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ElasticityofCalciumandCalcium-SodiumAmphiboles1
J.MichaelBrown*brown@ess.washington.edu2Fax:206543048934EvanH.Abramsonevan@ess.washington.edu56EarthandSpaceSciences7UniversityofWashington8Seattle,WA98195-1310910*Correspondingauthor1112
Abstract13
Measurementsofsingle-crystalelasticmoduliunderambientconditionsare14reportedforninecalciumtocalcium-sodiumamphibolesthatlieinthecomposition15rangeofcommoncrustalconstituents.Velocitiesofbodyandsurfaceacousticwaves16measuredbyImpulsiveStimulatedLightScattering(ISLS)wereinvertedto17determinethe13modulicharacterizingthesemonoclinicsamples.Modulishowa18consistentpattern:C33>C22>C11andC23>C12>C13andC44>C55~C66andfortheuniquely19monoclinicmoduli,|C35|>>C46~|C25|>|C15|~0.Mostofthecompositionally-induced20varianceofmoduliisassociatedwithaluminumandironcontent.Sevenmoduli(C1121C12C13C22C44C55C66)increasewithincreasingaluminumwhilealldiagonalmoduli22decreasewithincreasingiron.Threemoduli(C11,C13andC44)increasewith23increasingsodiumandpotassiumoccupancyinA-sites.Theuniquelymonoclinic24moduli(C15C25andC35)havenosignificantcompositionaldependence.Moduli25associatedwiththea*direction(C11C12C13C55andC66)aresubstantiallysmaller26thanvaluesassociatedwithstructurallyandchemicallyrelatedclinopyroxenes.27Othermoduliaremoresimilarforbothinosilicates.Theisotropicallyaveraged28adiabaticbulkmodulusdoesnotvarywithironcontentbutincreaseswith29aluminumcontentfrom85GPafortremoliteto99GPaforpargasite.Increasing30ironreduceswhileincreasingaluminumincreasestheisotropicshearmodulus31whichrangesfrom47GPaforferro-actinoliteto64GPaforpargasite.Theseresults32exhibitfargreateranisotropyandhighervelocitiesthanapparentinearlierwork.33Quasi-longitudinalvelocitiesareasfastas~9km/sand(intermediatebetweenthe34a*-andc-axes)areasslowas~6km/s.Voigt-Reuss-Hillaveragingbasedonprior35singlecrystalmoduliresultedincalculatedrockvelocitieslowerthanlaboratory36measurements,leadingtoadoptionofthe(highervelocity)Voigtbound.Thus,37formerusesoftheupperVoigtboundcanbeunderstoodasanadhocdecisionthat38compensatedforinaccuratedata.Furthermore,becausepropertiesoftheend-39memberamphibolesdeviatesubstantiallyfrompriorestimates,allpredictionsof40rockvelocitiesasafunctionofmodalmineralogyandelementalpartitioningrequire41reconsideration.42
KeyWords:elasticity;anisotropy;seismicvelocities;aggregateelasticity;43amphibole;hornblende 44
1. Introduction45
Amphibolesareabundantincrustaligneousandmetamorphicrocks.Theyexhibita46widerangeofcompositionsasaresultofextensivesolidsolutionbehavior,47accommodatingalloftheabundantcationspecies(silicon,aluminum,magnesium,48iron,calcium,sodium,andpotassium).Thestructurealsocontains~2wt%bound49water.Whensubducted,dehydrationreactionsinrockscontainingamphiboles50releasewateratdepththatprobablyaffectstheevolutionofmagmainarc51volcanismandislikelyassociatedwithintermediateanddeepearthquakes(Hacker52etal.2003a)aswellasseismictremor/slowslip(Audetetal.,2010).53
Sinceamphibolesareubiquitous,thedescriptionofthecrustalseismicstructure54requirescharacterizationoftheirelasticproperties(e.g.ChristensenandMooney551995,Christensen1996,Hackeretal.2003b,Barberinietal.2007,Tathametal.562008,Llana-FunezandBrown2012,Jietal.2013,Selwayetal.2015).However,57knowledgeconcerningtheirsingle-crystalelasticityandcompositionaldependences58hasremainedelusive.Inthepioneeringworkthatcontinuestobecited,59AleksandrovandRyzhova(1961a)reportedsinglecrystalelasticmodulifortwo60“hornblendes”ofunspecifiedcompositionbasedononlyslightlyover-determined61setsofultrasonicvelocitymeasurementsonmegacrystsunderambientconditions.62Aspreviouslydemonstratedinstudiesoffeldspars(Brownetal.2006;Brownetal.632016;andWaeselmannetal.2016),resultsfromtheseearlystudieshaveprovento64besystematicallyinerror.65
Thattheearlyultrasonicresultsunder-estimatevelocitiesmostlikelywasaresultof66opencleavagesurfacesandcracks.Alsocontributingwasaninadequatesamplingof67velocitiesasafunctionofpropagationdirection.Basedonthelackofreported68chemistryandprobablesystematicerrors,theseearlyresultsareconsideredherein69thecontextofhavingincorrectlyinfluencedvariousinterpretationsofcrustal70seismicstructurethatweregroundedonmineralproperties.Inparticular,inorder71tobettermatchlaboratorymeasurements,thecompensatinguseoftheupperVoigt72boundwhencalculatingaggregaterockvelocitieshasbeencommon.Incontrast,73WattandO’Connell(1980)demonstratedthat,inwell-characterizedandnearly74crack-freesamples,velocitiesintwophaseaggregatesfellwithintheHashin-75ShtrikmanboundswhichliebetweentheextremalVoigtandReussbounds(seealso76Wattetal.1976).77
Afewdeterminationsofsingle-crystalelasticpropertiesareavailablewithinthe78broadrangeofamphibolecompositions.Bezacieretal.(2010)gaveelasticmoduli79foracrystalhavingacompositionneartheglaucophaneend-member.Highpressure80x-raycellparameterdeterminationshavebeenreportedfortremolite,pargasite,81andglaucophane(Comodietal.1991)andforsyntheticglaucophane(Jenkinsetal.822010).83
Hackeretal.(2003b)compiledavailable(isotropicallyaveraged)elasticitydatafor84importantrock-formingmineralsincludingamphiboles.Theyexcludedthe85AleksandrovandRyzhova(1961a)moduliasprobablybeinginerrorandreliedon86theChristensen(1996)rockvelocitymeasurementstoestimatepropertiesofan87averagecrustal“hornblende”.Toconstrainpropertiesofotherend-member88
compositions,theyusedbulkmodulifromtheHollandandPowell(1998)89thermodynamicdatabaseplustheComodietal.(1991)compressionmeasurements.90Althoughanisothermalbulkmoduluscanbeinferredfrompressure-inducedstrains,91theshearmodulus,necessarytoestimatebodywavevelocities,cannotbe92determinedsolelyfromthehydrostaticx-raydata.Instead,Hackeretal.(2003b)93estimatedshearmodulionthebasisofthereportedbulkmoduliandanassumed94Poisson’sratio.Theynotedthatthiswasaremainingsourceofuncertainty.95
Asnotedinsection6.3,anisothermalbulkmodulusmeasuredunderhydrostatic96compression(whichisequivalenttotheelasticaggregatelower-boundReuss97average)issignificantlysmallerthantheappropriateVoigt-Reuss-HillorHashin-98Shtrikmanbulkmodulususedforcalculationofseismicvelocities.Bulkmodulifor99someamphibolesgivenbyHackeretal.(2003b)appeartorepresenttheReuss100bound.Theycombinedlowerboundmoduli(insomecases)inanupper-bound101Voigtaverageforcalculationsofvelocitiesinrocksasmixturesofminerals.Thus,102theaccuracyoftheiranalysisreliedonhowwellthetwoerrorsoff-seteachother.103
Hereelasticmoduliarereportedfornineamphibolesthatlieintherangeof104compositionscommonlyfoundincrustalrocks(Schumacher2007).Elasticwave105velocities(quasi-longitudinal,quasi-transverseandsurfaceacousticwaves-SAW)106weremeasuredusingImpulsiveStimulatedLightScattering(ISLS)(Abramsonetal.1071999).Ajointinversionallowedaccuratedeterminationofthe13elasticmodulifor108thesemonoclinicminerals.Thedependencesofmodulioncompositionare109determinedthroughlinearregression.Fromthese,relationshipstocrystalstructure110andseismicvelocitiescanbeexplored.Ultimately,moreaccuratepredictionsof111seismicpropertiesofrockscanbeundertakenonthebasisofmodalmineralogyand112elementalpartitioning.113
2. Amphibolechemistryandstructure114
AsreviewedbyHawthornandOberti(2007),themonoclinic(C2/m)calcium115(includingcommonhornblende)tocalcium-sodiumamphiboleshaveageneralized116formulaof117
A0-1B2C5T8O22(OH)2118
wheretheA-siteisoccupiedbysodiumandpotassiumorremainsvacantandtheB-119siteisoccupiedbysodiumorcalcium.TheoctahedrallycoordinatedC-sitescontain120iron(divalentortrivalent),magnesium,oraluminum(designatedasviAl).The121tetrahedralT-sitescontainsiliconandaluminum(typicallyuptotwoaluminumper122eightsites,occasionallymore,anddesignatedasivAl).Othercommonminor123chemicalcomponents(Ti,Mn,Co,Cr)arefoundinsizeandvalence-state124appropriatesites.FluorineandchlorinecansubstituteforOH-1.125
Thenamingconventionsassociatedwithchemistryofcalciumandsodium126amphiboles(Hawthorneetal.2012,seealsoLeakeetal.1997)areillustratedin127Figure1usingthreecompositionalvariables.Althoughcompletesolid-solution128substitutionispossiblewithinthiscompositionalspace,severalofthe129stoichiometriccompositionsaregivendiscretenames.Tremoliteis130[]Ca2Mg5Si8O22(OH)2(wherethebracketsdenotethevacantA-site)whilewinchite131
Figure1
hasonecalciumandonesodiumintheB-site.GlaucophanehasallsodiumintheB-132sitewithcoupledsubstitutionsofatrivalentcationinC-sitesrequiredtobalance133charge.Hornblendeisbothanend-memberinFigure1andisageneralizedtermfor134calciumamphiboleswithintermediatetetrahedralaluminumcompositions.In135addition,solid-solutionsubstitutionofironformagnesiumgivesrisetoironend-136membersforallphasesshowninFigure1withferro-addedtothename(e.g.ferro-137pargasite).138
AmphiboleshaveI-beamstructuresoftwodoubletetrahedralchainsthatare139bondedtoeachotherbyanoctahedralsheetcontainingfiveC-sitecations.TheI-140beamsareorientedalongthec-axiswithA-sitecations(whenpresent)bondingthe141I-beamsalongthea-axisandB-sitecationsservingtobondI-beamsalongtheb-142direction.Clinopyroxenessharesimilarchemicalvariationsinastructurethatis143closelyrelatedtotheamphiboles,bothbeinginosilicatesbutthepyroxeneshavea144singletetrahedralchainsalignedalongthec-axis.Thegeneralformulaofthe145clinopyroxeneisBCT2O6withtheBandCsitesbeingequivalenttothosefoundin146theamphiboles.Endmemberpyroxenesincludediopside(CaMgSi2O6)147hedenbergite(CaFeSi2O6),andjadeite(NaAlSi2O6).148
Havingawiderangeofsolid-solutionsubstitutionsforessentiallythesamecrystal149structure,amphibolesprovideanaturallaboratoryfortheexplorationofchemical150controlsonelasticity.Variationsinelasticmoduliareanticipatedfromchangesof151ionicsizesandcharges,asaresultofcationsubstitutionsintheA,B,CandTsites.152Comparisonsofelasticitybetweenpyroxenesandamphibolesprovidesfurther153opportunitiestoexplorefactorsinfluencingelasticbehavior.154
3. Samplesourcesandcharacterization155
Thesources,localities(whenknown),x-raydeterminedcellparameters,and156densitiesofnineamphibolesaregiveninTable1.Individualcrystalsas2-3mm157mineralseparateswereobtainedeitherfromdisaggregatedcrystallinerocksor158werebrokenofflargerpreviouslycollectedcrystals.Chemicalhomogeneityof159samplesfromeachsourcewasconfirmedbyapplicationofanalyticmethodstoall160individualsamplesusedinthestudy.161
MicroprobeanalysesforallsamplesarereportedinTableS1ofthesupporting162materials.Alackofchemicalzoningwasconfirmedinallcrystalsandthereported163analysesaretheaveragesofpointdeterminationsacrosseachcrystal.Structural164formula,basedonProbe-AMPH(TindleandWebb,1994),aregiveninTable2and165areplottedinperspectiveinFigure1.Samples1and2with~1sodiumintheB-site166areclassifiedascalcium-sodiumamphiboles.Theremainingsevensamplesare167calciumamphiboles.ThechemistryoftheglaucophanesampleusedbyBezacieret168al.(2010)andtheaveragecalciumamphibolecompositionreportedbySchumacher169(2007)arealsoincludedinTable2.AsnotedbySchumacher(2007),calcium170amphibolescoverawiderangeofintermediatecompositionswithinthe171compositionalspacedefinedinFigure1andhisreportedaverage(basedonover1721700publishedanalyses)maybeabiasedestimatorofanaverage“hornblende”in173crustalrockssincesampleswereanalyzedforspecificscienceinterestsratherthan174beingchosentobestrepresentcrustalchemistry.Nonetheless,thisaverage175
Table1
Table2
Figure2
providesareferencepoint,definingacommonhornblendecompositioninthe176followingdiscussion.177
Theninesamplesusedinthisstudyshowarangeofcompositions(Figure2)that178generallybracketstheaveragesreportedbySchumacher(2007).Thisis179prerequisitetodeterminingcompositionalcontributionstotheelasticproperties180withintheappropriateboundsofelementalpartitioningincrustalcalcium181amphiboles.182
4.Experimentalmethods183
Thefollowingconventionisadoptedtoalignthecrystallographicaxeswithrespect184totheCartesianaxesforthedescriptionoftheelastictensor.TheYaxisisaligned185paralleltothecrystallographicb-axisandtheZaxisisalignedparalleltothec-axis.186TheXaxisissetinthea*-direction(perpendiculartotheb-andc-axes).Elastic187moduli(stiffnesses)arerepresentedbythe6by6matrixCijusingtheVoigt188convention.TheinverseofthismatrixisthecompliancematrixSij.189
Theverticalsumofthefirstthreerowsofthecompliancematrixgivessixstrains,bi,190associatedwiththeapplicationofunithydrostaticstress.Basedon2/msymmetry191b4andb6arezero.Thesestrainscanbecastasa3x3symmetrictensorwhichgives192crystalcompressibilityunderhydrostaticstressatthelimitofzerostress.193
Threecrystalsforeachsamplewereorientedonanx-raydiffractometer.The194crystals,inselectedcrystallographicorientations,weregluedtoglassslides195(mechanicallyindexedtolaboratorycoordinates)whilestillattachedtothex-ray196goniometerhead.Samplesweresubsequentlygroundandpolishedontwosides197using¼microndiamondgritforthefinalpolish.Thefinalthicknessesofsamples198wereoveranorderofmagnitudegreaterthanthenominalacousticwavelength(~1992.5microns).Theorientationsofseveralsamples,re-checkedonthex-ray200diffractometeraftercompletionofthegrindingandpolishingprocess,agreedwith201theoriginalorientationtowithinfourdegrees.PriortomeasurementsofSAW202velocities,a40(+/-5)nmlayerofaluminumwasdepositedonthetoppolished203surface.Thismetalliclayerallowedcouplingoftheincidentlaserenergytothe204samplesurface(Brownetal.2006).Thechangeinvelocitycausedbythealuminum205layerislessthan0.2%andisaccountedforintheanalysisdescribedbelow.206
Bothbodywave(quasi-longitudinalandquasi-transverse)andsurfaceacoustic207wave(SAW)velocitiesweremeasuredusingthemethodofImpulsiveStimulated208LightScattering(ISLS)(Abramsonetal.1999).Theexcitationspotsizewasabout209200micronsandtheprobewasfocusedtoabout15microns.Sinceoptical210absorptionissmallinthesesamples,laserheatingisestimatedtobelessthan0.2°C211(Brownetal.1989).InthecaseofSAWmeasurements,theexcitationlaserintensity212wasreducedwellbelowthepowerrequiredtodamagethesurfacecoatingandthe213heatabsorbedbythethinaluminumlayerresultedinanegligiblechangesofsample214temperatures.215
Theopticalqualityofthenaturalsamplesinsomecasespresentedexperimental216challenges.Opticaldefects(includingcracks,cleavageseparations,andinclusions)217werecommon.Theseservedtoincoherentlyscattertheprobelaser,causing218
saturationofthedetector.Successfulmeasurementscouldbemadeifa~100219micronnearlydefect-freeregionwasavailable.Sincepolishedsurfaceswere220typicallygreaterthan1000micronsacross,regionsofadequatequalitycould221usuallybefound.Photomicrographs(supplementalmaterialsFigureS1)ofseveral222samplesillustratestypicalcrystalqualityfortheseexperiments.223
Allmeasurements(typicallybetween150and200individualvelocity224determinationspersample)arereportedinTableS2ofthesupplementary225materials.Furtherdetailsoftheexperimentsandthemethodsusedtodetermine226elasticparametersforlowsymmetrymineralshavebeendescribedforbodywave227measurements(CollinsandBrown1998)and,separately,forSAWmeasurements228(Brownetal.2006).Thethreeeuleranglesthatrelatethecrystalaxestolaboratory229coordinateswerealsooptimizedtoaccountfortheorientationerrorsintroducedby230thesampleprocessingstepsgiveninthefirstparagraphofthissection.The231numericalmethodsaredescribedinBrown(2016).232
NewinthisworkisthejointinversionofbothbodyandSAWvelocitiesasdescribed233inBrown(2016).Althoughacompletebodywavedatasetissufficienttodetermine234allelasticmoduli,asnotedabove,itproveddifficulttoobtainafullsetofvelocities235(quasi-longitudinalandtwopolarizationsofquasi-transversewaves)forall236propagationdirections.Insomedirections,internalflawsscatteredlightsostrongly237thatthebodywavesignalcouldnotberecoveredfromthebackground.Separately,238velocitiesforbothpolarizationsoftransversewavescouldnotalwaysbeobtainedin239anadequatenumberofpropagationdirectionsasaresultofsmallvaluesofeither240the(angle-dependent)piezo-opticcoefficientorabsorption.SAWvelocities,based241onlightcoherentlyscatteredfromapolishedsurface,couldbemorereadily242measuredforalldirectionsofpropagation.243
Thetypicalrmsmisfit(reportedinTableS2)obtainedthroughjointfittingofall244measurementsis~0.3%.Thisis10-12m/sforSAW(fornominalvelocitiesnear3-4245km/s)and16-20m/sforbodywaves(fornominalvelocitiesof6-9km/sforquasi-246longitudinaland3-5km/sforquasi-transversewaves).Suchmisfitsareessentially247identicaltothosepreviouslyreportedforindividuallyanalyzedbodywaveandSAW248datasets(e.g.CollinsandBrown1998andBrownetal.2016)andarethoughtto249representintrinsicrandomerrorsassociatedwithstudiesbasedonnaturalcrystals.250Thus,noadditionalsystematicerrorsappeartohavebeenintroducedthroughjoint251analysisofbodywaveandSAWvelocities.Furthermore,theconsistencyof252velocitiesmeasuredinseparatecrystalswithdifferentorientationsforsimilar253polarizationsandpropagationdirectionsandtheconsistencyofresultsforsurface254wavesthatprobedthetopfewmicronsofeachcrystalrelativetobodywave255measurementsthatsampledalargerinternalvolume,arguesthatthecrystalswere256adequatelyhomogeneousandthatthemoduliarerepresentativeofeach257compositionalsample.258
TheresultingelasticmoduliCijandtheirassociated2σuncertaintiesarelistedin259Table3forthenineamphibolesplusglaucophane.Thecompliancematrixelements260Sij(inverseofthematrixCij)andcompliancessums,bi,arelistedinTableS2.The261sumsarealsogiveninprincipalaxiscoordinatesofthehydrostaticcompressibility262ellipsoid.263
Table3
BodywavevelocitiesforglaucophaneinBezacieretal.(2010)werereanalyzed264usingthesamenumericaloptimizationmethodsusedhere.Optimizationofcrystal265euleranglesallowedreductionofvelocitymisfitfromthepreviouslyreportedrms266errorof43m/s(~0.8%scatterinvelocitymeasurements)to37m/s(~0.6%267velocityscatter).Someofthenewmodulidifferby~2GPa.Someuncertainties268givenbyBezacieretal.(2010)aresubstantiallydifferentfromthosereportedhere269(seefurtherdiscussioninBrown2016).Inparticular,itwouldappearthat270previouslyreporteduncertaintiesofsomeoff-diagonalmoduliwereunder-271estimatedandseveraldiagonaluncertaintieswereover-estimated.Moduli272uncertaintiesforglaucophaneareroughlytwiceaslargeasthoseforthecalcium273andcalcium-sodiumamphibolesasisappropriatefortheobservedlargermisfitto274measuredvelocities.275
5. Elasticmoduliandtheircompositionaldependence276
Elasticmoduliandisotropicbodywavevelocitiesareplottedasafunctionoftotal277aluminuminFigure3.Alsoshownarepredictions(describedbelow)basedonlinear278regressioninchemicalcompositionthataccountformostoftheobservedvariance.279Modulithatarenon-zerofororthorhombiccrystalsareshowninthetopthree280panels.Theuniquelymonoclinicmoduliareplottedinthelowerleftpanel.281AdiabaticbulkandshearmoduligivenasthemeanofHashin-Shtrikmanbounds282(Brown2015)areshowninthemiddlelowerpanelandtheresultingisotropic283compressionalandtransversewavevelocitiesareinthelowerrightpanel.284
Forallcompositions,therelativesizesofthemoduliremainconsistent.Thatis285C33>C22>C11andC23>C12>C13,andC44>C55~C66andfortheuniquelymonoclinicmoduli,286|C35|>>C46~|C25|>|C15|~0.Thesamepatternandroughlysimilarmoduliareapparent287forglaucophane.However,theC22,C33,andC23moduliofglaucophaneare288significantlystiffer.Asfurtherdiscussedbelow,thelargevalueofC35(comparable289totheoff-diagonalorthorhombicmoduliandlargerthanallothermonoclinic290moduli)isresponsibleforarotationofelasticextremainthecrystallographicplane291containingthea-andc-axes.292
Sixchemicalcontrolsonelasticityassociatedwithchangesincationchargesand293sizescanbeidentifiedaslikelytoproducesignificanteffects.Theseare(1)total294aluminumcontent,oritsseparatecontentineither(2)T-sitesor(3)C-sites,(4)iron295contentinC-sites(mainlyreplacingmagnesium),the(5)degreeofA-siteoccupation,296and(6)sodiumreplacementofcalciuminB-sites.Otherpossibilitiesthatareless297likelytohaveameasureableimpact(includingreplacementofOH-1withCl-1orF-1,298orchangesintheferric-ferrousironratio)couldnotbeinvestigatedusingthe299currentsamples.300
Moduliareassumedtobelinearinthesixcompositionalmetricsidentifiedabove.301Tremoliteisusedasthebasecompositionandchangingchemicalcontentisgivenby302theformulaunitmeasureslistedinTable2.Astandardstatisticalmeasure,theF-303test(Rencher2002),determinedthesignificanceoftheproposedmetricsthrough304stepwiseadditionandremovaloftermsusingMATLAB®functionstepwiselm.Only305threecompositionaltermswerefoundtohavesignificantimpactatthe95%306confidencelevel;thesearetotalaluminum,A-siteoccupancy,andironintheC-site.307
Figure3
Despitethedifferenceinchargeandionicradius,thesubstitutionofsodiumfor308calciumintheB-siteappearstohavenegligibleimpactonmoduli.Norwasthefit309significantlyimprovedbyallowingseparatecontributionsofaluminuminC-andT-310sites.Atthe95%confidencelevel,notestedparameterizationcouldreconcilethe311glaucophanemoduliwiththecalciumandcalcium-sodiumamphibolesmoduli.This312suggeststhattheelasticityofthefullysodiumamphiboledoesnotlieona313continuumoflinearsolidsolutionbehavior.Regressionsbasedoncompositional314“vectors”thatarelinearcombinationofcompositionalmetrics,assuggestedby315Schumacher(2007),werenotsuccessful.316
RegressionparametersandmisfitstatisticsarelistedinTables4and5.Blanksin317thetablesindicatenosignificantcontributionsforparticularterms.Forthethree318modulishowingsensitivitytoA-siteoccupancy,thealternativeparameterizations,319usingonlytotalaluminumandironcontent,arelistedwithconcomitantlarger320misfits.Asshowninthetables,mostregressionmisfitsarecomparableto321experimentaluncertaintybuttendtobeslightlylarger.However,furtherreduction322ofvariancebyallowingmoredegreesoffreedom(additionalcompositionalmetrics323ornon-lineardependencesofmoduliwithchemistry)isnotstatisticallyrobust.324
Predictionsformoduliandisotropicvelocitiesbasedonthethreecompositional325metricslistedinTables4and5arerepresentedinFigure3.Sincesample326chemistriesarevariable,predictedmoduliandvelocitiesdonot,ingeneral,lieon327theplottedlinesthatarebasedoneitheriron-freeorferro-equivalentminerals328(bothwithnoA-siteoccupation).Inallpanelspredictedmoduliandvelocitiesare329nearlywithinuncertaintiesofthemeasurements.330
Alldiagonalelasticmoduli(andtheisotropicaverageshearmodulus)decreasewith331theadditionofironandderivativesofthesemoduliwithrespecttoironaresimilar.332Sevenofthirteenmoduli(C11C22C13C12C44C55C66)increasewithaluminumcontent333whileC46decreaseswithaluminum.Fivemoduli(C33C23andthemonoclinicmoduli334C15C25andC35)havenosignificantdependenceonaluminum.Bothisotropicmoduli335(bulkmodulusandshearmodulus)increasewithaddedaluminum.OnlyC11C13and336C44aredependentonA-siteoccupancy;misfitsaresubstantiallylargerforthe337alternativeassumptionofnodependenceonA-siteoccupancy.Allthree338compositionalmetricsarenecessarytoadequatelypredictthevariationsofdensity339andtheisotropictransversewavevelocitieswhilealuminumandironcontentare340sufficienttopredictcompressionalvelocities.341
6. Discussion342
6.1Compositionalandstructuralcontrolsonelasticity343
InTable6theelasticmoduliofseveralamphibolecompositionsarecomparedwith344chemicallyrelatedclinopyroxene.Amphibolemoduliassociatedwithlongitudinal345stressesandstrainsinvolvingthea*-axis(ie.C11C12C13C55C66)areallsignificantly346smallerthancorrespondingclinopyroxenemodulibyapproximatelyafactoroftwo347whilemoduliassociatedwiththeb-andc-axes(C22C33C23C44)arenotablysimilar.348That(asshowninTable4)C11andC13increasewithincreasingA-siteoccupation349seemsreasonablesincecationsintheA-siteprovideadditionalbondingandthus350additionalresistancetocompressionalongthea*-direction.However,evenwithfull351
Table4
Table5
Table6
A-siteoccupations,theseamphibolemoduliremainsmallerthanthosefor352pyroxenes(thatlacktheA-site).Thereversalofsignfortheuniquelymonoclinic353moduli(C15C25andmostimportantlyC35)areresponsibleforamajorshiftinthe354orientationofanisotropybetweenamphibolesandpyroxenesthatisfurther355discussedinSection6.2.Withafewexceptions,amphibolesandthecompositionally356relatedclinopyroxenesshowsimilarpatterns:addedaluminumincreasessome357moduliandaddedironlowersthediagonalmoduli.Thesetrendsinamphibolesare358furtherexploredthroughcomparisonofvelocityanisotropyandthehydrostatic-359inducedstrainanisotropy.360
6.2VelocityAnisotropy361
Quasi-longitudinalandquasi-transversewavevelocitiesareshowninFigure4asa362functionofpropagationdirectioninthreeorthogonalplanes.Forcomparisonwith363currentamphiboledeterminations,velocitiesforanearlyiron-freechrome-364containingdiopsidebasedonmodulireportedbyIsaaketal.(2006)areincluded;365thediopsidemodulimorerecentlyreportedbySangetal.(2011)areinclose366agreement.VelocitiesbasedontheelasticmoduliofsampleIofAleksandrovand367Ryzhova(1961a)areincluded,asarevelocitiesforglaucophanebasedon368measurementsofBezacieretal.(2010).369
Quasi-longitudinalvelocitiesforbothdiopsideandtheamphibolesareuniformly370mostanisotropicintheX-Zplane(containingthea-andc-axes)andaremost371isotropicintheY-Zplane(containingtheb-andc-axes).Althoughnotfully372symmetric,quasi-longitudinalvelocitiesintheX-Zplaneareroughlyellipsoidal373(althoughdiopsidemaintainshighervelocitiesoverabroaderrangeofdirections)374withthesemi-majoraxisrotatedfromalignmentwiththec-axis.Thediopsidesemi-375majoraxisisrotatedclockwise(associatedwithpositivevaluesfortheuniquely376monoclinicmoduliC15andC35)whilethesemi-majoraxisforallamphibolesis377rotatedcounterclockwise(associatedwithnegativevaluesforC15andC35).378
Themaximumquasi-longitudinalvelocityforboththediopsideandtheiron-free379amphibolesis>9km/s.Allamphibolesaremoreanisotropicthanclinopyroxenesas380aresultofthesmallvaluesofmoduliassociatedwiththea*direction(C11C12C13C55381andC66).Thelowestquasi-longitudinalvelocityfortremoliteis~6km/sina382direction~20°counter-clockwisefromthepositivea*.Pargasite(withmore383aluminumandfulloccupancyoftheA-site)hasalargerminimumvelocityof~7384km/sinroughlythesameorientation.Glaucophanequasi-longitudinalvelocity385anisotropyisintermediatebetweentremoliteandpargasitewiththesemi-major386axislocatedclosertothec-axis.BasedonthecompositionalderivativesinTable4,387velocitiesforiron-richamphiboles(ie.ferro-actinoliteandferro-pargasite)are388substantiallylower(8.4km/sinthefastdirectionand5.3km/sintheslowest389direction)asaresultofsmallervaluesforthediagonalmoduliandlargerdensities.390
Thegreaterquasi-transversewaveanisotropyforamphibolesthanfor391clinopyroxeneisevidentinFigure4.Amphibolequasi-transversewaveanisotropy392rangesfromaminimumvelocityfortremoliteof3.7km/sandamaximumof5.2393km/s.Withinthea-bplanequasi-transversevelocitiesforthetwowave394polarizationsareequalinthea*-directionandshowthegreatestdifferenceintheb-395direction.396
Figure4
AsshowninFigure4,velocitiesbasedonthehornblendemodulireportedby397AleksandrovandRyzhova(1961a)donotcomparewellwiththecurrentamphibole398data.Quasi-longitudinalvelocitiesarebothsignificantlysmallerandhaveless399anisotropyasshownintheb-cplane.Themagnitudeofthevelocityanisotropyand400itsorientationrelativetocrystalaxes,asillustratedinthea-cplane,donotmatch401currentdata.Itislikelythatthesemoduli,likethemoduliforthefeldspars(as402previouslydiscussedinBrownetal.2016andWaesselmanetal.2016)arebiasedas403aresultofopencleavagesurfacesandcracks.404
Themoduliforamphibolesreportedhereandthepreviouslyreportedmodulifor405plagioclase(Brownetal.2016)andpotassiumfeldspars(Waeselmannetal.2016),406takentogether,showthatallmajorcrustalmineralphasesarehighlyanisotropic.In407fact,theyareasanisotropicasthesheetsilicatesphlogopite(Chhedaetal.2014)408andmuscovite(VaughanandGuggenheim1986).Thus,anypreferredorientations409ofmineralswillleadtorocksthatexhibitsignificantlyanisotropicvelocities.410
Intheabsenceofdatafromothersources,allpasteffortstounderstandcrustal411seismicanisotropyandtheanisotropymeasuredinrockswithpreferredcrystal412orientationshavereliedonthemodulireportedbyAleksandrovandRyzhova413(1961a,1961b).Apragmaticchoice,compensatingforthelowmodulihasbeento414usetheVoigtaverage(upperelasticaggregatebound)ratherthanthemore415appropriateHillorHashin-Shtrikmanaverage(seealsothediscussioninBrownet416al.2016relatedtoplagioclaseminerals).AsdemonstratedinFigure4,useofthe417thesemodulifailstoaccountforthefullanisotropyoftheamphibolesinrocks418containingcrystalpreferredorientations.Itwouldappearnecessarytorecalculate419propertiesofamphibole-richrocksonthebasisofmoreaccuratedeterminationsof420amphiboleelasticity.421
6.3Isotropicmoduliandbodywavevelocities422
Determinationsofbulkmoduliforseveralcalciumandsodiumamphibole423compositionsaresummarizedinTable7.Forthesehighlyanisotropicminerals,the424Reuss-boundbulkmoduli,measuredinisothermalcompressionexperiments,are425significantlysmallerthattheadiabaticHashin-Shtrikman(orHillaverageofVoigt426andReussbounds)moduliappropriateforcalculationofelasticwavespeeds.427CurrentReuss-boundadiabaticbulkmoduliarecorrectedtoisothermalconditions428usingthermodynamicpropertiessummarizedinHackeretal.(2003b)(areduction429ofabout1.5%).ThemeanadiabaticHashin-Shtrikman(H-S)moduliarenearly10%430greater.ThemodulideterminedunderisothermalcompressionbyComodietal.431(1991)appeartobettermatchthecurrentH-Sestimates.However,areanalysisof432thesedataforglaucophanegaveamodulusmoreinaccordwiththeReuss-bound433valuebasedonelasticmoduli.Jenkinsetal.(2010)alsomeasuredlatticestrainsin434twosyntheticglaucophanecrystalsto10GPaandreportedanisothermalbulk435modulusthatisonlyslightlylargerthantheReuss-boundestimate.436
InTable8currentvaluesoftheadiabaticshearmodulusforseveralcalciumand437sodiumamphibolecompositionsarecomparedwithHackeretal.(2003b).Their438compilation,basedonanassumedvalueofPoisson’sratio,consistentlyunder-439estimatestheshearmodulusforallamphiboles;thiswouldleadtoanunder-440predictionoftransversewavevelocities.441
Table7
Table8
Isotropicbodywavevelocities,compressional(Vp)andshear(Vs),forthecalcium442andsodiumamphiboles,listedinTable9,exhibitcleartrends.Increasingaluminum443contentleadstohighercompressionalvelocitieswhilealuminumcontenthasless444impactonshearwavevelocities.Increasingironcontentdecreasesboth445compressionalandshearwavevelocities.Thepriorestimatesarenotconsistent446withthesetrends.Thattheestimateforhornblende,asanaveragecalcium447amphibole,haslowerdensityandhighersoundspeedsthanthehornblenditeof448Christensen(1996)mightreflectacompositionaldifference.Withmorealuminum449andmoreiron,thepredictioncanbemovedintheappropriatedirectiontobetter450matchthehornblenditevelocities.451
Poisson’sratio,𝜎 = #$1 − '(
')
$− 1
*#,wasidentifiedbyChristensen(1996)as452
animportantdiscriminatorintheinterpretationofcrustalseismology.Thenear453constantvalueassumedbyHackeretal.(2003),whennoindependent454determinationoftheshearmoduluswasavailable,isnotsupportedinthecurrent455work.Poisson’sratiorangesfrom0.20to0.27.Itdecreasesstronglywithiron456contentandincreasesmodestlywithaluminumcontent.Thesodiumamphibole,457glaucophane,hasthesmallestvalue.458
Thecomparisonsmadeinthissectionsuggestthatthepredictedisotropicbody459wavevelocitiesofanaveragecalciumamphibole(hornblende)basedonthe460compositionaldependencesdeterminedhereareinreasonableagreementwith461laboratorymeasurementsonahornblenditeofunspecifiedcomposition.Prior462effortswerenotabletocorrectlydescribethevariationofisotropicelasticwave463velocitieswithintherangeofamphibolecompositionsfoundincrustalrocks.464
6.4Anisotropicstrainunderhydrostaticstress465
Projectionsontwoplanesofelasticcompressibilitiesunderhydrostaticstressare466showninFigure5usingthecompliancesums,bi,ofTableS2.Threeamphiboles467(tremolite,pargasite,andglaucophane)andoneclinopyroxene(diopside)are468plotted.Allamphiboleshavethemostcompliantdirectionalignedbetweena*-and469c-axes.Thesemi-majoraxisofglaucophaneisrotatedleastfroma*andtremoliteis470rotatedmost.Althoughthesyntheticglaucophaneaxescompression471measurementsshownearisotropicstrainintheb-cplane,theelasticmodulipredict472significantanisotropythatissimilartopargasite.Tremoliteintheb-cplanehas473intermediateanisotropy.Theweakerbondingofamphiboles(withvacantor474partiallyfilledA-sites)allowsgreaterstrainsinthegenerala-axisdirection.The475structurallyandcompositionallysimilarmineraldiopsideislesscompliantinthis476direction.Inthea-cplane,themostandleastcompressibledirectionsfordiopside477arerotatedbynearly90°relativetotheamphiboles.478
Thecomparisonsmadeinthissectionindicatethatcompressibilitiesunder479hydrostaticcompressioncalculatedonthebasisofelasticmodulimeasuredat480ambientpressureareingeneralaccordwithx-raymeasurementsmadeathigh481pressure.However,axescompliances(andtheresultingbulkmoduli)basedonx-482raycompressionmeasurementsaresensitivetotheformofequationsofstateused483tofitthedata(e.g.multipleentriesinTable7andinJenkinsetal.2010).Such484
Table9
Figure5
uncertaintycanexplaintheerrorinthecompositionalbehaviorthatwaspreviously485associatedwithamphibolesinthecompilationbyHackeretal.(2003b)whenthese486measurementsprovidedtheonlyinformationrelatedtoelasticpropertiesof487importantamphiboleendmemberphases.488
7. Summary489
Thefullsingle-crystalelasticmoduliofninenaturalcalciumtocalcium-sodium490amphiboleshavebeenmeasured.Inaddition,velocitiesofapreviouslystudied491naturalsodiumamphibolehavebeenre-analyzedwithinthecomputational492frameworkusedinthecurrentstudy.Cleartrendsinthebehaviorofmoduliofthe493calciumandcalcium-sodiumamphibolesasafunctionofcompositionhavebeen494identified.Alinearfitbasedonthreechemicalmeasures(totalaluminum,totaliron,495andA-siteoccupation)accountsformostofthecompositionally-inducedvariancein496moduli.SeparatingcontributionsofaluminuminC-andT-sitesisnotsignificantat497the95%confidencelevel.Alinearfitincompositioncouldnotreconcilethesodium498amphiboleglaucophanewiththeothercalcium-sodiumamphiboles.499
Theamphibolesandchemicallyrelatedclinopyroxenessharesimilarvaluesof500moduliexceptthatamphibolemodulirelatedtothea*direction(C11,C12,C13,C55,501andC66)areapproximatelyafactoroftwosmaller.Thisislikelyassociatedwiththe502partiallyoccupiedorvacantA-sitewhichisassociatedwithbondinginthea-axis503direction.IncreasingoccupationoftheA-siteincreasessomeofthesemoduli.In504contrast,thesubstitutionofsodiumforcalciumintheB-sitehasnosignificant505impactonmoduli.ThesubstitutionofironintheC-sitesdecreasesalldiagonal506elasticmoduliwhileleavingoff-diagonalmoduliunaffected.507
Theorientationofquasi-longitudinalvelocityextremainthea-cplaneis508significantlyrotatedbetweentheamphibolesandtheclinopyroxenes.This509differenceisassociatedwithalargenegativevalueofC35foramphibolesandalarge510positivevalueforclinopyroxenes.511
Sincethevariationofisotropicelasticbehaviorofamphiboleswithcompositionis512importantininterpretationsofcrustalseismology,parametersareprovidedthat513allowaccuratedeterminationoftheisotropicbulkandshearmoduliofcommon514amphibolesincrustalrocks.515
Itisnoteworthythatamphiboleshavehigherelasticwavevelocitiesandaremore516anisotropicthansuggestedbytheearlyultrasonicmeasurements.Infact,517amphibolesexhibitanisotropynearlyaslargeasthatobservedinsheetsilicatesand518thefeldspars.Ineffortstoreconcilelaboratorymeasurementsonrockswith519predictionsbasedonthesingle-crystalmodulireportedbyAleksandrovand520Ryzhova(1961aand1961b),thead-hocuseoftheupper-boundVoigtaverageis521common.Thisprovidedpartial,butinappropriate,compensationformodulisubject522tosystematicexperimentalbias.Furthermore,analysesbasedontheearliermoduli523failedtoaccountforthefullanisotropyofamphiboles.Thus,allpredictionsofthe524seismicresponseofrockswithpreferredcrystalorientationswillneedtobere-525evaluated.526
527
Acknowledgments528
SupportfromtheNationalScienceFoundationEAR-0711591enabledthisresearch.529Thefollowingstudentshelpedpreparesamplesandcollectdata:N.Castle,E.Chang,530S.Pendleton,K.Pitt,K.Straughan,A.Teel,andH.West-Foyle.Themicroprobe531analysesofS.KuehnerandN.Castleandx-rayanalysesofW.Kaminskywerevital532contributionstothiswork.B.W.Evanscontributedsamplesandmaintained533continuingdiscussions.TheRRUFFdatabaseandmaterialsprovidedbyR.Downs534arehighlyappreciated.ThisresearchwasinspiredbyacourseofferedbyN.I.535Christensenin1974ontheelasticityofmineralsandseismicstructureofthecrust.536Itwasco-attendedbyM.Salisbury,D.Fountain,andR.L.Carlson.Thescience537contributionsandcontinuedenthusiasmofthesecolleaguesisgratefully538acknowledged. 539
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Figures651
652
Figure1.Classificationandcompositionsofmagnesiumcalciumamphibolesplus653glaucophane(basedonLeakeetal.1997).Theaxisextendingtotherightgives654increasingaluminumintetrahedralcoordination(from0fortremoliteto(Al2Si6O22)655fortschermakite).OccupancyoftheA-siteby(Na+K)(from0to1)isshowninthe656verticaldirection.SubstitutionofNaforCaintheB-siteextendsintothefigurewith657fullreplacementofCabyNafoundinglaucophane.Namedstoichiometricend-658membercompositionsareidentified.Fullsolid-solutionreplacementofmagnesium659byironislabeledbyaddingferro-totheend-membernames(exceptionsferro-660actinoliteistheiron-bearingformoftremolite).Smallfilledandnumberedcircles661arecompositionsofthecurrentsamplesbasedonthechemistryprovidedinTable2.662Thelargegraycirclegivestheaveragechemistryforcalciumamphibolesreported663bySchumacher(2007).LinesprojectedtothezeroofA-siteoccupationare664providedasanaidinvisualizingthesamplecompositions. 665
Glaucophane
Taramite
3
Winchite
4
2
Tschermakite
Kataphorite
97 8
6
Hornblende
1
PargasiteRichterite
5
Tremolite
Edenite
Increasin
gA-siteOc
cupatio
n
666
Figure2.CompositionsoftheamphibolesamplesinformulaunitsreportedinTable6672.Plottednumberscorrespondtothesamplenumbers.Thegraycircleisthe668averageofcalciumamphibolesreportedbySchumacher(2007).Thetoptwopanels669showfrontandsideprojectionsofthecompositionsillustratedinFigure1670(tetrahedralcoordinatedaluminumversusA-siteoccupationandsodiumintheB-671sitevsA-siteoccupation).Thelowerleftpanelshowsthenumberofironatomsper672formulaunitvstetrahedralcoordinatedaluminum.Thelowerpanelontheright673showsoctahedral-coordinatedaluminumversustetrahedral-coordinatedaluminum. 674
0 0.5 1 1.5 2ivAluminum
0
0.2
0.4
0.6
0.8
1
Na
+ K
in A
1
2
34
5
6 7 89
0 0.2 0.4 0.6 0.8 1Na in B
0
0.2
0.4
0.6
0.8
1
Na
+ K
in A
1
2
34
5
6 789
0 0.5 1 1.5 2ivAluminum
0
1
2
3
4
5
Iron
1
2
3
4
5
6
7
8 9
0 0.5 1 1.5 2ivAluminum
0
0.2
0.4
0.6
0.8
1
viAl
umin
um
1 2
34
56
7
8
9
675
Figure3.Elasticmoduliandvelocitiesofamphibolesasafunctionoftotalaluminum.676Filledsymbolsarecurrentexperimentalresultswith2suncertaintiesshownwhen677largerthantheplottedsymbol.Differentsymbolsareassociatedwithparticular678moduliaslabeledineachpanel.PointswiththeXsymbolsaremoduliandvelocities679forglaucophane(Bezacieretal.2010).Opensymbolsgivethepredictions.As680indicatedonlyintheupperleftpanel,solidlinesarepredictionsforincreasing681aluminumcontentinaniron-freemineralwithnoA-siteoccupation.Dashedlines682(whenpresent)givethepredictedferro-equivalentbehavior. 683
0 0.5 1 1.5 2 2.5 3
Total Aluminum (atoms/F.U.)
100
120
140
160
180
200
220
240
260
Mod
ulus
(GPa
)
C33 iron-free
ferro-equivalent
C22
C11
0 0.5 1 1.5 2 2.5 3
Total Aluminum (atoms/F.U.)
35
40
45
50
55
60
65
70
75
80
Mod
ulus
(GPa
)
C44
C66
C55
0 0.5 1 1.5 2 2.5 3
Total Aluminum (atoms/F.U.)
35
40
45
50
55
60
65
70
75
80
Mod
ulus
(GPa
)
C23
C12
C13
0 0.5 1 1.5 2 2.5 3
Total Aluminum (atoms/F.U.)
-30
-25
-20
-15
-10
-5
0
5
10
Mod
ulus
(GPa
)
C46
C15
C25
C35
0 0.5 1 1.5 2 2.5 3
Total Aluminum (atoms/F.U.)
50
55
60
65
70
75
80
85
90
95
100
Mod
ulus
(GPa
)Bulk Modulus
Shear Modulus
0 0.5 1 1.5 2 2.5 3
Total Aluminum (atoms/F.U.)
4
4.5
5
5.5
6
6.5
7
7.5
8
Velo
city
(km
/s)
Compressional
Transverse
684
Figure4.Selectedinosilicateelasticwavevelocitiesasafunctionofpropagation685directioninthreeorthogonalplanes.ForeachplanenormaltoaCartesianaxislight686circlesrepresentvelocitiesof2,4,6,8,and9.5km/s.Theorientationsof687crystallographicaxesareshown.Thicklinesarevelocitiesbasedontheelastic688moduliandpropagationdirections.Theinnerthicklinesarequasi-transversewave689velocities,theouterthicklinegivesquasi-longitudinalvelocities.Toprow:diopside690velocitiesbasedonIsaaketal.(2006).Secondandthirdrows:calciumamphibole691end-membervelocitiesbasedonthecurrentwork.Dashedlinesinsecondroware692quasi-longitudinalvelocitypredictionsbasedontheelasticmoduliforahornblende693reportedbyAleksandrovandRyzhova(1961a).Bottomrow:glaucophanevelocities694basedonBezacieretal.(2010). 695
c
aa*
X-Z Plane
Diopside
c
b
Y-Z Planeb
a*
X-Y Plane
c
aa*
Tremolite
c
b
b
a*
c
aa*
Pargasite
c
b
b
a*
c
aa*
Glaucophane
c
b
b
a*
696
Figure5.Selectedinosilicatestrainellipsoidsunderhydrostaticstressprojectedon697thecrystallographica-candb-cplanes.Thicksolidline:diopsidefromIsaaketal.698(2006),solidline:tremolitefromcurrentwork,greyline:pargasitefromcurrent699work),dashedline:glaucophanebasedonBezacieretal.(2010),dottedline:700glaucophanebasedonJenkinsetal.(2010). 701
aa*
c
b
c
Tables702703Sample Unitcell
volume(A°3)
a(A°)
b(A°)
c(A°)
b (°)
Densitykg/m3
Source
1 905.7 9.88 17.98 5.27 104.6 3027 unknown2 914.3 9.87 18.03 5.31 104.6 3255 unknown3 901.9 9.80 17.98 5.29 104.9 3162 GoreMountainNY,collectedbyB.W.Evans4 913.6 9.83 18.08 5.32 104.9 3293 unknown5 908.4 9.84 18.02 5.29 104.6 3038 LakeWenatchee,WAcollectedbyB.W.Evans6 918.1 9.91 18.08 5.31 105.1 3213 RRuff.info#60029R.Downs7 934.0 9.96 18.21 5.33 104.9 3418 RRuff.info#60044R.Downs8 907.2 9.86 17.99 5.30 105.3 3163 RRuff.info#60632R.Downs9 895.2 9.80 17.92 5.28 105.2 3190 Unknown704
Table1.Amphibolesampleinformation.Samplesof“unknown”sourcewere705obtainedasmineralseparatesfromrocksofunknownorigin.Unitcellvolumesand706cellparametersarefromxrayanalysis,densitiesarecalculatedbasedonunitcell707volumesandthemicroprobedeterminationofchemistry.Individualcrystalsfrom708thesamesourceexhibitedvariationsincellparametersofabout0.01A°.Basedon709thisvariabilityanuncertaintyintheunitcellvolumeis0.3%.Thedensity710uncertainty,accountingforchemistryandvolumeuncertainties,is0.5%. 711
712
Table2.Microprobechemicalanalysisofninecalciumandcalcium-sodium713amphibolesinformulaunits(basisof22oxygens)usingProbe-AMPH(Tindleand714Webb,1994)plusthechemicalanalysisofglaucophane(GL)reportedinBezacieret715al.(2010)andtheaveragecalciumamphibole(HBL)asreportedbySchumacher716(2007).Seesupplementaltableforweight%oxidesmeasuredbymicroprobe717analysis. 718
Sample 1 2 3 4 5 6 7 8 9 GL HBL Structural Formulae Si 7.859 7.023 6.182 6.513 7.718 6.698 6.571 6.152 5.898 7.76 6.458 Aliv 0.141 0.977 1.818 1.487 0.282 1.302 1.429 1.848 2.102 0.24 1.542 Alvi 0.032 0.064 0.553 0.464 0.113 0.259 0.015 0.683 1.047 1.76 0.470 Ti 0.013 0.109 0.112 0.173 0.002 0.060 0.239 0.319 0.021 - 0.123 Cr 0.000 0.002 0.001 0.000 0.035 0.000 0.000 0.002 0.211 - 0.001 Fe3+ 0.097 0.769 1.263 0.727 0.494 0.365 0.559 0.000 0.157 - 0.718 Fe2+ 0.287 1.601 0.150 2.004 0.244 1.526 3.298 0.815 0.592 0.92 1.201 Mn 0.028 0.157 0.012 0.030 0.040 0.049 0.205 0.007 0.013 - 0.034 Mg 4.543 2.298 2.910 1.601 4.072 2.741 0.684 3.105 2.958 2.34 2.453 Ca 1.243 0.993 1.503 1.639 1.741 1.706 1.528 1.798 1.892 0.06 1.752 Na 1.236 1.674 0.683 0.577 0.146 0.805 0.981 0.696 0.789 1.90 0.480 K 0.264 0.265 0.088 0.095 0.009 0.341 0.339 0.370 0.071 - 0.121 F 0.937 0.695 0.000 0.000 0.035 0.732 0.000 0.263 0.005 - 0.000 Cl 0.002 0.015 0.000 0.008 0.000 0.030 0.000 0.005 0.000 - 0.000 OH* 1.060 1.290 2.000 1.992 1.965 1.238 2.000 1.732 1.995 2 2.000 Total 17.743 17.932 17.275 17.310 16.895 17.852 17.848 17.795 17.752 - 17.353 Site Occupancy (Ca+Na) (B) 2.000 2.000 2.000 2.000 1.887 2.000 2.000 2.000 2.000 1.96 2.000 Na (B) 0.757 1.007 0.497 0.361 0.146 0.294 0.472 0.202 0.108 1.90 0.248 (Na+K) (A) 0.743 0.932 0.275 0.310 0.009 0.852 0.848 0.864 0.752 - 0.353 Mg/(Mg+Fe2) 0.941 0.589 0.951 0.444 0.943 0.642 0.172 0.792 0.833 - 0.671 Fe3/(Fe3+Alvi) 0.750 0.923 0.696 0.610 0.814 0.585 0.974 0.000 0.130 - 0.604
719
1
2s
22s
32s
42s
52s
62s
72s
82s
92s
Gl2s
C11
119.20.8122.7
0.9133.6
0.9122.8
0.9108.6
0.7131.1
0.9122.7
0.9141.6
0.9148.7
1.0121.5
1.6C12
47.50.7
50.00.3
50.91.0
51.81.3
48.40.9
53.21.0
52.61.4
57.11.4
56.51.1
44.42.0
C13
41.20.744.3
0.843.1
0.745.9
0.737.7
0.647.2
0.847.5
0.849.6
0.848.8
0.737.4
2.3C15
-1.70.3
-1.40.3
-0.80.3
-0.70.3
1.00.3
-1.00.3
-2.00.3
-0.20.3
0.30.4
2.71.0
C22
182.21.3184.6
1.2193.4
1.2189.3
1.3191.6
1.4186.6
1.2178.6
1.2197.8
1.3204.6
1.3229.7
2.3C23
58.20.859.5
0.958.3
0.962.3
0.959.2
0.960.3
0.960.8
0.960.9
1.061.9
1.175.8
2.4C25
-5.60.9
-7.11
-10.81.1
-7.01.0
-5.61.1
-8.51.2
-9.50.9
-10.91.3
-6.91.3
-4.92.6
C33
228.01.5223.7
1.5225.8
1.4222.9
1.4230.8
1.5224.3
1.4216.6
1.3225.4
1.6232.1
1.5256.2
2.8
C35
-30.60.5-29.4
0.5-30.3
0.4-30.0
0.4-29.6
0.5-30.3
0.5-31.0
0.4-31.4
0.5-28.8
0.5-23.9
1.5C44
75.60.670.5
0.575.5
0.671.5
0.577.0
0.672.5
0.667.5
0.575.8
0.676.7
0.679.3
0.9C46
4.70.4
5.30.4
3.80.3
5.40.4
7.90.5
4.40.4
6.30.4
3.30.4
1.90.4
9.30.9
C55
45.90.342.5
0.347.5
0.346.8
0.350.0
0.346.5
0.339.7
0.349.9
0.354.1
0.352.9
0.7C66
49.20.445.9
0.350.4
0.346.2
0.448.6
0.448.0
0.440.8
0.351.7
0.452.9
0.451.3
0.6
Table3.Elasticmoduli(inGPa)ofam
phiboles.The2suncertaintiesincludemisfitstovelocitiesanduncertaintyinsam
pledensities.Thecolum
nlabeled“Gl”givesre-analyzedmodulianduncertaintiesforglaucophanebasedonvelocitiesreportedbyBezacieretal(2010).
ModulusGPa
dM/dAlGPa/atom
dM/dAGPa/atom
dM/dFeGPa/atom
ExperimentalUncertainty
GPa
RegressionMisfitGPa
C11 107.2 10.6 13.3 -2.9 1.0 1.1
109.1 11.4 1.0 5.0C12 47.1 2.8 1.2 1.6
C13 36.7 2.6 6.5 0.8 1.2 39.6 3.1 0.8 2.4C15 -0.8 0.3 0.9C22 185.9 6.2 -3.8 1.2 2.7C23 60.0 1.0 1.4C25 -7.8 1.2 1.9
C33 231.6 -3.4 1.5 2.1C35 -30.2 0.5 0.8
C44 78.5 0.7 -3.0 -2.4 0.6 0.2 78.0 -2.5 0.6 1.1C46 6.7 -1.3 0.4 1.1C55 48.0 2.1 -2.6 0.4 1.9C66 50.3 1.7 -2.8 0.4 0.8K 84.5 4.3 1.2 1.7
G 57.5 2.0 -2.2 0.8 0.8720
Table4.Linearregressionparametersforamphiboleindividualelasticmoduliand721themeanofHashin-Shtrikmanboundsfortheadiabaticbulk(K)andshear(G)722modulus.Basemodulifortremoliteareinthefirstcolumnofvalues.Derivativesare723inunitsofmoduluschangepersubstitutionalatomintheformulaunitrelativeto724tremolite;thealuminumcontentvariesfrom0to>3,theA-siteoccupationranges725from0to1,andironintheC-sitecanrangefrom0to5.OnlyC11,C13,andC44havea726statisticallysignificantdependenceontheA-siteoccupation.Analternativefitwith727nodependenceonA-siteoccupationisprovided(withaconcomitantincreasein728misfit).Thelasttwocolumnsgiveexperimentalandregressionmisfits. 729
730
M dM/dAl dM/dA dM/dFe
ExperimentalUncertainty
RegressionMisfit
Density(kg/m3) 2974 42 6 58 15 7 2928 76 158 15 157
Vp(m/s) 7380 100 -181 47 40Vs(m/s) 4446 50 -113 -138 24 26
4379 46 -137 24 44731
Table5.Linearregressionparametersfordensities,andcompressionaland732transversewavevelocitiesoftheamphiboles.Derivativesareinunitsofchangeper733substitutionalatomintheformulaunitrelativetotremolite;thealuminumcontent734variesfrom0to>3,theA-siteoccupationrangesfrom0to1,andironintheC-site735canrangefrom0to5.Densityandtransversewavevelocitieshaveastatistically736significantdependenceontheA-siteoccupation.Analternativefitwithno737dependenceonA-siteoccupationisprovided(withaconcomitantincreaseinmisfit).738Thelasttwocolumnsgiveexperimentalandregressionmisfits. 739
740 C11 C12 C13 C15 C22 C23 C25 C33 C35 C44 C46 C55 C66DiopsideIsaaketal.2006
228 79 70 8 181 61 6 245 40 79 6 68 78
Tremolite 107 47 37 -1 186 60 -8 232 -30 79 7 48 50 HedenbergiteKandelinandWeidner1988a
222 69 79 12 176 86 13 249 26 55 -10 63 60
Actinolite 93 47 37 -1 167 60 -8 215 -30 67 7 35 36 JadeiteKandelinandWeidner1988b
274 94 71 4 253 82 14 282 28 88 13 65 94
GlaucophaneBezacieretal.2010
122 46 37 2 232 75 -5 255 -24 80 9 53 51
Di72Hd9Jd3Cr3Ts12CollinsandBrown1998
238 84 80 9 184 60 10 230 48 77 8 73 82
Tr72Ac9Pg19 122 48 44 -1 185 60 -8 228 -30 74 6 46 48741
Table6.ComparisonofamphiboleandclinopyroxeneelasticmoduliinGPaunits.742AmphibolemoduliarecalculatedusingparametersgiveninTable4. 743
tremolite ferro-actinolite
hornblende tschermakite pargasite glaucophane ferro-glaucophane
Current:Reuss 78(1) 78(1) 88(1) 88(1) 94(1) 88(1) H-S 85(1) 85(1) 93(1) 93(1) 99(1) 96(1) C91 85 97 96
88(6)
J10 92(2) H03 85 76 94 76 91 96 89
744
Table7.Bulkmoduli(GPaunits)forselectedamphiboles.“Current:Reuss”are745isothermalvaluesusingtheparametersinTable4andfromBezacieretal.(2010).746Anadiabatictoisothermalcorrectionwasappliedtotheadiabaticmoduliusingthe747thermodynamicpropertiessummarizedinHackeretal.(2003b).H-Sarethe748averageofadiabaticHashin-Shtrikmanbounds.Inthecurrentwork“hornblende”is749acompositionbasedontheSchumacher(2007)averagecalciumamphibole.The750Comodietal.(1991)(C91)andJenkinsetal.2010(J10)valuesarebasedonhigh751pressureisothermalxraycompressionmeasurements.Comodietal.reported752valuesbasedonlinearfitstothedata.ThesecondestimateintheC91rowisthere-753analysisgivenbyJenkinsetal.usingasecond-orderfinite-strainequationofstate754usingEoSFit5.2(Angel2001).Inthelastrow(H03)isothermalmoduliaretaken755fromTable1ofHackeretal.(2003b).Uncertaintiesforthecurrentworkarefrom756TableS2.Theuncertaintiesformodulibasedonaxescompressionmeasurements757arereportedbyJenkinsetal.(2010). 758
759
tremolite ferro-actinolite hornblende tschermakite pargasite glaucophane ferro-glaucophane
Current 58 47 57 62 64 64 H03 49 44 55 44 53 56 52
760
Table8.Adiabaticshearmoduli(GPaunits)forselectedamphiboleend-members.In761thecurrentwork“Hornblende”designatesamineralcompositionbasedonthe762Schumacher(2007)averagecalciumamphibole.ThetoprowlistsmeansofHashin-763ShtrikmanboundsbasedonTable4andonBezacieretal.(2010)forglaucophane.764ThebottomrowlistsvaluesfromTable1ofHackeretal.(2003b). 765
766
tremolite ferro-actinolite
hornblende pargasite tschermaktite glaucophane ferroglaucophane
LiteratureVpkm/s 7.1 6.3 7.20 7.3 6.7 7.6 7.0Vskm/s 4.1 3.6 4.12 4.1 3.8 4.3 4.0
Poisson’sratio .25 .26 .26 .27 .26 .26 .26Densitygm/cc 2.98 3.43 3.25 3.07 3.04 3.01 3.30
CurrentVpkm/s 7.4 6.5 7.22 7.7 7.6 7.5
Vskm/s 4.5 3.8 4.21 4.5 4.5 4.6 Poisson’sratio .21 .27 .24 .23 .23 .20 Densitygm/cc 2.97 3.26 3.18 3.11 3.06 3.07
767
Table9.Isotropiccompressional(Vp)andshear(Vs)velocitiesanddensitiesfor768selectedamphiboles.LiteraturevaluesarebasedonthecompilationofHackeretal.769(2003b).CurrentvaluesarebasedonTable5forcalciumamphiboles(usingthe770averagecalciumamphiboleofSchumacher(2007)forhornblende)andBezacieret771al.(2010)forglaucophane.Sincehornblenditevelocitieswerereportedtofour772significantfiguresbyChristensen(1996)moreprecisionisprovidedforthe773hornblendetableentries.774
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