Solid StateCommunications,Vol. 23, pp. 759—764,1977. PergamonPress. Printedin GreatBritain

LATFICE RELAXATION AND THE ENERGYOFMDUNG IN DILUTE ALKAU-ALKAU ALLOYS

G. Solt

CentralResearchInstitutefor Physics,1525 Budapest,P.D.B.49,Hungary

and

A.P.Zhernov

I.V. KurchatovInstitutefor Atomic Energy,Moscow,USSR

(Received2 May 1977byA. Zawadowski)

Theenergy of solution for a homovalentimpurity in asimplemetaliscalculated,with accounttakenof the staticdisplacementfield aroundthesolutedatom.The asymptoticbehaviourof the distortingforceandtheatomic displacementsareexpressedin termsof the pseudopotentialsforhostandimpurity atoms.Numericalresultsfor the heatof mixing indilute alkali—alkali systemsshowa delicatebalancebetweenthevolumemisfit termon the onehand,andthe electronegativityandrelaxationtermsonthe otherhand.The relaxationenergyprovesto be of the sameorderastheheatof solutionitself. Thevaluesand thetrendsin theheatof mixing comparereasonablywell with theempiricaldata,exceptforlithium-basedalloys,and thepredictedvariationof thebulk moduluswithconcentrationagreeswith the experiment.

ALTHOUGH differencesin electronegativityandin wherec = cB is theconcentration,thefirst term is theatomic sizeare empirically well knownto bethe energyperion within a pieceof puremetalA homo-essentialfactorsin determiningalloyingbehaviour geneouslyexpanded(or compressed)to haveaunit cell[1, 2], a quantitativeandmoreor lessab initlo descrip- volume~2o(insteadofitsactualequilibriumvolume ~ZOA)

tion of thealloying processin metallic systems,evenfor andthesecondtermstandsfor contributionsnotcon-simple (s—p)metalsandin the dilutecase,becomes nectedwith latticedistortion,while theselatterareunavoidablyrathercomplicated.Only for the studyof groupedinto thelastterm.In a dilute alloy (c ~ 1) thehomovalentimpuritiesin simplemetalscanthe new displacementsof the impurity ionsfrom theregularstandardperturbationtreatmentof the polarized lattice sitescanbeneglected.Then,in caseof disorder,electronliquid [3] be extendedin a straightforward one hasway, in the hopethat thesecondorder(linearscreening) F, — b ~approximationalreadygives abasicallygood description. I~.E

1,= c B A —— ~ (j~12_ IVA(G)I)~GIn thepresentwork thetotal energyof a metal ~ 2 G

containinghomovalentimpuritiesis calculatedobtain- c(1 — c)ing therebythe displacementfields,the limiting partial — I AVq I

20q (2)heatof mixing andothercohesiveparametersfor the q G

alloy. Thepresentmethoddiffers from earlierwork whereG is a vectorof thereciprocallattice,i~is the[4—6]in includingthe energycontributionfrom the meahpotential i~= (1 — C)VA + CVB, andb standsforlattice relaxation,giving thusa morecompletetreatment the averageof thenon-coulombicpartof thepotentialof the energyto beinvestedon accountof thesize i r I Ze~\differencebetweensolutedandsolvent.Wenotethat b. = ~ J V

1(r) + — dr (I = A, B),the latticestrain in alkali alloyswas calculatedearlier r /[7] by asimilar methodto studyelectricalresistivity. N is the numberof ions, ~v = VB — VA and~ is defined

Let thebareionic potentialsfor solventA and throughthe staticdielectric functionsolutedBbe VA andVB, andlet ~2obe the average 2

volumeperion in thealloy. In thesubstitutionalcase, ~ = £~ (1 — l/e~).thetotal energyperion in secondorderperturbation 4iretheoryis Fore(q) the form proposedby GeldartandTaylor [8]

E(c,~2~)= EA(fZO)+ + ~E (1) was used.Thelattice distortiongives rise to the

759

76(3 LATFICE RELAXATION IN DILUTE ALKALI—ALKALI ALLOYS Vol. 23, No.10

relaxationenergy Herec(l—c)f~ h~&VGt2(PGJI~VqI2~O

~‘id == — ~ Re {v~(q)~vq[S~(u)— S~(0)]SB(Q)}~q 2 ~ G øq ~ dqj

a

and the term+AEAØs) (3)= (1 — C){EA(~TZo)— EA(~

7OA )}wherethe structurefactorfor thedistortedlattice+ C{EB(flo) —EB(L20B)} (10)

S~(u)= ~ ~ will be calledthe“volume preparationenergy”asit is

thework necessaryto expand(or squeeze)the twois a functionof the staticdisplacementsu

1 of the atoms piecesof metalsA andB from their normaldensitiesfrom theidealsite at I, Sq(O)is the structurefactorfor 1/~oAand l/~zoB,respectively,to havethe actualthe ideal lattice,andSB is the analogousexpressionbut numberdensityof the alloy.with summingoverthe impurity sitesonly. Further, The first two contributionsin (9)were studied~EA is theenergyneededto reproducetheabove in earlierwork [4, 5]. The remarkablecancellationS~(u)(i.e. the abovestatic displacements)in apieceof betweentheseterms[5] underliestheimportanceofpuremetalA. taking into account~Er, which,on physicalgrounds,

Now, themethod [9, 10] of expanding~Er in isexpectedto be roughlyproportionalto ~ Thepowersof u1, stoppingat secondorderand findingthe numericalcalculationconfirmedtheseexpectations.energyminimum with respectto the displacements Still generally,(5) leadsto a purelyradialforceleadsto field, with asymptoticoscillationsfor largeII’ I = II — LI

1#L duetothesingu1arityin~atq=2k~,= ~ (4) 1’ girz

2[VAE&V (P/P)2] sin (2kFl’)a F(l’) ~ -~— ~

where!= L is an impurity site,Fia = ~ F(l — L)a, andthe forcefield due to animpurity at L is wherep = (q2f4ire2)(e

0—1) andP0 is the Lindhard

function. In a first approximationonecanneglectg~C1F(l — L)a = ~ q~vA(q)L~vQ~q sin [q(l — L)J (5) besideI in (6), to obtain for thestatic displacements

while the staticdisplacementfield canbe expressedas ~ —~ ~ ~ g~(q)Q~(q)sin [q(l — L)] (11)qEB

U?a = — ~ (I + gAH)j;i1,~,g~~,z”a”Fra’. (6) wheretheq — s aresummedwithin the Brillouin zone,i’a’i”a” and

HereI is the identitymatrix,Q~(q)= ~ (q+ G)~v~(jq+ GI)L~v1Q+G~ç~I(Iq+ GI).

aF~ G (12)Hiai’a’ =

alc, With this approximationfor Uj, onegetsfrom (4), (5)

andgA is thestatic Green’sfunction [101 for the pure and(11)solvent, _c(l —c) ~ ?~(q)Qa(q)Q~(q). (13)

= 2 ~gi”~,,ra’ = ~‘(ci) ~ (7)EB Forlargedistancesfrom theimpurity the displacement

~- e~(q,X)e~~(qX) field follows, of course,the inversesquarelawas(8)g~~(q)= MA ‘~ predictedby thecontinuumtheory [11],

with the usualnotationfor lattice frequencies,polar- (~-~)~ (1’ = 1—L) (14)ization vectorsandionic mass.

From(1), (2) and(4)—(7) thevolumeis determined wherethe amplitudeC, at agiven direction,is determinedby aE(c, cz0)/a~z0= 0, while theheatof mixing is by

0A’ VB and4 (q-~0). Insteadof quotingtheobtainedby subtractingfrom (1) the concentration somewhatlonggeneralformula,we noticehereonly thatweightedaverageof the energyperion of the pure

on approximating~by theGreen’sfunctiong~,of ancomponents, isotropiccontinuum[10], it reducesto

= + ~id + ~r (9)

Vol. 23, No.10 LATFICE RELAXATION IN DILUTE ALKALI—ALKALI ALLOYS 761

lul \

005 ~““It2_’~~~.0~ -~

D3O

~fl20

d~to~e~ [~/~II I I I1 2 3

Fig. I. Thecalculatedradialforce field F(l) (in units of c11a

2)and theradialdisplacementfield u(l) (in unitsof thelatticeconstanta) vs thedistancefrom a potassiumimpurity in sodium.Forcalculatingu(l) the continuumGreen’sfunctionwasused.The asymptoticsfor u(l) is shownby the dashedline, thearrowsindicatethe positionsof thefirst two neighboursin the bcclattice.

C = ZAz(q = 0) 1’ ThiscontinuumGreen’sfunctionapproximation41rc~4k • ‘r~-~~) (15) wasusedto calculateA.E, andhencethelimiting partial

heatof mixing, i.e. theenergyneededto solvejust oneconfirming theintuitive ideathat thestrengthof the ion B at zeropressure,displacementfield hasto beproportionalto thedifferenceof the averageelectron-ionpotentialandto the hAB = I ~ (17)the compressibility(‘- 1/c

11)of the host.In thesimplest ~ c(1 — c)but informativemodel whensolventandsolutedionsare characterizedonly by an emptycore radiusR~and For thenumericalcalculationsthemodel potential [12]

—— r>ro{(R~)

2— (R~)2}. rR~,the amplitudeC is seento beproportionalto { Ze2The forceanddisplacementfields areshown in v(r) = (18)

Fig. I. for thediluteNa(K) alloy. In calculatingu? the Ze2—U— T’~T0

approximation~-~g~,jwas usedin (6), leadingto To

purely radialdisplacements.For AEr, againwith the was used,with theparametersr0 andu adjustedsoastosamedevice reproducethedensityandc,A for thepurecomponents.

ThecontributionstohAB areshown in Table1 for— c(1 — c) r {qv~(q)Av~~ }2 dq. (16) sometypical cases.Therelaxationenergyis seento take

r — 4~.2cAb

762 LATFICE RELAXATION IN DILUTE ALKALI—ALKALI ALLOYS Vol. 23,No. 10

Table1. Contributionsto the limiting partial heatof m~j!Jg[equations(9)and(17)], in 1O~ry,for selecteddilutealkali alloys. The partial limiting volumechange Afl0, thechangein the atomicradii AR andthederivative.ofthe bulk moduliB at c = 0 aregivenin unitsof thedifferencesin therespectivequantitiesfor thepurecom-ponents.Theuppernumberin bracketisan empiricalestimate[15] andthe lowerone is thevaluefor the liquidalloy [16] at 384K In caseofstrictly linear lawsfor all concentrations,thenumbersin the last threecolumnswouldbe±1

~ F AE~~ { AE~ AE,d 1 h (~~)AB (A1~)AB 1 (dB\oy [cO —c~J~~o[c(1 —c) c~O c(1 —c)j~..~o A(B) I~OA ~OBI IRA RBI IABI~~dc,Lo

2.1Na(K) 36.8 —14.3 —20.4 (4.6) 0.77 0.97 —1.90

(2.8)

4.4K(Na) 18.6 — 4.3 — 9.9 (3.8) —1.19 —0.98 0.62

(1.7)

0.8K(Rb) 2.4 — 0.5 — 1.1 (0) 0.96 1.02 — 1.26

(0.4)

0.6Rb(K) 2.0 — 0.5 — 0.9 (0) — 1.05 —0.96 0.79

(0.4)

Table2. Calculatedvalues,in iO~ry, for the limitingpartial heatsof mixing in alkali—alkali solidalloys. Theempiricaldata [15] andthosefor the liquid alloy [16] are in bracketsasin Table1. Thepotentialparametersr0 andu arealsogiven

(Li) (Na) (K) (Rb) (Cs)

Li —10.8 —36.4 — —

= 1.5589 (12.9) (49.5) (63.2) (78.5)u=C.+08 — — — —

Na 0.2 2.1 7.4 17.3= 2.1731 (9.1) (4.6) (6.9) (9.9)

u = 0.480 — (2.8) (5.5) (6.2)

K 16.9 4.4 0.8 4.2= 3.099 (25.1) (3.8) (0) (0)

u = 0.624 — (1.7) (0.4) (0.5)

Rb 22.4 6.9 0.6 0.9= 3.4863 (29.7) (4.6) (0) (0)

u = 0.697 — (2.6) (0.4) (— 0)Cs 32.8 12.6 2.5 0.8

= 3.9966 (32.8) (6.1) (0) (0)u =0.768 — (1.3) (0.2) (—‘0)

off some20—30%of the volumepreparationenergy. — (d&~0\ — ~1OA[a2 (AEp + AEr)1

The significanceof this becomesespeciallyclear when (A~2o)AB — — ~ aca~2O .IC=O

realizingthatAEr isof the sameorderofmagnitudeas (19)the heatof mixing itself, or evengreater than that.

Thepartialvolume changedueto theimpuritieswas Comparingtheinitial slopesof ~ (c) with I ~1OA — ~‘ZoBIdeterminedfrom it is seenthat11

0(c)startsin all casesconvex,indicating

Vol. 23,No. 10 LATFICE RELAXATION IN DILUTE ALKALI—ALKALI ALLOYS 763

volumecontractionon alloying. Moreover,Vegard’slaw anequallystrongrelaxationof thehost. [In fact,Li(Rb)for theatomic radii (i.e. for the latticeconstant)is andLi(Cs) areevenoutsidethescopeof the presentnearlyfulfilled, althoughtheobservedsmall negative scheme,sincero of theimpurity would belargerthandeviation from this rule in K(Rb) andK(Cs) alloys [13] theWigner—SeitzradiusR of thecompressedstate.] Aswasnot found.On the otherhand,the limiting variation for Li(Na) andLi(K), themodelpotentialsfor theseof the bulkmodulusB with concentration,for Na—K impuritiesmayleadto an underestimateof AE1,,, atalloys,indicates(lastcolumnof Table1) that thealloy very largecompressions[12], while theharmonicstartsto be“softer” thana linearlaw for B(c)would approximationfor AEr alsogetsincreasinglyworseforpredict, in agreementwith the experimenton theliquid largesizedifferences.Na—K system[14]. Wenotealsothat theempiricalrulehAB > ~ if

Incidentally,wenotice that the simplelinear ~2OB>f2

04, is confirmedin mostcases,thoughthisrelationshipbetweenthe strengthof the displacement agreementmaybe fortuitousin view of thesubtlenessfield and~ holdingin themodel of a sphericaldefect of this asymmetryeffect.in anelasticmedium [11] is recoveredhereasa In conclusion,the calculationof the heatof mixing“zero-thorder” approximation.In fact,by ignoring andothercohesiveparametersfor homovalentdilutebothA.E~andall termsofsecondorder in thepotential alloysof simplemetalsseemstobe within the scopeofin AE~,(19) isseento reduceto the appropriatecon- the pseudopotentialtreatment,if in calculatingthestanttimesC in (15). volumemisfit energytherelaxationof thehostaround

The resultsfor hAB are summarizedin Table2 and the impuritiesis takeninto account.To achieveanyfor comparisonboththe estimate[15] basedon an reasonableaccuracy,however,themodel potentialsempiricalcorrelation[2] anddatafor the liquid alloys must give a reallygood descriptionof both thepure[16] areshown. solventandimpurity. In mostof the alkali—alkali sys-

ThepredictedhAB.s,exceptfor Li(Na) andLi(K), tems,wherethe abovecriteriacanbe fulfilled, theare all positive,sensiblyclosein generalto the empirical numericalresultsturn out to bereasonable,despitethevaluesand follow the trends in a given row or column. extremesmallnessof theheatsof solutionin this case.Realizingthat theseheatsof solutionsare on the Whetherthe discrepanciesfor Li(Na) andLi(K) areMillirydberg scale,smallerby an orderof magnitudethan dueto thefact that lithium is the leastgood“pseudo-thecohesiveenergiesof the purecomponents,the overall potentialmetal”amongthe alkalis,or the toolargeagreementmaybe regardedasgratifying. sizeeffecthere requiresa differenttreatment,is to be

Yet,while hAB is expectedto bepositiveevery- studied,togetherwith the role of theanisotropyandwhere(evenfor Rb—Csthedata [16] suggesthAn>0 thecontinuumapproximationin calculatingthe relax-for thesolid phase),thecalculatedvaluesfor Li(Na) and ation energy.Li(K) arenegative.The errormaylie in that theuseof alocal potential is muchworsefor Li than for theother Acknowledgements— The authorsare gratefultoalkali metals,or in the very largesizeeffect in these Prof. Yu. Kagan for suggestingtheproblemandforcases.In thecalculationthis appearsasa verystrong severalenlighteningdiscussions.Oneof us(S.G.)is

indebtedto Prof. P.F. deChatel (Amsterdam)forcompressionof the solutedmetal(AE~,),followed by sendinghim the dataquotedasreference[15].

REFERENCES1. HUME-ROTHERYW., inPhaseStability in MetalsandAlloys (Editedby RUDMAN P.S.,etal.). McGraw-

Hill, New York (1967);HODGESC.H. & STOTTMJ.,Phil. Mag. 26,375(1972).

2. MIEDEMA A.R., DE BOER F.R. & DE CHATEL P.F.,Phys.F: Met. Phys.3, 1558 (1973).

3. BROVMAN E.G.& KAGAN Yu., DynamicalPropertiesofSolids,(Editedby HORTON G.K. &MARADUDIN A.A.), Vol. 1. North-Holland,1975.

4. INGLESFIELDJ.E.,J.Phys.C: SolidStatePhys.2, 1285 and1293 (1969).

5. HAYES T.M. & YOUNG W.H.,Phil. Mag. 21, 583 (1970).

6. TANIGAWA S. & DOYAMA M.,J. Phys.F: MetalPhys.3,977(1973).

7. POPOVICZ.D.,CARBOTFE J.P.& PIERCYG.R.,J.Phys.F: MetalPhys.3, 1008 (1973).

8. GELDART D.J.W.& TAYLOR R.,Can. J. Phys.48, 167 (1970).

9. KANZAKI H.,J. Phys.Chem.Solids 2, 24(1957).

764 LATFICE RELAXATION IN DILUTE ALKALI—ALKALI ALLOYS Vol. 23,No.10

10. FLYNN P.A. & MARADUDIN A.A.,Ann.Phys.18,81(1962).

11. ESHELBYJ.D.,J.AppLPhys.25, 255 (1954).

12. BROVMAN E.G.,KAGAN Yu. &HOLAS A.,FYz. Tverd. Tela12,1001(1970).

13. KRASKO G.L & GURSKI Z.A.,FYz. Tverd.Tela 14,321(1972).

14. ABOWITZ G. & GORDONR.B.,J.Chem.Phys.37, 125 (1962).

15. MIEDEMA A.R., DE BOERF.R. & BOOM R. (private communication).

16. YOKOKAWA T. & KLEPPAOJ.,J.Chem.Phys.40,46(1964).