LETTER Finite-Strain Analysis of Relative Compressibilities

American Mineralogist, Volume 76, pages 1765-1768, 1991

LETTER

Finite-strain analysis of relative compressibilities:Application to the high-pressure wadsleyitephase as an illustration

Rq,vN{oNo JuNr,oz Department of Geology and Geophysics,University of California, Berkeley,California 94720, U.S.A. Ronnnr M. HIzBN GeophysicalLaboratory and Center for High PressureResearch, Carnegie Institution of Washington, 5251 Broad BranchRoad NW, Washington,DC 20015-1305,U.S.A.

Ansrnl,cr Finite-strain analysisof recentlypublished hydrostaticcompression measurements shows that the isothermal bulk modulus of B-(Mg,Fe)rSiOowadsleyite is independent of composition [.00 > M/(Mg * Fe) > 0.75] within +2.5o/o.The averagebulk modulus over this compositionrange, Kor : l7 1.0 (+ 0.6) GPa, agreeswell with the value of 172.5(+ 1.0) GPa obtained by Brillouin scattering.This offers a significant crosscheckbetween the elastic moduli determined by independentmethods on an important mineral phaseof the Earth's mantle.

fNtnonucrroN DrscussroN The application of single-crystalX-ray diffraction tech- The Taylor expansionof enthalpy in the Eulerian mea- - niques to several specimensthat are being compressed sure of finite strain,/: l(y/VJ-2/3 ll/2, resultsin the within a single,hydrostatic pressurechamber offersa pre- isothermal equation of state, or pressure-volume(P- Z) cise method for determining the relative compressibilities relation of different crystals(Hazen, I 9 8 I Hazen and Finger, I 982; ; P -- 3Korft + zf)'zs(r + af + .. .) (l) McCormick et al., 1989;Hazen et al., 1990).In this way, it is possibleto resolve the effectson the equation of state in which Ko, is the bulk modulus and a : 3(K'o, - 4)/2 of varying composition, nonstoichiometry, site disorder- (Birch, 1952, 1978).Here, subscriptszero and Zdenote ing, or other subtle factors. Hence,the finite-strain theory ambient and isothermal conditions, respectively, and of equations of state has been extendedto the analysisof prime indicates differentiation with respect to pressure. such data (Jeanloz and Sato-Sorensen,1986; Jeanloz, Clearly, if X'o, : 4 the third-order coefficient (of strain I 99l). energyin strain) a : 0. The purpose of this note is to illustrate the finite-strain Birch ( I 978) has introduced the normalized stress,.F : analysisof relative compressibilitiesby applying it to the P/I3f(l + 2J)' 51,in order to analyzecompression data recently published data for the high-pressure mineral as a function of strain, / A plot of .F vs. /highlights the phase, 0-(Mg,Fe)rSiOowadsleyite (Hazen et al., 1990). constraints that the data place on the terms in the equa- The results turn out to be significant, not only as an il- tion ofstate because lustrative example but also becausethey addresstwo em- F:Ko,(l+af+...) (2) pirical conclusions that have been emerging from re- searchin mineral physics:(l) that finite-compressionand from Equation I . Thus, the intercept of such a plot yields acoustic (infinitesimal-compression) elasticity data are Kor, and if F is linear in /a third-order equation of state internally consistentwhen reducedby the Eulerian finite- is adequate;that is, coefficientsbeyond a in the bracket strain formalism (Birch, 1952, 1977, 1978;Jeanloz and of Equation 2 contribute insignificantly to the pressure Knittle, 1986;Jeanloz, 1989) and (2) that the bulk mod- over the compressionrange ofthe data. ulus is typically insensitive to composition acrossMg-Fe The measurementsof Hazenet al. (1990)for synthetic solid solutions for oxides and silicates (Anderson, 1976, wadsleyiteof compositionsMg/(Mg + Fe) : 100,92,84, 1989;Jackson et al., 1978).In particular,the new data and75o/oare summarized in this manner in Figure l. The help validate the mutual reliability of Brillouin-scattering averageof the zero-pressurevolumes, determined before and single-crystal,static-compression measurements and and after compression,is used to calculatethe strain, and 'are especiallyvaluable in offering all ofthese crosschecks the published uncertaintiesin the pressure(+ g.g5 6pu, for a mineral phasethat is thought to be abundant within and in the compressedvolume are propagatedto obtain the transition zone of the Earth's mantle (e.g., Jeanloz the error bounds on /and -F. Becausethe zero-pressure and Thompson, 1983;Gwanmesia et al., 1990b). volumes were found to be reproducible to within much 0003404x/9l/09 l 0-1765s02.00 t765 t766 JEANLOZ AND HAZEN: COMPRESSIBITIES OF HIGH-P PHASES

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z ++ 0 0.002 0.004 0.006 0.008 0.010 0 0.002 0.004 0.006 0.008 0.010 Eulerian Strain f EulerianStrain f Fig. l. Hydrostatic compressiondata of Hazen et al. ( 1990) Fig. 2. Ratios of normalized pressuresfor wadsleyite com- plotted as Birch's (1978) normalized pressureFas a function of positions Mg,oo(filled circles), Mg*n (open triangles), and Mg,, the Eulerian strain measure/ The bulk moduli of Mizukami et (open circles), relative to the Mge2composition, are shown as a al. (1975) Qargeopen circle) and Sawamoto et al. (1984) (open function ofstrain. The symbols are the sameas in Figure l, and square),obtained from static-compressionand Brillouin-scatter- a dashed line at F(x)/F(92) : 1.0 is shown for reference. rng measurementson end-mernberB-MgrSiOo, are shown at zero strain for comparison. A second-order Eulerian finite strain equation of state based on the Brillouin-scattering results is indicated by the horizontal line. F(i)/F(J) : K"r(i)/lQ,O) x + a,.f,* ...)/(l + a,f,+ ...)]. (3) lessthan their estimateduncertainties (Hazen et al., 1990), [(l the uncertainties in Zo are not included in the error esti- This is a measured quantity in the static-compression mates for the strain and normalized pressure. experiments,being completely determined by the volume As is evident from Figure l, the hydrostatic compres- strains of the two samples at a given pressure(Jeanloz sion data are entirely compatible with the determination and Sato-Sorensen,1986; Jeanloz, l99l). That is, the of Mizukami et al. (1975)by staticcompression of Ko,: pressureneed not be measured,in principle, as long as it 165 (+ 40) GPa. Also, the new measurementsare in good is known to be uniform acrossthe samplesi and 7 (for a agreementwith the value establishedby Sawamotoet al. given experiment). Hydrostatic conditions of stress are (1984) of Ko, : 172.5GPa: the adiabaticbulk modulus identical to a state of uniform, isotropic stress. obtained liom Brillouin scatteringis converted to the iso- The ratio of normalized pressuresis shown in Figure thermal value using the data listed in Jeanloz and 2, in which the Mgn, composition is taken as the reference Thompson (1983). Averaging of the elastic moduli prob- and the uncertainties quoted by Hazen et al. (1990) have ably contributes the largestambiguity in this acousticde- been propagated through. Two immediate conclusions termination of Ko., so t 1.0 GPa is a reasonableestimate from this plot are that Ko. is constant to within +2.5 of its uncertainty. Finally, the hydrostatic compression (+ 1.5)o/ofor all four compositions and that no variations measurementon the Mgl00 sample (Fig. l) is in agree- are resolvable in K'o, or higher order derivatives of the ment with the bulk modulus values Ko. : 164.4-166.8 bulk modulus. The latter point is demonstrated by the (t 0.8) GPa derived from the ultrasonic velocities vari- fact that least-squaresfits to the ratios in Figure 2 show ously reported by Gwanmesiaet al. (1990a, 1990b)for no statistically resolvable dependenceon strain. From polycrystalline B-Mg,SiOo. Equation 3, this implies that the third-order (and higher) The straiglrt line shown in the figure is appropriate for terms are insignificant in evaluating the relative com- a second-orderequation of state(a : 0, K'or: 4).In fact, pressibilities from these data. there is no statistical justification for incorporating any Also, if one assumesthat Ko. does not vary with com- higher order terms when reducing this data set. Least- position and takes a -- 0 (K'or: 4) for the Mgn, compo- squaresfits of the data of Hazen et al. (1990) in terms of sition, which is a good approximation accordingto Figure F vs. /yield slopes that are indistinguishable from zero l, then a deviationof +0.03 from F(x)/F(x: 92) : 1.00 (although a slope may be apparent on casualobservation, for strains,f = O.OO8Srequires variations in K'0, as large none is warranted when the data are properly weighted). as +4.7 from the value of 4. This is an implausibly large This conclusion supports the assumption of Hazen et al. variation in K'o, with composition since 2 - K'o, < 6 for that K'or: 4 in their Birch-Murnaghan fit of the data. most materials(e.g., Birch, 1977;Jeanloz, 1989). Thus, In order to evaluate the relative compressibilities among small-strain (/ < 0.01) measurementsare insensitive to samples of composition I and J, one simply forms the reasonable variations or uncertainties in K[. (Jeanloz, ratio of the normalized pressures l99l), and the presentdata must be analyzedassuming JEANLOZ AND HAZEN: COMPRESSIBITIES OF HIGH-P PHASES r767 that K'o, is independent of composition. As already im- plied, a second-orderequation ofstate is adequate. 175 With theseassumptions, the bulk modulus ratio Kor@)/ I Brillouin Kor(x:92) is simply given by the weightedaverage of I Aug. F(x)/F(x:92), ((s')/ (s 'z)where ( denotesthe average T t ) F ' and s is the estimated uncertainty for each value of f : vl 70 rhis Study F(x)/ F(x: 92). The resultsfor the compositionsMg,oo, q I Mgro, and Mgrr, relative to the Mgn, value, are 0.950 (+ 0.017),0.972 (! 0.014),and 0.975(t 0.013),with un- x given (s z) tz:. certainties by Becausethese values are J obtained from different strain measurementsthat are col- lected simultaneously at each pressure,they should be Wadsleyite free of most systematicbiases that can influence the fitting of static-compressiondata for individual samples. 0 0.1 0.2 Except for the Mg,oovalue, the bulk moduli are indis- Composition Fe/(Mg + Fe) tinguishable from the Mgn, value atthe 2o level. Overall, the variation amounts to less than 5olofor the different Fig.3. Summaryof the zero-pressure,isothermal bulk mod- wadsleyiteas a functionof composition. wadsleyite compositions, and factors other than compo- ulusof d-(Mg,Fe)rSiOo The trianglesindicate the resultsobtained from the hydrostatic sition, such as variations in intersite disordering, could compressionmeasurements according to the analysisof Hazen ',rdthin be influencing the relative compressibilities this et d. (1990)(open) and the presentanalysis (filled). In both cases, range. K'or: 4.0is assumed;tahng K'r, : 4.7 (Gwanmesiaet al., I 990b) Consideringthat the unit-cell volume of wadsleyitein- afectsthe derived1(o.values by <1.8 GPaover the strainsof creasesby 1.50/oas the Mg content decreasesfrom 100 to the measurementsof Hazenet al. The averagevalue (and un- 75o/o(Jeanloz and Thompson, 1983;Hazen et al., 1990), certainty)obtained for all compositionsin the presentstudy is a decreasein bulk modulus of -6 (+ 3)o/omight be an- givenby the horizontaldashed line. For comparison,the open ticipated over the same composition range,assuming a circle,open square,and crossesshow, respectively, values for the Mg end-memberdetermined by Mizukamiet al. (1975),Sa- logarithmic volume derivative of bulk modulus @K/n/ wamotoet al. (1984),and Gwanmesia et al. (1990a,1990b). @V/n*o - -4 (! 2). As noted above,the presentdata are consistent with pressure derivatives (for constant composition)K'or: -(d ln Kr/lln l)r:o: 4. Instead, both the presentanalysis and that ofHazen et Because we have found that the bulk modulus of al. (1990) show that the bulk modulus of wadsleyitedoes g-(Mg,Fe)rSiOois independentof Mg-Fe composition, an not decreaseas the Mg content decreases.If anything, averagevalue of Ko' : 171.0(+ 0.6) GPa can be derived there is a slight increase,as indicated by weighted fits of from the hydrostatic compression data of Hazen et al. the normalized pressures,4 for each composition (Fig. (1990).The quantitativeagreement with Ko,: ll).J (+ 3):Ko,: 164.8(+ 2.3),174.5(+ 1.8),169.3 (+ 1.4),and 1.0) GPa obtained by Brillouin scattering (Sawamoto et 170.I (+ 1.4)GPa for Mg,oo,MBrr, Mgro, and Mg^ sam- al., 1984) is a significant validation of the reliability of ples, respectively (lo estimated uncertainties).This is in the two methods. complete agreementwith previous findings for other Mg- In comparison, the recently reported ultrasonic mea- Fe oxide solid solutions; the bulk modulus is essentially surementson polycrystalline wadsleyiteyield bulk mod- constant or increasesslightly (-2-l2o/o) with increasing uli -2.5-3.9o/o lower in value. This differenceexceeds the Fe substitution(Jackson et al., 1978). estimated uncertainty of the data (-0.50/0, Gwanmesia et As is evident from Figure 3, thesevalues of bulk mod- al., I 990a) and suggeststhat porosity still affectsthe poly- uli differ in absolute magnitude from those quoted by crystalline measurementsat pressuresof 2-3 GPa. In- Hazen et al. (1990). The main reason for this difference deed, Ki, = 4 implies that the bulk modulus changesby is that Hazen et al. did not use Zo in their least-squares an amount comparable to this discrepancy(2-4Vo) over fits of the data to the Birch-Murnaghan equation of state. l-2 GPa. Therefore, the estimate by Gwanmesia et al. In the presentanalysis, we incorporate the observedval- (1990b) of the pressurederivative of the bulk modulus ues of Zo in the data reduction, and as noted by Hazen may be somewhat high. et al., the resulting bulk moduli are systematicallyhigher by - 5 GPa. This illustrates how elasticity parameters AcrNowr,rncMENTs obtained from finite-compressionmeasurements are sen- We have benefited from discussions with E. Knittle, M. Kruger, C. sitive, at the level of a few percent as is being considered Meade, and B. O'Neill. This work was supported by the National Science here, to the detailed assumptionsthat are made in fitting Foundation. the data (e.g., weighting of data, assumptions regarding fixed parameters).Specifically, it is important to include RerrnnNcns crrED the constraint of Vo,when this value is well determined, Anderson, D.L. (1976) The 650 km mantle discontinuity. Geophysical in deriving equationsof statefrom pressure-volumedata. ResearchL.etters, 3. 347-349. I 768 JEANLOZ AND HAZEN: COMPRESSIBITIES OF HIGH-P PHASES

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