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&centsserontaussiconsid&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