REASSESSMENT OF FISSION FRAGMENTANGULAR DISTRIBUTIONS FROM

CONTINUUM STATES IN THE CONTEXTOF TRANSITION-STATE THEORY

Louis C. VAZ and John M. ALEXANDER

Departmentof Chemistry,State Universityof New York at StonyBrook,StonyBrook, New York 11794, US.A.

INORTH-HOLLAND PUBLISHING COMPANY - AMSTERDAM

PHYSICSREPORTS(Review Sectionof PhysicsLetters)97. No. 1 (1983) 1—30. North-Holland PitblishingCompany

REASSESSMENT OF FISSION FRAGMENT ANGULAR DISTRIBUTIONS FROMCONTINUUM STATES IN THE CONTEXT OF TRANSITION-STATE THEORY

Louis C. VAZ and JohnM. ALEXANDERDepartmentof (‘hemistry.State University of NewYork at StonyBrook. StonyBrook.New York 11794, U.S.A.

ReceivedDecember1982

Uontents:

I Introduction 3 7. Reassessmentof thetheory for high spin nuclei 22

2. Transition-statetheory 4 8. Summars 263. Someimportant resultsfrom the earlywork 7 Note addedin proof 27

4. The spin windowfor fission 12 References 28

5. Resultsof theanalysisin terms of K0

2 15

6. Effective momentsof inertia (.$t~

5)comparedto calculationsfor saddle-pointconfigurations 20

Abstract:

Fission angular distributions have been studied for yearsand have been treated as classic examplesof transition-statetheor~.Early workinvolving compositenuclei of relatively low excitationenergyE* (~35MeV) and spin I (~25h)gave support to theory anddelimited interestingpropertiesof the transition-statenuclei. More recent researchon fusion fission and sequentialfission after deeply inelastic reactionsinvolvescompositenuclei of much higher energies(~200MeV)and spins(~l00h).Extensionof the basic ideasdevelopedfor low-spin nuclei requiresdetailedconsiderationof therole of thesehigh spinsand, in particular,the“spin window’ for fission. We havemadeempirical correlationsof cross

sectionsfor evaporationresiduesand fission in orderto get a descriptionof this spin window. A systematicreanalysishasbeen made for fusion

fission inducedby H, He andheavierions. Empirical correlationsof K~(K~= J~~5T/h

2)are presentedalongwith comparisonsof .I,~to momentsof

inertiafor saddle-pointnucleifrom therotating liquid drop model, This model givesan excellentguidefor the intermediatespin zone (3)) ~ I s~6.5),

whilestrong shell and/orpairing effectsareevidentfor excitationsless than~35 MeV. Observationsof stronganisotropiesfor sery high-spinsystems

signal thedemiseof certain approximationscommonlymadein the theory,andsuggestionsare madetoward this end.

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L.C. Vaz and f.M. Alexander, Reassessment of fission fragment angular distributions 3

1. Introduction

Researchrelatedto fission-fragmentangulardistributionsandcorrelationscan be groupedinto fourstagesor aspects:(1) Understandingof the principal driving forcesat modestexcitation andspin, (2)Studiesof propertiesof the transition-statecomplexes,(3) Measurementsof sequentialfission fragmentcorrelationsas a probe of deeplyinelasticcollisions, and (4) Studiesof intrinsically unstablecompositesystemsof high spin and/orZ2JA.

The first stagebeganin 1952with the observationof angularanisotropiesin photofissionby Winhold,Demos and Halpern [Wi 52]; correspondinganisotropieswere soon reportedfor fission inducedbyneutronsandchargedparticles[Di 53, Br 54, Co54, Co55, Br 55, He 56,Co58, Pr58,Bi 59, Si 60, Ne61,Va 65]. Aage Bohr [Bo 55] then sketchedout an extensionof the transition-statetheory [Pe32, Ey 35,Ev 35, Bo39] to addressangulardistributions.This theory was subsequentlydevelopedfor continuumstatesby Strutinski [St56], Halpern and Strutinski [Ha58] and Griffin [Gr 59]. The phenomenologyatthis stagewas reviewedby Halpern [Ha59], andthe theory was extensivelydiscussedandexplainedbyWheeler[Wh 63].

Stagetwo, the propertiesof the transition-statecomplex,was led by Huizengaandothersin a seriesof studiesof fission inducedby beamsof H, He andneutrons[Va61, Hu 61, Hu 62, Ch62, Ba63, Gi 64,Le 65, Si 66, Re66]. They observeda strongdependenceof fission anisotropyon Z2/A which was to beexpectedif the transitionstatewas identified with the saddlepoint of a nuclearliquid drop [Co62,Co63, St 63]. Such a strong Z2/A dependencewas not expectedif the transitionstatewas identifiedwith the scissionconfiguration [Er 60, St 63]. Momentsof inertia for the transition-statenuclei werethen extractedalong with somefeaturesof the pairing interactions[Va61, Gr62, Gr63, Br 63, Br 65,Si 66,Br 68,Hu 68,Mo 69, Ip 71,Br 74, Ig 77]. A verycompletereviewof thefirst two stageswasgiven in1973 by VandenboschandHuizenga[Va 73].

Stagethree,the useof fission as aprobe of deeplyinelasticreactions(DIR), began,for our purposes,in 1977,with the out-of-planeangular-correlationstudiesof Vandenboschandcoworkers[Dy 77]. Thesestudiesaddressthe magnitudeand the alignmentof angularmomentumin strongly dampedreactions[Dy 77, Wo78, Sp78, Ha 79, Dy 79, Pu 79, Go 79, Br 82, Mo 82]. (There is also an extensiveseriesofstudiesof reactionmechanismsvia linear momentumtransferas inferred from measurementsof thefolding anglebetweentwo fission products.This very important applicationhasbeen very useful incurrentresearchbut is not discussedhere[seefor exampleVi 82, Du 82,Ga82a,Ly 82, Ki 82].) Someofthe importantmechanisticquestionscurrentlyunderinvestigationfor DIR havebeendiscussedrecentlyby Vandenbosch[Va 82], Specht[Sp811 andby Moretto [Mo 81]. In this contextwe will be concernedwith only two aspects:(a) the empiricalpropertiesof the transitionstatenucleithat areneeded,and(b)somepossiblemodificationsof the theoreticalconceptsthat may be necessaryfor very high spins.

Stagefour was initiated by Back et a!. in 1981 with observationsof strongfission anisotropiesin thefusion-like fission of 32S+ Au, Th,U andCm [Ba 81]. Basedon the rotating liquid drop model (RLDM)[Co74], the compositenuclei formed were expectedto havenearly sphericalsaddle-pointshapesandthereforevery weak fission anisotropies.This work andsimilar findings from sequentialfission [Sp81]haveindicatedthe needto reexaminethe applicability of the simplest form of transition-statetheory(and the RLDM) to theserapidly spinningandeven inherentlyunstablenuclearsystems.

Recentresearchon fission correlationshasfocusedon the third and fourth stages,andonemay getthe impression from thesestudies that the basic theoretical framework, as well as the empiricalparameters,are well known. Indeedit is true that the theoreticalequationshave, through the years,beenreexaminedandrefined [Wi 56, Gr 59, Er 60, Le 62, Hu 68, Mo 75, Ba 78, Br 791, andthereare a

4 LU. Vaz and f.M. Alexander, Reassessment of fission fragment angular distributions

large numberof experimentalpaperscontainingmore than a hundred angulardistribution measure-mentsfor fission after completefusion. However, it is not the casethat a unified understandingof theseresultshasbeenachievednor is it a straight-forwardtask to organizeall thesestudiesin acoherentway.Major difficulties lie in the estimationof angularmomentumdistributionsandtemperaturesappropriatefor each fissioningnucleus.The temperatureproblemhasbeenmentionedmanytimes,but the problemof the “spin window” for fission hasbeen treatedin detail in only a few very recentinvestigations[Vi 80, An 801.

In this paperwe havereanalyzedthe extensiveset of fusion-fission angulardistributions. For thispurposewe developedempiricalcorrelationsof evaporationresidueandfission cross sectionsin orderto approximatelower andupper limits for the spindistributionsof thefissile nuclei (i.e. “spin window”).Correspondingmodificationshavebeenmadein the equationsfor fission anisotropyin order to deducetheparametersK~and~efl- Tablesandgraphsaregiven for theempirical parameter,K~(K~=

as discussedin section2). With various assumptionsabout the temperatureT, we havealso estimatedthe effective momentsof inertia ‘~efi and havemadecomparisonsto thosecalculatedfor saddle-pointshapesby use of the RLDM [Co741.

In section 2 we review the basic theoretical framework, and in section 3 we look at someof theimportantconclusionsfrom the earlywork. In section4 we discussthe problemof the spin window forfission and presentour empirical approachto the definition of its limits. We give the results of ourreanalysisfor K~in section 5, and in 6 we makecomparisonsof .1f~values to thosefrom RLDM.Finally in section7, we makesomecommentsandsuggestionsconcerningthe approximationsmadeinthe theory andin the analysis.

2. Transition-statetheory

Thebasicstructureof transition-statetheory wasworkedout about50 yearsagoto addresschemicalreaction rates[seefor example Pe32, Ey 35, Ev 351. Its applicationto nuclearfission was made in aclassicpaper by Bohr and Wheeler [Bo 39] in the context of the liquid drop model of the nucleus.Transition-statetheory can be said to be an approximation to a more general diffusion theory asdiscussedin detailby Kramers[Kr 40]. The relationshipbetweenreactiondynamics,asdescribedin theKramersapproach,andvarious stationaryapproximations(e.g., transition-statetheory)is currentlyanactiveareaof study [Ho 83, Sc 82, Mo 82a].

Fig. 1 outlinesthe essentialelementsof the theory for angulardistributions as developedby Bohr.Halpernand Strutinski andothers [Bo 55, St 56, Wi 56, Ha58, Gr 59, Wh 631. The basic assumptionisthat the fission angulardistribution is “frozen in” at a certaintransition-stateconfiguration(normallytakento be axially symmetric),in the classicaltreatment,fission occursexactly alongthe symmetry (orbody) axis, andthe statisticalfeaturesof the transition-statecomplexprovide the driving forcesfor theorientationof thisbody axiswith respectto the spinaxis. Conservationof angularmomentumin a givennuclearreactionprovidesa space-fixedaxisand putsconstraintson the projectionM of the vector I onthis axis. In the analogousquantummechanicaltreatmentone addsthefeatureof a probability functionP~K(O)for the separationdirection with respectto the space-fixedaxis (denotedby the angleO). Thisfunction is given in eq. (1) by the squareof a rotationalwave function d~K(O)I2,which is characterizedby threequantumnumbers:I (total angularmomentum),M (projection of I on the space-fixedaxis)and K (projection of I on the symmetry axis). The energyof this rigid rotor has a term (h2K2/2.9’

11)correspondingto rotation aboutthe symmetryaxis and anotherterm (h

2R2/2~)for rotation about an

L.C. Vaz and f.M. Alexander, Reassessmentoffission fragment angular distributions 5

g~ili4~m~7he~42~42e~uj~4~uIa4e044e/.aelOm~

‘k!’ body P,~K(ë)~(2t+1)Id~K(8)I2 (1)~ /1 “~ ~ ~2 R2 ~2 K2

E~~t + 22 (2)

__________________•pace +~axis 29~ 2I~ff

—t —t —,

I W~or6)czfd�fdK[Tj (e)p (E~)<)PI~K] (4)

00

° ~om T1 (�){~ � independent of K (5)

p (Es. K) ~ exp (E~/T)exp {{E— tEro(tBt (1) —es] / T } (6)

~ exp[_~ K2/29eff T]

i Se1 I~oio~r:o~weie~xmea~ ~ e~4e4~4C~~I~24tG(xoy)~+M~I

WI,9eff T1~’1~ exp(2psin2 4i) (7)

it ‘~4SIsi#t~4~SI~#e�zloi~tch~&o~a~eme#e4:T &~u#4o~m4I~ep~.te(iioz) ~ M 0

~ T~6~~ exp(—psin28) J’0(ipsin

26) (8)~ ~4i~I~/49~ff T ~(I /2K

0)2 (9)

Fig. I.

axis perpendicularto this symmetry axis (R is the componentof I perpendicularto K). The totalrotational energy in eq. (2) is proportional to K2 through an effective moment of inertia, 3~I/ =

— ~ The classicalvalue of ~ is zero for a sphereand increaseswith increasingdeformation.

The othersymbolsin fig. 1 areas follows: total excitation energy,E; fission barrierB1 (with subscript

to specify the transition-stateconfiguration);the thermalexcitation at the transitionstate,E1.Two simple idealizedcasesfor fixing the spaceaxis in an experimentalsituationare as follows: (1)

Perfect alignment of the spin I along the spaceaxis (M = I). This case can be approximatedinsequentialfission reactionsby a coincidencerequirementthat locatesthe spaceaxis (oz) perpendicularto the reactionplane(xoy). (2) Uniform distributionof I in the (xoz)plane(M = 0). This latter caseisapproximatedin fusion fission where the spaceaxis is given by the beamdirection (oy) and I isuniformly distributed in the plane perpendicularto the beam. (Several papershave consideredtheeffectsof deviationsfrom theseidealizedcasesbut we will not include them here.)

The angulardistributions W~(por 0), in eq. (4), are then takento be proportionalto a sum (orintegral) over K and e that contains threefactors:(a) the transmissioncoefficient T1(~)for passagethrough the transition statewith collective energys~in the fission mode; (b) the level density at thetransitionstatep~(E~,K); and(c) the probability distribution P’MK(fp or 0) from eq. (1) (or its classicalanalogue,cos~ = K/I). At this stageaseriesof simplifying approximationsareoften made;we will firstgive the whole seriesandreexamineit later: (1) transmissioncoefficients T1fr) are takento be sharplycut off, eq. (5); (2) the fissionbarrier B1(I) is often takenas spin independentB1(I = 0),both in eqs. (5)and (6); (3) the fission barrier B1(I) is taken as independentof K; (4) the constant temperature

6 LU. Vaz andf.M. Alexander,Reassessmentoffission fragmentangulardistributions

approximationis used for the level densityat the transition state,p~(E5.K) exp(E,/T).(In section7we reassesssomeof theseapproximations.)

With theseapproximationsoneobtainsa Gaussiandistribution for K (eq. (6)) that is folded with theprobability function P~IK(tpor 0) in eq. (4) to obtain the angulardistribution or correlation W~(çoor0). Finally, sumsare madeover spin (I) of the fissile nucleusand its projections(M) on the spacefixedaxis.

First considerthe classicallimit for the caseof perfectalignment (coincidenceexperiment)given byM = I and cos~ = K/I. Replacementof K by I cos~ in eqs. (4) and (6) leadsdirectly to eq. (7) [Br 791.As shown in fig. 2 this very simple classical approximation(eq. (7)) gives an excellentmatch to thequantumform of eq. (4) that usesthe full rotationalwave functions, d~=J.K(~).To obtain an equationfor the caseof fusion fission (M = 0), one must make an integrationof eq. (7) to allow all azimuthalorientationsof I in the plane(xoz).This leadsdirectly to eq. (8) whereJo(i p sin

2 0) is the zerothorderBessel function of imaginary argument.The more common route to eq. (8) has been to use asemiclassicalapproximation to the functions d~l=i.K(O)l2 in eq. (4). These paths are completelyequivalent.

In this theory,it is the parameterp or h2I2/4~~0Tthat controls the strengthof the anisotropy.Fig. 3

showsthe family of angulardistributions for fission after fusion to give an ensemblepossessingoneuniquevalueof p. For fission following fusion to form a very fissile compoundnucleus(high Z

2IA). onemustmakea weightedsumover the completedistribution of I for this system.Fig. 4 showsthe familyof angulardistributions calculatedfor this sum from zero spin to maximum spin Im (for a sharp-cuttriangularspin distribution). In section4 we reexaminethe assumptionsimplicit in this integration,butfirst let us reviewsome of theearlywork that gaveconfidencein the basicmechanisticframework of thetheory.

ANGULAR CORRELATIONSWITH RESPECT TO THE SPIN AXIS

I I I lj I I

w~.~(~I~fexpEK2/2I<i]I2I÷1 lid ~I c ~

60 W~.

1i~1a exp [lr2sIn24,I/2K~} ~

2 II I- K0 ~t76.1 -

10 Angular Distribution ForFission Fragments (for on unique I)

00 I I i~_i~±~-i_~ 90 80 70 60 50 40 30 20 tO 0

90 80 70 60 50 40 30 20 10 0

~ (deg) cm. (deg.)Fig. 2. Comparisonof the classical expression(solid line) of eq. (7) Fig. 3. Calculatedangulardistributionsfor fission fragmentsof uniquewith that involving thefull rotationalwave functionsin eq. (4). (The spin (or uniquevalue of p). Eqs. (8) and (9) of fig. 1.latter calculationswere kindly made for us by R. Vandenbosch.)Ineachcasethevalueof K~was 176.1.

L.C. Vaz and f.M. Alexander, Reassessment offission fragment angular distributions 7

Angular Distribution For10 Fission Frogments(with a distribution nI)

p~(Im/2Ko)2

:2p xl/Sexp(_xsint9)J

0(issjn29) t~o

8 W18) erf(,/2i)

90 80 70 60 50 40 30 20 10 06c.m. (deg.)

Fig. 4. Calculatedangulardistributionsfor fission fragmentsobtainedby summingover a classicalsharpcutoff distribution of I (from 0 to I.,).

3. Someimportant results from the early work

In the theory outlined above, a measuredangulardistribution resultsfrom the combinedeffectsofthe angularmomentumdistribution, the effective moment of inertia, and the temperature.Often it isconvenient to consider the combined roles of ‘9eff and T, and thus one defines the parameterKo2J~e

8TIh2,

— ~2T2/A 11 T’— if /~f/’ \2Ji — ft 1m/’t~1’eff~ — t~.Im/~.1~O)

Coffin and Halpern [Co581, studiedaseriesof fission reactionsinducedby 2H and4He, for which theyarguedthat values of ~ and T must be comparable.In eachcasethe anisotropyfor 4He inducedfission is larger than for 2H, as shown in fig. 5, and the ratio is close to that expectedfrom simpleclassicalestimatesof the ratiosof I~.This resultsupportedthe role of I~in the theory.

I I I

R226~ 43NeV~ PARTICLES•2.0 ‘~ ~209 ° 22 MeV DEUTERONS 0

I 8 “\,~Th232

~~:r Th232

t,a - ~ L~ ~235

0NP

237pZ39

36 37 38

Z~’AOF THE COMPOUND NUCLEUSFig. 5. The fission fragment anisotropy asa function of Z2/A ofthe compound nucleusformed. The measuredpoints are labelledaccordingto the targetnucleus.The data are thoseof Coffin and Halpei-n [afterCo58).

S LU. Vaz and f.M. Alexander.Reassessmentof fission fragmentangulardistributions

It is very difficult to focuson the role of excitationenergyor temperatureseparatelyfrom that ofspin. Changesin projectileenergyinvariably lead to changesin spin as well as initial excitation energy.In addition, the relevant temperatureis that for the transition state, and energy lost by prefissionneutronevaporationcan reducethis temperaturefrom that of the initial compoundnucleus.Figure 6from Leachmanand Blumberg illustratesseveral very nice sets of results that bearon this problem[Le 65]. The generaloverall increaseof anisotropywith incident energy(fig. 6) is primarily the resultofincreasingangularmomentum.This conclusionresultsfrom the correlationbetweenthe anisotropydatafor neutronand4He inducedfission with their respectivecalculatedvaluesof (J2~,

The very interesting stair-stepbehavior is interpretedas a reflection of the temperatureat thetransition state.For neutron energiesof <5 MeV, only first step fission can occur, and the initialexcitationenergy(lessthanthefission barrierenergy)fixes the temperature.Thereforefrom 2 to 5 MeVincidentneutronenergythe temperatureis increasingalongwith the meanspin. Henceoneexpectsthevalueof p to changeratherlittle, andthe observedanisotropyis essentiallyconstant.A steepincreaseisobservedfor neutron energiesnear 6 MeV where one expectsthe onset of the reaction (n.nf) orsecond-chancefission. Neutronemissionseemsto haveoccurredprior to arrival at the transitionstate.andthecorrespondingreductionin temperaturehasgiven a significant increasein the anisotropy.Fromthis study one concludesthat a preciseunderstandingof the competitionbetweenfission and evapora-tion is essentialto separateK~into its componentsT and~ Only after such a dissection can oneexplorethe propertiesof the transitionstatecomplex.

Vandenbosch,Warhanekand Huizengahavemadea careful study of fission anisotropiesas related

NEUTRON ENERGY (MeV)0 2 4 6 8 10 12 14 16 18It II I I I I I

0 SIMMONS a HENKEL + u236

•PROTOPOPOV a EISMONT~ OBLUMBERG BLEACHMAN

1.3

~I.2

0 1 iW PRESENT WORK~ ~‘j ~n+Pu239Z i.o 4 U CATCH FOILS

S SEMICONDUCTOR ~a + U236DETECTORS

I I i ,.._I~. I I I IIi ~ I II I

96/

80a + U236(SQUARE WELL MODEL I /

64 Rt1,5A”3+1.2)F //

~ 32

ALPHA PARTICLE ENERGY (MeV)

Fig. 6. Anisotropy data of fragmentsfrom thecompoundnucleus2~°Puand, below, orbital angularmomentumcalculations after Le 651.

L.C. Vazand f.M Alexander, Reassessment offission fragment angular distributions 9

to the fissility of the compoundnucleus[Va 611. Some of their experimentalresultsareshownin fig. 7andtheir inferencesof K~in fig. 8. In fig. 7 oneseesfrom 238Thto 238U to 233U a reductionof anisotropyand a disappearanceof the stair steps.This observationcorrelateswith the strong increasein fissilitywith increasingZ2/A as demonstratedexperimentallyby a strong reduction in the (4He, xn) crosssections[Va581. From the anisotropiesfor 4He+ 233U theseauthorsobtainedvalues of K?~with onlysmall corrections for fission after the first chance. The trend of values for K~(for both 4Heandn inducedfission) is shownby the solid line in fig. 8. This solid line in fig. 8 wascomparedto twotheoreticalcurvesandsuggestedtwo importantconclusions.First considerthe dot-dashedcurvewhichvaries with excitation energymuch more slowly than the data. This curve was calculatedwith theassumptionthat the transition state is definedby the scissionconfiguration of two touching spheres[Er 60, St 63]. The disagreementprovidesa strong argumentagainstthe identification of the scissionconfigurationas the transition-statecomplex.The dashedcurvewas calculated(afternormalizationtothe point at 33 MeV) with eqs. (10) and (11) and the assumptionthat the transitionstateis the saddlepoint:

K2o=,9effT!h2, (10)

T = at12(E— B1) 1/2 (11)

The disagreementat low energieswascited at that time asevidencefor nuclearpairing.More recentlyithasbeenconsideredas possiblya reflection of theshell structurethat gives a doublehump in the fissionbarrier [Ra70, Va 73, Va 73a]. If the transitionstatecorrespondsto amoreextendedsaddlepoint then~ would beespeciallylarge or K~would be especiallysmall. Both the effectsof pairing andshellsareexpectedto dissolvewith increasingtemperature.

It is conceivablethat thereis a spectrumof transitionstateshapesandthat memoryof this spectrumpersiststo the scissionpoint where it is correlatedwith the massratio of the fragments.A numberofexperimentshavesearchedfor such an effect [Co54, Co55, Fl 64, Vi 65], andsomeof the resultsareshown in fig. 9. For

4He+ Pb/Bi the fragmentanisotropyis found to beindependentof massexceptfora 10% reductionfor the extrememassratio of 1.52 that correspondsto very low yield. The yield mass

~ ~238i

3O ________________________24 28 ‘ 32 ‘ 36 40 00 4 8 12 16 20 24 28 32 36HELIUM ION ENERGY EX-Bf(MeV)Fig. 7. Energy dependenceof the fission fragment anisotropiesfor Fig. 8. The solid line curve a is thedependenceof Kl vs. E0—B1asreactionsof

4He with threetargets.The ordinate is the ratio of the obtainedfrom neutron-inducedfission of Pu andhelium-ion-induceddifferential cross sectionat approximately92°(center of mass).The fission of B3U. The verticalbarsrepresentestimatesof theuncertaintystandarddeviations areapproximatelythe size of thesymbols.The in this dependence.The dashed curve b is the prediction of thesolid lines are smoothcurvesthrough the experimentaldata (after Halpern—Strutinskii,Griffin saddlepoint hypothesis.The dot-dashed

[Va61]). curve c illustrates thedependencepredictedby theEricson scissionpoint hypothesis(the ordinate representso~+o~ratherthan Kl inthis case)[afterVa 61).

10 L.C. Vaz and f.M. Alexander.Reassessmentoffissionfragmentangulardistributions

I I .80 , I

a b

:.: 1 .60 ‘ ~ 48Mev4He + 238U -

2.1~: ~ ~

I .6 I I I I I 22 M~V6 -f

.0 t.t I.e 1.3 1.4 1.5 1.6 1.00 -

N

1 IN2 I I I I1.0 1.2 1,4 1.6 1,8

Fig. 9. (a) Fission-fragmentanisotropy as a function of mass split for 42 MeV helium-ion-inducedfission of206Pb and 209Bi [after Fl 64]. (b)

Anisotropy [o~(0°)/u(90°)]asa function of massratio(MH/ML) for 2~Ubombardedwith 22MeV ‘H ETh 24MeV 2H — •~and48 MeV 4He — D [afterVi65].

curvesare symmetricandrathernarrowfor thesecases.For 238U fission by H and He, thereis a smallbut significant dependenceof anisotropyon massratio. In thesecasesthe massdistributionsare broadand mass asymmetric[Su571. It is thought that the higher anisotropiesfor mass asymmetricproductsderive from an admixtureof first-chancewith multiple-chancefission, i.e. an effect of temperature.Theyield of the mass symmetric productsdecreasesrapidly with decreasingexcitation energy, and thisdecreasestrongly discriminatesagainstsecondandhigher chancefission [Su~71.

Huizengaandcoworkersmadean extensiveseriesof measurementsin the 1960’sdirectedtowardthedetermination of the momentsof inertia of the transition-statenuclei [Va6l, Hu61, Hu62, Ch62,Ba 63, Gi 64, Re661. They usedempirical informationon fission-evaporationcompetitionto correctforfission after the first chance[Va 581, and optical-model calculationswere used to estimatethe spindistributions of the fissile nuclei [Hu 62). An illustrative graphfrom their work is shown in fig. 10. (In

9sph~~

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1.0

xFig. 10. Comparisonof saddledeformationsproducedby deuteronexcitation(shadedrectangleandopencircle) andby helium ion excitation(closedcircle) as a function of therelative fissionability parameterx = (Z2/A)I(Z2/Z)~~,.One deuteronpoint representsextremevaluesobtainedfromseveralsetsof transmissioncoefficients using the Bi209(d. f) anisotropydeterminedin [Ba63]. The otherdeuteronpoint (open circle) is from theNp(d, f) anisotropydataandtheassumptionof 100% 1st chancefission (hencetheupper limit in .1’,

551f,4).Thehelium ion points arefrom dataofChaudhryet al, [Ch63]. Temperaturesatthesaddleconfigurationswerecomputedwith the Fermi gaslevel densityparametera = A/8. Theoreticalpredictionsof thecritical deformationatthesaddle-point(—) andatthescissionpoint (———)arebasedon thechargedliquid-dropmodel calculationsbyCohenand Swiatecki [Co631, andby Strutinski [St631 [afterBa 631.

L.C. Vaz and IM Alexander, Reassessment offission fragment angular distributions 11

this figure, asin figs. 18—21 later, valuesof ieff aregiven as ratiosto ~-~sph (or ~, themomentof inertiaof a rigid sphere.)The flat dashedline was calculatedwith theassumptionthat the transitionstateis thescissionconfiguration [Er 60, St631; the solid line was calculatedfrom the (non-rotating)liquid dropmodel for saddle-pointconfigurations[Co631. It is this strong dependenceof the dataon Z2/A that hasgiven major experimentalsupportto the conceptthat fission angulardistributions are frozen in at (ornear)the saddle-pointconfigurations.

The basicconceptof transition-statetheory stimulatedthe applicationof sequentialfission studiesasa probe of spin transferin deeply inelastic reactions.If the choice of angulardirections in fission isindependentof the mode of formation of the fissile nucleus, then one can use measuredangularcorrelationsas a reflection of the spin vectors generatedby “tangential friction” in the primary DIreactions.To accomplish this purposeone needs a solid footing of empirical values of K~fromanisotropymeasurementsfor fusion fission [Dy 77]. In addition it is desirable(possibly not completelyessential)to havea well understoodtheoreticalmodel for the transitioh-statenuclei, and the rotatingliquid drop model seemsvery promising. In fig. 11 we give an outline of the logic of the threeinterdependentinformationchannelsthatare employed.Onepurposeof this paperis to reexaminestepII of the logical schemein fig. 11—namelythe systematicsof K2

0 from fusion fission.Since the completionof many of the pioneeringexperimentsmentionedaboveand their review in

1973 [Va 73], therehavebeena numberof importantdevelopments.Many new measurementshavebeenreportedfrom fission inducedby

4He at high energy[Ka 66, Mo 69, Cl 75] aswell asby heavyions[Vi 63,Ka 67, Ko 71, An 80, Va 80, Vi 80, Ba 81, Va 82]. Comparedto 40 MeV4Hebeams,thesenewermeasurementswith heavy ion beamshaveextendedthe region of excitation energyfrom ~35MeV to~200MeV andspin from ‘~20hto ‘~100h.Liquid dropmodelcalculationshavealsobeenmadefor the

°7h’~�e9#d~s4e#sde.et9#~sma~sCha..uselj.

iSe~1sseelàzl~4 (e.~6XMe~J%+_f~+&*)

(f)iWea~ee.seKE,Z,A oj psoie&de-li~e~sa9me#d(Kr’)ad~s/~4~2,~oe#u,cIe~ (Bi).12)Mea~w~ed-oj-p~a#zeco~e/alio.a~(3)’~4oscmssji~sas~& K~/o~

~ oiW~r2oued#uwlee~maiched fzs ~, ~, ~.

(4)L~.s~sact<J2>~soørp <J2)/4K~

(5) °7es~DI ~eacle�usmodel psedicl.iss#sd j~ss.dp~clSaf404.

.

114 ià,r~4~g(e.~.f5OMeVNe+Ta_.BI*)0) Meavses’ze ca~s~eel.~zsd Sm: p(2)’~4om/KIS~m o2o&t secli~u,s~ <J2

(3)sacl K0

2 /som p°<J2)/4K~(4)~’sose‘~cse ad.mode1

9~&T:E°aT2

(5)&~lsa&.~ /4om K~ ~9effTtñ

(6Y7ed ‘~eaetio.n.model peedictisut~/04 9effdeepmodelcal~ela~

(I) 4~SSIUS40.54104 Bf (I)

(2)Saddl.e-p&el ~thcsf2e9eff Cl/ ~

=~° ~ T~N-Z/A

E~ot 12 r 1,92

y0 E0AT/3hll78T2]

Fig. 11.

12 LU. Vaz andf.M. Alexander,Reassessmentof fissionfragmentangulardistribution.s

spin dependenceof i~0andB1 [Co741. Thesecalculationsindicatea strongdependenceof fissionabilityon spin that has been qualitatively verified by measurementsof cross sections for fission andevaporationresidues[Ze74]. In the next sectionswe examinesomeimplicationsof thesemore recentinvestigationson the interpretationof fission angularcorrelations,both in terms of K~(asneededforapplicationsto sequentialfission) andin termsof 95,~(asneededfor testsof RLDM).

4. The spin window for fission

Heavy ion beamshaveled to an enormousnumberof studiesof the role of angularmomentuminnuclearreactions.Classification of spin zonesin thesereactionsis generallymadeby referenceto thecrosssectionsfor evaporationresiduesER. fusion fission FF, deeply inelastic reactionsDIR and quasielastic reactionsQE. Thesegroupsarethought to follow in order (ER, FF, DI and QE) of increasingimpact parameter(or entrancechannel 1) as shown in fig. 12. The correspondingspin windowboundariesfor fission,

1ER to ‘cril, can be estimatedfrom measurementsof the crosssectionsforevaporationresidues°ER andfusion fission am~.

If onehas measuredthesecrosssectionsalongwith the fission angulardistribution then the naturalinferenceis that the appropriatespin zonefor fission is approximatelyfrom ‘ER to lcrjI. Thus a morereasonablerelationshipbetweenp value and anisotropycan be obtainedby changing the limits ofintegrationusedfor fig. 4 (from zeroto p) to Prnin to Pmax [Prnin= (IERI2K

0)2 andPmax= (lcritI2Ko)2]. We

haveperformedtheseintegrationsand show in fig. 13 somevalues of W(O)IW(90°)as a function of

~

Ci~~ +~ 4~ir~,(4~ti) I 0 0 ~

EiotinpIt for E*. 115 tle.V 3 ~

Z5Z N~V~Ne+~yb ~ * ~ 2 ~

3’4ON~V404 +~‘~Sn, H8 ~ i4-~ ~ ~ 0.0 0.1 0.2 0340506 OJ 08 O~t.O

F,g. 12. Fig. 13. Three sets of values for the anisotropy W(O)/W(90°)for

various values of p as a function of PmInIPmox Thesecurveswere 0

calculatedaswerethosein fig. 4 but with two sharpcutoff limits, P,~fl

andPma~

L.C. Vaz and f.M Alexander, Reassessment of fission fragment angular distributions 13

PminlPmax.The morecompletefamilies for PrninlPrnax = 1 weregiven in fig. 3 andthosefor Pmin/Pmax 0in fig. 4.

For many reactions,resultshavebeenreportedfor fission anisotropies,but the correspondingcrosssectionsfor ER and fission havenot been measured.In order to extract a value of p from thesemeasurementswe havedevelopedan empirical schemefor estimatingPrnin/Prnax [or (Imin/Imax)2 or(1ERI1CrII)

2l. Valuesof ‘crit are estimatedfrom the fusion crosssection codeFRANPIE [Va 81a] whichutilizes empirical fusion barriers from Vaz et al. [Va 81] and (for high energies)the critical radiusapproachof Galin et al. [Ga74]. Forenergiesmorethan 10% abovethe barrierthis procedurehassolidempiricalsupport [Ga74, Ta75, Va 81, Ra821.

Valuesof ‘ER can alsobe ratherwell estimatedfor thosecaseswith substantialfission crosssections.The approachis shownin fig. 14. A typical set of experimentaldata for 0~ERand °fis is shown in fig.14(a).The associatedvaluesof ‘ER, as shownin fig. 14(b), exhibit aratherflat plateaufor energiesabovethat for the onsetfor fission. Theseplateauvaluesof 1ER havebeenobtainedfrom manystudiesof °ER

in the literature [Sc76, Oe80, Da77, Ca80, Ga75, Br 76, P178, Ra82, Lo 80, De77, Le 82, Hi 82,Ge80, Bl 82, Si 821. They areshown as a function of Z2IA in fig. 14(c), andwe haveusedthem in theanalysisto follow. As shown in fig. 14 anddiscussedin refs. [Ra82, Al 82 andBl 821, theplateauvaluesof ‘ER are not strictly independentof energyor entrancechannel.However,our own numericalstudiesindicatethat the sensitivity of the analysisto this uncertaintyis rathersmall.

Canoneget informationon the fuzzinessof the window for fission?Miller andcoworkersdevelopeda procedurethat doesindeedallow for a moredetailedview of the shapeof the window [Au 68, Ko 71,Ze74]. A very recentstudy by Leigh, Hinde and othershasused this method to obtain the resultsshownin figs. 15 and16 [Le 82, Hi 821. Theseresultsarecompletelyconsistentwith thosefor 1ER, shown

II 56 70 79 92 106• “1critt’~’~100 404r+’I°Pd

40Ar +107.9Ag80 n

~ 60~ ~ I I I I I I

~:: ~ (1ER+1)2 (C)

1200 55 72 83 ~ ~ ~ (MeVI 6040Ar + “°Pd ‘~ FUSION 50 ~

00~ ~-m’v \0~1000 / I— U

4~/t~F~SON 20

200 / •n A~I I I I I 0 I I I I I I

140 160 180 200 220 240 260 280 26 28 30 32 34 36 38 40Elab (MeV) Z~’A

Fig. 14. (a) Experimentalexcitation functions for fusion fission and for ER (datafrom [Br76, De77, Ca801). (b) Values of ‘ER deduced from

thevaluesof ~ER. (c) Thevaluesof IER at theplateauvs. Z2/A for thecompositenucleus(seetext for references).

(4 LU. Vaz andf.M. Alexander,Reassessmentoffissionfragmentangular distributions

1/ECM(19F)

80 100 12p(10~MeV’)

~ ~ : ~ ~

70 80 90 100(10 MeV~) / ‘18

l/EcM(30Si) ~-_ 20 III

(b) 0 I I I’ I

0.8 1.0 (b)110Er+30Si —-

55 1._ -r 080.6 ~0 r’

45 ,,‘ ~ ~ l~J0.6 /0.4 40 , Q.~ /

35,’ ,P 45 0.4 /0.2 9’ 0 To+ F 02 ~ ——E

5°73MeVE553MeV

—VARYING

60 - ~ 0 20 40 60 80E~(MeV) L(fl)

Fig. 15. (a) Measuredfusion excitation functions.The curvesare fits Fig. 16. (a) Calculatedangularmomentumdistributions for formingto the datausing theparabolicbarrierapproximation.(b) Experimen-

2151Pb with I7SEr+ ~°Si(dash-dot)and 151Ta+19F (dashed).The fulltal total fission probabilities; the curves are statistical-modelcal. curve shows the additional angular momentum brought ~nb~theculationswith at/a

0 = I and B,= (1.83. The relatively poor fit at heavier projectile. The full fine curve IS the angular momentumbombardingenergiesnear the Coulomb barrier arises from the sen- distribution of evaporation residuesfollowing the

30Si reaction.(b)sitivity of thecalculationto the input angularmomentumdistribution, Statisticalmodel calculationsof the total fission probabilities com~andthepossibility of shell andpairing effectsbecomingimportantfor paredwith theexperimentaldata. The full curve is for E

0 varying inlow L andE,. The verticalarrowsindicatethe Laax, for eachreaction thesamemannerasthedata, whilst the othersarefor fixed E. [after[afterLe82]. Le 82].

in fig. 14(c),but in addition theygive an indication of the shapeof the fission probability versusspin. Inour analysis we have used the sharp cutoff approximation for integration between ‘ER = ‘mn and1crit = ‘max. This approachcan be expectedto be less preciseas 1~approaches‘crii (or for very smallfission crosssections).

What is the qualitativerole of this spin window for fission? Fromfig. 14(c)we seethat the valuesof1ER only becomeappreciablefor Z2/A ~ 36. Therefore the early estimatesof (12) are not seriouslyaffectedby this modification for fissioningspeciesof Z~ 90. However, interpretationof the resultsforfission of Au, Pb, Bi etc. is certainly affected;we now estimateappreciablylargervalues of (J2) Thesenew estimateslead to a substantialreductionin g~

0IS1~fffor the points for low Z2/A (or x) in fig. 10.

More importantly,however,this procedureallows us to seekfor a coherencebetweenK?~and S100/S

tefffor fission inducedby light andheavyion projectilesas afunction of both angularmomentumandZ2/Afor the fissile nucleus.Such a coherencehas heretoforebeenelusive[seefor exampleVi 63, Cl 75, andBa 781.

L. C. Vaz and f.M Alexander, Reassessment offission fragment angular distributions 15

5. Resultsof the analysisin terms of K02

In table 1 we list the resultsof our reanalysisof fusion-fissionangulardistributions.Reactionsarelisted in orderof Z2/A for the initial compositenucleus.Columns1 and2 give the projectile energiesandestimatedvalues for ‘cr15 (oi~Im) respectively.For thosereactionswith measuredcrosssectionsforER and/or fission we have used those values for determination of Prnin!Prnax. Otherwise we havecalculated1~,as describedin section4 and/ortaken‘ER from fig. 14(c).Fromtables(similar in contentto fig. 13) we haveused the reportedanisotropiesto determinep [p = (Im/2K

0)21 which is given in

column 3. For somecasesthe value of 1ER from fig. 14(c) is largerthanthe value of lcrii~This signifiesthat we are consideringan energybelow the achievementof the plateaushown in fig. 14(b). For thesecaseswe assumethat the fission reactionsskim off the creamof the I wavesnear ‘crit andhencewe usePrninlPmax = 1. In columns4—7 respectively the values are given for K02 and .9sp/~eff.In this sectionweconcentrateon the values of K02 becausethey are less complexandpossibly more straight-forwardforempirical usein sequentialfission studies.We havegroupedthe resultsin arbitrarybins of Z2/A anddisplayedthem in fig. 17. In the pastit hasbeenconventionalto plot K02 versusthe squareroot of the“excitation at the saddlepoint” E [E*u2= (E — Erot — B

1(I = 0))h/2]. For a Fermi gasthevalueof T isproportionalto E*t/

2 and since K02 = 9~ffT/h one hopesto obtain a straight line in K20 versusE*tl’

2.

This choice is clearly basedon the hope for first chancefission andthe approximationsthat B1(I) and

ie0 do not vary with spin. As theseaspectsseemquestionablewe choosethe moresimple plot of K02againstgross initial excitationenergy.

The primary questionaddressedby fig. 17 concernsthe basicassumptionof statisticalequilibrium in

200 (a) I I I

100

200 (b)• .‘

tOO 33.5-340

200 _—J~--*- I I101

200 Id) I I I

£ 35.5-36 2

100

200 Cc)36.3-36.8

100 1?~~’ I I I I I

0 20 40 60 80 100 120 140EXCITATION ENERGY(MeV)

Fig. 17. Valuesof Kl from table 1 versusgrossinitial excitationenergy.Symbolsfor variousreactionsareindicatedon fig. 18. Upperlimits indicatethepossiblepresenceof fission aftertransferreactionsalongwith thefusion fission. Solidlines aredrawn to guidetheeye.

16 LU. Vaz andf.M. Alexander,Reassessmentoffissionfragmentangular distributions

Table IResultsof thereanalysisof fission anisotropies

E10b IC,II E105 ‘~riI

(MeV) (h) ~b d 2~j 3d (MeV) (6) ph K~ 1’ 2~6Li + iSiTa [Vi 801; Z2/A = 30.90 4He+ 207Pb[Mo691;Z2/A = 33.4 (contd)

94.4” 33 2.16 126 1.29 1.18 1.05 42.0” 20 1.26 84 1.18 0.99 ((.6884.2” 31 1.63 147 1.03 0.93 ((.81 41.0” 20 1.33 76 1.27 1.07 ((.71)74.8” 29 2.18 102 1.38 1.24 1.05 40,0” 19 1.26 77 1.23 1.02 ((.65

39.0” 19 1.18 78 1.18 ((.97 0.596Li + ‘9tPt Ni 80]; 72/A= 32.2 38.0” 18 .24 71 1.28 1.04 ((.64)

944W 31 2.15 112 1.68 1.54 1.40 37.5” 18 (.30 66 1.35 1.09 ((.6184.2” 29 1.53 138 1.28 1.17 1.04 370W 18 1.35 SI 1.42 1.14 0.6174.8” 27 1.52 125 1.32 1.19 1.04 36.5” 17 1.35 59 1.45 1.16 (L59

36.0” 17 1.51) 52 1.62 1.29 ((.62H

2+204T1 [Ba63]; P/A = 32.6 355W 17 1.43 53 1.57 1.24 ((.55

21.0” 9 (L54 44 2(16 1.47 ((.78 35.0” 17 1.44 SI 1.62 1.27 ((.5334.5” IS 1.52 47 1.72 1.33 ((.49

4He + ‘97Au [Ch62, Cl 75]; P/A = 32.6 340” 16 1.43 48 1.65 1.27 0.42l40.0~ 39 2.13(96) 179 1.21 1.13 1.04 33.5” IS 1.46 46 1.72 1.30 ((.3342.8W 20 1.44 76 1.34 0.98 ((.56 330” IS 1.69 38 2.02 1.52 (L4442.7” 20 1.37 79 1.29 ((.94 ((.53 32.5” 15 1.65 37 2.02 1.4941.2” 20 1.25 82 1.20 ((.85 ((.41 32.0” 15 1.71 35 2.12 1.5538.8” 19 1.24 74 1.27 ((.87 ((.31 31.0” 14 1.86 29 2.43 1.7236.8” 18 1.37 (ii 1.48 (1.96 ((.1235.!” 17 1.38 54 1.57 ((.96 4He+206Pb[Mo69]; Z2/A = 33.633.3” 16 I.!)) 60 1.36 ((.76 50.0” 23 1.12 98 1.22 ((.96 ((.70

45.0” 2! (.35 57 (.26 ((.95 ((.606Li + ‘94Pt [Vi 80]; Z2/A = 32.8 42.6” 20 1.39 78 1.35 0.99 ((.56

94.4” 31 2.02 119 1.53 1.40 1.25 41.0” 20 1.42 7! 1.43 1.02 (L5184.2” 29 1.71 123 1.39 1.26 1.11 400W 19 1.30 74 1.35 (L94 (L4374.8” 27 1.79 02 1.58 1.42 1.23 39.0” 9 1.33 69 1.40 0.96 (L37

38.5” 18 1.39 64 1.50 1.02 ((.36‘H

2+2116Pb [Ba63];Z2/A = 33.1 38.0” 18 1.45 61) 1.58 1.06 ((.32

21.0” 9 ((.48 49 1.71 1.21 ((.30 37.5” 18 1.32 54 1.46 (L97 ((.2237.0” 18 (.35 61 1.51 0.98 ((.26

4He+ 2~°TI[Ch62]; P/A = 33.1 36.5” 17 1.32 SI 1.51 0.9742.8” 20 1.35 81 1.31 1.01 ((.58 36.0” 17 1.33 58 1.55 (L9839.6” 19 1.49 64 1.56 1.15 ((.52 35.5” 17 1.28 59 1.50 ((.9335.7” 17 1.39 56 1.64 1.12 35.0” 7 1.43 SI 1.72 1.04

34.5” IS 1.42 50 1.74 1.036Li + 97Au [Vi 80]; Z2/A = 33.1 34.0” IS 1.58 43 1.98 1.15

94.4” 29 2.11 103 1.82 1.67 1.50 33.5” 16 1.57 42 2.01 1.1384.2” 27 1.75 107 1.64 1.50 1.32 33.0” 15 1.64 39 2.15 1.1874.8” 25 1.61 97 1.70 1.54 1.33 32.5” 15 1.65 37 2.21 1.17

32.0” IS 1.90 3! 2.60 .3211.0+ 51Ta [Va80[; Z2/A = 33.3 31.5” IS 1.69 33 2.36 1.152l5.0~” 81 6.50(35) <254 >0.92 >0.86 >0.81 31.0” 14 1.91 28 2.74 1.26

30.0” 4 2.53 19 3.84 1.52I~ + 209Bi [Ba63]; Z2/A = 33.4 29.9” 13 6.09 7 9.81 3.0321.0” 9 (L43 55 1.57 1.25 (L62

4He+ 2°’PbICh 62]; Z2/A = 33.64He+207Pb[Mo691; Z2/A = 33.4 42.8” 20 1.29 85 1.25 (L92 ((.53

5(1.0” 23 1.17 120 (1.96 (L85 ((.66 42.3” 20 l.37 78 1.34 ((.98 ((.5348.0” 23 .18 112 1.00 ((.87 ((.66 39.8” 19 1.28 74 1.33 0.94 ((.4146.0” 22 1.18 1(15 1.03 (1.89 ((.66 38.1” l8 l.39 63 1.52 1.03 ((.3244()W 21 1.28 90 1.16 ((.99 ((.71 36.0” 17 1.36 59 1.55 0.9943.0” 21 1.28 85 1.18 1.00 ((.70 33.9” IS ((.67 102 ((.84 ((.49

L.C. Vaz and .LM Alexander, Reassessment offission fragment angular distributions 17

Table 1 (continued)

E’,,, EiCb l~,,,(MeV) (h) ~b Kl

1d 2d 3d (MeV) (h) ~b Kl 1d 2d 3d6L1 + 208Pb [Vi 80]; P/A= 33.8 ‘4N + mAu[Vi 631;P/A= 35.1 (contd)

944W 33 2.15(77) 130 1.43 1.32 1.20 96.7 43 4.45(28) 108 1.49 1.35 1.17

84.2” 31 2.65(82) 94 1.84 1.69 1.52 83.1’ 31 3.15(40) 77 1.88 1.68 1.3974.8” 29 2.35(88) 92 1.74 1.58 1.40

‘4N+ ‘t7Au [Ka67];P/A= 35.14He+209Bi[Ch62];P/A = 33.9 145.0 66 6.15(18) 177 1.17 1.10 1.0142.8” 20 1.10 99 1.01 0.76 0.48 110,0 52 5.35(23) 131 1.33 1.22 1.0842.6” 20 1.13 96 1.05 0.78 0.49 102.0 47 3.70(26) 154 1.08 0.98 0.8640.8” 20 1.13 89 1.09 0.79 0.44

ii 2095 , -395W 19 1.08 87 1.07 0.76 0.37 1~ I

38.7” 19 1.13 80 1.14 0.79 034 114.0 56 4.53(19) 173 1.10 1.00 0.9138.0W 18 1.10 79 1.13 0.78 0.29 95.9 51 4.20(21) 155 1.08 0.96 0.8537.4” 18 1.03 81 1.08 0.72 0.22 91.9 48 4.10(22) 144 1.13 1.00 0.8736.6” 17 1.07 75 1.14 0.75 0.22 80.4 42 3.75(26) 118 1.26 1.08 0.9236.0w 17 1.03 75 1.12 0.71 69.5 33 2.65(32) 109 1.22 1.00 0.8035.1” 17 1.14 64 1.27 0.78 62.7’ 27 2.70(40) 69 1.77 1.41 1.0634.2W 16 1.12 62 1.29 0.76 iH+222Th [Ba631; P/A = 35.433.1” 15 1.05 58 1.31 0.71 21.0” 8 0.39 42 2.73 2.28 1.6632.4” 15 0.95 63 1.17 0.6031.3” 14 0.93 59 1.20 0.54 160+i97Au [Vi63]; P/A ~35530.7” 14 0.72 72 0.94 0.36 166.0 74 6.95(10) 199 1.09 1.01 0.93

143,0 66 6.50(12) 168 1.17 1.07 0.974He+209Bi[Ka66, Cl 75]; P/A = 137.0 65 5.85(12) 185 1.02 0.93 0.83140.0” 39 1.95(61) <201 >1.13 >1.06 >1.00 117.0 53 5.45(14) 133 1.28 1.14 0.99115.0” 35 1.85(68) <173 >1.17 >1.09 >1.01 84.3’ 17 2.35(44) 34 4.01 3.13 2.60110.0” 35 1.83(69) <167 > 1.18 >1.09 >1.00100.0” 33 1.78(73) <157 >1.18 >1.09 >0.99 i2C+2OcBi1Ka67l;P/A= 35.890.0” 31 1,75(77) <144 >1.21 >1.10 >0.99 81.0 35 2.85(12) 111 1.17 0.95 0.7380.0” 29 1.67(82) <132 >1.22 >1.09 >0.9670.0” 28 1.51(87) <130 >1.13 >0.99 >0.85 i2C+ 209Bi [Vi 63]; P/A= 35.860.0” 26 1.40(94) <121 >1.08 >0.92 >0.76 125.0 58 4.70(7) 179 1.05 0.95 0.8550.0” 23 1.03 135 0.86 0.69 0.53 112.0 55 4.40(8) 172 1.00 0.89 0.7840.0” 19 1.01 95 1.01 0.74 0.42 105.0 51 4.35(8) 155 1.05 0.92 0.80

97.4 47 4.15(9) 136 1.13 0.97 0.82°B+~Au[Vi63]; P/A = 33.9 89.7 42 3.85(10) 117 1.22 1.03 0,84114.0 55 5.52(43) 141 1.35 1.23 1.12 81.0 35 3.13(12) 101 1.30 1.06 0.82100.0 53 5.05(46) 139 1.27 1.14 1.0185.6 45 5.00(53) 105 1.54 1.35 1.17 4He+232Th[Va61,Re66]; P/A= 35.980.1 42 4.50(57) 100 1.55 1.35 1.15 42.8 20 1.15(21) 95 1.41 1.20 1.0069.5 34 3.00(70) 162 1.40 1.19 0.98 29.9 13 0,73(33) 61 1.71 1.29 0.86

i2C+ iO7Au [Vi63]; P/A= 34.6 ‘H2+

220U [Ba63l;P/A= 36.0

125.0 58 6.15(30) 137 1.38 1.27 1.16 21.0 8 0.48(42) 35 3.45 2.91 2.20118.0 55 6.25(31) 123 1.47 1.34 1.22112.0 55 5.50(31) 141 1.24 1.13 1.02 ‘H

2+~U[Vi65]; P/A= 36.0108.0 53 4.85(32) 149 1.15 1.04 0.93 24.O~ 10 0.33(34) 76 1.58 1.33 1.0181.0 36 4.25(47) 80 1.75 1.50 1.2569.9’ 26 2.72(67) 62 2.03 1.69 1.33 160+

208Pb [Va 80]; Z2IA = 36.2215.0~” 92 4.67(2) <453 >0.59 >0.54 >0.51

‘4N+ ‘97AU [Vi63]; P/A = 35.1145.0 66 6.15(18) 177 1.17 1.10 1.01 22Ne+iO7AU [Ka67l;P/A = 36.2127.0 59 5.90(21) 150 1.27 1.19 1.07 190.0’ 88 10.00(2) 196 1.19 1.12 1.02123.0 60 5.65(20) 159 1.17 1.08 0.97 177.0’ 83 8.05(2) 214 1.04 0.98 0.88107.0 51 4.65(24) 140 1.22 1.12 0.98 154.0 74 6.65(2) 206 0.99 0.92 0.81

18 L.U Vaz andf.M. Alexander,Reassessment offission fragment angular distributions

Table I (continued)

E,,6 ‘cr11 Ei,b ~

(MeV) (6) pb K~ 1d 2d 3d (MeV) (6) ~b Kl 1d 2d 3d

4He + 230Th [Re66]; P/A = 36.2 160 + 2mBi [Ka67]; P/A = 36.842,8’ 20 1.15(9) 95 1.39 1.17 (1.96 134.0 63 4.60 219 ((.83 ((.74 ((.65

112.0 48 3.65 162 0.96 (1.82 (1,68‘H

1+220U [Vi 661; P/A = 36.2 94.0 29 2.47 91 1.44 1,15 ((.85

22,0’,’ 6 0.15 69 1.61 1.27 ((.914He+2SiPa [Re661; P/A = 36.8

+ 209B1 [Vi 63]; P/A = 36.3 42.8’ 21) ((.82 131 1.02 0.86 ((.70145.0 66 4.60 238 0.87 0.81 ((.73127.0 61 4.25 225 0.84 0.76 0.67 4He+230U[Re66]; Z2/A =

123.0 59 4.09 218 0.85 0.77 0.68428q 20 ((.79 136 1.02 ((.87 0.72

116.0 55 4.07 191 0.92 0.83 (L72110.0 52 3.80 179 0.94 0.84 1)72 ‘H2 +

237Np [Ba63, Le 651; P/A = 37.0106.0 49 3.65 168 0.98 0.87 0.74 21.0 9 ((.36 64 1.93 1.63 1.2299.5 44 2.90 172 0.91 0.80 ((.67 14.5 5 0.26 26 4.13 3.28 2.0483.1’ 28 2.35 87 1.54 1.30 ((.97 13.5 4 ((.33 14 7.48 5.87 3.46

‘H2 + 230 J [Ba63, Le 65[; P/A = 36.3 ‘H2 + ~

5U [Ba63]; P/A = 37.321.0 9 ((.47 50 2.38 1.99 1.46 21.0 9 ((.37 63 1.92 1.56 1.2014.5 5 0.35 20 5.04 3.93 2.2713.5 4 0.33 15 6.71 5.15 2.76 ‘He + 233U [Va61, Le 65, Re66]; Z2/A = 37.3

42.8’ 20 0.73 145 (1.93 0.80 0.64‘He + 238~[Ka 66, Cl 75]; P/A = 36.5 40.0’ 19 ((.83 III 0.93 0.99 ((.76140.0’ 40 1.24 <331 >0.84 >0.80 >0.76 38,O~ 18 ((.82 100 1.25 1.06 (1,79l15.0’ 36 1,12 <301 >0.83 >0.78 >0.74 36.0’ 17 0.73 99 1.22 1.01 0.73110.00 36 1.18 <275 >0.88 >0.83 >0.79 340’ 15 ((.70 88 1.32 1.08 ((.741(10.0’ 34 1.14 <254 >0.91 >0.85 >0.80 3

2,3~ 14 11.63 85 1.32 1.07 0.7090.0’ 32 1.18 <2)7 >1.00 >0.93 >0.87 29.1)’ II ((.55 64 1.61 1.25 0.6985.0’ 31 1.19 <202 >1.04 >0.97 >0.90 270’ 9 0.49 50 1.96 1.47 0.6680.0’ 30 1.10 <209 >0.97 >11.90 >0.83 25.0’ 7 0.35 39 2.36 1.69 ((.4270.0 28 1.18 172 1.09 1.00 ((.91 23,0q 3 0.33 7 12.62 8.5460.0 26 1.18 143 1.20 1.08 ((.97

‘H2+239Pu [Ba63]; P/A = 37.450.0 23 1.05 136 1.13 0.99 ((.87

21.0 9 0.40 58 2.17 1.78 1.3740.0 19 0.98 97 1.37 1.16 0.9630.0 13 0.70 64) 1.82 1.41 1.00

‘He + ~‘7Np[Re66]; Z2/A = 37.423.3” 5 ((.34 18 4.89 3.27 1.13 42.8’ 20 ((.67 158 0.88 ((.75 ((.62

‘He+230U [Vi65[; Z2/A= 36.5 ‘He+242Pu[Re661; Z2/A = 37.548,0’ 23 1.1(1 120 1.25 1.09 ((.94

428q 20 ((.65 162 ((.89 ((.76 0.64

‘H2+230U [Ba64];P/A = 36.6 ‘He+2~Pu[Re66[; Z2/A= 37.8

21.0 9 0.45 52 2.27 1.88 1.35 42.8’ 20 0.65 162 0.88 ((.76 ((.64

H2+

233U [Ba63]; P/A = 36.8 41-Ie + 243Am[Re66];P/A = 38.121.0 8 0.42 48 2.50 2.02 1.51 42811 20 0.58 181 0.80 ((.68 ((.57

160 + 209Bi [Vi 63]; P/A = 36.8 4He + 241Am [Re66[; P/A = 38.4

166(1 74 5.49 249 0.87 0.80 0.73 42,8’ 20 0.6(1 175 ((.81 ((.69 ((.56143.0 65 5.45 199 0.96 0.87 0.78137.0 65 4.33 245 0.75 0,67 0.60 4He+2”Cm [Re66]; P/A = 38.7119.0 53 3.67 198 0.83 ((.73 0.62 42.8’ 20 0.55 189 ((.77 0.65 ((.54117.0 52 4.10 168 0.96 0.84 0.71102.0 39 2.85 137 1.04 0.87 0.69 ‘He+249Cf [Re66[; Z2/A= 38.784.3’ 7 1.70 9 12.73 9.65 6.03 42.8’ 20 0.43 237 0.63 1)54 0.46

LU. Vaz and f.M. Alexander, Reassessment of fission fragment angular distributions 19

Table 1 (continued)

EI,b1~,il Ei,b

(MeV) (6) pb K~1d 2’ 3d (MeV) (6) pb Kl 1d 2d 3d

i6ç~~23Ei,j [Va80];Z2/A = 39.4 325+232Th[Ba81];P/A = 42.’215.0” 103 2.94 <911 >0.34 >0.33 >0.31 218.0’ 93 3.93 <551 >0.42 >0.38 >0.34

160 + ~U [Ka671; P/A= 39,4 32S + ~U [Ba81]; P/A = 43.2

114.0” 46 2.80 <192 >1.08 >0.99 >0.91 218.001 99 3.11 <794 >0.29 >0.26 >0.23

32~~~Au [Ba81];P/A = 39.4 325+2~Cm[Ba81]; P/A = 44.8

218.01 92 10.77 198 0.94 0.84 0.73 218.0” 86 3.11 <603 >0.38 >0.34 >0.30

‘Values of la,,, calculated from O~ER+ 0~t,= IT”X(lcrii +1)2 when possible. Otherwise they were estimated as described in section 4 [Va 81, Va 81a].bFrnm measured fission anisotropy as described in the text concerning fig. 13. Parenthesized numbers give values of [(Imin/ImaxX100)]that were

used (see section 4). Where no parenthesized number is given we have used Imj,/Im,.x = 1 for Z2/A <36, and Imjfl/Im,~,= 0 for Z2/A >36.Cl is the moment of interia for a rigid sphere of radius constant 1.2249 fm [Co74].dTemperature obtained from the following equations with the assumption of first (1), second (2), and third chance (3) fission, respectively:

E1= (A/8)T

2 andE~= E — Emt — Bf(Imin)’’ ~ S,—2xTwhereS,is theneutron separationenergy andx = 0, 1 or 2 for first, secondand third chancefission. Values of E,,

1 andB1 were calculated via RLDM [Co74].‘Valueof P/A for the initial composite nucleus.‘Large uncertainty in I due to the entrance channel barrier (not included in figures).“Possible systematic errors in ~ as lEE> l~m.iLow value for I,,, makes classical treatment questionable (not included in figures).‘Large values of ‘m imply an important effect of the rotational motion on the fission barrier.‘Special constraint has been imposed to select out or correct results to first chance fission. (See original reference.)“Possibility of sequential fission after direct reactions (limiting values indicated).‘Probability of extensive sequential fission after direct reactions (limiting values indicated).

the transition-statetheory:What is theevidenceconcerningthedependenceof 1(02 on entrancechannel?If the fission anisotropiesdependon target-projectilecombinationthen the fission fragmentdirectionsmust result from individual dynamic trajectoriesrather than from a statisticalequilibrium. An overallglance at fig. 17 indicates a reasonablyconsistent correlation between the results from differentreactions.A closerlook indicatesseveralsignificantdeviationsfrom this overall consistency.

For excitation energieslessthan ‘=35 MeV, the deviationsmay result from structuraldifferencesthatare not adequatelyseparatedby the rough groupsin Z

2IA. Oneobviouspossibilityhereis the role ofthe high spin for 209Bi (9/2). We havemadeno correctionfor the coupling of targetspin with that fromthe entrancechannel orbital motion. For 40 MeV 4He this effect hasbeen estimated(for randomcoupling) to be ‘=10% or about one third of the differencebetween2°9Biand 2°6Pb.If the fissionreactionsselectivelyemphasizethe higher spin couples,thenthe effect could be muchlarger.A secondpossibility is the likelihood that shell andpairing effects areimportant;they alsomaywell vary with Zand/or A of the compoundnucleus. If the nuclearexcitation is quite small indeedthen the fissionprocessmust be describedby discretesums over the relevant individual states,andin this casetheequationswe useareinadequate.

Fig. 17(b and c) indicates anotherclass of differencesthat dependon the entrancechannel.Forexcitationsgreaterthan 35MeV thereseemsto be a consistentdifferencein 1(02 values from differentreactions.Someof thesedifferencesmaysimply be experimentalproblemsor contaminationby fissionafter direct reactions.For energiesgreater than 70 MeV for 4He+ 2°9Bi,Bimbot and LeBeyec have

20 LU. Vaz and f.M. Alexander,Reassessmentof fissionfragmentangular distribution.s

suggestedextensiveoccurrenceof fission after direct reactions[Bi 711. However the presenceof suchdirect reactionsis not evidentin the momentumtransferstudyof Meyeret al. [Me 791. Although theseargumentsarenot compellingwe tend to give moreweight in fig. 17(b) to the resultsfrom the heavierprojectiles.There is no obviousreasonto prefera given set of resultsin fig. 17(c). It would appearthatif one needsvalues of K02 for 33.5< Z2/A <35.4 to better than±20%,then moremeasurementsmustbe made.

Our conclusionfrom the overall patternof fig. 17 is that the valuesof K~can probably beconsideredto be essentiallya smoothfunctionof excitationenergyandZ2/A.The uncertaintiesin this function aremore than one would desire in some regions, and there may be some differencesdue to specificstructuraleffects and reaction mechanisms.Nevertheless,the pattern as it standsnow, does give areasonablebasefor usein sequentialfission studies,with the usualcry that additionalexperimentswithgood precisionareclearlydesirable.

The major problem with this kind of correlation is that it does not exposethe role of angularmomentumin a direct way. In the next sectionwe examinetheseeffectsin the context of the rotatingliquid drop model.

6. Effective momentsof inertia (‘5eø) compared to calculationsfor saddle-point configurations

Fig. 10, takenfrom IBa 631, makesthe very interestingpoint that the empirical values of “eO are astrong function of Z2/A. This point was strengthenedby additional measurementsin [Re661 and

I I T 1 I T

3.0 - ~AuPb,Th,UCm+SoTo,Pb,U +0•Bi+0

o oBi+B,C,N,Neo £Ho,Au+B,C,N,0,Ne

25 + 020�Pb+He•X÷He

++ XTo,Pt,Au,Pb+Li+ +X+H

.5 U + Ar2.0 icr,t’21 ~‘ a •

‘~0~~~

\x ~ S

j”- cot-65 ~x a 0 00) 15

S- ~ 0

~II .X .~S5j~‘~ •. V

10

0 - a 0

“— ~ 0

0.5 g~. —~,... ~ 0

(a)1(b) ~(c)1~Id)4(e)4,,, Ct) ~ ~ 0

N 0

I 1. I I I I I I I I

313233343536373839404142434445Z

2/A 0~

Fig. 18. Values of ~ (l,~ is for a sphere; 5~= ,‘ — lii) for first-chance fission versusZ2/A. Lower limits are indicated for those reactionsthat may well include significant amounts of fission after transfer reactions. The lettered bands refer to separateframes in figs. 17 and 19. Dashed

lines were calculated for RLDMsaddle.point shapes. Some data points are off the scaleof this figure but table 1 gives thecomplete set,

L. U. Vaz and f.M. Alexander, Reassessment of fission fragment angular distributions 21

[Ka 67]. As emphasizedby theseworkers,this trendarguesfor the identificationof saddle-pointshapeswith the transition-stateconfigurations.In fig. 18 we give the samekind of display for the collection ofresults from table I. Dashedlines are drawn from RLDM calculationsfor the mean squarespin

1/r2\ — j1\12l.\’ I — k2)’crii

Thereis clearlyan overall trendfor decreasingvaluesof ~9sP/~eff with increasingZ2/A.However, thescatteris very large, andit is desirableto try to identify explicitly the role of angularmomentum.Wehave groupedthe data in bins of Z2/A as indicatedby letteredbandsin fig. 18 that correspondtoseparateframesin figs. 17 and 19. For thosereactionsthat may well be affectedby fission after partialmomentumtransfer,we haveindicatedK02 as an upperlimit in fig. 17 and~sp/~eff as a lower limit in fig.19. (Presumablytransferreactionswould reducethe anisotropycomparedto purefusion fission.)

The attemptof fig. 19 is to bring out the role of angularmomentum.As mentionedearlier (in thecontext of fig. 6), increasingspin is generallyaccompaniedby increasingenergyand thereforetheireffectscannotbe separatedin a totally unambiguousway. To obtain ~ from K02 we haveestimatedthetemperaturefor first, secondand third chancefission andhavegiven the resultsin table 1. Fromstudies

- FIRST CHANCE FISSION z~ _________________________________________

(a) IER>326 308-334 I I I I I I I

2 Icril’21 ~ -01 I I I I ~ ~ -3(b) IER~25 335~34Q• I

~He+x 33

3 ~ I I ~ ~ts’ 15 crit~~~ I I I ‘ I -

~ I I ~ ~¾~:~~ -

~ (d) 355 -36.2~._. 3+ 11:~.scIi I I I I I

LI~ ‘~s “ ‘1t’50£ riI ~ ‘ ‘0o~p—~.~ a~ .-..-~, ‘

2i~ V

I I t ‘3 Q.’O---~_ 0 -~ e 363-368 ~ ~‘---. 0

2’~. . -.--‘.-..~ °B,’

2C,’4N

1

160,32S+XI ‘~ ~ 11 ___________________________________________

0.5 .-‘-.-.-. . C I I I I I II Icrit’65 =

2 ‘.l• 370-448

o V~, . A0 V 1 £A.~•,

0.5’ -=ô~1~t& .

2 =1’~ ~ ---------- IZ /A’39~5 \, ‘

4N, 160, 22Ne,32S+x

0.1 I ‘ I 0 I I I I I I I IO 20 40 60 80 tOO 120 140 30 31 32 33 34 35 36 37 38 39 40

, ,~, z2,A“crit ‘‘‘

Fig. 19. ~ versus !CIiI for various values of Z2/A. Seefig. 18 for Fig. 20. ~ versus Z2/A for various values of ~ for variousmeaning of the symbols. Dot-dashed lines were calculated [Co74] for values of P/A. See fig. 18 for meaning of the symbols.Dot-dashedRLDMsaddle point shapes. lines were calculated[Co74] for RLDMsaddle point shapes.

22 LU. Vaz and f.M. Alexander,Reassessmentoffissionfragmentangulardistributions

of light particles in coincidencewith fission, thereis evidencefor significant evaporativecooling beforescission[Ch70, Fr 75, Lo 80, Ga81, Ga82, Al 82, Ki 82, Ri 82]. But it is very difficult to tell if thiscooling occursbefore or after arrival at thetransitionstate.For initial excitationsgreaterthan ~5()MeVthe temperaturechangecausedby severalprefissionneutronsis not very severe,but for lower energiesthereare substantialuncertainties.For simplicity in fig. 19 we show resultsonly for the assumptionoffirst chancefission; uncertaintiesdueto prefissioncooling will be discussedalongwith fig. 20 later.

The data points for ~sp/~en in eachgroup of fig. 19 are essentiallyindependentof 1crii exceptforlcrO ~ 35. For thosestudiesof fission at low spin inducedby H and He, there is a distinct increaseof~sp/~eff with decreasing‘crii~ This is the effect found by Simmonset al. ISi 601, extensivelyconfirmedbyothers,and discussedin connectionwith fig. 8. Proposedexplanationsinvolve pairing energyand/orshell effects possibly related to the double-humpedfission barrier jGr 62, Gr 63, Br 63, Ke 64, Br 65,Si66, Hu68,Mo69, Ra70, Ip7l, Va73,Va73a,Br74, 1g77 andNote addedin proof].

The dot-dashedcurvesfrom RLDM are very flat for 0 < Icrji ~ 65~at high spin they decreaseas thefission barrier is erodedby the centrifugalforces.The very interestingexperimentalresult (emphasizedby Back et a!. [Ba81] for S + Au, Th, U, Cm) is that observedfission anisotropies(or the respectivevaluesof ~sp/~eff) do not decreaseaspredictedby RLDM at thesevery large spins.This effect hasalsobeenobservedindependentlyin measurementsfor 22Ne+ Au [Ka 671, ~ + ~U IVa 80] and ~mAr~238U [Le 81].

The moststriking featureof the wholepatternof fig. 19 is the rathersmall sensitivity of .~ to either1cr0 or Z2/A (excludingthe zonefor ‘~rii~ 35). This point is mademore clear in fig. 20 wherewe plot

versusZ2/A for severalvaluesof ‘cr11. If we focuson the resultsfor first chancefission, then forZ2/A = 32 we have ~ 1.4. and for Z2/A = 37 we have .1..~/.~~~O.9—withscarcelyany changefrom ‘crit = 21 to ‘cr11 = 65. This trendis somewhatlessdependenton Z2IA than the RLDM calculations(dot-dashedlines). The valuesof ~ are also~().9evenfor the larger ‘cr11 valueswhereexpectationsfrom RLDM are for a decrease(fig. 19(d) andf)).

Onemay summarizefigs. 19 and20 as follows: (I) The RLDM calculationsfor saddleshapesgive agood representationof the ~ valuesfor 25 ~ ‘cr11 ~ 65. (2) Sharpincreasesin ~ for ‘cr0 ~ 25 seemto call for shell and/orpairing effects. (3) RLDM seemsto overestimate~ for Z2/A ~ 33.4 (butthis may be partly due to an overestimateof (J2) in this analysis).(4) For ‘cr11 ~ 80 the RLDM saddle-point shapesdo not account for the strongobservedanisotropies(or .i~t1/~~.jt0.9 in fig. l9(d and f)).The comparisonof the data to the lines indicates the generaloverall utility of the RLDM model, butsomeimprovementsare clearly desirable.

7. Reassessmentof the theory for high spin nuclei

Let us reviewbriefly the marriageof RLDM with the transition-statetheory in fig. 1 and with theapproximationsdescribedin section2. It seemsto us that thereare severalplaceswherethis menageatrois can be expectedto be unsatisfactory.In fig. 21 we reproducethe very elegantrepresentationof ~

for equilibrium and saddle-point nuclei from Cohen, Plasil and Swiatecki [Co74]. They havecharacterizedeachmodel nucleusby a CoulombenergyE~,a surfaceenergyE~anda rotationalenergyEroi. Then they solved the equationsfor the resulting stable equilibrium shapesand unstablesaddle-point shapes.The results for ~ are given in terms of the dimensionlessquantitiesx (Coulomb

L. C. Vaz and f.M Alexander, Reassessment offission fragment angular distributions 23

I I I I I I I

4.0

3.512

300.64

4825 040 032 024

9/-9sp20 6

012

1.5008

- 06~ oO.O

1.0 __________________

4002 DI

32 .24 .16 0.12

O~5 -72 084 0. 046 040

I I I I I0.0 01 0.2 0.3 0.4 0.5 06 07 08 0.9 1.0

a

Fig. 21. The principal moments of inertia (in units of ..~, the moment of inertia of a rigid sphere)plotted against x (after [Co741). The labels refer totherotationalparameter y. The moments of inertia about the axis of rotation are always greater than 1.

parameter)and y (rotationalparameter):

E~Z2____

(12)~O ~2l~crot I

~ ~:~i_ 1.7812 (13)

wherethe neutronexcessI is expressedas

I = (N — Z)/A.

No constraintwas imposedon the orientationof the symmetry axis with respectto the spin axis; it wasallowed to seek its energeticallypreferredorientation. Naturally the liquid drops prefer to elongateawayfrom their spin axisor alongthe direction K = 0 for prolateand K = I for oblateshapes.

In the traditional view of the role of theseshapesin transition-statetheory, one considersa verysimplified two steppicture.First the nuclei seekout their saddle-pointshapes,andErot (from eq. (2) infig. 1) becomesoperative.Then the orientationprobability is reflectedby the magnitudeof Eroi in the

24 LU. Vaz and f.M. Alexander.Reassessmentof fissionfragmentangulardistributions

level density (eq. (6) in fig. 1). As one invokes the valuesof ‘jeff from RLDM one is assumingthat thesaddle-pointshapesdo not care aboutthe orientationof thesymmetryaxiswith respectto thespin axis.This would seemto be a reasonableapproximationif the rotational energyplays a very small role withrespectto the Coulomb energy.Such a situationmay well apply for the n, H and He inducedfissionreactions(x 0.7 andy 0.003) that were first investigatedtwenty to thirty yearsago.It doesnot seemreasonableto expectthis to be a good approximationfor high spin nuclei.

In fig. 22 the variousmeasurementsthat havebeenmadeare displayedin termsof propertiesof therespectivecompositenuclei (via liquid-drop variablesx and y). This pattern (also used in [Ba8l~) isvery useful for assessingqualitatively the relative roles expectedfor Coulomb and centrifugal forces.The early studies(n, H and He induced fission) were for low-spin compoundnuclei or y ~ 0.003~themorerecentstudiesfor very high spin systemsevengo well abovethe line for zerofission barrier~otherstudiesof heavy ion and4He inducedfission fill in partsof the intermediateregion. It seemsto us thatan understandingof the spin dependenceof fission angularcorrelationsrequires a new look at theshapeswe might expect for transition statenuclei as a function of the combinedroles of both theorientationK and the spin I.

There is anotherobviousconsequenceof an orientationdependencefor the transitionstateshapesfor high-spinnuclei,namelya variation in the fission barriers.If the centrifugalforcecausesa significantreductionin the fission barrier for perpendiculardistortionsthenit will surelyinhibit or raisethe barrierto distortionsalongthe axis. Thereforethe disappearanceof the fission barrier from an RLDM saddle-point calculation does not actually imply a prediction of isotropic fission. One could elaboratethetransition-statetheory by calculatinga seriesof shapefamilies, such as in fig. 21, eachwith a separate

I • F I I0.11

010

£

009(a) Ib) (C) (dl t.) (ft

b $1 Ill -‘40.08:.:: ~ V

::a “N

: O6~~O~

Fig. 22. For eachexperimentin table I we plot a pointfor eachcompositenucleusat its value of x and its maximum valueof y [x andy aregiven ineqs.(12) and (13)1. The symbolsareindicated in fig. 18 and the line wascalculatedfor fission barrierdisappearance[Co74].

L. C. Vaz and f.M Alexander, Reassessment offission fragment angulardistributions 25

orientation(or K/I). In addition onecould calculatethe barriersassociatedwith eachof theseshapes.Armed with thesemultidimensionalvalues ..~eff(I,K) and B~(I, K), eq. (4) could be reevaluatedstill inthe spirit of statisticalequilibriumbetweensaddle-pointtransitionstates.

With this ratherdecidedlymorecomplexapproachto the theory,onemight takethe attitudethat thevery conceptof the transitionstatehadbeeninvalidated.In this view only full dynamicalcalculations,of extremelygreatcomplexity, would be satisfactory.One could not then assumethat the directionalchoicesin fission were independentof the modeof formation, and sequentialfission after DIR mightfollow very different trajectoriesfrom compositenuclei madein fusion-fissionreactions.Our view isthat one shouldnot take this pessimisticattitude until the evidencefor it is absolutelycompelling. Infact thereis substantialqualitativeevidenceto the contrary.

Out-of-planecorrelationsfor sequentialfission havebeen measuredby the Specht group for anumberof reactionsincluding U + Zr, U, Cm [Sp78, Ha79, Sp811. Thesereactionsproducevery highZ fissile nucleipresumablywith very substantialspins.Someof their resultsareshownin fig. 23. In thelower frameof fig. 23 theyshow inferredeffectivevaluesof K

0

2 for Z < 105 that arevery similar to thosefor Z> 105. This trend is consistent with that shown in fig. 19(d and f), namely the apparentindependenceof ~9~spI~eron l~ for highvaluesof l~.To make this comparison more quantitative, onewould haveto assertcompleteconfidencein the sticking condition for spin transferin DIR. It seemsprematureto go so far at this time, but, in any case,the comparisonprovidesqualitativeevidencein

S

4

‘5:. ,

40 ~-~‘~It/cking

20

0.

~(0(theory) ~_...—

t__.60 <I> (sticking

~‘40 ~ I‘~ ~. ~

90 100 110charge of fissioning nucLeus Z

Fig. 23. From top to bottom: FWHMof the out-of-plane fission fragment angular distributions, average oriented spins (Ii) compared to thesticking-model dependence and deduced effective values for K0 compared to the liquid-drop model prediction: 0 7.5 MeV/u ~U on

0)Zr, A7.5 MeV/u on ~U, • 7.5 MeV/u ~Cm, total-kinetic-energy loss >150MeV[afterSp 81].

2b LU. Vaz andf.M. Alexander.Reassessmentoffission fragmentangulardistributions

favor of the basicnotion of the independencehypothesisof the equilibrium theory evenfor transuranicnuclei of very high spin.

Let usreturn to fig. 20 andtherole of prefissionneutronemission.The pointsshownwerecalculatedwith the assumptionof first-chancefission; the numbers2 and3 indicateposition shiftsfor second-andthird-chancefission respectively.The effects are substantialespeciallyfor the relatively low excitationenergiescharacteristicof ‘cr11 ~ 30. As mentionedbefore,thereis considerableevidencethat abouthalfthe neutronemissionoccursprior to fission— both for 4He and heavy-ioninducedfission [Fr 75, Ga81.Ga821. If most of this emissionoccursbefore arrival at the transition statethen there are substantialdifferencesbetweenthe data andthe dot-dashedlines from RLDM. Thesedifferencesare not in thedirection that could be accountedfor by the orientationdependenceof ‘9e~(LK) or B

1(I, K). There isno experimentthat hasso far been able to distinguishthe detailedtime scalesfor this cooling process.Thus we cannotmakea completelysatisfactorytest of the saddle-pointhypothesis.

For most purposessuch a test is not really the most essentialmechanisticquestion; the morefundamentalconcernis the validity of the transition stateconceptitself. Wheeler [Wh 63] and Ericson[Er 60] emphasizedlong ago that increasingexcitation and shorterdecay lifetimes of the compoundnucleusweakenthetheoreticalexpectationfor K freezingpreciselyat thesaddlepoint. In this spirit theidentification of either saddle-point or scission shapeswith the transition statesbecomessomewhatirrelevant.Empiricalcorrelationsof the typeshown in figs. 17, 19 and20 (and fig. 23 at some later time)can provide strong tests of the independencehypothesis;that is the real essentialidea of the model.Furthermoreit is theseempiricalparameters(K~or ..~c0) that we mustuse in applicationsof the angularcorrelationtechniqueto study otherreactionmechanisms.More sophisticatedcalculationsof ~cff and B~

as a function of orientation would surely be interesting,especiallyas a guide toward a qualitativeunderstandingof the anisotropiesat high spin. But to us the most fundamentalquestioninvolves thetest for independenceof K~on entrancechannel(save for specific nuclearstructureeffects). Such testswould of necessitylead to many morefootprintson fig. 22 as well as to a reexaminationof the raggedfeaturesof figs. 17. 19 and20.

8. Summary

For thirty yearstransition-statetheory has provided the basicframework for understandingangulardistributionsin nuclearfission.The majorassumptionis that the anglesof separationare determinedbya statistical equilibrium at the transition-stateconfigurations. Since 1952 a very large number ofmeasurementshas been made to test the theory and ito determinepropertiesof transition-statecomplexes.

In recentyears it hasbeen clearly demonstratedthat the fission barrier decreaseswith increasingangularmomentum.This leadsto a rather strong dependenceof fission probability on spin that hasgenerallynot beenincluded in analysesof fission angulardistributions. In short, the spin window forfission hasbeenessentiallyignored.We haveusedmeasuredcrosssectionsfor fission and evaporationresiduesto develop an empirical method to define this spin window. Armed in this way we havereanalyzedthe ratherlargedataset on fission anisotropies.

The resultshavebeenpresentedin tablesandgraphsof K~ versusE5, ~efi versus lcrii and ~cff versus

Z2/A. The overall patterngives a strong voteof confidencefor the basicframework of transitionstatetheory.Importantproblemsdo remain,however,as well as gapsin theexperimentalinformation.These

L.C. Vaz and f.M. Alexander. Reassessment of fission fragment angular distributions 27

difficulties shouldbe attackedby morefusion-fissionmeasurementsif one is to utilize sequentialfissionstudiesas a preciseprobeof spin transferin DIR.

Recent studiesof compositenuclei of both high spin and large Z2/A show much greater fission

anisotropiesthan expectedfrom the simplestconsiderationof RLDM saddle-pointshapes.We suggestthat the approximationscommonly made in transition-state theory for high-spin nuclei may beresponsiblefor this discrepancyfor compositenuclei of high spin. When rotational energiesare largeone must consider perturbationsof the transition-statecomplexesdue to the orientationsof thesymmetry axis as well as the gross angularmomentumitself. Such considerationshave not yet beenreported.

Acknowledgements

We wish to acknowledgevery stimulatingdiscussionswith N. Ajitanand,JohnCole, K. Dietrich, E.Duck andR. Vandenbosch.

Noteadded in proof

Since the submissionof this paperfor publication, there have beenthree new studiesof fissionangulardistributionsfrom compositenuclei of very high spin (H. Rossner,D. Hilscher, E. Holub, G.Ingold, V. Jahnke,H. Orf, J.R. Huizenga,J.R. Birkelund,W.U. SchroderandW.W. Wilcke, Phys. Rev.C (in press 1983), 220 MeV 20Ne+ ‘65Ho, ‘~Au,2°9Bi; B.B. Back, R.R. Betts, K. Cassidy,B.G. Glagola, i.E. Gindler, L.E. Glendenin and B.D. Wilkins, Phys. Rev. Lett. 50 (1983)818, 160+ 238U and 32~+ 208Pb; and K.T. Lesko, S. Gil, A. Lazzarini, V. Metag, A.G. SeamsterandR. Vandenbosch,Phys. Rev. C. (in press 1983), “°Ar+238U). We havenot beenable to include thenew results in the table, but haveplotted the points from thosestudiesin figs. 18—20. In a separatepaper(submittedto Z. Phys.A) we havetried to put this new informationwithin the perspectiveof thisreview,andthereforewill addonly a few wordshere.

Our view is that the rigid rotor theory,as describedin section2 and criticized in section7, is reallynot applicableto thesevery high-spinsystems.Therefore,for this domainwe prefer to give rather littleweight to RLDM andinsteadto look for regularityin thevariationsof empiricalvaluesof K02 versusE*

and .~sp/~1eff versus ‘crjt• In this spirit we seefrom figs. 19 and20 that ~sp/~eff is a very slowly varyingfunction of both Z2/A and ‘CrlF (for ‘CrlF ~ 30). There is certainly a lot of scatter,especiallyin fig. 19(f).but our feeling is that most of this arisesfrom sequentialfission after transferandincompletefusionreactions.The techniqueusedby Leskoet al. for 340MeV 40Ar + ~3~Uwas successfulin rejecting suchreactions,but their measuredanisotropy W (160°)/W(90°),is so large that only a lower limit could beset for the valueof p (and hence~sp/~9~eff). This kind of studyis requiredif we are to get a true test ofthe entrancechannel dependenceof fission anisotropies.Only from such tests will we be able toestablish the domain of applicability of an equilibrium model. And only in this domain can weconfidently use sequentialfission as a probeof spin transferin deeply inelasticreactions.

An interestinganalysisof fission anisotropiesfor low-energyfission of low Z nucleihasrecentlybeenpublishedby Ignatyuketal. (A.V. Ignatyuk,K.K. IstekovandG.N. Smirenkin,Soy.J. Nucl. Phys.36(1982)32). They give a detaileddiscussionof the effectsof pairing and collectiveenhancementsof the leveldensities.

28 LU. Vaz andf.M. Alexander,Reassessmentof fissionfragmentangulardistributions

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