Temperature Thermodynamic Properties of Forsterite - CiteSeerX

JOURNAL

OF GEOPHYSICAL

RESEARCH,

VOL. 96, NO. B7, PAGES 11,805-11,816, JULY 10, 1991

High-TemperatureThermodynamicPropertiesof Forsterite PHILIPPE GIIJ.F.T

Laboratoirede MindralogiePhysique,Universit•de RennesI

PASCALRICHET AND FRANCOISGUYOT Institutde Physiquedu Globe,Paris

GUILLAUMEFIQUET Laboratoirede MindralogiePhysique,Universit•de RennesI

The high-temperature thermodynamic propertiesof forsteritcwere reviewedin the light of a new

determination of theisobaric heatcapacity (Cp),upto 1850K, andRaman spectroscopic measurements, upto 1150K and10GPa.TheCp measurements andavailable dataonthermal expansion (tz)andbulk modulus(K) showthat the isoch0ricspecificheat (Cv) exceedsthe harmoniclimit of Dulong and Petit above 1300K. This intrinsicanharmonic behaviorof Cv can be modeledby introducinganharmonic

parameters ai = Olnvi/•T)V whicharecalculated fromthemeasured pressure andtemperature shiftsof the vibrationalfrequencies. Theseparameters are all negative,with absolutevalueslower for the stretching

modes oftheSiO4tetrahe..dra (ai • - lx10'5 K'l) than forthelattice modes (ai • - 2x10 -5K-I). Through therelation Cp= Cv+ •"KTVT,thecalculated anharmonic Cv andthemeasured Cparethenused to determinethe-temperature dependences of the thermalexpansion and bulk modulffsof forsteritc,up to 2000 K, in agreementwith recentexperimental results.Finally, all thesedata point to an inconsistency for the G 'rtineisen parameter of forsteritc,wherebythe macroscopic parameter?'= tZVKT/Cvcannotbe evaluatedsimply at high temperatureby summationof the individual isothermalmode Grfineisen parameters TiT= KT (c)lnvi/c)P).

1. INTRODUCTION

A great many mantle minerals are not amenable to

thermodynamic measurements becauseof their instabilityat moderate temperaturesor of the minute amountsof material

done within the framework of the quasi-harmonic approximationwhose limitations at high temperaturesmust be assessed. As a matterof fact, the role of anharmonicitycan be investigated rather simply from the limited spectroscopic information that can be obtainedfor mantle minerals [Gillet et

that can be synthesized. To setupthermodynamic modelsof al., 1989]. mantlephaseequilibria,useis thusmadeof spectroscopically In thispaperwe aremainlyinterested in theisobaric(Cp) determinedheat capacitiesand entropies[e.g., Kieffer, 1979]. and isochoric (C v) heat capacities, thermal expansion The difficulty with this approach,however,is that there are coefficient(a) and isothermalbulk modulus(KT) of forsteritc. virtually no measurementsfor testing the validity of the Through calculationsof the heat capacityfrom spectroscopic methodunderthe high-temperature conditionsof the mantle. data, we first show that anharmonicity contributes Forsteriteis probablythe mineralthat hasbeen studiedthe significantlyto Cv and the other thermodynamic propertiesof most extensively.The purposeof this contributionis thus to forsteritcabove 1300 K. Then we discussthe consistencyof considerforsterite as a model mineral to investigatethe theavailable Cp,a andKT dataat hightemperature. Finally, consistency of the thermodynamic and spectroscopic datafrom we discuss the Grtineisen parameter of forsterite, in both ambientup to highpressures andtemperatures. Our purposeis macroscopic and microscopic terms. To complement the not to model mineral propertiesin the most comprehensive available experimentaldata base neededfor our purpose,we way, as done for example in the lattice dynamicsstudy of have also redeterminedthe heat capacityof forsteritebetween forsteritc by Rao et al. [1988], and the reasons for our 800 and 1800 K and measuredthe pressureand temperature simplified approach are twofold. First, experimentslike dependencesof the Raman-activevibrational frequenciesof inelastic neutron scatteringare not feasible on mineralsfor forsterite. whichthermodynamic measurements are lacking.Thuswe will 2. EXPERIMENTAL METHODS consideronly spectroscopic datathatcanbe gatheredon highpressurephases.Second,lattice-dynamics calculationsare still Calorimetric

measurements

were

made

with

the

ice

Papernumber91JB00680

calorimeter, high-temperatureequipment and experimental proceduresdescribedby Richet et al. [1982], with the slight modificationsreportedby Richet and Bottinga [1984]. About 5-6 g of forsteritc, contributingfrom 60 to 65% to the total

0148-0227/91/91JB-00680505.00

measured heat content, were run in a Pt-Rh 15% crucible.

Copyright1991 by the AmericanGeophysical Union.

11,805

11,806

GILLET ET AL.' HIGH-TEMPERATURE PROPERTIES OF FORSTER1TE

Measurements underthe sameconditionson oc-A120 3, the calorimetric standard, suggest instrumental inaccuraciesof about 0.2 and 0.5% for the relative enthalpies and heat capacities,respectively[Richet et al., 1982]. The forsterite specimen used for calorimetric measurements,of industrial origin, was given by O. Jaoul (Universit6 Paris XI). Its composition, as determinedfrom electron-microprobeanalyseswith the automatedCAMEBAX microprobe of the Universit6 Paris VI is 0.11 (2) wt %

175

170

165

160

A120 3, 42.61 (3) wt% SiO2, 57.36 (4) wt% MgO and 0.05 (2) wt % CaO, total of 100.15 (6). This compares favorably with the nominal composition,namely 42.70 and 57.30 wt% SiO 2 and MgO, respectively. The lattice

parameters a = 4.760(1),b = 10.201 (2) andc = 5.985(1)/•

[]

[] m • []/m

/ 155

150

n

Orr

were derivedfrom an Xray powderdiffractionpatternin which [] This work the reflections of only forsterire were apparent.Hence, the ! I ..... ! , , . ß , i ..... 145 molar mass used was the nominal one. Spectroscopic 700 1000 1300 1600 1900 measurementswere performed on a syntheticspecimenof similar quality. T (K) The Raman spectrawere recordedwith the multichannel microprobe(Microdil 28, from Dilor) of the MLRO serviceof Fig. 1. Mean heat capacityof forsterite.Open squares'resultsof the University of Nantes. The light was collected in the Orr [1953],referredto 273 K withtheH298 -H273 of Robieet al. backscatteringdirection through a Leitz UTK40 or UTK 50 [ 1982]; solid squares'this work. objective(focal distanceof 8 or 18 mm; numericalapertureof 0.63 or 0.32, respectively). The spectrawere obtainedfrom about 10 accumulationslasting each 20 to 40 s, and peak A simultaneous least-squares fit was made to the

positions wereidentified to within+ 1cm-1. Torecord a experimental Cp andenthalpy datato obtainthefollowing maximum number of bands at high temperatures,small single equationsof the form recommended by Haas and Fisher [1976], crystals were inserted with different orientationsinto the Bertnan and Brown [1985] and Richet and Fiquet [1991], heating stage of a microscope.Temperatureswere measured respectively: within a few degreeswith a Pt-PtRh 10% thermocouple.A diamond-anvil cell was used to compressthe sample in the Cp=297.570 - 55.684 10-3T- 5.187 05/r2 200-I.tm hole of a stainless-steelgasket to record the high-2731.7 T0'5 pressurespectra.A 4:1 ethanol-methanolmix was used as a + 19.79110-6 T3, (1) pressure-transmitting medium for all experiments, and pressureswere measuredwith the ruby-fluorescence method.

Cp= 241.777 + 2119.3/T 0'5+ 1.5657 106/T 2

3. CALORIMETRY

- 4.7258108/T 3,

(2)

In spite of the considerablegeophysicalimportanceof Cp= -402.753 + 74.290 lnT+87.588 103/T forsterire, calorimetric measurementshave long been scarce 25.913 106/T 2 and the relative enthalpiesof Orr [1953] up to 1800 K were the + 25.374108/T 3, (3) only available high-temperature data. Our experimental relative enthalpies (Table 1) are about 1% lower than the results of Orr, except at the highest temperatureswhere the At low temperatures, theadiabatic Cp of Kelley[1943], by thoseof Robie et difference tendsto decrease.This is apparentin Figure 1 where down to 50 K only, have beensuperseded both data sets are plotted in the form of mean heat capacities, al. [1982] for the wider temperatureinterval 5-380 K. Our results join smoothly with both sets of measurementswhich Cm= (HT-H273)/(T-273). are in mutual agreement,as indicatedby the deviationsof the values given by equations(1)-(3) from the experimentaldata TABLE 1. RelativeEnthalpyof Forsterite (Table 2). Theseresultsalso agree,to within their greatererror . (kJ/mol> margins, with the Differential ScanningCalorimetry (DSC) Run T,K HT-H273 measurementsof Watanabe [1982] andAshida et al. [1987]. The heat capacitiesreported in these different studiesare BD.8 787.9 75.718 plotted in Figure2. Up to 1600 K, the consistencyof all the BD.3 922.3 99.519 BD.11 977.0 109.12 data is to within + 1% in the temperature rangeof the various BD.9 1087.0 128.41 measurements.Above 1600 K, the anomalouslyhigh heat BD.4 1123.3 135.64 capacitiesgiven by the equationreportedby Orr [1953] and (1) BD.2 1264.2 160.30 result from the analyticalforms of theseequationsthat do not BD.5 1374.2 180.44 BD.1 BD.7

1615.6 1493.8

225.85 202.44

BD.10 BD.12

1722.1 1847.1

246.15 270.75

allow high-temperature extrapolations. As discussed by Richet and Fiquet [1991], equation (3) is to be preferredfor this purposewhereas (2) gives apparentlytoo low values because, in contrast to (1) and (3), it does not reproduce the

11,807

GILLETETAL.:HIGH-TEMPERATURE PROPERTIES OFFORSTER1TE

TABLE 2.Average Absolute Deviations (AAD)ofValues Given byEquations (1),(2)and (3) From TheExperimental CpandEnthalpy Data AAD, %

Reference

Kelley [19431 Robie etal.[1982] Watanabe [1982] Ashida etal.[1987] 0rr[1953] ThisWork

Property

N

Cp Cp Cp Cp

3 22 8 10

H H

16 !1

AT,K

(1)

276-295 270-380 350-700 280-700

398-1808 782-1847

0.31 0.08 0.43 1.!0 0.82 0.05

(2) .....

0.39 0.18 0.63 0.75 0.75 0.15

6) 0.23 0.06 0.33

0.84

1.15 0.06

temperatures is apparentin Figure3 and the negative

210

variations withtemperature of thefrequencies arelinearwithin

experimental uncertaintie• (Table3 andFigure 4). Therelative

I

frequency shiftsplott• •d in Figure 5 as a function of wavenumbershowthat the internalmodesdependless on

190

temperature thanthelatticemodes. Thisis alsoapparent in Table3 whichliststl•e• ' corresponding isobaric Grtineisen

I

,

parameters:

'

170

•qp= - 1/a (c)Invi/•T)p 150 oOø! .!

130

400

ß'- Watanabe

which were calculated with the 1-bar, 298 K thermal expansioncoefficient. At room temperature, infrared measurementsup to

-

Ashida

42.5GPahavebeenreported by Hofrneister et al. [1989],and

=

This Work

El

/

.........

800

1200

(4a)

1600

orr

2000

the availableRaman data [Bessonet al., 1982;Gillet et al.,

1988; Chopelas, 1990] extend up to 22.7GPa. For intercomparison purposesand for estimatingbetter the

T (K)

uncertaintieson the shifts, we have also recorded the room-

temperature Ramanspectrum of forsterite up to 5.5GPa. Fig.2. Heatcapacity of forsterite asreported byOrr [1953], Robie etal. [1982],Watanabe [1982], Ashida et al. [1987],andin this work.

Measurements were madewith both increasingand decreasing

pressures ortemperatures. No..•yst9resis wasobserved. Typical examples of the spectra at pressure aregivenin Fi•gure 6. Qualitatively, thetwofrequency ranges foundunderambient

conditionsare still observedat pressure,with only linear experimental data [o within their errormargins.This is increases of the vibrational frequencies with pressure

apparent in Table2 for the measurements of Robieet al. (Table3). [1982]andours.In thefollowing sections, theheatcapacity Above 10GPa, Chopelas[1990]andHofmeisteret al. minorchanges in the vibrational will thus be obtainedfrom (3), with an uncertaintythat is [1989] haveinterpreted pattern in terms of a phase transition and we willthusrestrict estimatedto be lower tha0 1% up to 2000 K at least. ourselves to pressures lowerthan10GPa whereour highpressure results agree withthemoreextensive dataofChopelas 4. SPECTROSCOPY

[1990],seeTable3 andFigure 7. Therelative changes in frequency of the internal modes of the Silo 4 tetrahedra depend As shownbelow, accuratecalculations of the heat capacity clearlylesson pressure thanthoseof thelatticemodes, as requirecomprehensive high-pressure and high-temperature previously pointed6ut by Hofmeister et al. [1989]and spectroscopic data.Previous Ramanandinfraredmeasurements on f0rsteritehavebeenreviewedby Hofmeister[1987]and Chopelas [1990].This is alsoshownin Table3 by the isothertna.1 Grtineisen parameters: McMillanand'Hofmeister [1988].Briefly,thehighvibrational correspofiding

frequencies, between 800and1000 cm -1,have been assigned to combinations of symmetricand asymmetricstretching modesof the SiO4 tetrahedra.A gap in the range800 -

•T = KT (3lnvi/3P)T

(4b)

650cm-1 separates these modes fromthelower-frequency part which were calculatedwith the 1-bar, 298 K bulk modulus.

of thespec•um, whichincludes thebending modes of theSiO4 tetrahedra and the lattice modes involving the MgO6

5. ANHARMONIC!TYANDHEAT CAPACITY

octahedra.

To fill in the lack of high-temperaturemeasurements, we

haverecorded theRamanspectrum of forsterite upto 1150K at

ambient pressure. The usualbandbroadening at high-

MacroscopicEvidence

The isochoricheat 9apacity(Cv) is relatedt? the measurable isobaricheatcapacityby:

11,808

GILLET ETAL.'HIGH-TEMPERATURE PROPERTIES OFFORSTER1TE LASERBEAM II b

LASERBEAM//• T= 300 K

T= 300 K

i

i I

i

I

t

•

I

I

i

i

_

I

i

i

I

I

i

i i

•00K i ,

I

,

I

I

,,

-•

'I

I

i

i

i i

i

111o

i i i

I

,

1000 900

'

I

'

I

I

800

'

700

I

I 600

500

•,00 300 200

I

!

1000 900 800 700

WAVENUMBER ([m-•)

!

__1

I

I

i

,

600 500 •00 300 200 WAVENUMBER I[m-•)

Fig.3.Raman spectrum offorsterite atthree temperatures with thelaser beam parallel toeither thea orthebaxis. The

intensity differences result from polarization effects inherent tothe collecting geometry ofthe microRaman setup. Cv:Cp-7va2K t.

(5)

Anharmonic Calculation of theHeatCapacity Heat capacitycalculations are usuallymadewithinthe Departure of Cvfromthelimitof Dulong andPetit,namelyharmonicapproximation, i.e., with the assumption that 3R/g atom (R = gas constant)is a traditionalmeasureof vibrationalfrequencies are independent of temperature. intrinsic anharmonicity. In Figure8 wehaveplotted Cv as Regardless of the detailsof thiskindof analysis, the main calculated fromequations (5)and(3)andthemost recent high- pointis thatthecontribution of eachacoustic or opticmodeto temperature measurements of ocandKT byKajiyoshi[1986] the heat capacityis in fact that of an harmonic,Einstein

andIsaaketal. [1989].TheDulong-and-Pefit limitispassed at

oscillator:

about1300K andtheexcess Cv overthislimitreaches about5

%at2000 K.Inother words, the TVa2KT term ofCpaccounts Cvi h=k(hvi/kT) 2exp(hvi/kT) /[exp (hvi/kT) - 1]2 (6)

for only part of the vibrational anharmonicity. As stated previously by Anderson andSuzuki[1983]fromolderdata, wherethefrequency vi is independent of temperature, andh and

forsterite thusclearly displays intrinsic anharmonicity athigh

temperature, a conclusion also reached when one uses other

k are the Planck and Boltzmann constants.

As shown in Figure4 for forsterite, one observesthat

valuesof ocsuchasthoseplottedin Figure2 of Isaaket al. vibrational frequencies dovarywithtemperature. In thequasi[1989].Suchaneffectis notapparent in simple compounds harmonicapproximation, thesevariationsare simply likeMgOandCaOforwhich theDulong andPetitlimitforCv accounted for by allowingthefrequencies to be temperature

isnotexceeded athightemperature [Anderson andZou,1990]. dependentin equation(6). Thus, in both the harmonicand Anexplanation is thatin these oxydes thephonons areonly quasi-harmonic approximations, thehigh-temperature limit of

weaklycoupled whereas, in complex structures likeforsterite,Cv is 3Rig atom sincethe high-temperature limit of an strongcoupling(and thus intrinsicanharmonicity) is Einsteinoscillatoris k. In otherwords,it is assumedthat the expected. anharmoniccontributionto Cv is negligible,and that

GILLET ET AL.: HIGH-TEMPERATUREPROPERTIES OF FORSTER1TE

11,809

TABLE 3. Pressure andTemperature Shiftsof VibrationalFrequencies, andAnharmonicParameters

vi , cm '1

(o%il•t,cm'l/K •T*

•T'}'

•p']'

ai •, !0-5K-1

•T;e

967 920 882 856 825 610 593 585 548

0.66 0.38 0.44 0.49 0.48 0.70

0.7 0.4 0.5 0.5 0.5 0.7

0.6 0.6

-0.027 (2) -0.019 (2) -0.023 (2) -0.016 (1) -0.017 (1) -0.013 (1) -0.012 (1) -0.013 (1) -0.011 (1)

1.07 0.82 1.00 0.72 0.79 0.82 0.85 0.85 0.77

0.66 0.53

443

1.60

426 414 376 341 334 307 244

1.41 0.99 1.25 1.87 1.16 1.63 1.21

1.8

-0.029 (2)

2.52 (36)

-2.38(7!)

-0.018 -0.022 -0.019 -0.025 -0.019 -0.023 -0.018

t.62 2.04 1.94 2.82 2.18 2.87 2.13

-0.58(17) -2.74(82) -1.79(54) -2.51(75) -0.81(24) -3.22(96) -3.10(93)

232

0.67

-0.011 (1)

1.82 (27)

-3.00(90)

183

2.09

-0.012 (1)

2.51 (37)

-1.12(34)

171{} 99õ

1.47 1.04

-0.012(2) -0.008(2)

2.60(62) 2.99(97)

-3.0(15) -5.0(25)

0.54 0.49 0.54

1.4 1.3 1.8

0.7

(2) (2) (2) (2) (2) (2) (1)

(16) (14) (16) (10) (10) (15) (13) (13) (13)

-1.07(32) -1.16(35) -1.46(44) -0.60(18) -0.81(20) -0.31 (9)

-0.51(15) -0.63(19)

(30) (30) (35) (43) (39) (46) (27)

* DatafromChopelas [1990] •'Results fromthepresent study, Uncertainties forai obtained froma = 2.7+ 0.2x!0-5K-1, a 15% uncertainty on 7•Tandthe reporteduncertainties on the •p g Data fromBessonet al. [1982]

õ Acoustic modes' 71TfromChopelas [1990] and•p calculated fromtheacoustic dataoflsaaketal. [19891.

1000

700

950

+-------+-+ w A V 900

I

+ +---.+.

500

E N

U M B E R S

850

400 ø'•'•'••••-'-'•"••----••-

300

800

-

I

I

I

I

I

I

I

I

100

200

300

400

500

600

700

800

.

200 900

TEMPERATURE (øC)

I 0

100

I' 200

I

I

-I

I

I

300

400

$00

600

700

800

TEMPERATURE (øC)

Fig. 4. Temperature dependence of thevibrational frequencies. The symbolsizeis similarto theerrorsof the data.

anharmonicity (often

called extrinsic anharmonicity)

1987; Rao et al., 1988; Choudury et al., 1989] have been

contributes to Cp in (5) throughthe thermalexpansionperformedup to 1000 K with either the harmonicor quasicoefficient a only. For forsterite, Kieffer modeling [Hofmeister, 1987; Chopelas; 1990] or lattice dynamicalcalculations[Price et al.,

harmonic assumptions.In view of the excess C v over the Dulong-and-Pefitlimit shownby forsteritc,a calculationof the intrinsic anharmonic contribution to C v itself must be

11,810

GILI•T ET AL.: HIGH-TEMPERATURE pROPERTIES OF FORSTER1TE

[ (31nvi/aT) (10-5 K-l) 4

P=1.2

•.

internal modes

2r• lattice modes tt •ttt • "0

200

400

600

86•3

1000

frequency (cm-1) P= 3

Fig. 5. Relative changesin internal and lattice vibrational frequencies withtemperature. The absolute valuesarerepresented, but all the shiftsare in fact negative.

1

performed. Gillet et al. [1989] pointed out that this contributioncan be determinedsimply from high-pressure and .,

high-tempera.ture spectroscopic measurements. To demonstrate this result in a more rigorous way, ...wewill characterizethe intrinsic anharmonicity of a vibrational mode i by the

I

I

variation of vi withtemperature at constant volume. Wewill use for this purposethe anharmonicparametersai introduced by Mammoneand Sharma [1979]'

ai = Olnvi/•T)v = otKT Olnvi/•P)T + Olnvi/•T)p.

P= 5.2 GPa

(7)

i I

I

The parameterai canalsobe expressed as

I

ai = ot (•iT- •P)

(8)

and thusit can be determinedfrom the pressureand temperature

dependences of vi through(4a)-(4b). The•se parameters at 298 K and ! bar are listed in Table 3 for forsterite, along with their uncertaintiesas obtained from the spectroscopic results. Note in Figure 9 that the absolutevalues of these parametersare lower for the internalmodesof SiO4 tetrahedra than for the lattice modes.

I

900

800

600

!

I

!

•

500

L•O

300

200

.

WAV ENUMBER (cm-")

,

As we will shownow, the usefulness of theai parameters is that the isochoricheat capacityof an ensembleof anharmonic

Fig. 6. Ramanspectrum of forsteriteat threepressures.

oscillatorsis simplyrelatedto the harmonicheatcapacities'

Cv= Y•Cvih(1- 2aiT)= Cvh- TZ 2aiCvih

(9)

whereCvh, theharmonic partof theheatcapacity, canbe

Spectroscopic measurements showthat Olnvi/•)T)v generally differ from zero. Thus we will use (11) but we take into account

the variationsof the frequencieswith temperatureat constant volume when inserting(11) in (10) to obtain:

obtainedwith a suitablemodel in which the room-temperature andpressurefrequenciesare used.To demonstrate equation(9), U = U0 +Z Uih(1-aiT) we begin with the internal energy of the ensemble of oscillatorswhich is related to the partition functionsof the where the harmonicvibrationalinternalenergyis individual oscillators(Zi)by:

U= U0 + Z kT20lnZi/aT)v

(10)

where U O is the lattice cohesiveenergy. For an harmonic oscillator, the partition function is' Z i = exp(-hvi/2kT)/[1- exp (-hvi/kT)]

(11)

Uih: hvi {1/2- 1/[1- exp(hvi/kT)]}

(12)

(13)

The anharmonicCv is thus

Cv : OU/aT)v

: Z {Cvi h(1- aiT)2- T Uih[ai2 + (Oai/•T)v]} (14)

GILLET ET AL.: HIGH-TEMPERATURE PROPERTIESOF FORSTERrYE

ai(10'5K '1)

(31nvi/3P) (10 4GPa 4)

2,0

11,811

-2 internal modes

internal modes

1.0

-3

lattice modes

lattice modes

200

400

60O

80O

1000

-6

frequency(cm-1) Fig. 7. Relative changes in internal and lattice vibrational frequencieswith pressure.Open circles:Chopelas [1990]' solid circles'

i

0

this work.

ß

200

i

ß

i

400

ß

600

i

800

ß

!

lOOO

frequency(cm-1) Fig. 9. Anham•onicparametersof the Ramanmodesof forsterite calculated at 298 K and 1 bar.

In the next section, we evaluate the various terms of (14) to reducedto (9) up to a few thousanddegrees.To show that the termisnegligible, wenotethatthetermT Uih isabout show that this expressionreduces numerically to the much second simpler equation(9). Gillet et al. [1989] obtained previously T2Cvi h at hightemperatures. Finally,(aai/aT)vcanbe determined from for Cv:

Cv= Z Cvih - Z aiU• - TZ aiCvih.

(15)

(aai/aT)V = (aai/aT)P + crKT (aai/aP)T.

(16)

of the Equation (14) is more rigorous than (15) because it From our results which show linear variations incorporates the contribution of the zero-point energy and frequencieswith pressureand temperature, we estimatethat the takes consistentlyinto accountthe variation with temperature twoderivatives of theRHSof (16)aresmaller than10-9K-1. of the vibrationalfrequencies.Numerically,however,(15) also This makes it these terms negligible in (16). In summary,the reduces to(9)because Uih donotdiffermuch fromTCvi h at parametersai mustbe consideredas temperatureindependentin high temperatures.For practicalpurposes,(9), (14) and (15) (9), a conclusion similar to that obtained by Gillet et al. thus give nearly the sameresults. [1990] for quartz from detailednumericalcalculationsof the ai

parameters. Numerical

Calculations

From the spectroscopicdata, one finds that all the a i

Application to Forsterite All optic vibrational modes are not Raman or infrared active. For practical applications,it is thus necessaryto use an averaging schemefor the observedfrequencies.In Kieffer's model the contributionto Cv of a set of m optic continuais given by

parameters areoftheorder of-10-5K-1 (cf.Table 3).Hence, the first term of the right-hand side of (14) can be safely

(J/m01/r)

190

Vui

m

Cv = 3 nR

170

x2exp(x) dx

N

i=

(Vui- Vli) [exp(x)- 121

! Vli

m

150

= 3 nR

E i=

Cv

ih,

(17)

1

with x = hvi/kT, n is the numberof atomsin the mineral formula,R is the gasconstant,ni is the numberof modesin the

130

ith continuum, N is the total number of vibrational modes, and

vQandVuiarethelower andupper cutoff frequencies ofthe

11o

'

300

i

500

'

i

700

'

I

900

'

I

'

I

'

I

'

i

•

I

1100 1300 1500 1700 1900

Temperature(K) Fig. 8. Isobaricand isochoricheat capacityof forserite:C : given

byequation (3).and Cvcalculated from relation (5)with ø•Pand KT as givenby Isaak et al. [1989].

it"continuum, respectively.

Using Kieffer's [1979] model with severaloptic continua, Hofmeister [1987] has proposeddifferent approximationsfor the densityof statesof the optical modesof forsterite.Two of these models are shown in Figure 10, along with a third one which we have set up to accountfor the threedifferentrangesof

11,812

G!LLET ETAL.:HIGH-TEMPERATURE PROPERTIES OFFORSTERHZ

MODEL I

-!.5 -2.5

-!

-0.5

(d/mo]/K) 115

&82 505

227 2t, t,

646

825

ANHARMON I C MODELS

975

WAVENUMBER O g P limit MODEL ii •60

-2

HARMONIC MODELS

-!

i45

825

/•B2 505

105

975

WAVENUMBER (cm-t)

•30

MODEL III

ISOCHORIC

SPECIFIC

HEAT

-!.6 -! 65

t6 !

i

6/,/,

105

825

300 I

500

700

goo

]•00

•300

TEMPERATURE

9?5

i500

•700

•900

(K)

WAVENUMBER (cm-t)

Fig.10.(a) Models ofdensity of states consistent withthespectroscopic data(acoustic modes notshown). Models I and !II areadapted fromHofmeister [1987]. Thelowest mode at105cm-tistaken fromtheinelastic neutron scattering data

ofRaoetal. [!988].Thenumbers in theboxes represent thenumber ofmodes in thatcontinuum. Thenumbers above the

boxes refertothemean ai parameter forthose modes. (b)Harmonic andanharmonic Cvcalculated withthedensities of statesshown in (a).

frequencies apparent in Figures 5 and7. Thenumerical data relevantto heatcapacitycalculations with thesethreemodels are includedin Figure10. The remaining data,namelythe

6. CONSISTENCY OFTHERMODYNAMIC DATA

Thedataplottedin Figure11 illustrate theaforementioned

uncertainties on the thermal expansion coefficient of [1979] and Hofrneister [1987]. As usual,the resultsare forsterite.Publishedvaluesdiffer by 30 % between300 and sensitive functions of the cutoff frequencies at low 1400 K and the differencesreach 50 % when extrapolating

acousticvelocitiesandmolarvolumes,weretakenfromKieffer

temperatures, but the mainresultis thatall models yield theseresultsup to 2000K, To improvethermodynamic Fei andSaxena [1987a]andSaxena[1988,1989] harmonic heatcapacities differing by lessthan2 and0.5% at modeling, pointedout thatequation (5) provides a simplemeans of 300 and1000K, respectively (Figure10 andTable4). consistency between thermal andvolume properties. With the anharmonic contribution, the isochoric heat ensuring ForMgO,forexample, Saxena[1989]usedexperimental Cp capacity is valuesto optimizesimultaneously oLKT andCv. (In passing, we notethatsatisfactory resultscouldbe obtained onlywhen Cv wasallowed to differfromtheDulong-and-Petit limit,i.e., whensignificant anharmonicity wasassumed.) i=] Basically, wewill followthesameapproach, withthemain Forevaluating (18) anaveraging scheme mustalsobeusedfor differencethat we greatlyreducethe numberof adjustable by considering Cv asa knownquantity obtained theai parameters. To ensure consistency withthecalculationparameters

Cv=3nRZ Cvih (1-2 aiT).

(18)

of the harmoniccontribution to Cv, theai wereaveragedover from anharmonicvibrational modeling.The optimization in detailby Tarantolaand the samefrequency continua as usedin thevariousKieffer methodfollowedhasbeendescribed Valette [1982] and Sotin [1986]. The startingparameters are models.Calculationsshow that the anharmoniccontribution

uncertainties becomes significant at about1300K (Figure10 andTable4) listedin Table5, alongwith all the estimated for the thermalexpansion coefficient. and reaches about 7 % at 2000 K, with values that are whichare the greatest expressions havebeenusedfor •x andKT. insensitive to theaveraging scheme, andbringsthecalculatedVariousanalytical Theresults depend littleontheformof theseequations andthe Cv to within1% withthevaluesobtained frommeasurements followingsimpleexpressions havebeenfoundsatisfactory: with equation(5).

GILLETET AL.: HIGH-TEMPERATURE PROPERTIES OF FORSTER1TE

11,813

TABLE 4. Isobaricand isochoricheatcapacityof forsterite(J/molK).

T(K)

300 600 900 1200 1500 1800 2000

Cp

Cv

exp

exp 't'

119.2 158.2 171.4 180.4 188.2 195.7 199.6

117.9 154.7 165.6 170.6 175.0 178.4 180.0

calculated Cvwith models ;e

1h

1anh

116.8 154.4 164.9 168.9 170.9 172.! 172.3

117.7 156.9 168.9 172.8 178.4 180.8 181.2

2h

2anh

120.1 155.4 165.3 169.2 171.1 172.1 172.3

121.1 158.2 169.8 175.4 178.4 181.5 182.1

3h

3anh

116.7 154.2 164.8 168.9 170.9 172.1 172.3

117.5 156.9 169.1 174.7 177.6 180.9 181.6

* Values obtained withequation (3). t Calculated fromequations (3)and(5),thebulkmoduli ofIsaak etal.[1989] andthethermal expansion of Kajiyoshi [ 1986].

;• Calculated fromharmonic (h) or anharmonic (anh)vibrational modeling. Thenumbers referto thevarious densitiesof statesof Figure 10.

•

TABLE 5. ThemaalExpansionandBulk Modulusof Forsteriteup to

Oo-SK

2000 K From Data Inversion

1

Inverte d Values

Input Data* T.K 300 600 900 1200 1500 1800 2000

/

(1% Uncertainty)

and Uncertainties

cp

cp

119.2 158.2 171.4 180.4 188.2 195.2 199.6

Cv 117.8 156.9 168.8 172.8 177.6 180.8 181.2

117.9 158.7 172.7 181.3 188.6 196.2 201.7

Cv 116.7 155.4 167.2 172.8 176.8 179.2 182.1

ß

Parameters •' 300

500

700

900

Apriori Values and Uncertainties

Aposteriøri Values and Uncertainties

i

1100 1300 1500 1700 1900

105a0 108al

Temperature(K)

3 (2) 1(2)

a2 Kr (kbar)

Fig. 11. Thermal expansioncoefficient of forsterite. Curve 1 (dashedline) from Suzukiet al. [1983]' Curve 2 (thin line) from Kajiyoshi [1986]; curve 3 (dottedline)from Fei and Saxena

1 (10) 1290 (20)

(VodK•/cma•/m0cbar/K) -0.23(20) ol)

[1987b]; Thick curve obtained in this work from data inversion with (5).

43.67 (30)

2.77(9) 0.97(9) -0.32 (54) 1277 (20)

-0.20(2) 43.55 (41)

* Input data forCpasgive• byequation (3),and Cv(anharmonic

vibrationalmodel 3 'of Figure 10) taken every 50 K from 300 to 2000 K.

a = a0+ alT + a2/T2 KT = KTO+ (dKT/dT)oT.

(19)

?Initialestimates ofa,KT,and(dKT/dT) taken from FeiandSaxena [1987b] andIsaak et al. [1989], respectively.

(20)

Comparisonof the derived a (Table 5) with available data in

Finally, the slight temperaturedependenceof the volume, which playsonly a minor role in (5), was accountedfor by

V = V0 (1 + I a dT).

Figure 11 shows very good agreement with the hightemperature observations and extrapolations of Kajiyoshi [1986]. The invertedvalues of KTO and (dKT/dT)o listed in (21) Table 5 are also in excellentagreementwith thoseof lsaak et al. [ 1989].

Finally, the consistency of our deriveddatacanbe checked given by equation (3) were used in (5), along with the independentlyby calculatingKT from the measuredKs of Isaak anharmonicCv calculatedwith the model #3 of Figure 10, to eta!. [1989],ourmeasured Cp,andourcalculated anharmonic adjustthe coefficientsof (19) and(20). Cv with: The slight curvaturewith temperatureof the adiabaticbulk modulus (Ks) observedby lsaak et al. [1989] cannot be KT= KsCv/Cp. (22) translated to the isothermal K T in view of the stated uncertaintiesof the other data appearingin equation (5). The moduli obtainedin this way are comparedin Table 6 and

In summary, numerical values of Cptaken at50K intervals as

11,814

GILLET ET AL.' HIGH-TEMPERATUREPROPERTIES OF FORSTERITE

TABLE 7. Griineisenparameters of forsterite

TABLE 6. Bulk modulusof forsterite(GPa)

macroscopic

T,K

Ks*

300 600 900 1200 1500 1800 2000

KT•'

128.7 123.8 118.4 113.0 107.6 101.9 98.2

KT •

127.1 122.7 116.8 109.4 101.5 94.2 89.3

KI•

127.7 121.7 115.7 109.7 103.7 97.7 91.7

microscopic

T,K

127.4 120.9 114.1 107.0 100.0 93.1 88.3

300 600 900 1200 1500

1.29 1.17 1.15 1.15 1.15 1.14 1.14

1800 2000

* Adiabatic datafromlsaaketal. [1989].

1.28 1.10 1.07 1.07 1.07 1.06 1.06

1.19 1.06 0.99 0.93 0.87 0.81 0.76

[1.22] [1.09] [1.02] [0.97] [O.92] [0.87] [0.83]

1.21 [1.24] 1.10 [1.13] 1.03 [1.06] 0.97 [1.O1] 0.91 [O.96] 0.85 [0.91] 0.79 [0.86]

1.36 !.29 1.23 1.16 1.09 1.03 0.99

[1.38] [1.32] [1.27] [1.21] [1.15] [1.10] [1.06]

t Calculated from KT=KsCv/Cp.

* Datafromlsaaketat. [1989],obtained from(23).

;eObtainedfrom data inversionwith (5).

J'Calculated fromequation (23)withthepresent or,KT,andCv.

õlsaaketal. [1989].

;eCalculated with equation(29) andKT=128GPa;all themodesare

Figure 12 with thoseobtainedby our inversionprocedureand the experimentalresultsof lsaak et al. [1989]. All the values agreeto within 2-4 % over the 300-2000K range.

used.The first value is obtainedwith all termsof the right-handsideof (29), the valuesin bracketsrepresentthe first term of the right-hand side of (29).

õSame asprevious column butthe•/Tofthestretching modes ofthe SiO4 tetrahedraare calculatedwith the Si-O bulk modulusof 190 GPa

reported byKudoh andTakduchi [1985].

• Calculatedwith equation(29) with the latticemodesonly (vi < 500

7. GRONEISEN PARAMETERS

cm-1).

Thermaland MicroscopicParameters The properties studied in the previous sectionsare also involved in the calculationof the thermalGrtineisenparameter, Ym= Z•qCvih/Z Cvih (25) y, which is extensively used for calculating the adiabatic gradientof the mantle.Macroscopically,y is definedby where the harmonic calculation is emphasized by the superscripth. (23) y= otKTV/Cv= V[c)P/t•U]v The parameterscalculated with (23) and (25) should be equal,but it hasbeenfoundthateqn(25) givesvaluestoo small and the values calculatedwith our anharmonicC v and the ct with respect to those obtained with (23) [e.g., Chopelas, and K T values obtainedfrom our data inversionare given in 1990]. Price et al. [1987] suggestedthat part of the Table 7. The results agree to within 10% with the figures discrepancyarises from the intrinsic anharmoniccontribution reportedby lsaak et al. [1989] or Whiteet al. [1985] whichare to Ym. Hence, in this sectionwe will first derive a general also included in Table 7.

relationshipbetweenYmand the individualmodeparameters in Originally, this parameter was introducedby Griineisen the case of an anharmonicsolid. But applicationof this [1912] in a microscopicway in the form of a vibrationalmode relationship to forsterite, for which a great many mode parameter •qT: parameterscan be determined[Chopelas,1990; this work] still •qT= - (01n vi/O lnV)T. (24) underestimatesy. We will finally discusspossiblereasonsfor the discrepancy between the macroscopic and microscopic We will note Ym the Grtineisen parameter which can be parametersand describea samplingmethodwhich would allow expressedin terms of these microcopicparameters.For the to obtain consistency between the microscopic and specialcaseof an harmonicsolid,one has[Slater,1939] macroscopicthermal parameters. Anharmonic

Griineisen

Parameters

The Helmholtz free energyof an harmoniccrystalis 1,04

F =Fo-kTZlnZi = FO + kT Z {(hvi/2kT )+ ln[1 - exp(hvi/kT)]}

ß

o

o

= FO+kr • Uih.

1,02 o

o

(26)

o o

o

Differentiatingwith respectto volume,one has 1,00

'

P =- (c•F&?V)T= PO+ (l/V)Z •'TUih, 0,98 300

'

ß

i

800

ß

i

1300

'

i

1800

Temperature(K)

(27)

where P0 is the pressureat 0 K. This expressionis in fact the Mie-Grtineisen equation of state. Differentiating further and making use of (12), in introducingintrinsicanharmonicityone obtains

Fig. 12. Comparison of the KT valuesof lsaaket al. [1989]with thoseobtainedfrom data inversionwith (5) (opencircles)or from (22) (solidcircles).Valuestakenfrom Table 6 arenom•alizedto the valuesof KTof lsaaket al. [1989].

(o•P/8I)v = (l/V)Z {•i'T[aiUih+ Cvih(1- aiT)] + (3•qT/3T)v Uih} = aKT.

(28)

GILLET ET AL.: HIGH-TEMI•RATURE PROPERTIESOF FORSTER1TE

11,815

Finally,the microscopic Gri•neisen parameteris thusgivenby:

SiO4 tetrahedra and a secondone related to lattice vibrations. The }tit of the lattice modeswould then be calculatedwith the mean bulk modulusof the lattice (in this case the crystalbulk 7m= 5•3'TCvih/Cv+ 5•3Tai(Uih- TCv?t)/Cv modulus or that of the Mg-O bond). We can then propose (29) + 5•7iT(•)•qT/•)T)v uih/Cv ß polyhedral Gr•ineisen parametersfor the internal vibrations This equation, which accountsfor anharmonicity,represents with the Si-O bond bulk •nodulus(KTSi_O). The 7iT for the so far the most general relation between the thermal and the internal modes would thus be written as microscopic Gri•neisen parameters. In the limit of high temperatures,this equationreducesto (31) ]qT =KTSi-O( • lnvi/•P).

7'm= 5•3'T/n+ T 5•(a•hT/aT)v,

(30)

The Grtineisenparmnetercomputedwith (29) and (31) agrees better with that obtained from (23) at 300 K, but it s.till andwe thusconcludeasusualthat 7mis the averageof the mode decreaseswith temperatureto reach0.86 at 2000 K (Table 7). Grtineisen parameters only if these are not functions of Thus a significantimprovement is obtainedin this way, for the temperatureat constantvolume. results(Table 7) agreeto within 10% between300 and2000 K with 7'as givenby (23). But for the momentthereare no clear Application to Forsterite theoreticalexplanations for this procedure.It couldbe justified In this section, we assessthe importanceof the various by the existenceof a sublatticewhich undergoes mostof the terms of (29) from the spectroscopicmeasurementson structuralchangesand in which rigid, isolatedSiO4 unitsare forsteritc. Only the optic modes are taken into account. embedded.In contrast,the classicalapproachfor calculating Because the data for all active modes are not known, we 7'mfrom the 71'Tshouldwork for •nineralshavingonlyonekind represent the16 modes lyingbetween 800and1000cm-1 by of polyhedra, or different polyhedra with si•nilar elastic the five well-determined Raman modes which are treated as properties, of which perovskite structures are the most Einstei n oscillators. Theremaining 65 modes aremodeled by important examples [Williams et al., 1987; Hemley et al., the other 13 observed Raman modes.

1989].

The major term of the right-handsideof equation(29) is the

We emphasize,however,that a differentexplanationcould

first one.The variationof (avi/o•P) T with temperature is be proposedfor the discrepancybetweenGrtineisenparameters unknownbecauseof the lack of spectroscopic measurements at

calculatedwith (23) and (29). Little is in fact known about the

simultaneously highpressure andtemperature. Hence, wewill variationof (3vi/•P)T with temperature.The availabledata

first assumethat this derivative is constantand evaluate (29) as

[Dietrich and Arndt, 1982] are sketchyand suggestthat this a functionof T from the ac.tualvaluesof KT andvi at different derivative decreaseswith temperature, in which case the temperatures.With this assumption,this term decreasesfrom disagreement couldbe worsestill. But moderateincreases of

1.22 at 300 K to 0.97 at 1200 K and 0.83 at 2000 K (Table 7). The two other terms of the right-hand side of (29) are negative and contributesignificantlyless, being of the order

this derivative are predictedif one assumesthat vibrational frequencies are mainlydetermined by thevolrune[Richetet al., 1991], and such increasescould account for the observed

of 10-1 and10-2 between 300and2000K, respectively (Table discrepancy at hightemperatures. Measurements of (3vi/gP)T

7). Their effect is to lower the valuescalculatedwith only the first term of the right-handside of (29) by 2-3% at 300 K and 10% at 2000 K. If the agreementbetweenthermalGrtineisen parameterscalculated with equations(23) and (29) is fair at 300 K, the parameter calculated with (23) reaches a hightemperaturelimit of about 1.15 [lsaak et al., 1989] or 1.06 (this work) whereas that obtained from (29) decreases continuously with temperaturedown to 0.8 at 2000 K. The disagreementat high temperaturesis thus real, even when consideringthe scatterin the reportedmacroscopic 7'[Boehler, 1982; Anderson and Suzuki, 1983; lsaak et al., 1989]. This difference can be attributed to several effects. First, too few modes would be taken into account in the calculation of

at different temperaturesare thusbadly neededto assessthe influence of this derivative on the various terms of (29).

In summary,the exampleof forsteritcshowsthat accurate

experimental Cpdatacanbeused toobtain reliable values ofo• andKT at hightemperatures fromtheroom-temperature values of theseparametersand an anharmonic calculationof Cv. Of

course, thisis notpossible formantle minerals whenCpisnot known experimentally.If, as is usuallythe case, a is also unknown, then the importance of the anharmonic

contributions to Cp makes it difficultto calculate Cp from spectroscopic data at high temperatures.In addition, the currentdiscrepancies betweenGrtineisenparameters calculated from macroscopic and microscopic data alsopreventsreliable

7'mwith (29), i.e., the 18 Ramanmodesusedwouldnot applicationof spectroscopic measurements to quantitative

represent correctly the optic modes of forsteritc. This modelingof thermodynamic propertiesof mantleminerals.As explanationappearsunlikely, however,becausesimilar results pointedout above,this discrepancyrequiresspectroscopic are obtainedwhenthe IR dataof Hofmeisteret al. [1989] for 15 measurements at both high temperatureand high pressurein additional modes are included in the calculations.

Anotherreasonfor the discrepancycould originatein the fact that forsteritchas strong(Si-O) and weak (Mg-O) bonds [Hazen and Finger, 1982; Kudoh and Takeuchi, 1985]. The existenceof contrastedbonds is partly reflected in the (a lnvi/c•P)Tvalueswhich are smallerfor the Si-O stretches than for the lattice modes (Figure 7). The above observations suggest,as emphasizedby Sherman [1980] in the case of molecularcrystals,the separationof the crystalvibrationsin two groups:a first one relatedto the internalvibrationsof the

order to be resolved. In addition, efforts should be made to

deviseways of determiningthe temperaturedependence of thermalexpansioncoefficientsfrom spectroscopic data. Acknowledgments. We thank Fredo Vaucelle and Christophe Sotin for help in computerprograming,B. Reynardfor fruitful discussions, O.L. Anderson,A. Chopelasand G. Helffrich for helpfulreviews.O. Jaouland Y. Gueguenkindly providedthe samples used in this study. Contribution CNRS-INSU-DBT 237.

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