10 ((s

IAEA-SM-174/12

TITLE: MODIFIEDDEFINITIONOF THESURFACEENERGYIN THELIQUID-DROPFORMULA

AUTHOR(S): H. J. Krappeand J. R. Nix

SUBMITTED TO: Third IAEA Symposiumon the ChemistryandPhysicsof Fission,Rochester,New York,Auguet13-17,1973

8V ucoptmco of thh artlcl. for publiwt;on, ths publislwtwognlzw tlw Govwnmwtt’s Oiwmwl fight. In any oopyrlghtsnd ttw Qwwwmnt @ndits Whorkd r.pmantstlwa hcwunmtdotod ri*t to mprdm In Mole or in part MM art1c14und.r w oop@#tt acur~ bv tha Wblldwr,

Th. LOO Alamw Sd.ntlflo Lab#@tow Wuwts that tkpublhhor Id.ntlw this ortlcla a work @ormd underthmmpkw of rho U. S, Atomlo Energy Comrnlalon,

of *ho Udvordty of Callfwnlato%AiAMOS,NIW MEXICO 07S44

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.

IAEA-SM-174/12

L

MODIFIEDDEFINITIONOF ~ SURFACEENERGYIN THE LIQUID-DROPFORMULA*

H. J. Wappe t

LawrenceBerkeleyLaboratoryUniversityof California

Berkeley,California94720U.S.A.

and

J. R, Nix

Los Alanm ScientificLaboratoryUniversityof California

Ias Alamo3,New Mexico 875b4 U.S.A.

ABSTRACT

When calculatedin the llquld-dropmodel,the deformationener~ ofstronglynecked-infissionor fusion configurationsohowe a epurlouelystrongdependenceon the detailsof the shape in the neck region. This 16 ● conse-quence of the Pmumed sharp surface in the liquid-dropmodel. Thie modelcan be Iqprovedby replacingthe surface-ener~ term by $he self-enar~ of● drop caueedby a short-rangetwo-particleinteraction. For ● Yuhwafunctionthe self-energyintegralcan be evaluatedanalytically‘fora fewImportant specialcontigurations, and it can be tram formedinto ● throe-dhenoimal integralfor arbitrwy ulally qnmetric shapes. A numericalcalculationis thereforeonly slishtlymore complicatedthan tho usualtrmtment of the Coulombenergy.

0~le work wau gupportedby the lJ,S. Atomichor~ Comtdsaion and thlIGo-anAcademl.cExchangeServicet llann-MoitrierInntit ut fb Kernforachung,Berlln,FodoralRepublicof Germany.Vi~Ltor to the LawrenceBerkeleyLaborato~, April 1YV?-S@ptOdMr1973.

In additionto the parametersin the conventionalliquid-dropmodel,the new definitionof the nuclearpart of the deformationener~ containsaparameterto speci~ the range of the Yukawafunction. Parametersfor thenew definition~e determinedfrom fisgion-b~rierhei@ts ~d interaction-barrierheightsthroughoutthe periodictable.

d

The influenceof the proposedchangein the liquid-dropformulaonthe stiffnessof sphericalnuclei,the ground-statedeformation,and theexistenceof shapeisomericstatesin lightnuclei is discussed. Fission- ~barrierheightsand saddle-pointshapesare determinedfor nucleialongtheline of beta-stability,and the staticinteractionpotentialbetweenheavyions is calculated.

1. INTRODUCTION

Considerableprogresshas been made in calculatingthe nuclearpotentialener~ of deformationas a functionof the nuclearshape and themass and chargenumbers by splittingit into a slowlyvaryingfunctionofthese quantitiesand a rapidlyfluctuatingpert. The latteris usuallycalculatedaccordingto a prescriptiongivenLy Strutinsky [1]. Here we willdeal onlywith the smoothpart. It is usuallyexpressedin terms of aBethe-lfeizsackertype of expansionin powersof A-1/3 and 12, for example[2-4]

.

‘LD= V(J 57 [Bcw-b27-Q#%q,‘1)~A+Ca A2’3B(~)+~ez8Vo 0

whom

Cv 2●av(l-KvI),

c, ● ●,(1 - K, 12) ,

-i .’.. .

Tho quanti~ d io the aurfaga-thicknoseparameterIn ● Fermi functionzhatcpeelflostho chargodistribution.Tho shape-dependmtfumtlon B~(~v) istho ratioof the surfaceema of the deformednucleusto that of the ophericalaucl.w, and Be(~) is the ratioof the Coulomb●.rgy Of tho dcfomed●quivahnt m’”it,rp-ourfacenucleusto that of the sphericalnucleus. Such aloptodermmm mpanmlon is valid only If all geometricaldimen~ions of the&op u. largecomparedto tho surface thickneos. This condltion Is uotsattafiodfor strongly necked-inconfigurationswith neck radiismallerthanabout2 fh, for ●xamplomound the ecianienregion In fissionor the pointOf fhet oontaotin heavy-lon reactions.

.;

d

-2-

One couldovercomethis difficultyby goingback to a constrainedself-consistentmicroscopiccalculationfor such configurations.But thatwouldbe a ratherinvolvedprogramfrom a numer?,calpoint of view. Therefore,it is desirableto constructa generalizationo~ the liquid4rop formula -stillon a purelyphenomenologicalbasis - which satisfiesthe followingconditions:

(1) For sphericalconfi~ations it shouldgive practicallythe sameresultas the old liquid-dropformula(exceptfcr very lightnuclei).

(2) In contrastto the usual suf’aceener~ it shouldnot be sensi-tive to hi@-multipolewiggleson the surfaceof the drop. The liquid-dropformulayieldsa spurious and undesirablesensitivityof calculatedfissionbarrierson unphysicalfine detailsof’the shape in the neck region.

(3) Betweentn separatedfragmentsthere shouldbe en attractivenuclearinteractionener~ besidesthe Coulombrepulsion. The range of thatforceshouldext~ndbeyondthe equivalentsharpradiusby roughly the rangeof the nucleon-nucleoninteraction.

(4) It shouldbe possibleto calculatethe new e~ressioa for gcnerslshapeswith reasonablecomputationaleffort.

we will show thaH

ne can satis~ theseconditionsby replacingthe . “surfaceenergyterm Cs A / Bs(~) W

D (2)

with the two phenamenologicalparametersV and Insteadof the singleliquid-dropparameterC~ and allowingfor 8 reno-aizatlo. of the volume- mener~ coefficientCv. The six-foldIntegrelIs to be takenover the mlumeof the equivalentsharp-surfacenucleuswhose shape can be parametrizedbyMY suitableset of deformationparameters;this volumeis epecifiedby thenuclear-radiudparameterrO.

20 SPECIALCONFIGURATIONS

We will discussthe resultsof thin replacementfor a fioquenceofshapesof Inc=eaeimgcmplexlty and show that these four coaditioneamfulfilled.

2.1. sDherical~hape

A strdghtforwardcalculationof the Integral(2) la sphmicalcoordlnateeyield~

[

411 3- 2R@

Emvo-T RO + 27@ - 2’na3+ 2na(R0+ a)2 e 1 9

-3-

whereR. = r@l/ 3 is the equivalentsharpradius. For a/R. << 1 the last two ~termsare negligible. The secondtermyieldsthe surfaceener~ if the inter- =IdactionstrengthV. is relatedto the semi-empiricalsurface-ener~constant :c* by ‘~u

<

**u

.~s firsttem gives a contribution“tothe volumeenergyand has to be compen-sated for by a renormalizationof C.<. This way WQ meet the firstof’the fourrequirementson E. The limita + O yieldsthe usual liquld-dropmodel.

2.2. Bubblenucleus

For a bubblenucleuswith innerradiusRl and outerradiusR2 one gets imom (2)

E “ - (~f% [: F)’- : (>)3 - (>Y - (>T + 2- p+ 1)2 .XT&l,al

Recentlythe bubble-nucleusmedelcameth~ applicationof the usualbe replaced~ this formula.

has been discussedfor R1 ~ a [5]. IL “-hisliquid-dropformula(1) ts doubtfuland should

2.3. Smalldistortionsabout a spherictishape

If the ~h8peis parametrizedby tha normalcoordinate for harmonicvibration aroundthe sphericalshape,

Ware m

(3)

(4)

m

-4-

Here Ig+l/2ad Kg+l/2 are modifiedBesseland Hankel

The simplestway to derivethis formulais to

‘d

2$

functions,respective~ [6]. A

uee the expansion

R(Q) r R.

for the tntegratlonvfth respectto r~.

with respectto r and ● similarone far t.leiate~ationIntegz@o of the type

1

f‘* P2(X)

-1

are evaluated~ uoe of the additiontheoremfm modifiedBemel function-[6],

1

“:(2’+’) “(x) *’h’’2(:)K’+’’2(~) “ “)

h ●xpansion of Eq. (4) In pow.rs of A“2’3 ytelds for tho ●tiftiessconmtmt for multipolovibration of ordor Q tho result

-5-

The f~st term is the well-knowncontribution.ofthe surfa e energyin the o ●

liquid-dropformula. 7There is no tezm proportionalto Al 3, which means~ically that the contributionto the eaer from the curvatureof thenuclearsurfaceis identicallyzero. FThis A /~ term is absentalso for moregeneralshapes[7],providedthat the smallestcurvatureradiusis largecqared to the range a. The term of orderAO reducesthe stiffhessforfinitevaluesof the range a$ this reductionbecomesrelativelymoreimportantfor lightnuclei. The last twa terms Ue negligiblefar low~tipele or&r8 ~, providedthat the nuclearradiusR. is lesge coqparedto●. For highertit ipolea,the e~ression (4) for the stiffnessconstatbecomesindependentof multipoleorder,becausefor largev

~EI tg to be contrastedto the quadraticincreasewith multipoleorder forthe stiftiesscoaetant calculatedwith the usual Mquid=drop mdel. It shOw8& tnsensitivityof the mdified liquid-dropformulato unphysicalfinewee of the surface. Thersforealso the secondof the four requirementson1!is satisfiedby (2).

The fisslkltypsrameterx is definedas the ratio of the CoulombenergyE$o) of a sphericalsharp-surfacedrop to twice the sphericalsurfaceenerg@ m Cg ● A2/3. The valueof the criticalfiasilltyICcri for which the

1spherelose~stabilityagain~tfissioncorrespondsto the po nt where ther8storir~force@mat P2 vibrationsvanishes. Additionof the Coulombcon-tributionto the deformationener~ (k)yields for E=2, m-O the result

(0)E’ 2

[

2 - 2X “9●2~+(Y(e 1-2Ro/a,

9@ - ‘~ 820

‘o

MO leadsto

-1- ~ (+’2+~ (e-mO/a)‘crito

(7)

kstead of the usualvalue 1.

-6-

ZOJO Two non-overlappin~spheres——

The nuclearinteractionenergyof two non+verlappingspheresof radii “RI ad R2 and center-~i-massdistanceD ~ RI + R2 followsfrom straight-for~ardintegrationof (2):

Eint= .-4 (~y %(> Cosh? ‘id 3(2 Cosh>- ‘inh:) * ● (8)

l?’orRl ~/a >> 1 thitireducestos

where2 is the distsncebetweenthe two sharp surfacessurfacetension.

This formulais a specialcase of a generaltheorem [8]which statesthat to order# the interactionener~ betwesntwo arbitrarilyshapedob~ectainteractingvla a shortrsngeforce (tihortcomparedto all curvatureradii)can alw~s be expressedin the form

The firstfactoris purelygeometricaland In the case of two spheresIn equalto 2nRlRQ/(Rl+R2). The quantitye(~) ia the interactionener~ per unit a{eaof two parallelinfinitesurfacesat distance~. Obviouslye(0) = - 2Y . AThmas-Fermi calculatiorA[8j of the functione(~)yields a resultthat can beapproxirltedroughlyby an e~onentlal functionof range ● - 1.4 fiu, that Is,

2.5. Non-overlalming enhericalnucleusand olimtly deformednucleue

The generalizationof Eq. (8) to the interactionener~ botwoenasphericalnucleuswith radiusRI and ● deformednuclms with radiua

[R=R21+B +corr F ‘1B Y (fl)is givento secondorder In B,ml?ldm #w

‘J

I

~2Rl

()(‘1 ‘1

R2 3- #Ph

‘int Lnt=.2c — — cosh— )()-sinl?— —

sr09

L a a

(9)

Here e and ~ are the angularcoordinatesof the body-fixedcoordinatesystemof the deformednucleusrelativeto the vectorjoinin~ the centersof mess ofthe two nuclei. The deformationparametersSRm refer to the body-fixedsystem.The otherquantitiesare givenby

sphaad Eint is the expression(8) for the sphericalcase. This formulaisobtainedeaeilyby use of the e~ansions (5) ~d (6)●

3S ~!SERMINATIONOF PARAMETERS

The shape-dependenttermsof the nuclearmacroscopicener~ calculated●ccordhg to Eq. (1) containa totalof fourparameters: the equivalentsharp-surfacenuclear-radiuyparameterr., the range of the Yukavafunction,the surface-energyconstantas for equalnunibe~sof neutronsandprotons,and the surface-as-try conc~ent~. Thenuclaar-radiueparameterr. ie knownacwrately fram●ckteri~ data;its value la thereforenot ad~ustedbe 1.16 fta from thesestudies [9].

Interaction-barrierheightsdependmainlyon

equivalentsharp-surfaceanalysesof electron-but is taken insteadto

rfiand the range a andmre veaklyon q and ~. Therefore,once rc Is fixed-therange a isdotmmiaed by a~usting to experimentalinteraction-barrierheights [10-25].TM resultingvaluoof 1.4 h is the same as the rangedeterminedfrom the*OVO mentionedThomas-Fermicalculations[8].

OnQoboth ro and ●

-ormiaod by ●justing toare known,the finaltwo parameter a and K8 aree~erimental fimion-burier heights?26-28].

-8-

Becausethesetwo parametersere highlycorrelated,theirindividualvaluesare determinedpoorly. For example,the valueof 4.0 determinedfor KS is ,uncertainby at least * 1.0.

The resultingvalue of 23.7 MeV for as is significantlyhigherthanthe valueof about 18 MeV obtainedin the usual liquid-dropmodelby adjustingto fissionbarrierheights [2,3]. It is on the otherhand only slightlylargerthanvaluesobtiainedby adjustingto nuclearground-statemassesalone [4]. The differencebetweenour value and the valuesof’Refs. [2,3]arisesbecausethe finiterange of the nuclearforcereducesthe effectivestiffnesswith respectto defamations. It ib thereforepossiblethat thesurface-enerf~constantis indeedlargerthan previouslybe.Lieved.

To mmmarize, the preliminaq velues chosenfor the four parametersare

.ro = 1.16 fm ;1

a =1. kf’m.

as = 24.7 MeV ,

(lo)

KS = 4.U

No attempthas been made so far to redeterminethe parametersin the shape-independenttermsof Eq. (1) afterreplacingth~ surfaceterm ~ the integral(2)●

4. INTERACTIONBARRIERS

The combinedactionof the Coulomband the nuclearforceusuallyyieldsa mwdmum in the interaction-energyas a functionof the ii~t~.cebetweentwoions. For symetric configurationsthis interactionbarrierdisappearsfora criticalfissilityof the combinedsystemof x = 6/5, at which pointtheCoulombrepulsioncan no longer‘becountar-balancedby the nuclearforceevenfor tva touchingspheres. Below this criticalvalue of x the height of thebarrieris oftenrepresentedIn the form n

‘lz2ez

reff(#3+ J&s) ●

Figure 1 shm re ~ as a ~ction of the char@s 21 8nd 22 of the two collidins ‘ ~ztons fornucleia ong Green’s approximationto the line of beta-stability[29].

EqM~ Ion (11) can be rewrittenin the foxm,-/:H.-

Z1Z2e2

‘L: ‘R1+R2+a+d s (12)

whered is the distancebetweenthe two nuclearsurfacesat which the totalinteractionenergyhas its mmcimum. The value of d is determinedeasilybyiterationfromthe equations

(2

~ e ‘1Z2dDD )1— +Etit(D) =0 ,

whereEint(D)is insertedfromEq. (8). A contourplot of d(Zl ,22)for nucleialongthe line of beta-stabilityis given in Fig. 2.

In Fig. 3 experimentalCoulombbawlers from reactioncross-sectionmeasurements[1o-18] and from elasticscatteringexperimentsare comparedwiththe predictionsof this theo~. The deviationsfrom the calculatedvaluesaresmallerthan t 5.4$ or 9 MeV i absoluteunits.

bThe data used n Fig. 3

includethe deformednuclei23 i[13-15,19], 232Th [10],md 16 ~ [12]. Forthesedefomed nucleiEq. (~) is used with ~~o = 0.27’7,0.248,and 0.319,respectively[30]. The orientationanglese and $ are taken equalto zero,whichgivesthe minimuminteractionbarrier. We have not taken into accountany shelleffectson interactionbarriersbecausethe Influenceof one poten-tialwell on the leveldensityaroundthe Fermi surfaceIn the otherwellis suppasedtG be vexy smellat the point of geometricalcentact or evenfartherout.

,

The distancebetweent;e two centersof mass is the only degreeoffreedomthatwe have consideredin calculatingInteract Lon barriers. Weare thus disregardingthe couplingof the relativemotionto the neck-healingor any intrinsicdegreesof freedomof the two ions. Elastic-scatteringdataon the otherhand we usuallyanalyzedin terms of opticalpotentials. Onlythe tail regionsof thesepotentialsare detezmin+dunambiguously,which oftenexcludesthe maximum. Moreoverthe opticalpotentialreflectsthe couplingof intrinsicdegreesof freedomto the relativemotion in an averageway,whereasthoseeffectsare completelyneglectedin our model.

Information on interactionbariq~ersis also ektractedfrom fusionreactioncroes sections. They are mostlyanalyzedin termsof transmissioncoeff~clenbscalculatedby assumingtransmissionof a realparabolicpotentialbarrier. This amounto to assumingan ingoing-waveboundaryconditioninsi~ethe potentialbarrier. It has been shown [31]thatloptical-modelpotentialsare not necessarily identicalwith potentialsto be used with an ingoing-wavebounda~ condition~ especiallywhen the imaginarypart is neglectedor notdeterminedin the lattermethod.

‘J-.4

-1o-

To overcomethe problemsconnectedwith the ambiguityin potentialfitsit is advisableto determinethe criticalangularmomentumA(E) as afunctionof ener~ from a phase-shiftanalysisof elasticor reactionscatteringdata [32]. Using the relation

A(.A+l) = (kRr)2(1 -W

for several energies,one can extractthe pair of valuesV(R=) and Rr, i.e.the interactionpotentialat the reactionradiusRthe mulmum of that potentialmightbe.

r9 independentof whereOf course,the assumptionhas been

made that the interactioncan be describedby an ener~-independent,localpotentialend that the processis purelydiffractive.

50 GENERAL SHAPES

The integral(2) can be reducedto the doublesurfaceintegral

by using the identity

. la——‘-”EL

aand applyingGauss’stheoremwith reopectto ; and ;’. For -ystemawithcylindricalsymmetry(13)reducesto the three-dlmenoionalintegrelin ‘cylindricalcoordinate

‘“-3}zfiz12;’R(z)‘R(z)- R(z’) CO~ $ -R’(z) (z-z’)] R(z’)

u-d

X [R(zt)- R(z) COO $ - Rt(zt)(z~-z)]~+~ 9d“

(14)

-11”

where

d = : [R2(z)+ R2(.zF) - 2R~z)R(zf) co8 $ + z2 + zt2 - 2ZZ@/2 ●

*

The functionR(z) @ves the shape in cylindricalcoordinates,and R?(z) Isthe derivativeof R(z) with respectto z. The z integrationsare takenbetweenthe zerosofR(z), The three-foldtntegral(14) in generalmust beevaluatednumerically,but this is only slighklymore complicatedthen theevaluationof the integralfor the Coulombenergy.

60 FISSIONEARRIERS

Figure 4 8howothe maximumandminimumradiiof saddle-pointshapesu functionsof the fissilityparameterx for variousvaluesof the range a.The remaining conatante-e held fixedat the valueedeterminedIn Ref. [3]oa the baaie of the liquid-dropmodel,that 1s, for zero range. The saddlepdnto are calculatedby u8e of the methodsof Ref. [33], withthe mrfaceenergy replacedby Eq, \14). The clansof’shapesinvestigatedIs that oftwo spheroidsconnectedsnmothlyby a quadraticsurfaceof revolution[33].Tharala a clemrtmdency to sore compactsaddle-pointshapeswith increasing afor fixed valuesof the otherconstants. The shiftof xcrit to valuesmailer than 1 as givenby (7) to secondorder in # 10 aleo clearlyseen.The criticalMslnaro-Gallonepoint (whereotabilit$againstmans aaymmetryislost) [34]firstmovesto slightlylargervaluesof the fissility x withincreasingran~e a. It reaches a maximumat a/rO * 0,7 and then it movesback to smallervaluesof x. Figure 5 gives the firnsion-barrierhei@t ●s afunctionof the fissilityparameter x for variouovalueeof the range ●,again for fix~dvalueeof the other conotanta. The burier h,?ghts are seente decraue drasticallywith increasing range.

Figure6 comparesthe calculatedmacroscopiccontributionto thefission-barrierheightwith ● erimentalvalues. The curvo Is calculated

7with the parameter of Eq. (10 , and thu experimantd data ropreoontbothreducedfia~fon-barrierheightsfor actinidonuclei [26,27 ] and ohell-correctedflsmlon-bmrlerheightsfor lighternuclei [28]. In the regionof fissilityparameter x between0.50 and 0.55 the experimental valuesare eyrtematlcally.omewhtithigherthan the calculatedcurve. It would be very desirabletohavo●~porimontaldata for stillll~hternucleiwith fiaailitypommotersmaller than 0.5.

Fi~uro 7 nhclwntho difference botwoenthe predictedbarrierheightIntho liquid-dropmodolwith the paremetorsot from Ref. [3]and In our modelwith tho parameter (10) alon~ the lin. of beta-stablllty.The finiterangeof the nuclear fore@ lowors the fioaion Mrrial d of nucleinear nilverby●bout 10 MeV relativo to those calculate with th. liquid-dropmodal and -hil’trn

#tho eritlcalIhminaro-Onllmepoint to Z /A = 23, in approximateagreemmntwithrecentuporim.ntal .Videnee[35]o

-12-

7 t POTENTIAL-EMLRGYSURFACESOF LIGHTNUCLEI

The generaltrendto decreasethe stiffnessof nucleivlth increasingrangeof the interaction a showsup especiallyfor ltghtnucleiwhere theradiusis no lon~eran orderof magnitudelargerthan a. A calculationofthe deformationenergyof 40Ca as a functionof the quadruple defamation~hows ti-,flt the shellcorrectionis more effectivein producinga secondminimumwith our expressionfor the macroscopicpart of the ener~ than withthe conventionalliquid-dropmodel, a~ seen in Fig. 8. This providesanatural interpretationof the rotationalstateoobeervedIn this nucleusandcertainotherlightnuclei [3’7].A similarstudyfor l%r shown that it-calculatedground-statequadruple moment is shifted mmewhat tow8rds lugervalues by the finite-rmge model, althoughft is stillnot u large u theexperimentallyobserved qua~ole momeni [38]. This remit is shown inFig. 90

8. SIJWARY

We have redefinedthe surface texm in the liquid-dropformulaso thatit can be used for configurationsin whichthe size of a curvatureradlueofthe nuclearsurfacebecomescomparableto tho mrfaco thickness. We haveshownthat the new versionof the liquid-dropformula yields ● weaker depen-denceof the deformationener~ on surfacewigglesof high multipoleorderthan the old model and generallyresultoin ● smallernuclearstiffi’ioss. Asa consequencethe shell correction producema larger ground-statedeformation@speciallyof some lightnuclels and there oeemoto appear ● secondminimum

in the deformation-ener~curveof 40Ca.

Saddle-pointshape; have been calculated, and they am lass necked-nthan in the usual 1iquld-dropmodel. The dapondenco of tha Businar~Gallon@point on the rangeparameterof our model haz been ctudied. We hmre derivedan explicitexpressionfor the nuclear interaction, ●ner~ botwem two non-ovarlapplngions and have calculatedInteraction-barrierheights. For veryheavy systomathe muimum In the Interactionenergytremfoms Into a pointof inflectionand the interactionener~ increasesmonotonloally withdecreasingdistancebetweenthe ions.

We determinedthe parameter of our model so that the reduced fissionhurrierheightsfor finsilityvalueolargerthan 0.5 and ●~erhental inter-

action-barrierheightsare reproducedon the ●verage. The fission barriersfor nucleiwith a nmaller f ie~ilityparameterare predictedto be lower thanthey ●re in the old versionof the liquid-dropformulawith parametora fromRef. [3].

We are Indebtedto W. J.dl~cu~sionnthroughoutthfswork

ACIOIOWLEDO~NTS

Swlatefik~andW. D.and to S. E. Koonin

Drouramto evaluatethe three-dimenolonalintegralfor the nuclearmacroscopic ~. -.

●nergy. We+altiothankJ. Randrupfor making hiu calculation availableprior-d

to publication,

●1-L

[2][31[4]

[51[6]

[71[81[91[10]

[11]

[12][131[14][151[16[171

[18]

H1920[21][22]

[23][24]

[25]

[26][27]

[28]

N2930[311[32][331[341[351[361[371[38]

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Handbookof mathematicalfunctions, <NationalBureauof Standards,Washington,D. C. (1964;.TSANG,C.F., BerkeleyReport, UCRL-18899(1969).RANDRUP,J., .SKLATECIIU,W.J,, to be published.MYERS,W.D., BerkeleyPreprint,13L-1259 (1972).BIMBOT,R., (3AUVIN,H., LE BEYEC,Y., IEFORT,M., PORILE,N.T.,TAM41N,B., Nucl. Physau89 (1972) 539,GAWIN, H., LE BEYEC,Y.; LEFORT,M., DEPRUN, C., Phys.Rev. Letters~(1972) 697.I@ BEYW, Y., I&FORT, M., VIGNY,A., Phys.Rev. c ~ (1971) 1268.WON(I,C!.Y., Oak Ridge Preprint(1972).SIKKEL4ND,T., Ark. ~o. ~ (1967) 539,VIOLA, V.E., mqmim), T., Phyo.Rev. 128 (1962) 767.BIMBOT,R., GARDES,l)., RIVET,M.F., Phye.Rev. c ~ (1971)2180.BIANN,M. , Bull.Am. Phyo. Hoc.~ (1973) 31; elightly revisedfinalvalue.are given in GUTBROD,H.H., WINN,W.G., BLANll,M., Rochesterpreprint,COO-3494-8(1972).LE BEYEC,Y., L#FORT,M., SARDA,M., NucloPhyo. A192 (1972)k05.LEFORT,M., NC3, C., PETER,J., TWIN, ‘B.,Nuclo Phya.Al

+(1972)485● ‘

OBST,A.W., Z&SHAN,D.L., DAVIS,R.H., Phye. Rev. C & (1972 1814.ORLOFF,J., DAEHNICK,W.W., Phyo.Rev. C ~ (1971) 430.ROBER!MON,B.C., SAMPLE,J.T., GOOSMAN,D.R. , NAGATANIOK. , JONES,K.W.,P~s . RQV. C & (1971) 2176.FLETCHER,N.R., ~ST, L., ~MPER, K.W., Bull.Am. Phy@. SOC. ~ (1971) 1149.BERTIN,M.Co , TABOR,S.L., WATSON,B.A., EISENOY., GOLDRING,0., Nucl.~s. A367 (1971) 216.GOLDRINO,G. SMRJEL,M., WATSON,B.A., BERTIN,M.C., TABOR@S.L., Physo&etters32B !1970)465sPAULI, H.C., LEDERGERBER,T., Nucl. Phys.~~ (1971)545.tiDRuP, J., mtwo co F., M)LLER, p., NILssON, s. c., mssoIu, s. E.,Bark@lay Preprint(1973).MORETTO,L.G., THOl@SOl:,S.O., ROUTTI,J., GA’ITI,R.C., PhyD. Letters”‘s811(1972) 4710GREEN,A.E.S., Nucleu Physics,McGraw-Hill,New York (1955) 185, 250bIhNER, KoEoG., VETTER,M., H6NI0 V. , NUC1. Data Tab. ~ (1970) 4950RAWITSCHER, G.H. , Nucl, Phye, 8

hh9661 3370

McINTYRE,JaAa, BNOER,9J., w o ILK., Phym ROY. 125 (Jt960584.NIX, J.RO,NUCI. PhYIBoN30 (1969) 241.BUSINARO,U.L., GALK)N’E,S., Nuovo Cim.~ (1957) 31SSMjTMSIRI , ‘r. , JOHANSSON,S .A. E. , Nucl . Phys● A167 (1971)976 3

kK)LLER,P., NIX, J.R. , Paper IAEA- SM-17k/202, theme procoodinga.‘:

SETH, K.K. , sm~ A- , ~~~OOU LO , NorQhwoatcm UniversityPraprint(1973).;;CKEI~Z , E , JARED,R.C., ‘FHOUPCON,S,O.,WILHELMJ,J.B., Phya. Rwt

‘h%

Letters~ \1970) 38~[39] BOMTERLI, M., FISET, E. O., NIX, JoR.,NO~N, J.L.,

Phy@.Rev. C ~ (1972) lo50*~

-14-

FIGURECAPTIONS

Fig. 1. Contourdiagramof’the effective nuclearradiuafunctionof the char~enumbersof the collidingionsvalleyof’beta-stability(definedb,yGremts formulasee Ref. [29]).

Fig, 2. Contourdiagremof the distance d betweenthe

parameter r=ff as afor nucleialongtheN-z = 0.4 Aq(A+200) ;

equivalentsharpsurfacesof twe spherical nuclei of charges z~mand Z at the peak of theinteractionpotential. The resulteare for nucleiJ ong the valleyofbet-stability. Beyondthe lfne definedby d = O the interactionbarrierhas nq maximum.

Fig. 3. Coqparfson of e~erimental and calculatedInteract Zoa barrier heights.The solidpoints are e~erirnentalvalues derivedfrom excitation functions(solidcircle [10],solid square [II],solfd diamond[12],solid ard-

Tpointingtriangle[13,14],solid downward-pointingtriangle[13,15, solldhexagon [16], solidpluo sign [17],and solid star [18]);the open pointsMO e~erlmental valueaderivedfrom elaetic-ecatterlngdat~ (opencirclo[191. open waare [201, open diemond [20,21],epen upward-pdntlngtri-angle [20,22], open dG~WUd-pOinth6 trimigle [20,23]~ men h-w [20s24]Dopen phM sign [241, md open star [25]).

Fig, 4. Saddle-pointshapesaa functione of the fissilityparameter SE&O)/ [2E$0)]for llquid-dropmodelparameter from Ref. [ 3] and a/r. = 0.0(0.2)1.2.The upperportionof the diagramgives the largestradiusof the mddle-pointshape in unitsof the radiusof the spherewith ●qual volume. Thelowerportion@vee the smallestradiue, The solid points giva thalocationof the Businaro-kllonepoint.

Fig. 5. Fission-barrierhqightin our model u ● functionof the flsoilltyparut.r x and tho range●/rO for liquid-drop-mdelparemeterafrom Ref.[ 3]. The solidpointm mark the Bueinaro-(hllonepoint.

Fig. 6. Comparison of thooroticalfission-barrierhelghtti(solidline) calcu-latedwith tho parameters(10) and experimentalbarrier hei hts corroctedfor eingle-particlaeffects. fThe circles [26] and ●quasws 27] am reducedfission-barri.r heights for actinldo nuclei,and th~ triangleo[28] areoh.11-corroctedtisaion-barrier hoighta fm lighter nuclei.

Fig. 7. Comparison of macroscopicbarrier heights calculated in our rnodol withtho paramet●ro (10)and In tho liquid-dropmodelwith parameters fromRef. [ 3] for nuclei alongtho line of bota+tabillty. Tho solidpoint.Indlcatotho Buinaro-(hllone point, The arrow show tho mass numbers ●twhichtho systemwould 10SO stabilitytowarda fission if shell oorrootionsworo not present, h-

Fig. 8, Deformation ener~ of 40Ca. The dashed llneoshuw tho macroscopiccontributionto the deformation●ergy in our model and in the oonventlonalliquld-drop-modgl.The solidcurve.give the totaldefcmmatlon•nw~Inoluding.ingla-particlocorrections;those correctior,sue calculated

*‘z

by US, of tha methodsMd parametoroof Ref. [36].Fig. 9. Atalouou#diagramto Fig. 8 for 102Zr;the singl~-particlocorrections

~

are calculatedby use of the methodsand parametor~of Ref. [W].‘J7~9

-ls-

Moss Number Al

I00

80

60

4C

2C

c

I00—- -— — — —

I I 1 I I

\

l\r@f~/(fro) = l.3 b

ro= L16 fmo=l.4fmOnm24.7 MeVK =4,0

.

Proton

60 80

Number 2,

250

200

I50

I00

50

80

‘>-,a’

-16-

. Mass Number Al

NN

Oc

8C

6C

4C

20

0

-++ ‘?O1 ‘?O1 2?012

N ro=[,[6 fma= L4fm

a$~24.7 Mev

\

l\

\

=4.0

d/(fm) =O.O\

9!

●

20 40 60 80

Proton Number 21

Figura 2

-17-

n

350

300

250

200

I50

●

✎

✌

●

●

1/9!5!5150 200 250 300 :

Calculated lnteroctlon-Barrier Height (MeV)

Figura 3

-18-

ou\*

Ea

z\c

● 0

E

a

20●

15●“

[c●

.

m

.

.

.

.

D

.

●

a/rO

.2249 fm?!9439 MeV

,

00●. ,

..

\\\-

10●

.

F7ssility Rmometer x

Fisura 4

-19-

0.25

0.15

0010

0.05

0.00

\ 0.2599-0.2151 x

#

ro=1.2249 fm.

.

●

✎

✎

m

0

.

.

.

.

.

,

●

,

.

.

.

b

.

●

as= 17.9439 MeV

K= L7826

Fissility Parameterx

.

-20-

0.05

0.04

o,o~

0.02

0.0 I

O.oc

‘o=1.16 fm

0= 1.4 fmb

OS = 24,7 MeV

K = 4,0

I

,A

,

~Lb—---L-~~1,50 0.55 0.60 0.65 0.70 ;

Fissility Porametcr x ~‘J

-21,-

t

9’s)1

m● -

L

smIs0m-u)u)

50

40

30

20

10

q

Z*IA1510152025303540 45

.

.

m

/

Finite mnge ofnuclear forcesincluded~0=1.16fm

a=l.4 fm

model

1 1 I 1 I

) 50 100 150 200 250 300 350

as=24.7#C=4mo

Mass NumberFigure 7

lASL AICOf I ICIA1

i

m

.

.

.

m

.

40Co

Liquid-drop

rangeofnuclearbrces included

Spheroid with

5 10 15

QuodrupoleMomentQ2 (Units of R02)

20

Figure8 LAN -AEC cl f tCIAl

.

!102 ~Zr

/.

/

/

/

Liquid-drop//

model/

//

/

h

m

h

1, 1forces included

Experimental

n‘o 5 10 Is 20

Quadruple Moment Q2 (Units of R02 )

25

-24-