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