G.Ruocco1,F.Sette2,R.DiLeonardo1,D.Fioretto3,M.Lorenzen2,M.Krisch2,C.Masciovecchio2,G.Monaco1,F.Pignon2,T.Scopigno4
12
arXiv:cond-mat/9911012v1 [cond-mat.dis-nn] 2 Nov 19994
DipartimentodiFisicaandINFM,Universit`adiL’Aquila,I-67100,l’Aquila,Italy.EuropeanSynchrotronRadiationFacility,BP220,F-38043,GrenobleCedex,France.3
DipartimentodiFisicaandINFM,Universit´adiPerugia,I-06100,Perugia,Italy.DipartimentodiFisicaandINFM,Universit´adiTrento,I-38050,Povo,Trento,Italy.
(February1,2008)
ThesoundattenuationintheTHzregionisstudieddowntoT=16Kinglassyglycerolbyinelasticx-rayscattering.Atstrikingvariancewiththedecreasefoundbelow≈100KintheGHzdata,theattenuationintheTHzrangedoesnotshowanyTdependence.Thisresulti)indicatesthepresenceoftwodifferentattenuationmechanisms,activerespectivelyinthehighandlowfrequencylimits;ii)demonstratesthenon-dynamicoriginoftheattenuationofTHzsoundwaves,andconfirmsasimilarconclusionobtainedinSiO2glassbymoleculardynamics;andiii)supportsthelowfrequencyattenuationmechanismproposedbyFabianandAllen(Phys.Rev.Lett.82,1478(1999)).PACSnumbers:78.35.+c,78.70.Ck,61.20.Lc,83.50.Fc
Oneofthemostimportantandstillunsettledsubjectsinthephysicsoftopologicallydisorderedsystemsregardsthemechanismsforthepropagationandattenuationofdensityfluctuations.Thepropagatingnatureofacousticwaves,asseenbyUltrasonicandBrillouinLightScatter-ing(BLS)measurementsintheMHzandGHzregionrespectively,hasbeenshowntopersistuptotheTHzregionbytheexistenceofalinearrelationbetweenthepeakenergy,E,andthemomentumtransfer,Q,oftheinelasticfeaturesobservedinthedynamicstructurefac-tor,S(Q,E),ofglasses[1].ThisresultistheoutcomeofextensivestudiesontheshapeofS(Q,E)performedus-ingMolecularDynamics(MD)simulations[2–5]andthenewlydevelopedInelasticX-raysScattering(IXS)tech-nique[1,6–8].ThislattertechniqueallowstostudytheS(Q,E)inthe”high”Qrange(Q≈1-10nm−1),thusincreasingbyabouttwoordersofmagnitudetheQval-uestypicallyinvestigatedbyBLS(Q≈0.01-0.04nm−1).IntheIXSandMDQ-range,besidethepersistenceofalineardispersionoftheacousticexcitationenergies,onealsoobservesaprogressivebroadeningoftheinelas-ticfeatures,whichisresponsiblefortheirdisappearanceatacertainQmvalue.TypicallyQmissometenthsofQM−thepositionofthefirstsharpdiffractionpeakinthestaticstructurefactor,S(Q)[1].Thestudyofthemechanismsleadingtothisdamping,and,therefore,theinvestigationofthesoundwavesattenuationattheseQ-values−characteristicofstructuralcorrelationsattheinterparticlelevel−isobviouslyofgreatinterest.
TheacousticexcitationsatfrequenciesintheTHzrange,asmeasuredsofaringlassesandglassformingliq-uidsbyIXS,havealinewidthparameterΓQwhichseemstoshowaQ2dependence[1].Moreover−inalltheIXSdatareportedsofar−ΓQ/Q2hasanegligibletemper-aturedependenceinawidetemperatureregionrangingfromvalueswellbelowtheglasstransitiontemperature,Tg,uptotheliquidphase[1].Atvariancewiththisbe-havior,aswellknown,thelinewidthoftheexcitationsin
1
theGHzregion,measuredbyBLS,showarelevanttem-peraturedependence,whichbecomesparticularlystronginthelimitofverysmalltemperatures[9–14].Thetem-peraturedependenceofthelinewidthintheGHzrangehasmotivatedmanytheoreticalstudies,leadingtodiffer-enthypothesesonthefrequency(orQ)evolutionoftheattenuationmechanisms[15].InthisLetterwereportanIXSstudyonthelowtem-peraturebehavioroftheexcitationslinewidthinglassyglycerol.Specifically,weconcentrateonthestudyofTHzexcitationsinthetemperatureregionwheretheBLSdataintheGHzrangeshowamarkedtempera-turevariation.Withintheerrorbar,thelinewidthmea-suredbyIXSistemperature-independentinthewhole0.1TgtoTgregion,whereas,inthissameregion,theBLSlinewidthincreasesbymorethanafactoroften.Thistwooppositbehaviorsindicatethatthereareatleasttwodifferentattenuationmechanisms:i)Oneofdynamicori-gindominantinthelowQ(lowfrequency)region,andii)Asecondone,dominantathighQ,whosetempera-tureindependencesuggeststhatitsoriginisduetothestructuraldisorderoftheglass.Theglycerolresultsareconfirmedbyasimilarsoundattenuationbehaviorfoundinvitreoussilica,asobtainedbytheanalysisofexistingBLS,IXSandMDdata.Theobservationoftwodistinctattenuationmechanisms,eachonedominantinadifferentQregion,impliestheexistenceofacross-overfrequency,whichliesinthe100GHzrangeforbothofthestudiedglasses.ItalsosuggeststhatthefrequencydependenceofthedynamiccontributiontothesoundattenuationagreeswiththeonerecentlypredictedbyFabianandAllen[16].TheexperimenthasbeencarriedoutatthenewveryhighenergyresolutionIXSbeamlineID28,attheEuro-peanSynchrotronRadiationFacility.Theincidentx-raybeamisobtainedbyaback-scatteringmonochromatoroperatingattheSi(111111)reflection[17].Thescat-teredphotonsarecollectedbyasphericalsiliconcrystalanalyzer,alsooperatingattheSi(111111)reflection[18].
Themonochromaticbeamhasanenergyof≈21,748eVandanintensityof2·108photons/s.Thetotalenergyres-olution−obtainedfromthemeasurementofS(QM,E)inaPlexiglassamplewhichisdominatedbyelasticscat-tering−is1.5meVfull-width-half-maximum(fwhm).Themomentumtransfer,Q=2k◦sin(θs/2),withk◦thewavevectoroftheincidentphotonandθsthescatteringangle,isselectedbetween2and4nm−1byrotatinga7mlonganalyserarminthehorizontalscatteringplane.ThetotalQresolutionhasbeensetto0.2nm−1.Energyscansaredonebyvaryingtherelativetemperaturebetweenthemonochromatorandanalyzercrystals.Eachscantookabout180min,andeach(Q,T)-pointspectrumhasbeenobtainedfromtheaverageof2to8scansdependingonthesampletemperature.Thedatahavebeennormalizedtotheintensityoftheincidentbeam.Thesamplecellismadeoutofapyrex-glasstube(4(10)mminner(outer)diameterand20mmlength),cappedwithtwodiamondsinglecrystalsdiscs,1mmthick,tominimizeundesiredscatteringsignals.Thecellhasbeenloadedwithhighpu-rityglycerolinanargonglovebox.IntheQ−Eregionofinterest,emptycellmeasurementsgavetheflatelectronicdetectorbackgroundof0.6counts/min.Thecelllengthwaschosentobecomparabletothex-rayphotoabsorp-tionlength,andmultiplescatteringwasnegligible.
ThespectrahavebeencollectedatT=16,45,75,114,145and167K,and,asexamples,thoseatT=16and167KarereportedinFig.1fordifferentQ-values.Thefulllinesarethefitstothedata,obtainedus-ingamodelfunctionmadebytheconvolutionoftheexperimentallydeterminedresolutionfunctionwithadeltafunctionfortheelasticpeakandaDampedHar-monicOscillator(DHO)modelfortheinelasticpeaks[1].ThismodelfortheS(Q,E)resultsfromtheas-sumptionthatthememoryfunction,mQ(t)[19],en-teringintheLangevinequationfortheconsideredQ-componentofthedensityfluctuation,hasatimede-pendenceas:mQ(t)=2ΓQ(T)δ(t)+∆2Q(T)[20].Thepresenceofthestructuralα-relaxation,observedintheliquidstateandfrozenintheglass,andofotherrelax-ationprocesseswithcharacteristictimesslowerthan≈1ps,isreflectedintheparameter∆Q(T),whosevaluede-terminesthechangeofthesoundvelocity,c,betweenthefullyrelaxed(co)andunrelaxed(c∞)limitingval-222
ues:∆2Q(T)=Q(c∞−co).TheparameterΓQ(T)de-terminesthewidthofthesidepeaks,i.e.thesoundwaveattenuationcoefficient,α=2πΓQ/hc.However,fitsmadewiththeDHOmodelorwithdifferentfittingfunctiongavevaluesfortheFWHMoftheinelasticpeaksconsistentamongeachotherwithintheirstatisticalun-certainties,indicating,therefore,theinsensibilityoftheresultstothespecificmodelfortheinelasticpeaks.Asitisevidentalreadyfromtherawdata,ΓQ(T)hasamarkedQ-dependencewhileitsT-dependence,ifany,ismuchsmaller.ThisisbetterseenbythedottedlinesinFig.1,whichrepresenttheunconvolutedinelasticpartof
2
S(Q,E).
ThevaluesofΓQ(T)resultingfromthefitoftheIXSdataofFig.1arereportedasafunctionofQinFig.2.InthesamefigurearealsoshowntheΓQ(T)obtainedfromIXSmeasurementsat175K[1],andthoseobtainedfromliteratureBLSspectrameasuredatQ≈0.03nm−1andtemperaturessimilartotheIXS’ones[9,21].Thisfiguredemonstratesthat,withintheerrorbars,ΓQ(T)isT-independentintheQregioncoveredbyIXS.Onthecontrary,ΓQ(T)showsamarkedT-dependenceattheQvalueoftheBLSmeasurements.TheseT-dependenciesareemphasizedinFig.3,whereΓQ(T)/Q2isplottedasafunctionofTforQintheBLSregion(Q≈0.035nm−1)[9,21]andforQintheIXSregion.Herethecrossedsym-bolsrefertoIXSmeasurementsatfixedQ(Q=2nm−1)andthefullsymbolstotheaverageofΓQ(T)/Q2overtheQ=2−4nm−1region.Thisfigureconfirmsthat,atthehighQvalues,ΓQ(T)issubstantiallyconstantevenatverylowT,whereas,atthelowQvalues,itincreaseswithtemperatureupto≈100K,whereitseemstoreachaplateau.ThefurtherincreaseaboveTgisduetotheα-relaxation;itisonlyseeninthelowQdataasitwouldaffectthehighQdataathigherT.
Thespecificdynamicmechanisms(anharmonicity,re-laxationprocesses,floppymodes,twolevelsystems...)attheoriginoftheacousticattenuationobservedattheBLS’Qvalues,aswellastheirtemperaturedependence,havebeenwidelyinvestigatedinthepast[15].Incon-trasttowhatitisfoundintheBLS’Qregion,thebe-haviorofΓQ(T)inthehighQregion,asreportedinFigs.2and3,showsthatherethesoundattenuationisnotdeterminedbytemperatureactivateddynamicpro-cesses.Consequently,inthisQrange,atvariancewiththecrystallinestatewheretheabsenceofdynamicpro-cesseswouldimplynosoundattenuation,intheglasstheobservednonvanishingvalueofΓQmusthavea”struc-tural”origin,i.e.itmustbeduetothetopologicaldis-orderoftheglassstructure.
Thepicturecomingfromthereporteddatasuggests
(D)S)
thatonecanexpressΓQ(T)=ΓQ(T)+Γ(
Q,where
(D)ΓQ(T)isatemperaturedependentdynamicpartand(S)
ΓQisduetotopologicaldisorder.TheQ-dependenceof(D)ΓQ(T)mustbesuchtobethedominanttermofΓQ(T)atsmallQ,whileitmustbenegligibleatlargeQ.Thisbehaviorisconsistentwitharecentcalculationofthedy-namic(anharmonicity)contributiontothesoundatten-uationbyFabianandAllen[16],whopredictaΓQ(T)∝Q2uptoaQcvalue,abovewhichΓQ(T)=const.Inthecaseofamorphoussilicon,Qchasbeencalculatedtobeinthe0.1nm−1range[16].Althoughthereisnosuchcalculationforglycerol,thepresentresultsindicatethatalsointhisglassthecrossovertakesplaceatQvaluesbetweentheBLS’andIXS’ones,i.e.intheνc≈100GHzfrequencyrange.Itisworthtonotethatarelax-ationprocesswithcharacteristictimeτ=1/2πνc≈2ps,
andresponsibleforalinewidthoftheorderΓQ/Q2≈0.2meV/nm−2,shouldalsogiveadispersionofthesoundvelocity,δc,givenbyδc/c=πΓQ/Q2τhc2≈1%,avaluetoosmalltobedetectablewiththeaccuracyachievableatpresent.Therefore,inthepresentcase,thechangeofsoundvelocitycannotbeusedtoestimatethecrossovervalueQc.
TheQ-dependenceofΓ(S)
Q,asalreadyobservedbeforeinmanyotherglassesandglassformingsystems[1],is
wellrepresentedbyaQ2law,Γ(
S)2
Q=DQ,intheQ
regioncoveredbyIXS.ThisQ2law,shownasafulllineinFig.2,however,cannotbeextrapolated(thinfullline)tolowQvaluesbecauseitwouldpredictwidthvaluesinexcesstothemeasuredΓQ(T).Thisobservationexcludes
thatΓ(S)
Q∝Q2inthewhole0.01-10nm−1Qrange.Un-derthehypothesisthatΓ(
S)Q=Qγ,andassumingthatat
thelowestmeasuredtemperatureΓ(
S)
withintheerrorbarsofQ≡ΓQ,onefinds
consistencyboththeIXSdataandtheBLSlowtemperaturepoint,asshownbythedashedlineinFig.2obtainedwithγ=2.5.Thisesti-mateofγisalowlimitingvalue,becausetheBLSlowTwidthcouldstillbepartiallyaffectedbyadynamiccon-tribution.Itisnotclear,however,whetherthehypoth-esizedpowerlawindeedprovidesagoodrepresentation
ofΓ(
S)
Q,somehintsonthisissuecanbegatheredbythestudyofotherglasses.
ThepreviouspictureforthesoundattenuationintheglycerolglassisfurthersubstantiatedbytheexistingIXS[22–24],BLS[10–13,25],MD[2,5]andPicosecondOpti-calTechnique(POT)[26],datainanotherprototypicalglass:vitreoussilica(v-SiO2).WereportinFigs.4and5theΓQ(T)valuesforv-SiO2inaformatequivalenttothatofFigs.2and3.InFig.4,theΓ1Q(T)valuesinthe1to5nm−region,asobtainedbyIXS,donotshowanyrelevantT-dependenceinthe300-1450Krange.AsaconsequenceofcontrastproblemsduetothelimitedenergyresolutionoftheIXSspectrometer,attempera-turesbelow300K,ithasnotbeenpossibletodiscrimi-natetheinelasticsignalfromthetailsoftheelasticone.ThemissinglowtemperatureIXSdataaresuppliedbyanextendedMDsimulationperformedintheharmonicapproximation(T=0K),whichprovidesaS(Q,E)line-shapeinexcellentagreementwiththeIXSdata,andcon-firmstheabsenceofanyrelevantT-dependenceofthesoundattenuationinthewholeT=0-1450KandQ=1-5nm−1ranges.Onthecontrary,asinglycerolandasemphasizedinFig.5,theBLSdataofv-SiO2datashowalargeT-dependence.Asinotherglasses,alsothehighQv-SiO2dataofΓQhaveaQ-dependencewellrepre-sentedbyaQ2law(fulllineinFig.4).Inv-SiO2arealsoavailableroomTdatainaQregionintermediatetoBLSandIXS(Q=0.03-0.4nm−1),asobtainedbyPOT.Thesedata,however,showarelevantinconsistencywithBLSdatameasuredinthesameQandTranges.
Thesimilarbehaviorbetweenv-SiO2andglycerolal-
lowstoformulatealsoforv-SiO2thesamehypotheses
ontheQandT-dependenciesofΓ(D)S)
Q(T)andΓ(
Q.Itis
worthtonote,however,thattheabilityofthepowerlaw
forΓ(
S)
Q(whichshouldhaveγ=2.6)topassthroughalltheIXSdataandthelowTBLSpointissubstantiallyworsethaninglycerol.Thereforethev-SiO2resultsclearlyindicatesthatthepowerlawhypothesisiswrong
andthatΓ(
S)
Qhasamorecomplexbehavior,namelyitis
∝Q2inthehighQregime(Q>1nm−1)andithasasteeperbehavioratlowQvalues.
Inconclusion,wehaveshownthatinglycerolandsil-icaglassesthedominantsoundattenuationmechanismhasadifferentoriginintheQregionspannedbytheIXSorBLStechniques.ThetemperatureindependenceoftheattenuationatlargeQ,correspondingtofrequen-ciesintheTHzregion,impliesthatitsoriginisstruc-turalanditisduetothedisorder.Onthecontrary,thewellknownstrongT-dependencefoundinthelowQre-gion,atGHzfrequenciesandbelow,impliesadynamicoriginofthesoundattenuation[15].Thesefindingsim-plyacross-overregionbetweenthetworegimes,whichshouldlieinthe100GHzfrequencyrange.TheoverallTandQ-dependenciesoftheattenuationconsideredhere
isconsistentwithadynamic(Γ(D)
Q)partthatcloselyfol-lowstheoneproposedbyFabianandAllen[16],anda
structuralpart(Γ(
S))thathasaQ2
QbehaviorathighQ
(Q>1nm−1)andasteeperQdependenceatlowerQ.Thehypothesisonacross-overbetweenthetwodifferentattenuationmechanismsconsideredherecallforfurtherstudies,wheretheQandT-dependenciesofthesoundwavesisthoroughlyinvestigatedusingtheBLSandIXStechniqueinawidernumberofglassmaterials.
[13]T.Scopignoetal.,tobepublished.
[14]J.Lorosch,M.Couzi,J.Pelous,R.VacherandA.Lev-asseur,J.ofNon-Cryst.Solids69,1(1984).
[15]S.Hunklinger,andW.ArnoldinPhysicalAcousticedited
byW.P.MasonandR.N.Thurston(Academic,NewYork,1976),p.155.
[16]J.FabianandP.B.Allen,Phys.Rev.Lett.82,1478
(1999).
[17]R.Verbenietal.,J.ofSync.Rad.3,62(1996).
[18]C.Masciovecchioetal.,Nucl.Inst.andMeth.B-111,
181,(1996);ibidemB-117,339,(1996).
[19]J.P.Boon,andS.Yip,MolecularHydrodynamics,Dover
PublicationsInc.,NewYork(1991).
[20]G.Monaco,A.Cunsolo,G.RuoccoandF.Sette,
preprint.
[21]W.T.GrubbsandR.A.MacPhail,J.Chem.Phys.100,
2561(1994).
[22]P.Benassietal.,Phys.Rev.Lett.77,3835(1996).[23]C.Masciovecchioetal.,Phys.Rev.B55,8049(1997).[24]T.Scopigno,Thesis,UniversityofL’Aquila(I),1998.[25]J.A.BucaroandH.D.Dardy,J.ofAppl.Phys.45,5324
(1974).
[26]T.C.Zhu,H.J.MarisandJ.Tauc,Phys.Rev.B44,4281
(1991);ibidem,inPhonon89,editedbyS.Hunklinger,W.LudwigandG.Weiss(worldscientific,singapore,1990),p.498andp.1430.
FIGURECAPTIONS
FIG.1-Inelasticx-rayscatteringspectrumofglycerolatT=16
and167KandattheindicatedQ-values.Thefulllinesarethebestfittothedataasdiscussedinthetext.Thedottedlinesrepresenttheuncovolutedinelasticcontributionstothefit.FIG.2-ThelinewidthparametersΓQforglycerolarereported
asafunctionofQattheindicatedtemperaturesintheIXS’(fullsymbols)andBLS’(opensymbols)Qregions.TheinsetshowsanenlargementoftheIXS’Qregion.ThefulllinerepresentstheQ2behavior,whichisthebestfittoΓQathighQ.AlsoshowninthefigureareextrapolationoftheQ2lawinthelowQrange(thinfullline)andtheQ2.5dependence(dashedline)indicatedbythelowTBLSdata.TheinsetshowsanenlargementofthehighQregion.FIG.3-TemperaturedependenceofΓQ/Q2inglycerolatQ≈
0.03nm−1(opensymbols),atQ=2nm−1(crossedsym-bols)andaveragedovertheQ=2−4nm−1region(fullsymbols).Theverticaldashedlineindicatestheglasstran-sitiontemperature,Tg=187K.FIG.4-SameasinFig.2butforvitreoussilica.DatafromMD
simulation(crossedsymbols)andfromPOT(stars)arealsoreported.HerethelowTBLSdataindicate(ifany)apowerlawwithexponentγ=2.6.TheinsetshowsanenlargementofthehighQregion.FIG.5-SameasinFig.2butforvitreoussilica.Theinset
showsanenlargementofthelowtemperatureregion.TheopencirclerefertoBLSdataatQ≈0.035(2and◦)andQ≈0.025(3)nm−1,thefullsquarestoIXSdataatQ=1.6nm−1,andthecrossedcircletoMDsimulationintheharmonic(T=0K)limit.TheinsetshowsanenlargementofthelowTregion.
4
604020T=16 KQ=2.0 nm 150-1T=167 KQ=2.0 nm -1100500120906030Q=4.0 nm -1Intensity ( counts )08060402003020100-10010Q=3.0 nm -1Q=3.0 nm -106040200Q=4.0 nm -120-1001020Energy ( meV )
- 1 -
1010
ΓQ ( meV )1
IXS0
T=16 K (This work) T=167 K (This work) T=175 K [1] T=20 K [9] T=170 K [21]1010101010
-1
BLS-2
Glycerol1011 Q (nm)-110-3
-4
ΓQ ( meV )10-510-110
-210
-1Q ( nm )
-1
10
0 10
- 2 -
2= γ52. =γ 0110
1
(2-4 nm)IXS-1 This work Ref. [6] Ref. [1] Ref. [7] Ref. [8] Ref. [9] Ref. [21]ΓQ / Q ( meV/nm )(2 nm)0
IXS-1Tg-210
(0.03 nm )BLS-1210
-1
Glycerol0100200300Temperature ( K )
- 3 -
101010
210
MDPOTIXSBLS T=0 K [2] T=300 K [26] T=1100 K [22] T=5 K [11] T=300 K [24]ΓQ ( meV )101010101010
-1-2-3-4-5-6
SiO2ΓQ ( meV )Q ( nm-1 ) 2= γ MD T=0 K [2] IXS T=300 K [23] IXS T=1100 K [22] IXS T=1375 K [24] 2.6γ =10-210-110-1
0101Q ( nm )
- 4 -
SiO2ΓQ / Q ( meV/nm )-210
0BLS Ref. [13] Ref. [11] Ref. [25] Ref. [24] Ref. [2] ΓQ / Q ( meV/nm )-221.51.00.50.0210
-1IXSMD0100200 T ( K )050010001500Temperature ( K )
- 5 -
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