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