M.PayamiFebruary6,2008
forTheoreticalPhysicsandMathematics,AtomicEnergyOrganizationofIran,
P.O.Box11365-8486,Tehran,Iran
Abstract
Inthiswork,wehaveusedtheliquid-dropmodelinthecontextofstabilizedjelliummodel,tostudythestabilityofZ-plychargedmetalclustersofdifferentspeciesagainstfragmentation.Wehaveshownthatontheonehand,singlyion-izedclustersarestableagainstanyspontaneousfragmentation,andontheotherhand,themostfavoreddecayprocessforthemisatomicevaporation.However,multiplychargedclustersofsufficientlysmallsizesmayundergospontaneousdecayviafissionprocesses.ComparingtheresultsfordifferentspeciesshowthatforfixedN,thelowerelectrondensitymetalclusterscanaccommodatemoreexcesschargesbeforetheirCoulombexplosions.Thiscomparisonalsoshowsthat,forfixedZ,theatomicevaporationwhichisthemostfavoreddecaymechanismforsufficientlylargeclusters,takesplaceatlowerNsforlowerelectrondensityclusters.
1
Center1Introduction
Studyofthestabilityofchargedmetalclustersagainstfragmentationisimportantinnano-sizedsystems.Inthiswork,wehavestudiedthebinaryfragmentationofN-atomZ-plychargedMZNclustersintheframeworkoftheliquid-drop[1]model(LDM)withstabilized[2]jelliummodel(SJM)coefficientsfortheenergies.Here,Z=1,2,3,4;andMcoversthespeciesAl,Ga,Li,Na,K,andCs.Here,inthecaseofAl,whenNisamultipleof3,itcorrespondstorealAlνclusterwithν=N/3.Theclustersareassumed
BB
tobesphericalwithsizesdeterminedbyR=N1/3rswithN≤100.ThequantityrsistheWigner-Seitzradiusoftheelectronsinthebulkmetal.ThepossibledecaychannelsofMZNare
Z−Z1Z1
MZN→MN−p+Mp
(1)
Here,pispositiveintegerasN,and0≤Z1≤[Z/2].Differentprocessesaredefinedby
differentvaluesofZ1.TheprocessinwhichZ1=0(oneofthefragmentsisneutral),iscalledevaporation.Theprocessesinwhich0 (2) issufficientforthedecayinginthatparticularchannel.Intheaboveequation,EZ(N)andE0(N)arethetotalenergiesofZ-plyionizedandneutralN-atomclusters,respectively.However,infissionprocessesanegativevalueforthedifferenceenergyisnotasufficientconditionforthedecayoftheparentcluster.Thisisbecause,thecompetitionbetweentheshort-rangesurfacetensionandthelong-rangerepulsiveCoulombforcemaygiverisetoafissionbarrier.Theheightsofthefissionbarriersarecalculatedusingthetwo-spheresapproximation[3].InFig.1,thefissionofaZ-plychargedN-atomclusterintotwoclustersofrespectivesizesN1,N2=N−N1andrespectivechargesZ1,Z2=Z−Z1isschematicallyshown.Qfistheenergyrelease,Bcisthefusionbarrier(orCoulombbarrier)whichisthemaximumenergyoftheCoulombinteractionoftwopositively-chargedconductingspheres,takingtheirpolarizabilitiesintoaccount.Bfisthefissionbarrierheightwhichisdefinedas Bf=−Qf+Bc. (3) 2 TheCoulombinteractionenergy,Ec,asafunctionoftheirseparations,d,fortwochargedmetallicspherescanbenumericallycalculatedusingtheclassicalmethodofimagecharges[3]. Themostfavoreddecaychannelinevaporationprocessesisdefinedasthechannelforwhichthedissociationenergyassumesitsminimumvalue D(N,p)=minD(N,p), p Z ∗ Z (4) andthemostfavoreddecaychannelinfissionprocessesisdefinedasthechannelforwhichthefission-barrierheightassumesitsminimumvalue, Bf(N,p∗)=min{Bf(N,p)}. p (5) TheenergyofanN-atomZ-plychargedclusterintheLDMisgiven[4,5]by EZ(N)=E0(N)+Z(W+ c 2(R+a) , (6) inwhichW,c,R,andearetheworkfunctionofthebulkmetal,thefinite-sizecorrectiontotheworkfunction,theradiusofthecluster,andtheelectroncharge,respectively.Forsimplicity,thepositionofthecentroidofexcesscharge,a,isneglectedinourcalculations.E0(N)istheenergyofaneutralN-atomclusterintheLDM,whichisgivenby B2B E0(N)=εN+4π(rs)σN2/3+2πrsγN1/3. (7) InEq.(7),thequantitiesε,σ,andγaretotalenergyperelectronofthebulk,surface energy,andcurvatureenergy,respectively.Wehavecalculatedthequantitiesc,W,σ,andγbyfittingtotheself-consistentKohn-Sham[6]resultsintheSJMfordifferentrsvalues[5].However,εiscalculatedusingtheSJMenergyexpressionforthebulksystem(Eq.(1)ofRef.[7]). 2Resultsanddiscussion Usingthemethodofimagecharges,wehavecalculatedtheCoulombinteractionenergyoftwochargedmetallicspheres,takingtheirpolarizabilitiesintoaccount.Thecalculationsshowthatthemaximumoftheinteractionenergy,Bc,isachievedforaseparationd0≥R1+R2.Fig.2(a)showstheCoulombinteractionenergyofanN1-atomclusterwithanotherN2-atomofrespectiveexcesschargesZ1=3andZ2=1.Theradiioftheclusters BB arecalculatedfromR=N1/3rs,asintheSJM.Inthisfigure,wehavetakenrs=2.07 3 whichcorrespondtobulkAl.WhenbothN1andN2aremultiplesof3,theresultscorrespondtorealAlclusters.ThevalueatthemaximumspecifiesthequantityBc,andthepositionofthemaximum,d0,istheseparationbetweenthecentersofthetwospheres.Asisseen,forfixedvaluesofcharges,Bcisthehighestwhenthesizesareequal.InFig.2(b),thesituationisshownforequalchargesZ1=Z2=2.Inthiscase,whenthesizesareequal,Bcismaximumasbeforebuthere,d0=R1+R2.AnotherfeatureshowninFig.2(c)isthat,whenbothchargesandsizesareasymmetric,Bcishigherifthesmallerchargecorrespondstothesmallercluster.InFig.2(d),wehavecomparedtheCoulombinteractionenergiesoftwoAlmetallicsphereswithrespectivesizesN1=2andN2=18fordifferentcharges.Itisseenthatkeepingthesizesfixedbutincreasingthechargeonanyoneofthem,withoutchangingthechargeontheotherone,increasestheheightofthemaximum. InourcalculationsweneedthevaluesoftheCoulombinteractionenergyatthemaximapoints,i.e.,theBcs.InFig.3(a),wehavecomparedtheBcvaluesforequallycharged 1/3 Z1=Z2=1,AlmetallicparticlesbutwithdifferentradiiR1=2.07N1andR2= 1/3 2.07N2.Thisfigureshowsthatwhenbothofthesizesaresmall,thebarrierishigherthanthecasewhenatleastoneofthemislarger.InFig.3(b),wehavecomparedtheBcsforsinglyionizedpairsAl,Na,andCswithdifferentvaluesofN1andN2.Inthealuminumcase,sincetheradiiofthespheresaresmallerthanthoseofNaandsoon,theirpolarizabilitiesaretherebysmallerandthustheAlvalueslieabovethatofNa,andsoon.Now,ifthechargeononeoftheclustersisincreasedwithoutchangingthatoftheotherone,thearrangementofthecurveswouldnotchangebuttheyshiftupwardaccordingtoFig.2(d). IntheenergycalculationsofEq.(6)weusetheresultsobtainedforcandWinRef.[5].ThevaluesofεarecalculatedusingEq.(1)ofRef.[7]forunpolarizedcase.σandγinEq.(7)areobtainedbyfittingtheresultsoftheself-consistentsolutionsoftheKohn-ShamequationsintheSJMforneutralclustersofsizesN≤100,anddifferentrs B values.ThevaluesofrsforAl,Ga,Li,Na,K,andCsare2.07,2.19,3.28,3.99,4.96,and5.63,respectively.InFigs.4(a)-(c)wehaveplottedtheSJMvaluesofε,σ,andγ,asfunctionsofrs,respectively. Toshowthevariationsofthedissociationenergy,DZ(N,p),fordifferentcharginganddifferentevaporationchannels,wehaveplotted,inFigs.5(a)-(b),thequantitiesD1+(N,p)andD4+(N,p),respectively,asfunctionsoftheneutralfragmentsize,p,forLi.AsisseeninFig.5(a),forsmallpvaluesD1+(N,p)hasanincreasingbehavior,thenafterpassingamaximum,itchangestoadecreasingbehavior,andfinally,forlargepvaluesitincreasesagain.Thisbehaviorhasitsrootsinthetradeoffbetweenthesurfaceenergy(thevolumeenergytermdoesnotchangeinthefragmentationprocess)andthelasttermofEq.(6). 4 Forsmallp,thesurfaceareadifference ∆S(R1,R)=4π 2R1 +(R− 3 32/3R1) −R, 2 (8) increases,andthenforintermediatepvalues,itassumesamaximum,andfinallydecreases. Theincreasingbehavioratthetailisbecause,withincreasingp,thesizeofthesinglyionizedfragmentdecreasesandhence,beyondacertainpvalue,thelastterminEq.(6)dominatesthesurfaceenergyterm.However,beyondaZvaluefortheparentcluster,thelasttermoftheEq.(6)dominatesthesurfaceenergytermforsmallerpvaluesbeforeDZ(N,p)reachesitsmaximumasinthecaseofFig.5(b).Infact,wehavefoundthatforLiZ70thistransitiontakesplaceatapoint1.6 22/3 DZ(N,p)→2πγrsp1/3+4πσrsp, (9) whichspecifiesthemostfavoredchannelinanevaporationprocesswithp∗=1.This meansthat,inevaporationprocesses,theparentchargedclusterpreferstoevaporatesingleatomsthanlargerneutralclusters.ThisfactisalsonumericallyshowninFigs.5(a)-(b).Ontheotherhand,thepositivityofDZ(N,p)inthewholerangeimpliesthattheparentchargedclusterisstableagainstanyspontaneousevaporation. InFig.6,wehaveplottedthemostfavoreddissociationenergies,DZ(N,p∗),ofNaclustersasfunctionsoftheparentsize,N,fordifferentZvalues.Theresultsshowthatforlargeenoughclusters,thedissociationenergyisindependentofZ,whichisconsistentwithEq.(9).However,forsmallclusters,thedissociationenergyincreaseswithcharge.Figures7(a)-(b)comparesthemostfavoreddissociationenergiesofdifferentspeciesforZ=1andZ=4,respectively.Itisseenthatdetachmentofasingleatomfromalowerelectrondensityclusteriseasier,i.e.,itneedslessenergy.Comparingthesetwofiguresalsoshowsthat,althoughthehierarchyisthesame,thedissociationenergiesareshiftedtohigherenergiesaswegofromZ=1toZ=4. Now,weconsiderthefissionprocesses.Wefirstconsiderthefollowingtwocases: 2+2+ Ga4+N→GaN−p+Gap,3+1+Ga4+N→GaN−p+Gap. (10)(11) Processes(10)and(11)describecharge-symmetricandcharge-asymmetricfissions, respectively.InFigs.8(a)and8(b)wehaveplottedtheirrespectivefissionbarriers, 5 Z,Z1Bf(N,p),forZ1=2,1asfunctionsofp,fordifferentsizesoftheparent.Asisseenin Z,Z1 bothofthesefigures,thereexistsaminimumsizefortheparent,Nmin,beyondwhichthefissionbarrierheightispositiveinallpossiblechannelsofthatparticularprocess.That Z,Z1 is,theparentclusterMZNwithN≥Nminisstableagainstanyspontaneousfissionviathatparticularprocess. Oneofthesignificantdifferencesinthecharge-symmetricandcharge-asymmetricfis-sionisthat[asseeninFigs.8(a)and8(b)],inthemostfavoredchannel,theproductshavemoreorlessthesamesizesintheformercase;whereas,inthelattercasethelesschargedfragmentassumesquitesmallersizes.Thatis,thecharge-symmetricfissionpro-ceedsmostlyviamass-symmetricchannel,andthecharge-asymmetricfissiondoesitviamass-asymmetricchannel.InFig.8(c),wecomparethemostfavoredsizes,p∗,forthecharge-symmetricandcharge-asymmetricfissionofNa4+N.Itisseenthatinthecharge-∗ symmetriccase,pincreaseswithNwhereas,inthecharge-asymmetriccaseitremainsmoreorlessconstant. Todecidewhetheraclusterwithagivensizeandcharge,MZN,isstableagainstanyspontaneousdecay,oneshouldcompareNwiththequantity ZNmin =max Z1 Z,Z1Nmin .(12) Z IfN≥Nmin,thentheclusterMZNisstableagainstanyspontaneousdecay.ToshowtheZ,Z1Z,Z1valuesofNminschematically,weplotthemostfavoredfissionbarriers,Bf(N,p∗)forallpossibleZ1values.Theevaporationprocessesneednotbeconsideredbecause,aswasshownbefore,intheLDMthereisnospontaneousdecayviaevaporation.However,inordertodetermineatwhatsizewhichprocessisdominant,weincludethemostfavoredevaporationprocessaswell.Itshouldbementionedthat,intheself-consistentKohn-Shamresults[8],becauseoftheshelleffects,thespontaneousevaporationisalsopossible. 4,Z1 InFig.9,wehavecomparedthemostfavoredfissionbarriers,Bf(N,p∗),ofGa4+N 4∗ forN≤150andZ1=1,2withthemostfavoreddissociationenergy,D(N,p).WehavelabeledtheintersectionpointsofthecurveswitheachotherandwiththehorizontalNaxisbyA,B,C,D,andE.AtpointA,thecharge-symmetricandcharge-asymmetricfissionstarttheircompetition;thuswenamethecorrespondingsizeasNs−a.ForclustersM4+NwithN However,spontaneousfissionviacharge-asymmetricprocessstillpersistsforNmin whichcorrespondstothepointC,thespontaneouscharge-.BeyondthesizeNminNmin 4,24,1 asymmetricfissionalsostops.Ourresultsshowthat,forsmallrsvalues,Nmin≤Nminwhereas,forrs>4.5,thereverseinequalityisatwork(asinthecasesofKandCs). 6 Z AccordingtoEq.(12),ifN>Nmin,thenanyspontaneousfissionstopsandthe 4,24,1 clusterM4+Nwouldbestableagainstanyspontaneousdecay.AtsizesNevaandNevawhichcorrespond,respectively,tothepointsDandE,competitionoftheevaporationwithcharge-symmetricandcharge-asymmetricfissionstarts,respectively.Beyondthesize ZNeva =max Z1 Z,Z1Neva ,(13) thedominantdecayprocessisevaporation.Ourresultsshowthat,atleastforrs≤7, 4,24,1 theinequalityNeva InFigs.10(a)and(b),wecomparethemostfavoredfissionbarrierheightsBf(N,p∗)2,1 andBf(N,p∗)fordifferentspecies,respectively.Thefiguresindicatethat,forlargeenoughclusters,thefissionbarriersarehigherforhigherelectrondensitymetals. Z InFig.11(a),wecomparethesmalleststablesizes,Nminfordifferentspecies.Lookingatthefigure,oneconcludesthat,forfixedN,thehigherelectrondensitymetalclusters(sayAl)haslesscapacityforchargingthanthelowerdensitymetalclusters(sayCs).Thisbehaviorisobservedforall1 Finally,inFig.11(c),wecomparetheNs−a,thesizeatwhichthecompetitionbetweenthecharge-symmetricandcharge-asymmetricfissionstarts.ThissizedecreasesfromAltoLi,andthenincreasesfromLitoK,andfinallydecreasesfromKtoCs.Intheplot,wehavetakenintoaccountthedecimalsfortheintersectionpointswhichcanberoundedtothenextnearestinteger. 3Summaryandconclusion Inthiswork,wehavestudiedthestabilityofZ-plycharged(Z=1,2,3,4)metalclustersofspeciesAl,Ga,Li,Na,K,andCsusingtheliquid-dropmodelwithstabilizedjelliummodelenergies.Fragmentationofclustersintosmalleronescanproceedviaevaporationorfission.Ourresultsshowthat,intheLDM,anychargedclusterisstableagainstspontaneousevaporation,andthemostfavoredchannelofinducedfragmentationdependsonthesizeandthechargeoftheparentcluster.Sufficientlysmallmultiplycharged 7 clusters,however,mayundergospontaneousfragmentationviadifferentfissionprocesses.Wehaveobtained,foreachspecies,thesmallestsizethataZ-plychargedclusterisstableagainstanyspontaneousfragmentation.Wehavealsoshownthat,forsufficientlylargeclustersthemostfavoreddecaychannelisatomicevaporation.Comparingtheresultsfordifferentspeciesshowthat,forfixedN,thelowerelectrondensityclustershavehighercapacityforcharging.Resultsalsoshowthat,forfixedamountofcharge,atomicevaporationdominatesatlowerNsforlowerelectrondensitymetalclusters. 8 References [1]M.Brack,Rev.Mod.Phys.65,677(1993). [2]J.P.Perdew,H.Q.Tran,andE.D.Smith,Phys.Rev.B42,11627(1990).[3]U.N¨aher,S.Bjørnholm,S.Frauendorf,F.Garcias,andC.Guet,Phys.Rep.285, 245(1997)andreferencestherein.[4]M.Seidl,J.P.Perdew,M.Brajczewska,andC.Fiolhais,Phys.Rev.B55,13288 (1997).[5]M.Payami,arXiv.org:cond-mat/0212160. [6]W.KohnandL.J.Sham,Phys.Rev.140,A1133(1965).[7]M.Payami,J.Phys.:Condens.Matter13,4129(2001).[8]M.Payami,arXiv.org:cond-mat/0305600. 9 Figure1:Fissionbarrierinthetwo-spheresapproximation.TheparentN-atomZ-plychargedclusterdecaysintotwoclustersofsizesN1andN−N1,withchargesZ1andZ2,respectively. Figure2:Coulombinteractionenergyoftwochargedmetalspheres,inelectronvolls,asfunctionofdistance.ThevalueofthemaximumisBc,andthelocationofthemaximumisatd0.(a)-thesphereshavecharges3and1,(b)-thesphereshaveequalchargesZ=2,(c)-thechargesofthetwospheresareexchanged,and(d)-thesizesarekeptfixedbutthechargesarechanged. Figure3:Coulombbarrierheight,Bc,inelectronvolts,(a)-fortwoequallychargedbutdifferentsizedpairsofthesamespecies,(b)-comparisonofBcfordifferentspeciesofequallychargedpairswithoneofthepairsizeskeptfixed. Figure4:Functionsofrs,(a)-totalSJMenergyperelectroninthebulkmetal,(b)-thesurfaceenergyobtainedfromfittingtotheKohn-Shamresults,and(c)-thecurvatureenergyobtainedfromfitting. Figure5:Dissociationenergy,inelectronvolts,fordifferentevaporationchannelsanddifferentparentclustersizes.(a)-singlyionized,(b)-4-plyionizedLiclusters. Figure6:Nadissociationenergy,inelectronvolts,asfunctionoftheparentclustersize,forthemostfavoredchannelofdifferentevaporationprocesses Figure7:Mostfavoredchanneldissociationenergies,inelectronvolts,ofdifferentspeciesfor(a)-singlyionizedclusters,(b)-4-plyionizedclusters. Figure8:Fissionbarrierheightsof4-plyionizedGaclusters,inelectronvolts,fordifferentparentsizesandfissionchannelsof(a)-symmetric,(b)-asymmetricprocesses. Figure9:Barrierenergies,inelectronvolts,inthemostfavoredchannelofdifferentsizesof4-plyionizedGaclusters.Z1=1,2correspondtoasymmetricandsymmetricfission,respectively,whiletheZ1=0casecorrespondtoevaporationprocess. Figure10:Mostfavoredfissionbarrierheights,inelectronvolts,ofdifferentspeciesfor(a)-3-plyionized,and(b)-doublyionizedclusters. 10 Figure11:Functionsofrs,(a)-theminimumstablesizefordifferentcharges,(b)-thesizesatwhichevaporationdominatesfission,and(c)-thesizesatwhichthesymmetricandasymmetricfissionsstarttheircompetitions. 11 d0BfQQcN,ZfN2,z28N1,z1ReactionCoordinateEnergyc7.5Al,Z1=3,Z2=1Q1=Q2=2(N1,N2)=(2,18)7(N1,N2)=(5,15)c(N1,N2)=(10,10)6.565.554.589d010111213141516d(a.u.)E(N1,N2,d)(eV)Bc12Al,Z1=2,Z2=2Q1=Q2=2(N1,N2)=(2,18)11(N1,N2)=(5,15)(N1,N2)=(10,10)10987678910111213141516d(a.u.)E(N1,N2,d)(eV)c7Al,N1=2,N2=18Q1=Q2=2(Z1,Z2)=(1,2)6(Z1,Z2)=(2,1)543278910111213141516d(a.u.)E(Z1,Z2,d)(eV)c14Al,N1=2,N2=18121086(Z1,Z2)=(1,1)4(Z1,Z2)=(2,1)(Z1,Z2)=(3,1)2(Z1,Z2)=(2,2)07891011121314151617181920d(a.u.)E(Z1,Z2,d)(eV)6Al(rs=2.07),Z1=1,Z2=15B(N1,N2)(eV)c4321N1=10161116N1=162126N1=313136N1=464146N26N1=1,Z1=1,Z2=15B(N1,N2)(eV)c4AlNaCs3210161116212631364146N20-2-4KNaLiCs(eV)-6-8-10GaAl-1223rs(a.u.)4561.81.6AlGa(eV/bohr^2)1.41.210.8100*0.60.40.2023LiNaK45Cs6rs(a.u.)87AlGa(eV/bohr)100*6543210LiNaK2345Cs6rs(a.u.))Ve()p,N(D5432N=701Li(rs=3.28)N=40N=20N=10001020304050607080p1+)Ve()p,N(D25201510N=705N=40Li(rs=3.28)N=20N=10001020304050607080p4+32.5Na(rs=3.99)Z=1Z=2Z=3Z=4D(N,p*)(eV)21.5Z10.500102030405060708090100N21.5D(N,p*)(eV)11+0.5Al00102030Ga40Li50Na60K70Cs8090100N)Ve()*p,N(D5AlGaLiNaKCs432100102030405060708090100N4+6N=60543210-1N=50N=40N=30B(N,p)(eV)fGa(rs=2.19)4,20510152025303540p1614N=60N=50N=40N=30)Ve()p,N(fBGa(rs=2.19)86420-2051015202530354045505560p12104,12520Na(rs=3.99)15p*Z=4,Z1=210Z=4,Z1=150051015202530354045505560N6Z1=24Z1=1Z1=0Ga(rs=2.19)B(N,p*)(eV)f24,Z10ECD-2BA-4020406080100120140N32B(N,p*)(eV)f13,10-1Al-2010203040Ga50Li60Na70K80Cs90100N43B(N,p*)(eV)f212,10-1Al-2010203040Ga50Li60Na70K80Cs90100N48Z=4Z=3Z=24338Nmin(rs)Z33282318138322.533.544.555.56r(a.u.)s)rs(avNe140130120Z=4Z=3Z=2110100908070605040302010022.533.544.555.56rs(a.u.)Z29K28(r)sAl27NaCsNs-aGa2625Li2422.533.544.555.56r(a.u.)s 因篇幅问题不能全部显示，请点此查看更多更全内容