Atomic Jump Processes in Self-diffusion
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Journal of Nuclear Materials 69 (k 70 (1978) 38-60 0 North-Holland Publishing Company ATOMIC JUMP PROCESSES IN SELF-DIFFUSION H. MEHRER Institut ftir theoretische und angewandte Physik der Universitiit Stuttgart Max-Planck-Institut ftir Metallforschung, Institut ftir Physik, Stuttgart, W. Germany The atomic jump processes involved in the vacancy mechanisms of self-diffusion in metals are reviewed with particular attention to disvacancies. The most important measurements which are helpful to separate mono- and divacancy contribu- tions - temperature, mass, and pressure dependence of the diffusion coefficient and correlation effects - are discussed. The recent experimental progress will be considered also. The extension of direct tracer studies to much lower temperatures has greatly increased the reliability with which monovacancy properties may be deduced. Amongst the indirect techniques like nuclear-magnetic-relaxation, Mijssbauer effect and quasi-elastic neutron-scattering, especially nuclear magnetic relaxation may be considered nowadays as a quantitative tool. In a discussion of individual metals the above mentioned topics will be illustrated by examples, with emphasis on those metals where a considerable deepening of our understanding of atomic jump processes has been achieved. Les processus de saut atomique qu’impliquent les mecanismes lacunaires d’autodiffusion dans les m&aux sont pass& en revue en portant une attention particuliere sur les bi-lacunes. Les mesures les plus importantes qui permettent de s&parer les contributions des monolacunes et des’bilacunes, les relations entre coefficient de diffusion et les variables tempkrature, masse et pression et les effets de correlation sont discuties. Les progris exp&imentaux ¢s seront aussi consid&s. L’ex- tension des Etudes directes par traceurs B des tempiratures beaucoup plus basses a augment6 considkrablement la pouibilitd de d6duire les propri&s des monolacunes. Parmi les techniques indirectes, comme la relaxation mag&tique nucldaire, l’effet Mllssbauer et la diffusion quasi-hlastique des neutrons, sp&ialement la relaxation magn6tique nuclhaire, peuvent 6tre consid&es maintenant comme des outiles quantitatifs. Dans une discussion concernant certains m&aux, les methodes mention&es cidessus seront ill&r&es par des examples en mettant l’accent sur ceux des r&taux pour lesquels an approfondis- sement considerable de notre comprehension des processus de saut atomique a 6th obtenu. Die atomaren Sprungprozesse, die beim Einfach- und Doppelleerstellenmechanismus der Selbstdiffusion in Metallen auf- treten, werden unter besonderer Beriicksichtigung des Doppelleerstell~nmechanismus betrachtet. Die wichtigsten Mes- sungen, die zur Trennung von Einfach- und DoppelleerstellenbeitrLgen hilfreich sind - Temperatur-, Massen- und Druck- abhiingigkeit des Diffusionskoeffizienten sowie Korrelationseffekte - werden diskutiert. Der neueste experimentelle Fort- schritt wird ebenfalls betrachtet: Die Ausdehnung direkter Tracer-Messungen zu sehr kleinen Diffusionskoeffizienten hat die VerlBsslichkeit, mit der Eiienschaften der Einfachleerstelle bestimmt werden kiinnen, stark erhiiht. Unter den indiiekten Techniken wie Kernspinrelaxation, MGssbauer-Effekt und quasielastische Neutronenstreuung kann insbesondere die Kern- spinrelaxation heutzutage als quantitatives Werkzeug angesehen werden. In einer Diskussion einzelner Metalle werden die oben erwghnten Punkte anhand von Beispielen erkXutert, wobei solche Metalle herausgegriffen werden, bei denen eine Ver- tiefung des VerstBndnisses atomarer Sprungprozesse erreicht wurde. 1. Introduction Self-diffusion in crystals is one of the most impor- tant manifestations of point defects in thermal equi- librium. In metals it is generally agreed that self-dif- fusion (and also diffusion of substitutional impurities) occurs by a series of exchange jumps of individual atoms with vacant lattice sites [ 11. For many years self- diffusion has been interpreted in terms of monovacan- ties alone. Whereas the monovacancy mechanism 38
description
self diffusion
Transcript of Atomic Jump Processes in Self-diffusion
JournalofNuclearMaterials69(k 70(1978)38-60 0North-HollandPublishingCompany ATOMICJUMPPROCESSESINSELF-DIFFUSION H.MEHRER I nstitut ftir theoretische und angewandte Physik derUniversitiitStuttgart Max-Planck-I nstitutftir Metallforschung, I nstitut ftir Physik, Stuttgart, W. Germany Theatomicjumpprocessesinvolvedinthevacancymechanismsofself-diffusioninmetalsarereviewedwithparticular attentiontodisvacancies.Themostimportantmeasurementswhicharehelpfultoseparatemono-anddivacancycontribu- tions-temperature,mass,andpressuredependenceofthediffusioncoefficientandcorrelationeffects-arediscussed.The recentexperimentalprogresswillbeconsideredalso.Theextensionofdirecttracerstudiestomuchlowertemperatureshas greatlyincreasedthereliabilitywithwhichmonovacancypropertiesmaybededuced.Amongsttheindirecttechniqueslike nuclear-magnetic-relaxation,Mijssbauereffectandquasi-elasticneutron-scattering,especiallynuclearmagneticrelaxation maybeconsiderednowadaysas aquantitativetool.Ina discussionofindividualmetalstheabovementionedtopicswillbe illustratedbyexamples,withemphasisonthosemetalswherea considerabledeepeningofourunderstandingofatomic jumpprocesseshasbeenachieved. Lesprocessusdesautatomiquequimpliquentlesmecanismeslacunairesdautodiffusiondanslesm&auxsontpass&en revueenportantuneattentionparticulieresurlesbi-lacunes.Lesmesureslesplusimportantesquipermettentdes&parer lescontributionsdesmonolacunesetdesbilacunes,lesrelationsentrecoefficientdediffusionetlesvariablestempkrature, masseetpressionetleseffetsdecorrelationsontdiscuties.Lesprogrisexp&imentaux¢sserontaussiconsid&s.Lex- tensiondesEtudesdirectespartraceursB destempiraturesbeaucoupplusbassesa augment6considkrablementlapouibilitd ded6duirelespropri&sdesmonolacunes.Parmilestechniquesindirectes,commela relaxationmag&tiquenucldaire, leffetMllssbaueretla diffusionquasi-hlastiquedesneutrons,sp&ialementlarelaxationmagn6tiquenuclhaire,peuvent 6treconsid&esmaintenantcommedesoutilesquantitatifs.Dansunediscussionconcernantcertainsm&aux,lesmethodes mention&escidessusserontill&r&espardesexamplesenmettantlaccentsur ceuxdesr&tauxpourlesquelsan approfondis- sementconsiderabledenotrecomprehensiondesprocessusdesautatomiquea6thobtenu. DieatomarenSprungprozesse,diebeimEinfach-undDoppelleerstellenmechanismusderSelbstdiffusioninMetallenauf- treten,werdenunterbesondererBeriicksichtigungdesDoppelleerstell~nmechanismusbetrachtet.DiewichtigstenMes- sungen,diezurTrennungvonEinfach-undDoppelleerstellenbeitrLgenhilfreichsind-Temperatur-,Massen-undDruck- abhiingigkeitdesDiffusionskoeffizientensowieKorrelationseffekte-werdendiskutiert.DerneuesteexperimentelleFort- schrittwirdebenfallsbetrachtet:DieAusdehnungdirekterTracer-MessungenzusehrkleinenDiffusionskoeffizientenhat dieVerlBsslichkeit,mitderEiienschaftenderEinfachleerstellebestimmtwerdenkiinnen,starkerhiiht.Unterdenindiiekten TechnikenwieKernspinrelaxation,MGssbauer-EffektundquasielastischeNeutronenstreuungkanninsbesonderedieKern- spinrelaxationheutzutagealsquantitativesWerkzeugangesehenwerden.IneinerDiskussioneinzelnerMetallewerdendie obenerwghntenPunkteanhandvonBeispielenerkXutert, wobeisolcheMetalleherausgegriffenwerden,beideneneineVer- tiefungdesVerstBndnissesatomarerSprungprozesseerreichtwurde. 1.Introduction Self-diffusionincrystalsisoneofthemostimpor- tantmanifestationsofpointdefectsinthermalequi- librium.Inmetalsitisgenerallyagreedthatself-dif- fusion(andalsodiffusionofsubstitutionalimpurities) occursbyaseriesofexchangejumpsofindividual atomswithvacantlatticesites[ 11.Formanyyearsself- diffusionhasbeeninterpretedintermsofmonovacan- tiesalone.Whereasthemonovacancymechanism 38 H. Mehrer/Atomic jump processesin selfdiffusion 39 indeeddominatesoverawidetemperaturerangeit hasbecomeclearinrecentyearsthatformostmetals adivacancycontributionisobservablenearthemelting temperature(forreviewssee,e.g.[2-5]).Therefore, whenwerelatemeasurabiequantitieslikethedif- fusioncoefficientorNMRrelaxationratestoatomic propertiesofthecrystal,welearnsomethingabout vacancy-typedefects. Inthegeneraldiscussionofsection2weconsider theatomicjumpprocessesinvolvedinthemono-and divacancymechanismofself-diffusioninmetallic structureswithparticularemphasisonrecentim- provementsofthetheoryofdivacancydiffusion.The tracerself-diffusioncoefficientduetomono-and divacancymigrationanditsdependenceontempera- ture,hydrostaticpressureandisotopicmasswillbe discussedindetailforcubicmetals.Insection3some remarkswillbemadeabouttherecentprogressin tracer-measurementsofverysmalldiffusioncoef- ficientsduetomicrosectioningtechniquesandthe resultingdeependingofourunderstandingofdif- fusionmechanisms.Section4containsacriticalsur- veyovervariousindirecttechniquesforthestudyof atomicjumpprocessesinself-diffusionincluding nuclear-magnetic-relaxation,Mossbauereffectand quasi-elasticneutronscattering.Owingtorecentim- provementsofthetheorynuclearmagneticrelaxation especiallymaynowadaysbeconsideredasareliable toolformeasuringself-diffusion.Insection5we turntoadiscussionofindividualmetalswithpartic- ularemphasisonrecentdevelopmentsandonthose metalsforwhichfairlydefiniteconclusionsondefect propertiesmaybedrawn. 2.Generaldiscussionofselfdiffusion 2. I _ General remarks and diffusionmee~~isrn Thetransport of matter whichaccompaniesthe motionofvacantlatticesitescanbedescribedbythe so-calledmacroscopiccoefficientofself-diffusionDSD. Itisrelatedtothemeansquaredisplacementofthe diffusingatomsandconsequentlytothejumpfre- quanciesandjumpdistancesoftheatomicjumps.Ac- cordingtorandomwaiktheory(see,e.g.f&7])we have (2.1) whereNisthenumberofdifferenttypesofjumps. r,(a: =1 ) .... iV)denotesthenumberofjumpsoftype (YmadebyanatomperunittimeandAx,thex-pro- jectionofthepertainingjumpdistance.Foradefect mechanismofdiffusionr,isgivenby r,=ca?,>cm wherec,denotestheatomicconcentrationofdefects presentatthermalequilibriuminaconfiguration whichpermitsano-typejumpofagivenatom.v,is thejumpfrequencyinvolved. Theself-diffusioncoefficientobtainedfromtracer experiments,DT, isdifferentfromDsD. Asfirst pointedoutbyBardeenandHerringIS]aquantitative measureofthisdistinctionistheco~e~tionf act or f .Itaccountsforthespatiaicorrelationbetweensuc- cessivejumpdirectionsoftraceratomsandleadstoa reductionofthetracerdiffusioncoefficientwith respecttothemass-transportcoefficient.Incubic crystalswehave DT=fz)=, 12.3) whereasinhexagonalcrystalstwotensorcomponents parallel(II) DT,I= f 11@D, (2.4a) andperpendicular(1) DT,~= fDSW (2.4b) tothehexagonalaxismustbedistinguished.Thecor- relationfactor(s)is (are)characteristicfora givendif- fusionmechanismandmaybecalculatedifthejump frequenciesoftheatomsinvolvedareknown.The methodsforcalculatingcorrelationfactorshavebeen reviewedbyLeClaire[9]andMehrer[lo]andwill notbediscussedhere.Correlationfactorshavebeen workedoutforalmostallcasesofpracticalinterest andwillbediscussedinsection2.3. 2.1 .I.Monovacancymechanism Inthermalequilibriumtheconcentrationofmono- vacanciesinamonoatomiccrystalisgivenby CIV =exp(Sj"v/k)exp(-~~~/kT)y(2.5) 40H. Mehrer /Atomicjump processes in self-diffusion withHFvandSrvdenotingtheenthalpyandentropy parameterscontainedinitbycomparisonwithexper- offormation. iments. Inanyofthethreecubic Bravais lattices themo- tionofthemonovacancyischaracterizedbythe monovacancyjumpfrequencytonearest-neighbour sitesinthelattice (2.6) whereH~vandSf;2r denoteenthalpyandentropyof motion,and$Vtheattemptfrequency.According to(2.3)theself-diffusioncoefficientoftracersmay bewrittenas NV=fivCIVhva2 ,(2.7) wherea isthecubiclatticeconstantandfivthemono- vacancycorrelationfactor(fiv=0.781inanfeeand fiv= 0.723inabeestructure[l11). Inthefeestructuretherearefourlatticesitesthat arenearestneighbourstobothsitesofavacancypair onadjacentsites(In-configurationofthedivacancy). Thismaybethereasonwhyintheliteratureithasbeen assumedthatthedivacancymigratesbynearest-neigh- bourjumpswithoutchangingitsconfiguration. Whereasthissimplemodelofdivacancymigration isindeedmostlikely,additionalpossibilities,e.g. additionalboundconfigurations,mayexist.Amore generalmechanismwhichincludesbounddivacancy configurationsatfirst-andsecond-nearestneighbour sites(withconcentrationsC:$andCiF)isillustrated intheupperpartoffig.1.Inthermalequilibriumthe divacancyconcentrationsarerelatedby Inhexagonal close-packedstructurestwodifferent jumpfrequenciesmustbeconsidered-one(v*v,A) forjumpswithinandanother(v~~,~)forjumpsobliqu tothebasalplane.Thecomponentsofthetracerself- diffusioncoefficientmaybewrittenas c: c*2v12=2c%v*v21> ( 2. 8)wherev2vfjdenotesthejumpfrequencieswhich transformthedivacancyfromanithnearesttoa jth nearestneighbourconfiguration.Usingthisrelation DTG =3;~ clvvlV,B C* and (2.7a) D;$=@f:vClV(3%V,A+ hv,da, (2.7b) witha andcdenotingthehexagonallatticeconstants. Thecorrelationfactorcomponents_f/?andf:vare functionsoftheratioVIV,A/YIV,Bandhavebeencal- culatedbyMullen[ 121(seealso[ 131). V2Vf 2 "2vrr I n2n 2.1.2.Divacancy mechanism Diffusionviaboundpairsofvacanciesismorecom- plexthanmonovacancydiffusion.Ingeneralseveral configurationsofthepairwithdifferentbinding energiesmaybepresentinthermalequilibriumand atomicjumpsovermorethanonetypeofsaddlepoint maycontributetoitsmigrationeveninthecaseof cubiclattices.Althoughvarioustheoreticalcalcula- tionsconcerningdivacancyconfigurationsandmove- mentshavebeenperformedtheyarebasedoninter- atomicpotentialsthatarenotsufficientlyreliableto permitadecisionastowhichoneofthevariouspos- sibilitiesprevailsinagivenmetaloreveninagiven structure.Thebestapproachmaythusbetoworkout theconsequencesofafairlygeneraldivacancyme- chanismforeachstructureandtodeterminethe 0.31 .2.I .6.8I .8.6.4.2 "2vrr)- vzytlv2v12Vzvrr Fig. 1. Divacancymechanismofself-diffusioninanfeelattice, correlation factor andmaximumisotopeeffect. H. Mehrer /Atomicjump processes in selfdiffusion41 thediffusioncoefficientfortracermotionbydiva- canciesmaybewrittenas T-22In f)sv-J aCzv(vav1t+v2v12)fzv.(2.9) Thecorrelationfactor.f2visafunctionoftheratio v2~ll/u2~12showninthelowerpartoffig.1. (For detailsofthecalculationsee[lo].)Inthecaseofthe simplemechanism(vav11 jumpsprevailing)fav approachesthevaluecalculatedearlierbyHoward [ 141,Bakker[15],andMehrer[ 161.However,as soonasthedivacancyhasanadditionalmigration modethetracermotionislesscorrelatedandfavis temperaturedependentinsteadofbeingjusta num- ber. Inthebeesfruccuretherearenolatticesitesthat arenearest-neighbourstobothsitesofavacancypair onadjacentsites.ThismeansthataIn-divacancycan- notevenmove(bynearestneighbourjumpsofthe individualvacancies)unlessadditionalnon-nearest neighbourcon~gurationsexist.MehrerEl 71con- sideredthreebounddivacancyconfigurationsatfirst-, ln2nI n 0.466 cl.33 .2.I .6.61.B.6.4.Z -3uL iV.?L%w-- Fig. 2. Divacancy rne~~an~rn ofselfdiffusionin a beelattice, correlationfactorandmaximumisotopeeffect. second-andfourth-nearest-neighboursites(concen- trationsC&,C$$andc*,$)andtheatomicjump frequenciesshownintheupperpartoffig.2.This fairlycomplicatedmodeofdivacancymigrationis notwithouttheoreticalsupport[l&20].Inthermal equilibriumthedetailedbalancingrelations 3GGv2~12=4C%2~21 (2.10a) and 12&2~24=GCy2v42 (2SOb) holdandallowthetracerdiffusioncoefficienttobe expressedintermsofthesecond-nearestneighbour configurationaccordingto[ 17 ]DTv=2 a2%@2V21+v2V24)f2V. (2.11) Thecorrelationfactorf2visshowninthelowerpart offig.2asafunctionofv2v21~~2v24. 0.3 00.20.40.60.87.00.80.60.L0.20 VZV,dRfvzv,*a-) -%,A.4fvrv,ns Fig.3. Divacancy mechanismof se~~iffu~onina hcplattice, correlationfactorsparallelandperpendiculartothec-axis[ 131. 42H. Mehrer /Atomicjump processes in self-diffusion Thedivacancymechanisminthehcpstructurehas beenconsideredbySteineretal.j13].Asshownin fig.3,twobounddivacancycan~gurationsmaybe distinguished:anA-configuration(concentrationC&) wherebothvacanciesoccupynrlrest-neighboursites inthesamebasalplanes,andaB-configuration(con- centrationC&V, wherethetwovacanciesarelocated inadjacentsitesoftwonei~bouringbasalplanes. Themigrationofthedivacancyasanentityinvolves fouratomicjumpfrequenciesshowninfig.3.The tensorcomponentsoftheself-diffusioncoefficientof tracermotionbydivacanciesmaybewrittenas[ 131 D#= c2C%v~~,~Bf8v (2.12a) and (2.12b) wherethecorrelationfactorcomponentsaremultival- uedfunctionsoftheatomicjumpfrequenciesshown inthelowerpartoffig.3. 2.2.Temperaturedependenceofself-diffusion Withinalimitedtemperaturerangeself-diffusion datamayoftenberepresentedwithsufficientaccuracy byanArrheniuslaw (2.13) whereboththepre-exponentia1factorDzffandthe activationenthalpyQeT;etakenas independentof temperature(kdenotesBoltzmannsconstant). Byinsertingeqs.(2.5)and(2.6)intoeq.(2.7),we obtainfortheactivationenthalpyofself-diffusionby monovacancies FM Q1v=H,vfHlV3 (2.14a) andforthecorrespondingpre-exponentialfactor (2.14b) ofcubiccrystals.Theinterpretationofmeasuredval- uesofQeffandDgff intermsofeq.(2.14)hassome- timesbeencalledthestandardinterpretationofself- diffusion. However,deviationsfromanArrheniusbehaviour appeartobeanalmostcommonfeatureofself-dif- fusioninmetals.Fortheso-calledanomalousbee metalslike/3-Ti, &ZrandV1 wherethedeviationsare fairlystrong,thishasbeenknownformanyyears(see, e.g.[l]).ConsiderablecurvaturesoftheArrhenius plothavealsobeenobservedforthealkalimetals (seesection5).Thesmallestcurvaturesarefoundin thefeemetals.However,theextensionofdiffusion measurementstolowertemperatureswiththehelp ofmicrosectioningtechniques(seesection3)andim- provementsoftheexperimentalaccuracyhavepermit- tedtheirobservation.Agoodexampleisprovidedby theself-diffusiondataonsilver,wherefourstudies ofthreeindependentgroupscoveralmosttenorders ofmagnitudeinthetracerdiffusioncoefficient [21-23,117]. Thereareseveralpossiblecausesforacurvatureof theArrheniusplotofbulkself-diffusion.Incubic metalsthemostimportantonesare* : (i)mono-and divacancycontributionstoself-diffusion,and(ii)tem- peraturedependenceoftheactivationparameters.In hexagonalmetalsthecomponentsofthetracerself- diffusioncoefficientevenforamonovacancyme- chanismwillingeneralnotobeyanArrheniuslaw. WhenthemigrationenthalpiesforA-andB-jumpsare different,weexpectfromeq.(2.7)deviationsdueto thesuperpositionoftwoArrhenius-termsineq.(2.7b) andduetothetemperaturedependenceofthecorrela- tionfactor.Inthefollowingsubsectionsweconfine ourselvestocubiccrystalsandconsidereachofthe abovementionedreasonsfornon-Arrheniusbehaviour insomedetail.Theextensiontohexagonalcrystalsis easilyperformed. 2.2.I.Simultaneousaction ofmono- and divacancies Whenbothmechanismsoperatesimultaneouslythe tracerdiffusioncoefficientisgivenby DT=D+D&.(2.15) Sincethemonovacancymechanismhasthelower activationenthalpyitalwayspredominatesatlower temperatures.WithincreasingtemperatureD&/DTv increases.WhereasL>Tv forcubicmetalsobeysan ArrheniusLaw, D&Z mayingeneralalreadybeasuper- * Atrivial cause fora curvatureof thekrrheniusplotatlow temperaturesis along shortcircuitslike grain bound- aries anddislocations. If highly perfect single crystals and/or themicrosectioningtechniquesdiscussed insection3 are used,theinfluenceofshortcircuitsmaybeeliminated. H.Mehrer/Atomicjump processesin self-diffusion43 positionofvariousArrheniustermswithslightlydif- ferentactivationenthalpiesandmayimplyatemper- aturedependentcorrelationfactor[seeeqs.(2.9)and (2.1l)].However,sincethedivacancycontributionis oftenonlya smallcorrectionterminDTitmaybe difficulttoresolvethedetailsofthedivacancyme- chanismfromananalysisofthetemperaturedepen- dence.Ontheotherhandcorrelationandmasseffects discussedinsection2.3aremoresensitivetosuch details. Forthosefeemetalswhereitissufficienttocon- siderthesimpledivacancymechanism,eq.(2.15) reducestoasuperpositionoftwoArrheniusterms DT=Dyexp(-g)+Diexp(-z),(2.15a) wheretheabbreviationsineq.(2.14)forthemono- vacancyparametersand Q2v =Wyv-H::+H%, D!:=4f2va2v$exp 2sF;+A&v+ s % k (2.1Sb) forthedivacancyparametershavebeenused.Hyv andHFvdenotethemigrationandbindingenthalpy ofanearest-neighbourdivacancy,Syvisthemigra- tionandAS,,istheassociationentropyofthediva- cancy.&isthepertainingattemptfrequency.The effectiveactivationenthalpydefinedbyQeffs-dIn DT/d(l/kZJisaweightedaverage D:vD;v Qeff=Qlv-+Qzv---, DTDT (2.16) oftheactivationenthalpiesofthetwomechanisms. 2.2.2.Temperaturedependenceof activation parameters Intheprecedingdiscussionwehaveimplicitly assumedthatdefectenthalpiesandentropiesare independentoftemperature.However,allequations remainvalidifthisassumptionisnotmade.Apriori, thereislittlereasontoexcludethepossibilityofa temperaturedependenceofthedefectparameters. Theonlythermodynamicrequirementisthatthetem- peraturevariationsofenthalpiesandentropiesare relatedaccordingto (E),=T(%lp. (2.17) Sinceeq.(2.17)definesa specificheat,atemperature variationofthedefectparametersisequivalenttothe statementthatthereisanadditionalspecificheat associatedwithdefectformationandmotion.Hence attemperatureswellbelowtheDebyetemperature, wherequantumratherthanclassicalstatisticsmust beused,defectparameterswillbetemperature dependent.AbovetheDebyetemperaturethedefect parametersaretemperatureindependentaslongas theharmonicapproximationcanbeused.Anhar- monicityeffectswhichmanifestthemselves,e.g.in thermalexpansion,giverisetoanincreasingrelaxa- tionofthedefectwithincreasingtemperature.This meansthatthedefectentropyandbecauseofeq. (2.17)alsotheenthalpymayincreasewithtempera- ture. Sincetheexpectedvariationsfornormal metals arerathersmallwemayexpandtheenthalpyina Taylorseries[3,4]as H(T)=H(TO) +cwk(T-TO) +Pk(T -TO)2 t... , (2.18) whereT,-,isareferencetemperatureandoand/3 arecoef- ficients. DeVries[24]hasstressedthepossiblesignificance ofthequadratictermineq.(2.18).However,theoret- icalestimatesbyLevinsonandNabarro[25],Giri- falco[26]andFlynn[27]indicatethatthetempera- turevariationoftheformationenthalpiesisvery smallforclose-packedmetals(typicallyoftheorder of0.01eVbetweenroomtemperatureandmelting point).Moreover,Franklin[28,29],whoincluded quantumandanharmonicityeffectsintothestatisti- calmechanicalapproachtoreactionratetheoryofdif- fusion,obtainedthatthepre-exponentialfactor0:of copperself-diffusionvariesbylessthan20%overa rangeoftenordersofmagnitudeinDT.Theassociated entropyvariationaccordingtoeq.(2.17)corresponds toavariationoftheactivationenthalpywhichisless than0.02eV.Aprocedurehowsuchsmallvariations canbeincludedintotheanalysisofdiffusiondata,if necessary,hasbeenworkedoutbySeegerand Mehrer[2]. RecentlyGilderandLazarus[30]claimedthatthe wholecurvatureintheArrhenius-plotofself-dif- fusionisexplainableintermsofasinglehighlyrelaxed vacancy-likedefectinwhichtheanharmonicityofthe 44H. Mehrer /Atomicjump processes in self-diffusion latticemodesgivesrisetoalargethermalexpansion ofthedefect.Positivethermalexpansioncoefficients ofthedefectwhichareasmuchas15timeslarger thanthoseofthecrystalitselfarepostulated.How- ever,inthepresentauthorsviewadecreaseofthe defectvolumewithrespecttotheatomicvolume shouldoccurratherthananincreasewhenthedefect configurationbecomesmorerelaxedwithincreasing temperature.GilderandLazarusarguethatthelarge defectexpansioncoefficientissupportedbytheob- servationthattheactivationvolumeofself-diffusion increaseswithtemperature.However,asoutlinedin section2.3,thiseffectcanbeexplainedinaquite naturalwaybythesimultaneouscontributionsof mono-anddivacanciestoself-diffusion. Inferromagneticmetals atemperaturevariation oftheactivationparametersmustbeexpecteddue totheinfIuenceofferromagneticordering.Incon- trasttothevariationsdiscussedabovethismaybea bigeffect.Clear-cutexperimentalevidenceforthis hasbecomeavailableonlyveryrecently,sinceprecise diffusionexperimentsintheferromagneticregion necessitateappropriatemicrosectioningtechniques. Anexampleisprovidedbythemeasurementsof Mehreretal.[31]onpureiron.Thediffusioncoef- ficientintheferromagneticregiondecreasesmore rapidlywithtemperaturethananArrheniusextrapoia- tionoftheparamagneticdatawouldsuggest.This meansthatonehastobecarefulifonecompares activationenthalpiesmeasuredwellbelowtheCurie temperaturewithmeasurementsintheparamagneti~ region. Ruthetal.1321haveproposedanexpressionin whichthedeviationfromanArrheniuslawisrelated totheferromagneticorder-parameterR.Formono- vacancydiffusion,whichcertainlypredominatesin theferromagneticregion,theirresultmaybewritten as DT =DTv=07exp[-QPva(lf~R2)/k7],(2.19) where@Fdenotestheactivationenthalpyinthe paramagneticregionand7isadimensionlessparam- eterwhichmaybedeterminedfromacomparison withtheexperimentaldata. 2.3.Correlation and the isotope effect Sincethecorrelationfactorisnotthesamefordif- ferentdiffusionmechanismsitsdeterminationmay helptoestablishthediffusionmechanism(seetable1). AccuratemeasurementsofDTandDSD accordingto eq.(2.3)areinprinciplecapableofgiving f=LPfP(2.20) andprovidingthisinformation.Anexampleforthis aremeasurementsonLidiscussedinsection5. Table1 Correlationfactorsandisotopeeffectforself-diffusioninmetallicstructures Face-centered-cubicBody-centered-cubicHexagonal-close-packed Parallelc-axisPerpendicularc-axis ~_.-_l. ..._.- fiV=0.723[llffivsee 112,131 Monovacancy fiv=0.781[ll] fivsee1121 EIV=fivAK,vEIV=fivAK,v.Eiv=ftvA&v,B Eivsee[44,13] simpledivacancy Divacancy fiv= 0.468[ 14-161 Ezv=f2vA&vWWfiv see fig.2 flv seefii.3 fh see fig.3 82v
