Physicsof theEarth andPlanetaryInteriors, 80 (1993) 37—48 37Elsevier SciencePublishersB.V., Amsterdam

The mixed F, T derivativesof elasticmoduli and implicationson extrapolatingthroughoutEarth’smantle

DonaldG. Isaak *

Centerfor ChemistryandPhysicsoftheEarth and Planets, Instituteof Geophysicsand PlanetaryPhysics,Unii’ersity of (‘uhfornia,Los Angeles,(‘A, USA

(Received17 November1992; revisionaccepted29 April 1993)

ABSTRACT

Compositionalprofilesof Earth’smantlerequireaccurateextrapolationof elasticity to simultaneoushigh pressureP andtemperatureT, usingthe first P and T derivativesof elasticity measurednearambientconditions.We presentthermody-namic relationsthat elucidateo

2KT/oPaT for mineralsfor which dataon thevolume dependenceof (~T/4P)~are available(KT is the isothermalbulk modulus and S indicatesconstantentropy). We find that ~2KT/aPaT is positive for almost allsolids and hasvaluesof (0.39±0.10, 0.33 ±0.09)X 10 K ‘for MgO and olivine, respectively.We suggestthat neglectof~2KT/~P~Tcouldcausetheolivine contentof the uppermantleto beoverestimatedby 3% at most,if second-orderpressureeffects are not considered.This result is basedon a comparisonof the longitudinal velocitiesof the a and ~ phasesofolivine at 400 km P. T conditions.We show that theisothermal~2KT/)P~T decreaseswith increasingP. Ab initio modelcalculationson MgO are consistentwith the results from thermodynamics.We show for MgO the theoreticalvaluesof(aKT/aP)T and(aG/8P)T, whereG is the isotropic shearmodulus,along a geotherm.

1. Introduction phases(Weidner, 1987; Bina and Wood, 1987;D.L. Anderson,1988; Duffy and DL. Anderson,

A correctunderstandingof the chemistryand 1989; Gwanmesiaet al., 1990; Isaak, 1992). Thestructureof the Earth’sinterior requiresaccurate first P and T derivativesat ambientconditions,information on the elasticpropertiesof minerals togetherwith an assumedequationof state,areovera rangeof simultaneouslyelevatedpressure usedto extrapolateP, T conditionsto a depthofP andtemperatureT. Comparisonsof laboratory 400 km (D.L. Andersonand Bass, 1984;Bassandelasticity datawith seismologicalobservationsare D.L. Anderson,1984; Weidner, 1986;Duffy andpossibleafterthe mineral datahavebeenextrap- D.L. Anderson,1989;Gwanmesiaet al., 1990).olated to the high P, T conditionsof the Earth. The importanceof higher-orderP, T termsSince the seismic wave discontinuities at about when extrapolating elastic properties measured400 km depthmay be usedto constrainthe olivine near ambientconditions to P. T conditionsap-contentof the uppermantle,accurateextrapola- propriateto Earth’smantle is not known. Experi-tions to P, T conditionsappropriateto this depth mental techniquesusingdiamond anvil cells andareextremelyimportant. shock waves (Williams and Jeanloz, 1991) have

Considerableeffort has been exerted to find improvedconsiderablyour understandingof manythe P and T derivativesof the isotropic bulk K properties of minerals at high T and high P.andshearG moduli of the likely mantlemineral However,this typeof datadoesnot providesuffi-

cient accuracyon the isothermal bulk modulus* Also atAzusa Pacific University, Departmentof Math and K

1 and its pressurederivative(~K~/aP)~simul-Physics,901 E. Alosta Ave., Azusa, CA, USA. taneously.Duffy and Ahrcns (1992) utilized the

0031-920l/93/$06.00© 1993 — Elsevier SciencePublishersB.V. All rights reserved

38 0.0. ISAAK

large F, T rangeaccessiblein shock studiesto We obtain (1) from the definition of the P, Tfind the mixed F, T dependenceof wave veloci- derivativeties for Mg2Si04. Theywereableto placeboundson this higher-ordereffect, but were limited in ~. =d

2KT [a(aK2/aT)~1

their interpretationby relatively large uncertain- ~T~P ~Pties and by the lack of information on the P

I ~ \1a(aK7jaT)~1dependenceof (aK5/iE)T)~and (aG/i~iT)~,where = — I — I (4)K~.is the adiabatic bulk modulusand G is the \ K~jL ~ iTisotropic shearmodulus.Other techniques,such From (2) we note that —aK3~T canbe identifiedas pulse ultrasoundor Brillouin scattering,have with (aK~/aT)~in (4), so thatprovided accurateelasticity data, but thesedataare restricted to low simultaneousP, T condi- a

2KT / ~ I a(~K~7)1

tions. Attemptsdirectly to measurethe mixed P, ~j~p = (5)T derivativesof elasticity of oxides and silicatesby ultrasoundexperimentshave resultedin large Differentiation by s~in (5), followed by factoringuncertainties(lsaakand Graham,1976). out commonterms,leadsto the threetermsof (1)

Our purposeis to understandthe higher-order enclosedin parentheses,when we also use theP. T dependenceof the pressurederivativesof identityelasticity. The bulk modulus is our focus, al- 6~=(d ln a/a In ~ (6)thoughsomeresultsalso pertain to G. AlthoughK7 and K5 are distinct in value, there is littledifference (2% or less) in (aKT/aP)T and(aK5/aP)1, and we representeither by K’. We 3. The mixed derivative a

2KT/8PaT at ambientrecast the problem of the mixed F, T derivative conditionsof K

7 by presentingrelationships from which~

2K3/~P~T is obtainedfrom measuredquantities Important information regarding a

2KT/aPaTother than a2KT/aPaT itself. is obtained by analyzing(1). First we note that

a2K1./aPaTis almostalwayspositive. This follows

from the fact that each parameterin (1) is apositive quantity. Thus, only if ~7—K’ is both

2. Thermodynamic relations negativeand has magnitudegreaterthan K cana

2KT/aPaT be negative. However, at ambientWe find a thermodynamicrelation conditions 6~ is usually larger than K’ (O.L.

a2KT/aTap=a6l(~T—K+K) (1) Andersonet al., 1992a). D.L. Anderson(1988)(his table I) tabulateddimensionlessparameters

that relatesthe mixed P and T derivativeof KT for 54 minerals at ambient conditions. We canfor a solid to the volume coefficient of thermal takeexpansiona andto the dimensionlessparameters Kç I aK

5\6T’ K’, and K. The dimensionlesstermsin (1) are (Ks}T K1. a~iT

definedby

= — ( 1/aK7)(aK3./dT)~ (2) theparametertabulatedby D.L. Anderson(1988),as representativeof K’ to within 2% or so. Thus,

and we find K’ <

5T (where 8T~{KT}P in the nota-

tion of D.L. Anderson)for all buteight of the 54K=(d In 6T/a In s~)T (3)

mtnerals. Of these eight, ~T — K’> —0.60 forIn the above expressioni~t is the ratio of the five, and ~T — K’ —2.0 for the other three.volume V to the volume l/

0at ambientpressure, Since K � 1.0 for all solids studied to date (seei.e. s~= V/Va. discussionbelow), we conclude that a

2K1./aP8T

PHYSICS OF THE EARTH AND PLANETARY INTERIORS 39

is almost certainlypositive at ambientconditions. (1992) gave m values for several materials.TheThree known exceptions are orthopyroxene, valuestabulatedby CB (1992)are from theorigi-TICdF3, and quartz. nal compression data of Boehler (1982) (A.

To use(1) to makequantitative assessmentsof Chopelas, personal communication, 1992). CBa

2KT/aPaT,valuesof K are required. Chopelas (1992) suggestedthat ln(aT/aP)5 varies linearly

andBoehler (CB) (1992)showedthat with V, rather than with In V as representedby

V a In C~ Boehler (1982). When fitting the compressional~m + — 1 (7) data, this difference in the assumedform ac-alnV T

counts for the differencesbetweenvalues of mwhere Ci,, is the constantpressureheat capacity and n (approximately10% or less) in the resultsand m is definedby of CB (1992)and Boehler(1982).

a ln(aT/aP) In Table 1 we show K for severalsolid corn-m = __________‘ (8) pounds calculated from (8) and (9) using the

T compressiondatagivenby CB (1992).WeincludeFrom (7) we have(a~T/ar,)l= m and NaF and quartz in Table 1 by assumingthe n

a In 1ST m values for m (Boehler, 1982). For NaCI weK = = — (9) smoothedthe K1(T) data of Yamarnotoet al.

aln~ ~ 8~. .

(1987) using a second-orderpolynomial fit in Tprovidedthat (a In Ce/aIn V)T is takenas mdc- before calculating 1St, rather than use thependentof V (O.L. Andersonet al., 1993). CB Yamarnotoet al. valueof 5~= 5.24 at room tern-

TABLE 1

CalculatedI

2KT/IPaT at ambient P, T for severalcompounds

(iO~K’ bT K’ d ‘~T/~] K I2KT/~)P0TPresentstudy Experiment/theory(i0-~K’) (l0-~K’)

NaCI 118 5.67 ~ 5.3 6.9 1.22(0.28) 1.1(0.4) 1.4 g 1.6NaF 98 5.80 a 5.0 a 9.5 1.64(0.38) 1.4(0.3) —

KBr 116 5.64 5.1 6.4 1.13(0.26) 1.1(0.3) —

RbCI 119 5.81 5.4 6.5 1.10(0.26) 1.0(0.3) —

MgO 31 5.26 ‘~ 4.2 7.0(1.6) 1.33(0.44) 0.39(0.10) 1.4 .0—0.5 ~, 0.3Olivine 27 5.94 C 5.2 7.8(0.7) 1.31(0.32) 0.33(0.09) —

Quartz 35 3.28 6.4 7.0 2.13(0.50) —0.11(0.07) —

Garnet — — — — — — 0.9(0.6)Perovskite — — — — — — —

MgSiO3(orthorhombic) — — — — — — 0.8

CaSiO3(cubic) — — — — — — 0.6

D.L. Anderson(1988).‘ Yamamotoetal. (1987). using smoothedK~(T)data.

O.L. Andersonet al. (1992a).Suminoand Anderson(1984), using averageof data provided.ChopelasandBoehler (1992), using m values(NaF andquartzuse n dataof Boehler, 1982).

Errorspropagatedfrom: a(±5%);~T(±6%); K’(±5%);thoseshownfor K.

Bands and Schuele(1965).Spetzleret al. (1972).O.L. Andersonand Andreatch(1966).Spetzler(1970).

k Isaaket al. (1990),electrongas (PIB) model.IsaakandGraham(1976).Wolf andBukowinski (1987), modified electrongasmodel (fourth-orderfit using their fig. 18 values).

40 0.0. ISAAK

perature.Their tabulatedvalue for 1ST is lower until such data are available. We consider thethan their unsmoothedK

7(T) databetween296 four sourcesof error discussedaboveto be mdc-and 338 K seemto allow, and is also lower than pendent and propagatethem accordingly (seethe 1S~valuethey apparentlyprefer(seetheir fig. Taylor, 1982,pp. 56—57).6). When 1S~is recalculatedin this way we find There are limited experimentaldataon K with

= 5.67 at 300 K and in good agreementwith which to compare our values in Table 1. Birch= 5.7 at ambient P, T conditions from the (1986)combinedultrasonic(Spetzleret al., 1972),

Spetzleret al. (1972)data. shock (Fritz et al., 1971), and piston cylinderThe errors indicatedfor K in Table 1 include volumetric (Boehlerand Kennedy,1980) datato

thosepropagatedfrom the original Boehler(1982) obtain an equationof state for NaC1 constraineddatafor n (see his table 2) andby assuminga 6% by all the availablecompressiondatathroughoutuncertainty in

1ST (Isaak et al., 1989). Boehier F, T space (25—500°C,0—30 GPa). The Birch(1982) provided experimentaluncertaintieson n (1986) results(see his tables6 and 10 for 1S~(~~))only for MgO and olivine. CB (1992) noted that imply that K 1.45 between25 and 500°C(O.L.the large uncertaintiesin m (or n), and therefore Andersonand Isaak, 1993) and is higher thanin K, for MgO and olivine result from small K = 1.22±0.28 that we obtainedin (9) using thetemperaturechangesin these materialsduring m (or n) and 1S~ data from CB (1992) andadiabaticcompression. Yamamotoet al. (1987), respectively.Thereasons

We mustconsidertwo other sourcesof uncer- for thesesmall differencesin K are unclear.Thetainty which are also included in the errors listed 6~(i~)found by Birch (1986) is constrainedby thefor K in Table 1. First we note that the form of shock data of Fritz et al. (1971), which go tothe equation used to fit the ln(aT/aF)

5 data pressuresbeyond30 GPa.The work of CB (1992)(Boehler, 1982; CB, 1992) can produce differ- is basedon the Boehler (1981) measurements.encesup to about 10%. Second,we emphasize Boehler (1981) used a maximum pressureof 5that Eq. (9) follows from (7) only if (~In C1~/ GPa.It seemsunlikely that 6T(s~)decreasesmorea In V).,. is independentof volume. CB (1992) rapidly with increasing P at pressureshighernoted that (~In C,,/a in V)T~I below the De- than 5 GPa. However, if this were the case thebye temperatureTD and is zero above T0. It is value of K derivedfrom the CB (1992) report isseenfrom vibrational spectroscopythat for sev- more representativeof K near ambient F, Teral mineralsthe changein (a ln Ce/a ln V)T is conditions.We also note that 1ST(s~)is an isother-smallif pressure(or volume)change(A. Chopelas, mal quantity; assumptionsare requiredin reduc-personalcommunication,1992). Although results ing the shock data included in the Birch (1986)from theoreticalstudieson MgO (O.L. Anderson compilationof 1S~(~~)to isothermalconditions.et al., 1993) also indicate that at high T the For MgO we find K = 1.33±0.44, which agreesapproximationthat (~ln Ce/aIn V)T is indepen- well with the 300 K valueof K = 1.4 givenby O.L.dentof V is very good,at ambientT the valueof Andersonand Isaak (1993). The results of O.L.K approximatedby (9) may be up to 20% too Andersonand Isaak(1993)arebasedon theoreti-high. A 20% increasein K correspondsto an cal calculations(electrongasmodel) (Isaaket al.,increaseof less than 10% in a

2KT/aPaT. Al- 1990). For the materials shows in Table 1, K

though it is difficult to establish preciselyhow generallylies within 1.1—1.6. Thereis not a largeclose (a In Ce/aIn V)T is to being independent variation in K from onematerialto the next.Oneof V at ambient temperaturefor minerals we exception is quartz, for which K > 2. Quartz isconsiderin Table 1, we assumeit results in an highly covalentwith a fourfold coordinationcon-uncertainty of 20% in K. It is clear that this figuration. Severalphysical propertiesof quartzuncertaintyillustratesthe needfor betterexperi- differ from trendsfound with many other miner-mentalunderstandingof the higher-orderdepen- als (D.L. Anderson, 1988). Although results fordenceof the heatcapacityon volume.Our stated quartz are included in Table 1, we emphasizeuncertainty in K must be consideredtentative that quartz is an exceptionto many of the ten-

PHYSICS OF THE EARTH AND PI.ANETARY INTERIORS 41

denciesfound in the propertiesof the otherma- (aKT/aT)J, at F = 0 and 8 kbar, whereastheterials in Table I. lower limit of zero is set from the observation

Values of K’ are also required to find that (aK7/aP)7 is unchangedat P = 0 as T goes1S

2KT/aPaTusing (I). We calculate a2K1./aPaT from 300 to 800 K. Isaak et al. (1990) reported

using the average of the K’ values (Table 1) a2KT/aFaT=0.27 x i0~ K~ for MgO directly

found in SuminoandAnderson(SA) (1984)when from their theoretical Potential InducedBreath-K’ is givenby SA (1984).The K’ valuesfor NaF ing (PIB) calculations.(PIB is a type of electron-and KBr are taken from the compilationof D.L. gasmodel. Isaaket al. (1990)usedquasiharmonicAnderson(1988). The precisionof a given mea- lattice dynamicsto account for the temperaturesurementfor K’ is usuallymuch betterthan the dependenceof the thermoelastic properties.)accuracyfound from comparingresults from dif- Their value for a2KT./aFaT is lower than foundferent laboratories.The uncertaintiesin K’ (Ta- here using (1), but is close to the stated uncer-ble 1) reflect these interlaboratory differences. tainties. Taken togetherwith the measurementsForNaCI, MgO, olivine, andquartz,all materials of Spetzler(1970) and the PIB calculations,ourfor which several measurementshave been re- results indicate that ~2K.J./~P~T for MgO isportedfor K’, we find K’ is contrainedto within nearer0.4 x 10~K’, rather than 1.4 X iü~about 5%. We take the K’ valuesfor the other K~’, as reported by O.L. Anderson and An-materialsin Table I to be similarly constrained. dreatch(1966).

We calculate ~2K1/~P~T from Eq. (1) for the We find that for MgO, olivine, and quartz at

materialslisted in Table 1 andinclude the results ambient F, T the a2K

1./aPaT valuescalculatedin Table 1. The estimatederrors in a,

6~,K’, from (1) are significantly lower (by more than aand K arepropagatedto a2KT/aPaT(see Taylor, factor of two) in magnitudethan those for the1982, pp. 56—57).There are data on a2KT/aPaT alkalide halides. The value of a varies mostfor NaCI and MgO with which we can compare among all the parametersin Table 1 and controlsour results. We find agreementwithin the cx- the variationof a2KT/aPaT from onematerial topresseduncertainty in the a2KT,/aPaT=(1.1± the next. We expect, therefore, that for other0.4) x i03 K I for NaCI using (1), when corn- mantle mineralsfor which a is significantly lesspared with experimentalvalues of (1.4±0.2)X than 100 X l0~’ K ‘, ~2K

1/~P~T is likely to heiO~K~(Bartelsand Schuele,1965) and(1.6± smaller in magnitudethan it is for the alkalide0.2) x i0~ K I (Spetzler et al., 1972). These halides.We suggestthat a valueof a

2KT/aPaTexperimental mixed derivatives are found from 0.4 x l0~ K’ is more representativeof materi-the differencesin the measuredK’ at two tern- als of geophysicalinterest than is 1.0 x l0~K I

peraturesnearambient F, T. The SpetzlerCt al. as recommendedby Isaak and Graham (1976)(1972)datashow little differencein a2KT/aPaTif based on their result of 0.9 ±0.6x 10~ K I

computedusingthe K’ valuesat 300 and 800 K, from garnet measurements.When uncertaintiesrather than using K’ at 300 and 550 K. The in thegarnetdatahavebeenaccountedfor, thereexperimentaluncertaintiesin a2KT/aPaTfor the is no conflict with our lower estimate forSpetzleret al. data are estimatedby assuming a2KT/aPaT.Our results imply that the 20% in-±0.07 in the reportedK’ (seetable6 of Spetzler creasein K’ over temperaturechangesof ~T=et al., 1972) at 300 and800 K. 1000 K for mantle minerals suggestedby Isaak

The a2KT/aPaTcalculatedby (1) for MgO is andGraham(1976) shouldbe reviseddownward(0.39±0.10)>< i0~ K I; this value falls within to less than 10%. More dataon a2KT/aFaTdi-the experimentalrangefound by Spetzler(1970), rectly, or on the parametersnecessaryto calcu-a2KT/aTaP=0.0—0.5X i0~ K1, but is consid- late a2K

1./aPaTusing(1), are requiredto verifyerably lower than 1.4X iO~ K’ reported by the validity of our suggestionthat a

2K.1./aPaT~

O.L. AndersonandAndreatch(1966).The upper 0.4x i0~ K- for most mantle minerals.limit of 0.5 x l0~ K’ from the Spetzler(1970) In contrast to the preceding discussion aredatais obtainedby comparingthe 300 K valueof resultsof theoreticalcalculations(modified dcc-

42 DO. ISAAK

tron gasmodel) (Wolf andBukowinski, 1987)that with temperature,and they estimatedthe magni-indicate a2K

1./aFaT for perovskiteMgSiO3 (or- tude of a2K.

1./aPaT by the approximationthorhombic) and CaSiO3 (cubic) are 2 and 1.5 a In K’times larger, respectively,than 0.4 x IO~ K’ — 1 (10)Mao et al. (1991) found that

1ST for MgSiO3 d In j

pervoskite is 6.5—7.5 at high T, and suggested suggestedfrom their analysis of Earth modelthat it maybe as high as 13.9 at ambientT. Thus PREM (Dziewonski andAnderson,1981)param-for MgSiO3 perovskite, 1S~ at 300 K is signifi- etersfor the lower mantle. In Eq. (10), p is thecantly larger than any of the other materials in density. Our resultsare in qualitative agreementTable 1. From (1) we expect that a

2K.1./aPaTfor with those of Duffy and Anderson (1989), i.e.

perovskitewill also be larger than it is for other ~2K.

1./~F~Tcan generallyhe takenas a positivematerials with similar values for a. Reported quantity, with quartz being an exception. Wevaluesof a measurednear roomtemperaturefor find, however,that (1) typically gives largervaluesMgSiO3 perovskite are 40 X i0~ K’ (Knittle for a

2KT/apaT and is closer to the availableand Jeanloz, 1986), 22 x 10~K~ (Ross and experimental values than (10). For instance,Hazen, 1989), and 15 x 10~K1 (Wang Ct al., a2K~/apaTfor NaC1 calculatedby (10) is about1991). The difficulty in specifying 1S~ accurately 0.4X i0~ K1, whereasour value is 1.1 )< I0~at 300 K (Mao et al., 1991) and the lack of data K~~. Similarly a2KT/aFaT for both MgO andpertainingto K for perovskite,precludeus from olivine averagedover 300—1800 K is 0.2x i0~quantitatively estimating a2K.

1./aFaT. We note, K I using (10), rather than near 0.4 x iO~andhowever, that the high value of

1ST’ rather than 0.3 x i0~ K1, respectively,found using(1). Thethat of a, may cause a2KT/aFaT for MgSiO

3 differencesin K’ at high T for MgO andolivineperovskiteto be more than twice as large as it is when using(10) and(1) are small enoughto befor olivinc andMgO. within the uncertaintyof the ambient K’, and

The compositionalmodel proposedby Duffy our results can be considered support for theand Anderson(1989) requires that K’ increase approachof Duffy and Anderson(1989).

(a) (b)

160 m~denvat~e(md) ~,nsla,it 60 ~ constani~

2500K~ ~ __ 50 2500K V ~ - -

120 15O0K~1~I ~ 30 /5O0K~/~~

~ W _______0 20 40 60 80 100 0 20 40 60 80 100Pressure (GPa) Pressure (GPa)

Fig. I. ~K dueto non-ZeroI2KT/2PIT at elevatedP, T conditions. Solid lines showconstantvalue of I2KT/2PIT. Dashedlines

show the easefor ~2KT/1P1T decreasinggraduallywith increasingP. so that at P = 100 GPa it is one-third of its ambientvalue.(a) 2KT/IPIT = 1.0 < l0~ K ‘ (alkalidehalides);(b) I2KT/IPIT = ((.33 X 10 ~ K ‘ (olivine).

PHYSICS OF THE EARTH AND PLANETARY INTERIORS 43

In Fig. I we illustrate the ~KT betweenwhen differencein longitudinal velocity V0 increasesbya

2K7./aPaT is assumedzero and then non-zero about 0.03 m s~ between these two phases.

whencomputing KT from F, T derivativedataat Since0.03 m s~is 3% of the 0.86 m 51 differ-ambientconditions.We considerthe casesof the ence in V~,betweenthe a and /3 phasesat 400non-zero a

2K7./aFaT=1.0 x i0~ K’ (Fig. Ia) km, neglectof the mixed P, T derivativefor K~

and 0.33 X i0~ K1 (Fig. ib). Figure lb shows may causethe olivine contentto be overestimated

that when extrapolating olivine (a2KT/aPaT= by about 3%. This difference is small compared0.33 X i0~ K~) to F, T conditions at 400 km with other uncertainties.For instance,Duffy and(13.5 GPa, 1800 K), neglect of a2KT/aPaT im- Anderson(1989)concludedthat the discontinuityplies that KT is underestimatedby about 6 GPa. in the shearwave J’~at 400 km supportsan upper

To use the discontinuitiesin seismicvelocities mantlewith 35—40% olivine, whereasthe dis-at 400 km to constrainthe olivine contentof the continuity indicates40—53% olivine. Furthermoremantle, the differencesin elastic propertieshe- uncertaintiesin the isotropic shearmoduli G oftween the a and /3 phasesof olivine must be the a and /3 phasesof olivine at 400 km condi-known at 400 km F, T conditions(see,for exam- tions may overshadowthe uncertaintiesin K. Inplc, Bina andWood, 1987, D.L. Anderson,1988, particular, information on (aG/aT)

1~of the /3Duffy andAnderson,1989,andGwanmesiaet al., phase, and a

2G/aPaT for both the a and /31990). We estimate here whether including phasesof olivine are required to improve esti-a2KT/aPaTcansignificantly effect the mineralog- mates of V~at 400 km for these two phases.ical modelsthatare basedon the V

1, discontinuity However, we conclude that by neglectingat 400 km. Data on /3 olivine at ambient condi- a

2KT/aPaT whenextrapolatingto 400 km condi-tions are available for all the parametersother tions the mineralogicalmodelsoverestimateonlythan K required to calculate a2KT/aPaT using marginally (at most about 3%) the amount of(1). For /3 phaseolivine at ambientconditionswe olivine requiredto satisfy the discontinuityin V

0.have

8T = 6.7 ±0.8 (Fei et al., 1992), a = 20.6 X106 K~ (Suzuki et al., 1980), and K’ = 4.8(Gwanmesiaet al., 1990). We estimate K to be ~. a2KT/aPaT at high pressure1.3 ±0.2 by taking the averageof the Table Ivalues(quartz excluded).We haveno reasonto Another important result found from analyz-believe that K for the 13 phaseof olivine deviates ing (1) relateschangesin a2KT/aFaT to P. Thesignificantly from valuesfound for other miner- dimensionlessparameterq definedasals; thereis remarkablysmallvariation in K fromone mineral to the next. Thus, we find — d In ya2KT/aPaT~o.44x i0~K’ for the /3 phaseof q = a In ~ (11)olivine using(1).

The approximateerror in estimating K1 — is generally takento be near unity for high pres-

KT at 400 km P, T conditionsdueto neglectof sure shock experiments(McQueenet al., 1970;~

2K.J./~P~Tfor bothphasesis about2.2 GPa, i.e. Duffy and Ahrens, 1992). Since q can also he[(0.44—0.33)x i0~ K1]~P~T,where ~P= identified as (O.L. Anderson et al., 1993)13.5 GPa and ~T= 1500 K. A similar difference a In C,~,in the K

5 values at the 400 km discontinuity q = — K’ + 1 a In V . (12)would be expectedowing to neglectof ~

2K5/~P~T

for both the a and /3 phasesof olivine. Thus where C,~,is the constantvolume heatcapacity,aK54 — K5,~(about 38 GPa; Anderson, 1988) at valueof q nearunity overa wide P rangeimplies400 km P, T conditionsis also underestimatedby that 1S~— K’ (~In C~/aIn V)T. The derivativeabout 2.2 GPa or 6% when only first P and T (a In Cv/a In V)T is not sensitive to F (O.L.derivatives are used to extrapolate.Increasing Andersonet al., 1993), which implies that ~ —

K54 — K5,, by 6% (or 2.2 GPa) implies that the K’ + K in (1) is also not sensitiveto P.

44 1)0. ISAAK

The term in parenthesesof (1) is not signifi- 5.5

cantly affectedby P. The markeddecreaseof awith increasingF (Chopelasand Boehler, 1989; 5.0 2500K MgO isot

O.L. Anderson et al., 1990) controls the pressure 2000K

dependenceof a2KT,/aPaT along an isotherm. 4 5 1500K

The relationshipbetweena and densityp (or ~ -

originally describedby O.L. Anderson(1967) is ~ 4 0given by (6). Equation (6) demonstratesthat 6~controls the rate of changeof a with ~ (or p). -

Along an isotherm for the caseof MgO we havea(P = 100 GPa) ~a(P = 0) (CB 1992; O.L. An-dersonet al. 1992a,b; Duffy and Ahrens, 1993). 3.0 I I I I

0 20 40 60 80 100Thus over this samepressurerangea2KT/aFaT Pressure (GPa)must decreaseto about one-thirdof its value atambientconditions,and thevaluesof a2KT/aFar Fig. 2. Pressuredependenceof isotherms of K’ for MgO

calculatedwith thePIB model. The convergenceof isothermsat ambient F, T shouldhe takenas upperlimits at high P illustrateshow K’ decreaseswith increasedP.

in P space.Although it is prematurequalitativelyto assessthe pressureeffects on a2K.~/aFaTup The T dependenceof isobarsof K’ and G’to 400 km for the a and 1~phasesof olivine, the from the FIB model calculationsis shownin Figs.tendencyof a2KT/aPaT to decreasewith a im- 4 and 5. The geotherms(Stacey,1977; O.L. An-plies that the effects of neglecting a2KT/aPaT derson,1981; DziewonskiandAnderson,1981)inwhen extrapolatingto 400 km F, T conditionsis Figs. 4 and 5 show how K’ and G’ deviatefromeven less than discussedin the previoussection. their ambient valuesas a geothermis followed.For F, T conditionspertinent to the lower man- For MgO, using the ambient K’ and G’ to cx-tie, a2K

1/aFaT takeson diminishing importance trapolateto 400 km F, T conditionsmeansthein relationto thevalue of K’. average K~-and G’ values will be underesti-

mated by about 5 and 10%, respectively.Thus,

5. Examples of (aKT/aP)T and (aG/aP)T at valuesof KT and G at 400 km conditionswill beelevated P and T from MgO model calcula- underestimated by about 1.5% (approximately3

GPa) and 2.5% (approximately 3 GPa), respec-tionsThe ab initio model calculationsof Isaaket al.

(1990) for MgO illustrate the tendency for 4 2500K

a2K

7/aFaTto decreasewith increasingP (shown2000K ~

in Fig. 2). The K’ values plotted in Fig. 2 are1500Kobtained from a fourth-order Birch fit of the

original free energyversusvolume calculations.3~1ooo:I

The distance between isothermsin Fig. 2 is ameasureof the averagea

2KT/aFaT andis seentobe about one-third as great at 100 GPa as atF=0.

A similar tendency is seen in the pressureI I I I I I I Idependence of a2G/aFaT, where G is the 0 20 40 60 80 100

isotropic shearmodulus(Hashim—Shtrikmanav- Pressure (GPa)erage). In Fig. 3 we plot isothermsof G’ from the Fig. 3. Pressuredependenceof isotherms of (1’ for MgOFIB calculations(Isaak et al., 1990),andfind that calculatedwith the PIB model. The markedconvergenceof

a2G/aPaTdecreasesevenmore rapidly with in- isotherms at high P shows that G’ decreasesmore rapidly

creasingP than doesa2K7./aFaT. than doesK’ with increasedP.

PHYSICS OF THE EARTH AND PLANETARY INTERIORS 45

5.0Ge~therms

Stacey [1977]cOntinent 0 GPaStacey [1977]ocean

35GPa

3.0 I I

0 1000 2000 3000

Temperature (K)Fig. 4. Temperaturedependenceof isobarsof K’ for MgO calculatedwith the PIB model. Geothermsillustrate that K’ isapproximatelyconstantthroughoutupper mantleP. T conditions,i.e. P � 13.5 GPaand T � 1800 K.

tively, if higher order than linear terms are ne- a2/aPaTor a2/aF2 is usedwithout the other, It isglected.Since even at ambient F, T conditions unclear whether the predominantupper mantleK’ is typically not known with an accuracy cx- mineral phaseswill behavesimilarly. If they do,ceeding5%, neglectof higher-orderP andmixed accurateextrapolationof data from ambient toF, T termswhen extrapolatingK.

1 is within pre- 400 km P, T conditionscan he achievedusingsent experimental uncertaintiesin K’. We em- terms linear in P and T only. Therefore, ourphasize that Figs. 2—5 include effects due to earlier results for olivine at 400 km conditionsa

2/aPaTand a2/aF2, and that these two deriva- basedonly on the a2K1,/aPaTterm musthe con-

tives tend to cancel each other in the upper sidered tentative until other higher-orderpres-mantle.Errorsin extrapolatingthe elasticproper- surederivativesaremore accuratelyknown.tics of MgO to depthsof 400 km are likely to be Finally, we observefrom Figs. 4 and 5 that forintroduced, rather than minimized, if either F> 15 GPa, both K’ and G’ for MgO decrease

with increasing F along the geotherm. This isanticipatedsince as depth increases,the mantle

Geotherms / F, T profile more closely resemblesan adiabat.Stacey [1977]continent ~~7o GPa Along the adiabat K’ and G’ are believed to

~:~

9~ran ~ decrease(Weidner, 1986).

-

O 2 ~ 6. Conclusionsand summary

1 We recast the thermodynamic parametera2KT/aPaTso that its valuecanhe obtainedfrom

othermeasuredquantities.Valuesof a2K3./aPaT

0 at ambient conditions are calculated for corn-0 1000 2000 3000 poundsftr which dataon the volume dependence

Temperature (K) of (aT/aF)5 are available.Our calculatedvaluesFig. 5. Temperaturedependenceof isobarsof C’ for MgO 2 . ~‘

of a K. /dPdT are in reasonableagreementwithcalculatedwith the P1B model. GeothermsIllustrate that (~like K’, is approximatelyconstantthroughoutupper mantle values obtained directly from measurements,P. T conditions,i.e. P� 13.5 GPaand T � 1800 K. where available.We find ~

2K1./~F~T to he posi-

46 Dci. ISAAK

tive for almost all minerals.Valuesof a2K.1./aPaT Model calculationsfor MgO show that both

for MgO and olivine at ambient conditions arc K’ and G’ are nearlyconstantalong a geotherm(0.39 ±0.10, 0.33 ±0.09)x l0~ K_t, respec- down to 400 km. The values of K’ and G’ fortively, and are significantly smaller(two to three MgO at 400 km are nearly identical to theirtimes) than for alkalide halides. These values ambient values. If other mantle minerals aredepend on the approximation that (~In C1,/ found to behavesimilarly we mustnote that bet-a In V)T is independentof V. Although this ap- ter accuracyis realizedwhen extrapolatingelasticproximation is very good at high T, it may be less propertiesto 400 km F, T conditions using thesecure at 300 K. The slope of the (~In C1,/ first P derivative, rather than including eithera In V)T versusV curve is probablyslightly nega- ~

2/~F~T or a2/aF2 without the other.tive at ambient temperature,implying that K anda2K~/aFaTare overestimatedby (9). A moreprecise assessmentof the effects of volume on Acknowledgments(a In C

1,/a In V)~from experimentaldata is re-quired to improve determinationsof K, and I thankO.L. Andersonof UCLA, who pointedtherefore,~

2K.J./~F~Tfor different minerals, out the thermodynamicrelationshipfor themixedWe show that a tends to control the magni- derivative, which is centralto this paper.B. Gra-

tude of a2KT/aFaT.A valuenear0.4x i0~ K ham of Penn State stimulatedmy interest in theappearsto be a good estimateof a2KT/aPaT for mixed derivative. T. Duffy of the Geophysicalmany oxides and silicatesof geophysicalimpor- Laboratory (Washington, DC) provided usefultance. MgSiO

3 perovskite,however,may not fit comments.A. Chopelas(Mainz) provided insightthis observation.Model calculationsindicate that into thermodynamicquantities.Support for thisa

2K.1./aPaT is near 0.80x i0~ K

1 for MgSiO3 work was provided through NSF grant EAR91-

pcrovskite.We find this valueto he larger than it 17280. IGPPContribution No. 3836.is for olivine andMgO becauseof the largevalueof

6T for perovskiteat 300 K.Provided that K for the /3 phaseof olivine is References

similar to that of other solids investigatedhere,neglect of the mixed F, T derivativeunderesti- Anderson,DL.. 1988. Temperatureand pressurederivatives

of elastic constantswith application to the mantle. J.matesKT: — K.

1~(or K54 — K5,.,) at 400 km F, T Geophys.Rca., 93: 4688—4700.

conditionsby about 6%. Estimatesof the olivinc Anderson,D.L andBass,J D.. 1984. Mineralogy andcompo-content that satisfy the V0 discontinuity could sitionof the uppermantle.Geophys.Res. Lett., 11: 637—

require a downward adjustmentof at most 3% 640.based only on these considerationsof K7 (or Anderson, O.L., 1967. Equation for thermal expansivity in

K5). These results highlight the need for more planetaryinteriors. J. Geophys.Res., 72: 3661—3668.Anderson,O.L.. 1981. Temperatureprofiles in theEarth. In:

datasince our conclusionsassumethe valueof KR.J. O’Connell andW.S. Fyfe (Editors), Evolution of thefor the /3 phaseof olivine by taking the average Earth. American GeophysicalUnion, Washington, DC,

of the materialsshown in Table 1. The justifica- pp. 19—27.

tion for usingthis assumptionis that K appearsto Anderson,O.L. and Andreatch,Jr., P., 1966. Pressurederiva-vary little from one materialto the next. Since a tives of elastic constantsof single-crystalMgO at 23°and

— 195.8°C.J. Am. Ceram. Soc., 49: 404—409.decreaseswith P the magnitudeof a

2KT/aPaT Anderson, O.L. and lsaak, D.G.. 1993. The dependenceofalso decreaseswith P andthe effectof neglecting theAnderson—Grüneisenparameter1ST upon compression~2K

1’/~F~Tis likely to be less than a 3% adjust- atextremeconditions.J. Phys.Chem. Solids, 54: 221—227.

ment in olivine content.We emphasizethat un- Anderson,O.L., Chopelas,A. andBoehlen,R., 1990. Thermal

certainties in (aG/aT)~for the /3 phaseolivinc expansivity versus pressure at constant temperature:are-examination.Geophys.Res. Lett.. 17: 685—688.

and in a2G/aPaTfor the a and /3 phasesare Anderson,O.L., Isaak,D. andOda, H., 1992a.High-tempera-

further uncertaintieswhich may overshadowthe ture elastic constantdata on minerals relevant to geo-

effectsof a2KT/aFaTon Vi.,, physics. Rev. Geophys.,30: 57—90.

PHYSICS OF THE EARTH AND PI.ANETARY INTERIORS 47

Anderson,O.L., Oda, H. and lsaak,D., 1992h. A model for ment. Nat. Bur. Stand. (U.S.), Spec. PubI., No. 326, pp.the computationof thermal expansivityat high compres- 201—208.sion and high temperatures:MgO as an example. Geo- Gwanmesia.GD., Rigden, S.. Jackson,I. and Liehermann,phys. Res. Lett., 19: 1987—1990. R.C., 1990. Pressuredependenceof elastic wave velocity

Anderson,O.L., Oda,H., lsaak,D.C. andChopelas,A., 1993. for ~-Mg~SiO4 andthecompositionof the Earth’smantle.

A thermodynamictheory of the GrOneisenratio: MgO as Science,250: 794—797.an example. Phys. Chem. Mineral., 19: 369—380. lsaak,D.C., 1992. High-temperatureelasticityof iron-bearing

Bands,R.A. and Sehuele,D.E., 1965. Pressurederivativesof olivines. J. Geophys.Res., 97: 1871—1885.theelasticconstantsof NaCI andKCI at295°and 195°K.S. lsaak,D.G. andGraham,E.K.. 1976. TheelasticpropertiesofPhys.Chem. Solids,26: 537—549. an almandine—spessartinegarnetand elasticity in the gar-

Bass, J.D. and Anderson, DL., 1984. Composition of the net solid solution series.J. Gcophys.Res., 81: 2483—2489.upper mantle: geophysicaltests of two petrological mod- Isaak, D.C., Anderson, O.L. and Goto. T., 1989. MeasuredcIa. Geophys.Res. Lett., 11: 237—240. elastic modulusof single-crystal MgO up to 180)) K. Phys.

Bina, CR. and Wood, B.J., 1987. Olivine—spinel transitions: Chem. Mineral., 16: 704—713.experimentaland thermodynamicconstraintsandiniplica- lsaak.D.C.. Cohen.RE. and MehI, M.J.. 1990. Calculatedtions for the the natureof the 401)-km discontinuity. J. elastic andthermal propertiesof’ MgO at high pressuresGeophys.Res., 92: 4853—4866. and temperatures.J. Geophys.Res.,95: 7055—7(167.

Birch, F., 1986. Equationof stateandthermodynamicparam- Knittle, E. and ieanloz. R., 1986. The thermal expansionofetersof NaCI to 300 khar in the high-temperaturedomain, silicate perovskiteand stratificationof the Earth’smantle.J. Geophys.Res.. 91: 4949—4954. Nature,319: 214—216.

Boehler. R., 1981. Adiabatsof ()T/OP)5. and GrOneisenpa- Mao. U.K.. Ilemley, R.J.. Fei, Y., Shu, J.F., Chen, L.C.,rameterof NaCI up to 50 kiloharsand 800°C.J. Geophys. Jeptheoat,AC. and Wu, Y., 1991. Effect of pressure,Res., 86: 7159—7162. temperature,and composition on lattice parametersand

Boehler, R., 1982. Adiabats of quartz, coesite, olivine. and densityof (Fe,Mg)Si03-perovskitesto 30 CPa. J. Geophys.magnesiumoxide to 50 kbar and 101)0 K and the adiabatic Rca.. 96: 8069—8079.gradient in the Earth’s mantle. .1. Geophys. Res.. 87: McQueen, R.G., Marsh, S.P., Taylor, SW.. Fritz, IN. and5501—5506. Carter,W.J.. l970. The equationof state of solids from

Boehler, R. and Kennedy, CC., 1980. Equationof state of shockwavestudies.In: R. Kinslcw (Editor), High-Velocitysodium chloride up to 32 kbar and500°C.J. Phys. Chem. Impact Phenomena.AcademicPress.San Diego, CA, pp.Solids,41: 517—523. 294—419.

Chopelas,A. andBoehler,R., 1989. Thermal expansionmea- Ross, N.L. and Hazen, R.M., 1989. Single crystal X-raysunementsat very high pressure,systematics,and a case diffraction study of MgSiO, perovskitefrom 77 to 40)) K.for a chemically homogeneousmantle. Geophys. Res. Phys. Chem. Mineral.. 16: 415—420.LeO., 16: 1347—13.50. Spetzler, H., 1970. Equationof state of polycrystalline and

Chopelas,A. and Boehler. R.. 1992. Thermal expansivityin single-crystal MgO to 8 kilohars and 800°K.S. Geophys.the lower mantle.Geophys.Res. Lett., 19: 1983—1986. Res., 75: 2073—2087.

Duffy, T.S. and Ahrens, T.J.. 1992. Sound velocity at high Spetzler,H., Sammis,CC. andO’Connell, R.J., 1972. Equa-pressureand temperatureand their geophysicalimplica- lion of state of NaCl: ultrasonic measurementsup to 8tions.J. Geophys.Res., 97: 4503—4520. kbar and 800°Cand static lattice theory. J. Phys. Chem.

Duf~.T.S. and Ahrens, T.J., 1993. Thermal expansion of Solids, 33: 1727—175)).mantle and core materials at very high pressures.Ceo- Slacey,F.D., 1977. A thermal model of theEarth. Phys. Earthphys. Res. Lett., 20: 1103—1106. Planet,Inter.. IS: 341—348.

Duffy. T.S. and Anderson, DL., t989. Seismic velocities in Sumino, Y. and Anderson,O.L., 1984. Elastic constantsofmantlemineralsand the mineralogyof the upper mantle, minerals.In: R.S. Carmichael(Editor), CRC’ HandbookofJ. Geophys.Res., 94: 1895—1912. Physical Propertiesof Rocks, vol. III. CRC Press,BOCa

Dziewonski, AM. and Anderson, DL., 1981. Preliminary Raton, FL. pp. 39—138.referenceEarth model. Phys. Earth Planet. Inter., 25: Suzuki, I., Ohtani, E. and Kuniazawa, M., 1980. Thermal297—356. expansionof modified spinel,~-Mg~SiO4. J. Phys. Earth,

Fei, Y.. Mao, H., Shu, J., Parthasarathy,C., Bassett,WA. 28: 273—28)).and Ko, J., 1992. Simultaneoushigh-P, high-T X-ray Taylor, iA., 1982. An Introduction to Error Analysis. Oxforddiffraction studyof ~-(Mg,Fe)2SiO4 to 26 CPa and 900 K. University Press(University ScienceBooks), Mill Valley,J. Geophys.Res., 97: 4489—4495. CA, 270 pp.

Fritz, J.N., Marsh, S.P. Carter, W.J. and McQueen, R.G., Wang, Y., Weidner,D.J., Liebermann,R.C., Liu. X.. Ko, S..1971. The Hugoniot equationof stateof sodium chloride Vaughan.MT., Zhao, Y., Yeganeh-Haeri,A. and Pacalo,in the sodium chloride structure.In: E.C. Lloyd (Editor). R.E.G.. 1991. Phasetransition and thermal expansionofAccurateCharacterizationof the High PressureEnviron- MgSiO., perovskite.Science,251: 410—413.

48 DO. tSAAK

Weidnen, D.J., 1986. Mantlemodel basedon measuredphysi- Wolf, G.H. and Bukowinski, M.S.T., 1987. Theoretical studycal propertiesof minerals.In: 5K. Saxena(Editor), Chem- of the structural propertiesand equationsof state ofistry and Physics of Terrestrial Planets. Springer, New MgSiO

3 and CaSiO3 perovskites:implications for lowerYork, pp. 251—274. mantlecomposition. In: M.1-I. Manghnani and Y. Syono

Weidner, D.J., 1987. Mineral physicsconstraintson a uniform (Editors), High-PressureResearch in Mineral Physics.mantlecomposition. In: M.H. Manghnani and Y. Syono American GeophysicalUnion, Washington,DC, pp. 313—(Editors), High-PressureResearchin Mineral Physics,Vol. 331.39. American GeophysicalUnion, Washington,DC, pp. Yamamoto, S., Ohno, I. and Anderson, O.L., 1987. High439—446. temperatureelasticity of sodium chloride. J. Phys.Chem.

Williams, Q. andJeanloz,R., 1991. Ultra-high-pressureexper- Solids, 48: 143—151.imental techniques.In: R.J. GaleandD.G. Lovening (Edi-tors),Molten Salt Techniques,Vol. 4, Plenum, NewYork,pp. 193—227.