CALPHADioZ.2,No.3,pp.227-238. DPergamon Press Limited, 1978. Printed in Great Britain. AMODEL FORALtOylNGEFFECTStNP~RR~NAGN~TlCMETALS MatsHillertandMagnusJar1 DivisionofPhysicalMetallurgy RoyalinstituteofTechnology S-10044STOCKHOLM 70 Sweden Abstract Amathematicalrepresentationofthemagneticspecificheat,recentlysuggestedbyInden, wasappliedtoironinanevaluationofthedifferenceinGibbsenergybetweenthefeeandbee states.Theresultingequationswerethenusedforatreatmentofalloyingeffectsinferromag- neticmetalsdueTVthechangeoftheCurietemperature.Theresultwasapproximatedinorder toconformtothesubregularsolutionmodel.Astrongasymmetrictermwasobtained. Introduction ItwaspointedoutbyZener(1)longagothattheeffectofanalloyingelementonthe magneticstateofaferromagneticbasemetalshouldresultinastrongthermodynamiceffect, Thiseffecthasrecentlyattractedconsiderableattention(2-4)buttheresultst&kecomplicated analyticalforms.Anattemptwillnowbemadetodevelopatreatmentinasimplerform.Inpar- ticular,anattemptwillbemadetoputtheresultintotheregularorsubregulartypeofre- presentation.Theworkwillbebaseduponatypeofdescriptionofthethermodynamicsofafer- romagneticmetal,recentfydevelopedbylnden(3)anditwiI1beappliedtoironandironbase atioys. Theregularandsubregularsolutianmodelsareusefultoolsforapproximatedescriptions ofthethermodynamicpropertiesofbinaryalloys.Theyareparticularlyvaluableinworkcon- cernedwiththecouplingbetweenthermodynamicsandphaseequilibriaandtheyhavefoundexten- siveuseinthatfield. DescriptionofGibbsenergyforiron ProvidedthatthetemperatureismuchhigherthantheDebyetemperature,iti spossibleto useapolynomialforthedescriptionofthedifferenceinGibbsenergybetweentwopossible state5ofapureelement, AG=A+BT4CTlnJ+DT2 flf Theferromagneticmetalsconstituteanimportantexceptionandithasbeencustomarytopresent theresultofevaluationsofGibbsenergyforironinTables(5-11).KaufmanandNesor(12)who usedanexpressionlikeeq.1hadtogivedifferentparametervaluesaboveandbelowtheCurie temperature. lnden(3)hasrecentlyshownthatthefollowingtypesofexpressionscanbeusedfora ratheraccuratedescriptionofthemagneticcontributiontothespecificheatofaferromagnetic metalA _1 cA ym! =KzRln-!.%? l -2 forT1 7-1 ThequantityTisdefinedasT/TCwhereTCistheCurietemperature. KiandKtaretwoconstants 228M. Hillert and M. Jar1 fortheelementAinitsferromagnetic(9)andparamagnetic(8)state. Byintegrationofthespecificheatlndenderivedexpressionsforthemagneticcontribu- tiontotheGibbsenergy.Thefinalexpressionsareverycomplicatedandinpartbasedupona powerseriesexpansion.inordertoarriveatasimplerexpressionitispossibletoexpandthe expressionsgivenbyeq.2and3inapowerseriesandtotruncatetheseriesbeforeintegra- tion.Theusefulnessofsuchaprocedurewillbeexploredinthepresentreport.Thefollowing approximationsofeqs.2and3willbeused. 2\A =2KaR(~3+,g/3+~15/5) m8 CA =2K;R(T-5+~-5/3+~-25/5) forT1 (4) (5) Theyapproximateeqs.2and3verywellexceptforatemperaturerangeveryclosetoT.Eqs.2 and3gotoinfinityatTwhichisinagreementwit 4and5approachfinitevglues,46/15RKaand $manytheoriesoforderingpheno &na.Eqs. tivealternativefromthispointofviea, 46/15RK fi. respectively,andmaythusbeanattrac- also.TheerghtatT,canbeadjustedbykeeping moretermsbutsomeoftheadvantagewitheqs.4and lndenevaluatedthenumericalvaluesofKa andK i? wi11beTestifmanytel;msareretained. KFe=0.714.Accordingtohisprocedureeqs.4affd5yifildthefollowingresult. forbeeironfindingKFe=0.554and T cCM m8 B;(=)-G;(B)=6$dT+7$dT=gR(K;+0.6K;). TC $(=) -H~~TC)=7mB TC cAdT=~~CK~ H,m(T,) -H;(D)=ILcrmdT=GRTCK; 0A (6) (7) (8) LetfbethefractionofthetotalmagneticenthalpywhichisabsorbedabovetheCurietempera- ture.Eqs.7and8thenyield, (9) Whenanalyzingthespecificheatdataforbeeiron,indenfoundf=O.4.Forfeemetalshefound f=O.28.Hesuggestedthatthefvaluedependsuponthestructureandproposedthat0.4could beusedforallbeemetalsand0.28forallfeemetals. Thetotalmagneticentropyisoftenexpressedbythefollowingexpressionwhere8,isthe meanatomicmomentexpressedinBohrmagnetons(9), q-4-St(O)=Rln(BA+1)(0) Combinationwitheq.6yields K;+0,6KB - 675 A5ii8 ln(OBA + If Bycombiningthiswitheq.9andusingthevaluef=O.4weobtainforbeemetals, Kfi = n(OBA+1) A 518. =0.64t71n(BA+1) 1125 (11 (12) K;=o.g80~(~AForbeeironthevalue f=O.28wouldyield Ki=0.426Bn(BA K;=.o46gn(BA +1)(3) of0BA=2.22yieldsKE=1.073andKi=O.7504.Forfeemetalsthevalue +1)(14) +1)(15) WhenevaluatingthemagneticcontributiontoGibbsenorgyweshalldefinethisquantityas zeroatTd.FortemperaturesabovetheCurietemperaturethefollowingexpressionisobtained IF A =*K;RTC[/10r4+[35T+/YOT241(17) BycontinuingtheintegrationbelowTConeabtainsthefolIowingexpressionfortheferromagne- ticto)state. om GA 8 =-KART&79/140-58~/125J-K;BTC[.4/6~~D/l~~~r6/600+71./120-51~~/675] (19) ThedifferenceinGibbsenergybatweehfoeandbeeironwiltnowbedescribedbyaubtr~ting tkemagneticcontributionforbeeIron,accordingtoerg*7or3,fromeq.,whfchcxmtains fouradjustabfeparameters.TheycanbedeterminedfromexperimentaT~nfor~tj~ontheeq~ilf- briumbetweenthetwophases.FromQrrandChipman(31)wecanchoosethe~~~~~~ngvaguerfor theequiJibriumtemperatures,eorrectsdtothefPTS68t~m~~rat~rescale,85and667K.From BraunandKoohthaatt(131wecanchoosesinenthalpyoftransformationof9iDJlmolat1185Kand -850J/ma\at166TK,Insertedinourequationsthesedatayieldthefollowingparametervalues, A=-5188.3,Bs45.79,C*-6.3andD=D,DD24,expressedinJ/maandK.AbovetheCurietemperature wethusobtain Belowt heCurietemperatureweobtain OGY Fe -G$*3883.4+36.07-6,3TnT+D.0D24T2 +$309[T4f6T~+TD/35~~+T16/600T~6)J&no (21 fn ho&rkeseexprassionsavalueofTC=lOQKmustbeused, Thefunctiongivenbyeqs.2Dand23wasevasratedn~~~~~~~~andinFigs,tand2itis comparedwiththeevaluationspresent&byOrrandChIpmanf]andbyKaufmanandBesot(2). Abovethetrans~~~rn~t~onpointatil85Kthenewevaluatianagreeswet1w3thGrrandChipman andbelowthetransformationitagreeswellwithKaufmanandNesor.Thisispartcuiarysatis- factorybecausetheybasedtheirevaluationonthepreviousevaluationbyKaufman,Clougherty andWeiss(14)whousedarealisticastIrnateoftheDebyetemperatureforfeeiron.Eqs.20and 21thusseemtogiveanadequaterepresentationoftheGibbs;energyforiron,Wwever,they shauldnotbeusc?dmuchbelowtheDebyotemperature. Previoustreatmentsofthemagneticaloyingeffect Longagoile-nerftfpointedoutthatanimportanteffectofalloyingadditionstoiranis causedbythechangeofthemagnetit:state,Hesuggestedthatthiseffectcouidbedescribed approximatelybyaparatfefdisptacementalongthetemperatureaxisofthemagneticpartof theGibbsenergyforIron,adisplacementcorrespondingtothechangeoftheCurietemperature. ThissuggestionWMfollowedbyHilert,WadaandWada(HWW)whoderivedthefollowingex- pressionforlowalloycontent,x,(151, Gze= xQS;edTC/dx (221 230 M. Hillert and M. Jar1 --Orr andChipmon -.-KoutmanawlNow Fig.1DifferenceinGibbsenergy betweenFCCandBCCironinthe temperaturerange1000-1800K accordingtodifferentworks. 4 --0~andChipman -.-Kc&nonandNuor 3 2 1 0 500lOOa Fig.2DifferenceinGibbsenergy betweenFCCandBCCironinthe temperaturerange300-1lODK accordingtodifferentworks. A MODEL FOR ALLOYING EFFECTS IN FERROMAGNETIC METALS231 Grnis a& OSrn themagneticcontributiontotheGibbsenergyofaniron-basealloyinthebeestate isthemagneticentropyinpurebeeiron. Z&erssuggestionwassomewhatarbitraryandcannotbestrictlyjustifiedonthermodyna- micgrounds.Aformallymoresatisfactorytreatmentcouldbebasedonanymathematiclmodelof themagneticpropertiesofiron.SuchtreatmentswererecentlydiscussedbyMiodownik(4).HOW- ever,apartfrcnntheworkofHWW notreatmenthasyetbeenputinanalyticalform. ThechangeinthemagneticcontributiontotheGibbsenergyduetotheadditionofan alloyingelementcanbewritteninthefollowinggeneralform AG;=G;-xAoG;-xBG;(23) Thesuperscriptmishereusedtodenotemagneticcontributionswhereasthesubscriptmdenotes molarintegralquantities.lnden(16)hassuggestedthatoneshouldusethesametypeofde- acriptionforthemagneticcontributioninanalloy,Gm,asinapureelementobyinsertingthe meanvalueoftheatomicmomentsforthemixtureofelementsinthealloyas tivewillbeexaminedinthepresentpaper. 8,.Analterna- Itisbaseduponaseparationoftheeffectsfrom differentelements.Forabinaryalloyweshallwrite AG; = xA(G; - "c;c,+ xg(G; - G;) (24) Thisequationcanbeusedindifferentways.FollowingZenersoriginalsuggestiononeshould neglectthedirecteffectofthealloyingelementanddescribethemagneticcontributionfrom thebasemetalbydisplacingtheGibbsenergyfunctionforthepurebasemetalalongthetem- peratureaxisbythesameaa-ountAT astheCurietemperaturehasbeenchanged. AG; = xAtoG;(T-AT)-G;(T)1(25) HWapproximatedthisexpressionbythefirstterminaseriesexpansionandassumedthatthe Curietemperaturevarieslinearlywiththealloyingcontent.Thismethodactuallyyieldsthe followingexpressionwhichhassometimesbeenused(17) AGE=-ATxAdoG;/dxB=xAxBoS;(T)dTCA/dxB(26) Themagneticalloyingeffectthustakestheformoftheregularsolutionmodelalthoughthe temperaturedependenceisquiteunique. Nishizawaetal.(18)haveemphasizedthatitmaybeessentialtoretainanotherterm. Thisisparticularlyevidentifonewantstocalculateamiscibilitygap.Theyderivedanequa- tionforthespinodalinthefollowingwaybyputtingthesecondderivativeoftheGibbsenergy equaltozero.Fromeq.25oneobtainsthefollowingexpressionforthecasewheretheCurie temperaturevarieslinearlywiththealloyingcontent, d2AGm m dG;(T-AT),,,d2'Gm(T-AT) -=2* A dx; (ddT)2 dTdXg+BdT2dxB Thefollowingequationisthusobtainedforthespinodal. d2GmdS;(T-AT) -= dx2, -2L+RT/xAxB-2'S;(T-AT)$$ - xA (daT)2=G B dTdx B (27) (28) whereLisaregularsolgtionparameterwhic,hmdescribesthenonmagneticdeviationfromideal solution.ThequantitydS/dTisequaltocITanditisthusevidentthatthelasttermin eq.28hasaverystronge e feetintheneighorhoodoftheCurielinewherethespecificheat6 goestolargevalues.Infact,Nishizawaetal.wereabletopredictthatamiscibilitygap, whichisprimarilyduetoachemcialeffectexpressablebyaregularsolutionparameterL,may developahornalongtheCurielineifintersectedbysuchaline. Newapproachtothemaqneticalloyingeffect Asalreadyemphasized,Zenerssuggestionwassomewhatarbitraryandanattrativealterna- tivewouldbetoinserteqs.17and19ineq.24whichcouldbedonebyusingtheindividual 232M. Hillert and M. Jar1 valuesofBforeachelementintheevaluationoftheKvaluesfromeqs.12and13butusing theCurietemperatureforthealloyineqs.17and19andassumingthatfineq.9isacon- stant.Admittedly,thetheoreticalbasisforsuchaprocedureco$dbeqtieestionedbutatleast itdoesnotviolatetherulesofthermodynamics.ThequantitiesGAandGinthealloywould dspendypontheconcentrationduetothekoncentrafigndripendenceoff3a!!dTC.21n2partlfular, dG,/dxBwillagaincontainatermwithcIsequalto Asaconsequence,thismodelmayalsopre8. sinceaGA/aT fr 1:*aGA/aT. tctthedeveiomentofahornalongaCurieline whichintersectsamiscibilitygap.Infact,Indensoriginalmodelwhichgivesinfinitespe- cificheatvaluesatTwouldpredictthatatleastaverythinmiscibilitygapshouldalways developalongthewho1Curieline.ti Inordertoapplythenewtreatmenttoaspecificcase,itisnecessarytoknowtheindi- vidualvaluesof@A and8,whereasmagneticmeasurementsonlyyieldtheaverageBforthealloy B =XABA+XBB8 Whenotherkindsofinfor~tionislackinganarbitraryassumptionmustbemade.Accordingto Bates(19)itisoftenreason?bietoputG,=Ofornonmagneticalloyingelementsinironandto treatGAasaconstant.ThesewerealsoZenersassumptions.Theywerenowusedforacompari- sonbetweentheHWversionofZenersmodelandtheneirmodelatlowalloycontents.Asa basisforthecomparisonthemagneticGibbsenergywasdefinedaszeroatTm.Theresultsare presentedinFig.3.Theyarequitesimilarbutthenewmodelpredictsaslightlysmalleref- fect. Miodownik(4)discussedtheshort-comingsoftheZenertreatmentandpointedoutthatit mightbeimportanttoincludetheeffectofchangesinthesaturationmagnetisation.Forthe alloysFe-It% CrandFe-lo%Cohemadeaquantitativecomparisonwithvaluesobtainedfromhis owntreatment.InordertocomparethetreatmentdevelopedinthepresentreportwithMiodow- nikstreatmentthenewtreatmentwasappliedtothesametwoalloys.Thecomparisonismade inFigs.4and5.FortheFe-11%Cralloythenewtreatmentwasappliedintwodifferentways. InModel1iswasassumedthattheGvalueoftheFeatomsisindependentofthecomposition andthattheBvaluefortheCratomsisalwayszero.Thisyieldsafairrepresentationofthe satusationmagnetisationoftheFe-Crsy5tem.Amoreaccuratedescriptionisobtainedwith -0.8xwhichmakesthebvalueforFeatomsdecreasefrom2.2forpureFeto1.4ata !!$t!EhdiifltioninCr.ThisvaluewasusedinMode12.Bothmodelswereusedwithavalue ofdT/dx=-700K/moiwhichisaveryroughapproximationanddoesnottakeinto.accountthe initr.EI?-1icreaseofTwhenCrisaddedtoFe.Thecurvegiven,fortheHWW treatmentinFig.4 wasalsocalculatedw$ththisvalue.FortheFe-lo%CoalloyModel1maynotbeveryrealistic sincetheGvaluefortheCoatomsisnotnegligible.Itwasfoundthatthesaturationmagne- tisationcurvefortheFe-CosystemcouldberepresentedratheraccuratelywithB=OB+1.9X, andaconstantvalueofB=1.7.ThecurvedenotedbyModel3inFig.5wasobtar *i8dwfeh thesevalues,withaCurl &temperatureof1200KforpurebeeCoandwithdTIdx=1050K/m01 forFerichalloys.Thelattervaluewasalsousedinthecalculationofth 5FcuvesfromHw treatmentandfromModel1.Ailthecurvesareverysimiiaranditmaythusbeconcludedthat allthemodelsareratherequivalent. Figs.4and5indicatethatthechangeinthesaturationmagnetisationhasnotadrastic effectbut inthecaseofFe-lo%Cotheeffectisappreciableatlowtemperatures. Sub-regularsolutionrepresentation ThenewmodelpredictsaverycomplicatedconcentrationdependenceoftheGibbsenergy throughtheconcentrationdependenceofGA,S,andTItissometimesusefultohaveapower seriesexpansionandformanyapplicationsthesubreformalismissufficient.It describestheexcessGibbsenergywithatermxx AB (Lattemptwillnowbemadeto approximatethenewmodelinaccordancewiththisforegionsaboveandbelowthe CurielinewillbetreatedseparatelybutanattemptwillbemadetodecreaseasmuchasPOS- siblethecreationofartificialdiscontinuitiesontheCurielinewhere$thetworegionsmeet. TheapproximationwillbedevelopedfortheArichsideofanA-Bsystem. ThevariableTCcanbeseparatedfromf3 lg.Thefollowingtypeofrelationcanbein e orS,inviewoftheformofeqs.12,13,17and reduced (30) wheregG(T)isobtainedbyinsertingeq. fromeqs.$2,13and19. 12ineq.17.Asimilarfunctiongo(T,)isobtained ThetreatmentwillbelimitedtocaseswheretheCurietemperatureandtheatomicmoments EE x A MDELFOR ALLOYING EFFECTS IN F~~O~G~T~CMETALS 233 -Thismodel --Subr r gut af apprax --~il~rt,~da,Wada -101 05001000 15002000 Temperature( K f Fig.3Comparisonbetweenthemagneticalloyingeffectattowalloy contentsaccordingtoHillert,WadaandWadaandthiswork. T~m~mture4 K 1 -lrnmW Temperature( K) Fig.4Comparisonofpredictionsofthemag- neticalloyingeffectinanFe-11%Cr Fig.5Comparisonofpredictionsofthemagnetic alloyingeffectinanFe-lo%Coalloy alloyaccordingtovariousmodels. accordingtovariousmodels. 234 M. Hillert and M.Jar1 varylinearlywithcomposition, TC=TCA (l+kxB)(31) GA=OGA GB=OGB where OB and factorii?eq. ln(BA+l) +ax B (32) +bx B (33) Garethevaluesatx50.Forpure6wethushavethevalueGR+b.Thesecond 30*givesthefollowinglowerseriesexpansion, axB .2x2 =In(ORA+axB+l)zln(OGA+l)+-- B OBA+l2(oGA+l)2 (34) andasimilarapproximationisobtainedforln(B,+l). byinsertingeqs.30and34ineq.24oneoltaine a2x2 AG;=xAln(BA+l)(g(TC)-g(TCA))+xAg(TC) B 2(Of3,+1)2 > +xBg(fC)~n(*~~+l) bXB -x~g(TCB)ln(~~+b+l)+xag(TC)r Be+ herpowersareexcludedsincetheydonotenterintothesubregularsolution orgshouldbeinsertedasgineq.35dependinguponwhetherTfallsaboveor belowtheCurietemperatureofeachterm(T m-Gm)andshouldthusgotozero ,TorT).Thefirsttwotermsineq.35come fromxA(Gfl 5st&8aliobBapproachespureA.Thispropertyisre- tained.1el&tthreetermscomefromx(G-mG) approachespureB.Thispropertyisnotpet!!ine8 andshouldthusgotozeroasthealloy becausetheseriesexpansionismadeinthe vicinityofpureAinsteadofB.Thisisnotaseriousdrawbacksinceeq.35isintendedtobe usedinthevicinityofpureA,only.Ontheotherhand,thepropertymayeasilyberestored bymultiplyingthelastthreetermsbyxA.Thisprocedurewillbeusedinthefoliowing,mainly becauseitwillhelpmakingtheexpressionsconformtothesubregularsolutionformalism. WhenintroducingtheconcentrationdependenceofTCiseqs.17aig19,allthehigher powersinTCwillfirstbeneglectedandonlytermsinTC,TCandTCwillremain. cm8= A -KBRT5,10T4 AC (36) GF-K;R[TC/2-2T/51A-KoR[T4/6T3+T/2-2T/3] cc (37) WhentheKvaluesarederivedforthisapproximationonefindsbythemethodusedbefore, K;= In(OBA+l) {+?j($l) =-#n(bA+l)=0.71431n(BA+1) Ko=ZKG=1 A 2A* 07141n(B+l) A (38) (39) ForbeeironthevalueofOG=2.22yieldsKo=l.2529andKG=G.8353. Theexpressionsgivenbfeqs. 36and39p reservetheAimportantpropertiesthatGmo andGmB, aswellastheirderivatives,havethesamevalueontheCurielinewherethetworegl\onsmeet, i.e.atT=T.WhenapproximatingthevariousTCtermsineqs.36and37bytruncatedpower seriesexpafisionsinxitwouidbedesirablenottodestroytheseproperties.Ifpossible,a methodofapproximatioRshouldthusbeusedwhichisexactatT=T ThetruncatedpowerseriesexpansionstakethefollowingforsdependinguponwhatpowersC onelikestoretain, A MODEL FOR ALLOYIEJGEFFECTS IN F~~O~G~T~~METALS235 (T,/T,,~~=(l+kx)=1 +5kxb5B*10(kxg?+LB(kxL@(40a) (TC/TC~)4=(l+kxg15=I+fjkxg+MRkxg(4fJb) (T,IT,,)-~= (l+kxg)-3=I-3kxg+6(kx,12fLa(kxB)* (TC/TC,.jW3=f l+kxg)-3=1-3kxN*MkxN (TCJTC&-~=(l+kx,)-3=f+Na TC/TCB =r+kx=1+0 cf- f3 (40dl (40e) f40fI (4Og) 1,M,Nand0arecorrectionparameterswhichcouldmaketherelationsexactiftheywere allowedtobefuntionsofthecomposition,x.Thoseexactfunctionscaneasilybecalculated fromtherelations.However,sincetheparam!tersarenotallowedtobefunctionsofx 8 inthe subregularsolutionformalism,onecouldinsteadgivethemconstantvalues,e.g.thevluezero. Ontheotherhand,anysuchvaluewouldintroduceanerrorontheCurielineexceptforone point.ItmaybeabetteralternativetolettheparametersvaryalongtheCurielineaccording totheirexactfunctionsbuttoaccomplishthisbytreatingthemasfunctionsofTinsteadof xB-Thiscanbedonebyinsertingtheapproximatevalue, kxg=i-1wheret=TIT CA f4Tf intheexactfuncrions.ThisrelationisexactatT=T terswi11thusgettheircorrectvaluesontheCurie1 IE inviewofeq.3&andr;bltheparame- e.ThepropertyGA=Gwillthusbe preserved.However,theirderivativeswithrespecttotemperaturewi11notbe&alsincean artificialtemperaturedependencehasbeenintroduced.Unfortunately,thereseemstobeno methodbywhichbathpropertiescanbepreserved.Thefollowingexpressionswereobtainedwith thismethod, Itisinterestingtonotethatallthesecorrectionparameterstakethevaluezeroat?=l,i.e. atT=T. T64fivetermsineq.35willnowbeevaluated.Thevaluesathightemperaturewiltbede- notedbythesuperscriptBbecausetheywiltdependuponthepropertiesoftheBstate.For thefirsttermweobtainfromeq.36byapplyingtheapproximationgivenbyeq.40a, AG;=x~o~~~(T~~-T~~~~~T4=-~~xgkRTGAOL~Ikl10+C3~kxgli10+4 $431 Thesuperscriptoindicate3thattheKvalueshoutdbecalculatedfromeq,38withthe3value characteristicofpureA,8,.Thevalueofthefirsttermineq.35atlowtemperatureiseva- luatedfromeqs.37and40d, 236M.Hillert and M. Jar1 AGa = 1 x KBR(T AA CA-TC)/2+xAK$T4/T;A-T4/T3+c3(TCATC)116 =-xAxBkRTCAIoK;/2+oK~[3-3_r4+(6cLa)kxB?4]/6~(44) Thesetwoexpressionshaveasfiriousdrawback.Eq.43isintendedf oruseathightemperatures andthecorrectionparameterLthengoestoveryhighvaluesaccordingtoeq.42a.Eq.44is intendedforuseatlowtemperaturesandthecorrectionparameterI.thengoestoveryhigh valuesaccordingtoeq.42d.However,thecorrectionswereintroducedsimplyinordertoavoid adiscontinuityattheCurietemperature.Itisthuspossibletotransferthecorrectionfrom oneoftheequationstotheotherwithachangeofsign,Suppose,forinstance,thattheCurie temperatureisloweredbythealloyingaddition,i.e.kisnegative.Itisthenconvenienrto includetheLtermfromeq.44ineq.43 AC!= -xAxBkRT~AtoK~~5+(10+L~)kxB]/,o~-4oK~LakxB~4/6}(451 ThetwoLtermscannowbeincludedinthecalculationabovetheCurietemperatureTCofthe alloyusingtheLvaluesgivenbyeqs.42aandd.However,asthetemperatureisincreasedto theCurietemperatureofpureA,TboththeLparametersgotozeroandthecorrectionterms canthusbeomittedaboveTthecreationofadiscontinuity. useofeq.45canthusbet&it Theinstructionfpr;he andLaaretakenfromeqs.42aanddforTT.ThesecorrectionswillthusbeincludedonlybetweenT tionmustCilsobemadeinthisregionbecausethealloyisinaBs F andTA&hercorrec- atebu FA;he ofpureAisa.Thefrstpart mBm13 feq.35shouldthencontainGmB(T)-Gnrr(T referencestate )OneshouldthusaddtheAfol~~ingAter )whereaseq.43 G(T :,~:~~a,~d,Br~dGen~~~;ak&i~~o*acco~t, LA toeq.45atthe AG* 1 =.x(Gm8- AA OGy)=xARTCA~oK~t5-4~-l/~4]/10+K;[3-4r+r4],6}(46) Formally,thiscorrectionaffectsthestandardstateforpureAratherthantheregularorsub- regular.solutionparameters. Ontheotherhand,allthecorrectiontermsshouldbeincludedineq.44iftheCurietem- paeratureisincreasedbythealloyingaddition,i.e.kispositive. AG;=-xAxBkRT~A{oK~[5-L~kxB/~4~/~0+oK~[3-3~4+(6~La)k~B~4]/6~-Ai+&(47) Inthiscase,thecorrectionshouldbeusedbelowtheCurietemperatureofthealloy,TC,and ~~LB~xp@=GforT