uniform region in PREM. Bullen’s seismological homogeneity parameter, h , shows a slight departure from the nominal value of unity for a homogeneous layer, attributed to the manner of parameterization, with a hint of a weakly superadiabatic temperature gradient ŽStacey, 1997.. These effects are small enough to have only a slight effect on equation-of-state fitting of r 0 and K 0 , although the effect on K 0X is more serious. Therefore, we must demand that any finite strain equation used for the lower mantle should be a good fit to the PREM tabulations of K and r . However, this still leaves a very wide range of alternatives. The slight curvature of P Ž r ., or K Ž r . relationships can be modeled in so many ways that they are uninformative. I have here introduced another constraint, K 0X s 4.21. Although it restricts the choice of equations somewhat, it still leaves five that fit the PREM lower mantle tabulation almost equally well. Assuming that we have correct values for the component minerals and a satisfactory approximation to the true composition, the security of this value of K 0X for the lower mantle is due to the circumstance that, at least at low pressure, ŽEK SX rET . P f 0, where K SX is defined as ŽEK SrEP . S , so that we can identify the value of K SX from low temperature laboratory data on a suitable mineral mix with the value of the foot of the lower mantle adiabat, without knowing the temperature. The problem of picking the correct value of K 0X is therefore reduced to calculating it for a mix of minerals whose individual properties are known. Intuitively, one might expect that since K 0X is close to 4.0 for both perovskite and magnesiowustite, this value would apply to a mixture of them. However, as shown in Sections 2 and 3 and Table 1, that is not so. A higher

value is required. Recognition that K 0X must be close to 4.21 for a plausible lower mantle mineral mix means that this is a requirement for any satisfactory lower mantle equation-of-state. I regard parameterfitting without this constraint as unsatisfactory to the point of being misleading, even acknowledging the uncertainty that arises from the remaining doubt about the precise value of K 0X for perovskite. The five equations that give good fits to the lower mantle, with the K 0X constraint, agree on the zero pressure values K 0 and r 0 ŽTable 3.. This means that we are simply fitting data to a graph with only modest curvature and details of the equations that match this curvature are unimportant. We cannot regard the fits as evidence for physical validity of any of the equations—they are best regarded as equally empirical. However, the agreement between them is strong evidence of the robustness of K 0 and r 0 . The precision with which we can match PREM appears to be greater than the certainty with which PREM models the mantle itself. The biggest uncertainty in r 0 and K 0 appears to arise from the imprecise value of K 0X for perovskite, but the uncertainty estimates in Eq. Ž51. appear large enough to cover this. The identification of plausible lower mantle mineralogy relies on K 0 and r 0 , but an important unknown is temperature. In this respect the calculations presented here differ too little from an earlier analysis ŽStacey, 1996. to justify a revision and the mineralogy that I assumed in preparing the tables was adopted from the earlier paper. However, I have revised slightly upward the estimate of potential temperature, T0 s 1725 K. This appears to accord better with the thermal properties calculated in Section 5 and with the temperature at 670 km depth estimated by Boehler and Chopelas Ž1991., although their uncertainty estimate, "300 K, cannot be improved on this basis. Interest in mantle tomography will subject the reliability of the calculated values of d S , or equivalently ŽEK SrET . P , to critical assessment. At zero pressure we have three semi-independent ways of estimating this from data in the tables. Table 1 gives the 290 K value ŽEK SrET . P s y0.0131 GPa Ky1 , although this estimate is dominated by the laboratory value for perovskite, which is not precisely known. At the potential temperature, T0 , Table 5 Žwith K 0 s

F.D. Staceyr Physics of the Earth and Planetary Interiors 106 (1998) 219–236

203.1 GPa. gives ŽEK SrET . P s y0.0235 GPa Ky1 . Assuming T0 s 1725 K, the difference between K S at T0 and at 290 K by Table 1 gives the average over this range ²ŽEK SrET . P : s y0.0188 GPa Ky1 . These numbers are reasonably consistent because the value of ŽEK SrET . P must fall off below the Debye temperature, justifying the assumption that d S is approximately constant at P s 0. That equation-of-state calculations of d S give a reliable value at Ž P s 0, T s T0 . adds confidence to the lower mantle values in Table 5. At high pressures d S is calculated from Eq. Ž53., in which q appears subject to the greatest uncertainty, although the estimates of g and q are linked ŽEq. Ž54.., so that uncertainty estimates are not independent. An important component in the equation for q ŽEq. Ž59.. is KK Y , which is sensitive to the precise form of the equation-of-state. But the lower mantle fits restrict K 0 K 0Y to a limited range about y6.0. Taking the results in Table 3 to indicate an uncertainty of "0.5, the corresponding uncertainty in q is "0.3. Applying this uncertainty to d S at 1900 km depth Ž r s 4471 km., where Robertson and Woodhouse Ž1996. reported evidence that d S ™ 1, we obtain d S s 1.9 " 0.3. Given that, at that depth, K X s 3.27 is known rather well, the condition d S s 1 would require g y q s 1.27. Since g is closely linked to K X ŽEq. Ž55.., the values in Table 4a,b give rather little scope for error, so that d S s 1 implies negative q, which is clearly implausible. The suggestion ŽSu and Dziewonski, 1997. that d S - 1 in the lowest mantle is even less plausible. These are now even more compelling objections to a simple thermal interpretation of lower mantle velocity anomalies than when difficulties were first pointed out ŽStacey, 1992, 1995.. The simple binary mix of minerals that I have used to model the lower mantle makes possible the quantitative conclusions but we may question the extent to which the properties of the real mantle are compromised. A better answer may be possible when we have improved data, especially for Ca perovskite, and for the effects of iron, but the equation-of-state calculations are really only influenced by the value of K 0X , to which the calculations are anchored. A change in K 0X by 0.1 varies the estimates of K 0 by approximately one of the standard deviations quoted in Eq. Ž51. and r 0 by less than half a standard

233

deviation. Similarly, the thermal properties in Tables 4 and 5 are changed only in detail and the essential conclusions are unaffected. In this paper, I have emphasized finite strain equations that are constrained to fit the parameter K`X s 1.425. Although for the lower mantle fits as presented in Table 3 this was evidently not necessary, I maintain that it is a useful component of a finite strain equation for the lower mantle because we have a very reliable value of K`X from Eq. Ž45.. It influences the estimates of KK Y and hence q and d S in the lower mantle even though K 0 and r 0 fits are not obviously affected. For this purpose we must disallow consideration of equations such as Eq. ŽA3. ŽBirch, fourth order. which fit the lower mantle K, r data well enough but give K`X higher than lower mantle values of K X . A post-Birch approach to finite strain is urgently needed. Eqs. Ž46. and Ž50. are presented as contributions to the necessary discussion.

Appendix A. Finite strain equations used in compiling Table 3 Entries Ži. and Žii. in Table 3 are discussed in Section 4. This appendix gives the basic details for the other nine entries, which are presented to provide comparisons and to give an indication of the robustness of the values of K 0 and r 0 calculated from Ži. and Žii.. Stacey et al. Ž1981. give further details of these nine and some others. Žiii. and Žviii.. Birch Ž1952. argued that an expansion of Helmholtz free energy, F, as a polynomial series in ´ s ywŽ rrr 0 . 2r3 y 1xr2 is strongly convergent, where ´ is referred to simply as ‘strain’ F s c2 ´ 2 q c3 ´ 3 q c4 ´ 4 q . . .

Ž A1.

The second order theory considers only the first term in Eq. ŽA1., the third order theory two terms and so on. The third and fourth order theories are commonly used in geophysics. Historically, the theory began with Love Ž1927. and was extended by Murnaghan Ž1951., but the modification applied by Birch Ž1952. made it essentially different from the Love–Murnaghan approach. However, Birch appealed to the authority of the earlier authors for the theoretical justification of his

F.D. Staceyr Physics of the Earth and Planetary Interiors 106 (1998) 219–236

234

definition of ‘strain’, that when presented in three dimensions it is invariant with respect to bodily rotations or changes in coordinate axes, although Love Ž1927. pointed out that in this respect the theory is not unique. The convergence argument has appeared relevant to geophysics because c 3rc2 s Ž4 y K 0X . and for many minerals K 0X f 4. However, for the lower mantle fit in Table 3, c 4rc2 is of order unity and the convergence is not intrinsic to the expansion of F as in Eq. ŽA1. but requires ´ < 1. Thus, the theory cannot be used for extrapolation beyond the range of data to which it is fitted. This is particularly obvious when K`X is calculated ŽTable 3.. The Birch expansion is formally equivalent to assuming an atomic potential function of the form 2

f s yA1 Ž arr . q A 2 Ž arr . 6

4

8

q A 3 Ž arr . q A 4 Ž arr . q . . .

Ž A2.

where the series stops at A 2 for the second order theory, A 3 for the third order theory and so on. Eq. ŽA2. has no theoretical basis. The Birch approach must be judged not by its pedigree but by the precision of its fit to observations, as for any other empirical relationship. It happens that the fourth order theory gives a good fit to the lower mantle, but the third order theory, a very poor one. Evidently the K`X extrapolation is too remote to affect the lower mantle r and K fits noticeably, although it is more significant if K X is considered. In numerical work on the Birch theory, I have found it convenient to represent P, K, etc., as power series in rrr 0 s x. For the fourth order theory Ps

9 16

This is more readily differentiated to obtain K, Žd Krd P ., etc., than expressions in terms of ´ . For the third order theory K 0 K 0Y s y35r9y Ž K 0X y 4 . Ž K 0X y 3 .

so that D s 0 and the other expressions in Eq. ŽA4. simplify. ŽFor the second order theory C s 0 also.. In fitting the fourth order theory to the lower mantle tabulation the volume-averaged lower mantle data point was assumed, as for Eq. Ž46.. This was not needed for the third order theory. Živ. The atomic potential function due to Morse Ž1929.

f s A exp 2 f Ž 1 y rra . y 2 exp f Ž 1 y rra .

K 0 Ž yx

Aqx

7r3

3

Byx Cqx

11r3

was developed in an early quantum-mechanical theory of molecular vibrations. It was applied to solidstate problems by Slater Ž1939.. r is the atomic separation and Ž rra. s Ž r 0rr .1r3 s xy1r3. Differentiation gives f s Ž K 0X y 1. and Ps

3K0

Ž K 0X y 1 .

A s K 0 K 0Y q Ž K 0X y 4 . Ž K 0X y 5 . q 59r9

¶

B s 3 K 0 K 0Y q Ž K 0X y 4 . Ž 3 K 0X y 13 . q 129r9

•

C s 3 K 0 K 0Y q Ž K 0X y 4 . Ž 3 K 0X y 11 . q 105r9 D s K 0 K 0Y q Ž K 0X y 4 . Ž K 0X y 3 . q 35r9

ß Ž A4.

Ž A7.

4

Žv. The Rydberg potential

f s A 1 y f Ž 1 y rra . exp f Ž 1 y rra .

Ž A8.

is an empirical modification of the Morse potential which Rydberg Ž1932. found gave a better fit to spectroscopic observations. In this case f s Ž3r2.Ž K 0X y 1. and P s 3 K 0 Ž x 2r3 y x 1r3 . 3 2

Ž K 0X y 1 . Ž 1 y xy1r3 .

Ž A9.

Žvi. The Brennan–Stacey equation K0

Ps

where

x 2r3  exp 2 Ž K 0X y 1 . Ž 1 y xy1r3 .

yexp Ž K 0X y 1 . Ž 1 y xy1r3 .

D.

Ž A3.

4 Ž A6.

=exp 5r3

Ž A5 .

ž

K 0X y

5 3

½ ž

= exp

x 4r3

/ K 0X y

5 3

/Ž

1 y xy1 . y 1

5

Ž A10.

follows from the free volume formulation of the thermodynamic Gruneisen parameter, g , with the ¨ common assumption g A ry1 ŽBrennan and Stacey,

F.D. Staceyr Physics of the Earth and Planetary Interiors 106 (1998) 219–236

1979.. However, as we now realise, both the free volume formula and the assumption gr s constant are flawed, so that Eq. ŽA10. is properly regarded as another empirical equation. Žvii. The Davydov potential ŽZharkov and Kalinin, 1971.

f s Ž Aarr y B . exp f Ž 1 y arr .

Ž A11.

is generically related to the Morse and Rydberg potentials. In this case fs

3 4

½

Ž K 0X y 3 . q Ž K 0X q 1 . K 0X y

ž

5 3

1r2

/ 5

as a representation of electrostatic attraction has generally been followed, as in the analysis leading to this entry in Table 3. Then K 0X s Ž n q 7.r3 and 3K0 X Ps Ž x K 0y4r3 y x 4r3 . Ž A17. X 3K0 y8 Žxi. The Meyer potential

f s yAarrq B exp Ž yfarr . Ž A18. combines the Madelung-style 1rr electrostatic attractive term with an exponential repulsion more compatible with quantum mechanics than the powerlaw repulsion of the Mie potential. f is uniquely but not simply related to K 0X

Ž A12. fs

and Ps

3 2

Ž K 0X y 1 . q

3 2

K 0X 2 y

14 3

K 0X q

19

1r2

3

Ž A19.

3K0

Ž f q 2.

and

= x 4r3 q fx y Ž f q 1 . x 2r3 exp f Ž 1 y xy1r3 . Ž A13.

Ps

3K0

Ž f y 2.

K X s Ž K 0X y K`X .

K0 K

q K`X

Ž A14.

is of interest as the first use of K`X in a finite strain equation. Keane Ž1954. recognised that K 0X ) K`X ) 1, but had no experimental or theoretical evidence that gave a tighter constraint on K`X and so it was simply a parameter to be fitted to compression data. Now that we have a direct method of finding K`X for the lower mantle, this was used in fitting Keane’s equation, which gives K 0X

X

x K` y 1. y X2 Ž

K`

ž

K 0X K`X

/

y 1 ln x

Ž A15.

The very poor fit indicates that Eq. ŽA14. is not a good representation. However, the spirit of Keane’s idea lives on in Eq. Ž50.. Žx. The Mie potential m

 x 2r3 exp

f Ž 1 y xy1r3 . y x 4r3 4

Ž A20.

Žix. Keane’s equation

PsK0

235

f s yA Ž arr . q B Ž arr .

n

Ž A16.

with nB s mA and usually with m s 1, has been widely used because of its mathematical simplicity. With m s 2, n s 4 this gives the second order Birch equation and other exponents have been suggested but the proposal by M. Born that m s 1 be favoured

References Anderson, O.L., Isaak, D.G., 1995. Elastic constants of mantle minerals at high temperature. In: Ahrens, T.J. ŽEd.., A handbook of Physical Constants: 2. Mineral Physics and Crystallography. American Geophysical Union, Washington, pp. 64– 97. Bass, J.D., 1995. Elasticity of minerals, glasses and melts. In: Ahrens, T.J. ŽEd.., A Handbook of Physical Constants: 2. Mineral Physics and Crystallography. American Geophysical Union, Washington, pp. 45–63. Birch, F., 1952. Elasticity and constitution of the earth’s interior. J. Geophys. Res. 57, 227–286. Boehler, R., Chopelas, A., 1991. A new approach to laser heating in high pressure mineral physics. Geophys. Res. Lett. 18, 1147–1150. Brennan, B.J., Stacey, F.D., 1979. A thermodynamically based equation of state for the lower mantle. J. Geophys. Res. 84, 5535–5539. Budiansky, B., 1970. Thermal and thermoelastic properties of isotropic composites. J. Composite Mater. 4, 286–295. Dziewonski, A.M., Anderson, D.L., 1981. Preliminary reference earth model. Phys. Earth Planet. Int. 25, 297–365. Dziewonski, A.M., Hales, A.L., Lapwood, E.R., 1975. Parametrically simple earth models consistent with geophysical data. Phys. Earth Planet. Int. 10, 12–48. Haddon, R.A.W., Bullen, K.E., 1969. An earth model incorporating free earth oscillation data. Phys. Earth Planet. Int. 2, 35–49.

236

F.D. Staceyr Physics of the Earth and Planetary Interiors 106 (1998) 219–236

Hashin, Z., Shtrikman, S., 1963. A variational approach to the elastic behaviour of multiphase materials. J. Mech. Phys. Solids 11, 127–140. Hill, R., 1952. The elastic behaviour of a crystalline aggregate. Proc. Phys. Soc. A 65, 349–354. Jackson, I., Niesler, H., 1982. The elasticity of periclase to 3 GPa and some geophysical implications. In: Akimoto, S., Manghnani, M. ŽEds.., High Pressure Research in Geophysics. Center for Academic Publications, Tokyo, pp. 93–113. Jackson, I., Rigden, S.M., 1996. Analysis of P-V–T data: constraints on the thermoelastic properties of high pressure minerals. Phys. Earth Planet. Int. 96, 85–112. Keane, A., 1954. An investigation of finite strain in an isotropic material subjected to hydrostatic pressure and its seismological applications. Aust. J. Phys. 7, 323–333. Kennett, B.L.N., Engdahl, E.R., Buland, R., 1995. Constraints on seismic velocities in the earth from travel times. Geophys. J. Int. 122, 108–124. Kumazawa, M., 1969. The elastic constant of polycrystalline rocks and non-elastic behavior in them. J. Geophys. Res. 74, 5311–5320. Love, A.E.H., 1927. A Treatise on the Mathematical Theory of Elasticity, 4th edn. Cambridge Univ. Press, Cambridge, 643 pp. Miller, M.N., 1969. Bounds for effective bulk modulus of heterogenous materials. J. Math. Phys. 10, 2005–2013. Montagner, J.-P., Kennett, B.L.N., 1996. How to reconcile body wave and normal mode reference earth models. Geophys. J. Int. 125, 229–248. Morse, P.M., 1929. Diatomic molecules according to the wave mechanics: II. vibrational levels. Phys. Rev. 34, 57–64. Murnaghan, F.D., 1951. Finite Deformation of an Elastic Solid. Wiley, New York, 140 pp. Robertson, G.S., Woodhouse, J.H., 1996a. Ratio of relative S to P velocity heterogeneity in the lower mantle. J. Geophys. Res. 101, 20041–20052.

Robertson, G.S., Woodhouse, J.H., 1996b. Constraints on lower mantle physical properties from seismology and mineral physics. Earth Planet. Sci. Lett. 143, 197–205. Rydberg, R., 1932. Graphische darstellung einiger bandspektroskopische ergebnisse. Z. Phys. 73, 376–385. Slater, J.C., 1939. Introduction to Chemical Physics. McGraw-Hill, New York, 521 pp. Stacey, F.D., 1992. Physics of the Earth, 3rd edn. Brookfield Press, Brisbane, 513 pp. Stacey, F.D., 1995. Theory of thermal and elastic properties of the lower mantle and core. Phys. Earth Planet. Int. 89, 219–245. Stacey, F.D., 1996. Thermoelasticity of ŽMg,Fe.SiO 3 perovskite and a comparison with the lower mantle. Phys. Earth Planet. Int. 98, 65–77. Stacey, F.D., 1997. Bullen’s seismological homogeneity parameter, h , applied to a mixture of minerals: the case of the lower mantle. Phys. Earth Planet. Int. 99, 189–193. Stacey, F.D., Brennan, B.J., Irvine, R.D., 1981. Finite strain theories and comparisons with seismological data. Geophys. Surv. 4, 189–232. Su, W.-J., Dziewonski, A.M., 1997. Simultaneous inversion for 3-D variations in shear and bulk velocity in the mantle. Phys. Earth Planet. Int. 100, 135–156. Suzuki, I., 1975. Thermal expansion of periclase and olivine and their anharmonic properties. J. Phys. Earth 23, 145–159. Thomsen, L., 1972. Elasticity of polycrystals and rocks. J. Geophys. Res. 77, 315–327. Watt, G.P., Davies, G.F., O’Connell, R.J., 1976. The elastic properties of composite materials. Rev. Geophys. Space Phys. 14, 541–563. Yeganeh-Haeri, A., 1994. Synthesis and re-investigation of the elastic properties of single crystal magnesium silicate perovskite. Phys. Earth Plant. Int. 87, 111–121. Zharkov, V.N., Kalinin, V.A., 1971. Equations of State for Solids at High Temperatures and Pressures ŽTranslated from Russian.. Consultants Bureau, New York, 257 pp.