Microscopic description of cluster radioactivity in actinide nuclei

17 Pages·2011·2.73 MB·English

PHYSICALREVIEWC84,044608(2011)
Microscopicdescriptionofclusterradioactivityinactinidenuclei
M.Warda*
KatedraFizykiTeoretycznej,UniwersytetMariiCurie–Skłodowskiej,ul.Radziszewskiego10,PL-20-031Lublin,Poland
L.M.Robledo†
DepartamentodeF´ısicaTeo´rica(Mo´dulo15),UniversidadAuto´nomadeMadrid,E-28049Madrid,Spain
(Received5July2011;revisedmanuscriptreceived10August2011;published13October2011)
Cluster radioactivity is the emission of a fragment heavier than an α particle and lighter than mass 50.
The range of clusters observed in experiments goes from 14C to 32Si while the heavy mass residue is always
a nucleus in the neighborhood of the doubly-magic 208Pb nucleus. Cluster radioactivity is described in this
paper as very asymmetric nuclear fission. A new fission valley leading to a decay with large fragment mass
asymmetrymatchingtheclusterradioactivityproductsisfound.Themassoctupolemomentisfoundtobemore
convenientthanthestandardquadrupolemomentastheparameterdrivingthesystemtofission.Themean-field
Hartree-Fock-BogoliubovtheorywiththephenomenologicalGognyinteractionhasbeenusedtocomputethe
clusteremissionpropertiesofawiderangeofeven-evenactinidenucleifrom222Rato242Cm,whereemission
oftheclustershasbeenexperimentallyobserved.Computedhalf-livesforclusteremissionarecomparedwith
experimentalresults.Thenoticeable agreementobtainedbetweenthepredictedpropertiesofclusteremission
(namely,clustermassesandemissionhalf-lives)andthemeasureddataconfirmsthevalidityoftheproposed
methodologyintheanalysisofthephenomenonofclusterradioactivity.Acontinuousfissionpaththroughthe
scissionpointhasbeendescribedusingtheneckparameterconstraint.
DOI:10.1103/PhysRevC.84.044608 PACSnumber(s): 23.70.+j,25.85.Ca,27.90.+b,21.60.Jz
I. INTRODUCTION value.Ontheotherhand,α decayenergeticsisdominatedby
thehugebindingenergy(ascomparedtoneighboringnuclei)
Theemissionofαparticlesandnuclearfissionarethetwo
oftheαparticle.
dominantspontaneousdecaymodesofheavyandsuperheavy
A common aspect of fission and α decay is that the
nuclei.Inbothcasestwonucleiareproduced.Inαdecayone
dynamical evolution from the parent nucleus to the daughter
4Henucleus(αparticle)isemittedoutoftheparentnucleusand
is not favorable energetically, although the Q value of
theremainingnucleonsformaheavymassresiduewithN −2
both reactions is positive. Therefore the quantum mechanics
neutrons and Z−2 protons. In contrast to the huge mass
mechanismoftunnelingthroughapotentialbarrierisrequired
asymmetryofαdecay,twonucleiofsimilarmassarecreated
to explain both types of decay. As tunneling probabilities
in nuclear fission. A large variety of isotopes are produced
dependexponentiallyonthewidthandheightofthebarrierthe
in spontaneous fission with masses covering the range from
expected half-lives can span a wide range of many orders of
A∼70 to A∼190. In many heavy nuclei the dominant
magnitude. This peculiarity makes understanding fission and
decaychannelcorrespondstoasymmetricfissionwiththemost
αdecayverychallenging.
probablemassofheavyfragmentA∼140andthemassofthe
In 1984 Rose and Jones [1] observed for the first time
lightoneintherangefromA∼100toA∼120dependingon
the emission of a 14C nucleus from a 223Ra probe. This
themassofparentnucleus.Symmetricfissionisalsopossible
discoveryrepresentedamilestoneinthedescriptionofnuclear
insomenucleiwiththemostprobabledivisionintotwosimilar
radioactivity as it bridged the gap between the α emission
fragments. Daughter nuclei lighter than A∼60 have never
radioactivity and the standard fission reaction. Since then,
been observed in any fission experiment. Therefore there is
cluster radioactivity (CR) has been found in twelve even-
a clear distinction between α emission and fission regarding
even isotopes [2–21] and seven odd-even isotopes (see, e.g.,
the mass of the lighter products as it shows a gap of light
referencesinRefs.[22,23])intheactinideregion.Theyrange
nuclei with A∼10–50 that cannot be produced in either of
from221Frupto242Cm.Theemissionof14C,20O,23F,24–26Ne,
the two mentioned decay channels of any heavy nuclei. The
28–30Mg,and32,34Sihasbeenobserved.Thecommonfactorof
source of the observed differences can be easily explained
allclusterradioactivityeventsistheheavy-massresiduewhich
from basic nuclear properties, namely, the energetic balance
is in the neighborhood of the doubly-magic 208Pb. This fact
ofthetworeactions.Fissionisfavorableenergeticallybecause
allowsustobettercharacterizeCRas“leadradioactivity”and
of the linear decreasing behavior of the binding energy per
indicatesthestronginfluenceofshelleffectsonthenatureof
nucleonformassnumberslargerthanA∼60(theironpeak)
thisphenomenon.
that prevents fragments with mass numbers lower than that
Experiments aiming to find CR in the distant region of
the neutron-deficient Ba isotopes have been described in
Refs. [24–26]. In this case another doubly-magic nucleus,
*[email protected] namely,100Sn,canbeconsideredastheheavyresidueandthe
†[email protected] carbon isotopes around 12C are expected to be emitted. The
0556-2813/2011/84(4)/044608(17) 044608-1 ©2011AmericanPhysicalSociety
M.WARDAANDL.M.ROBLEDO PHYSICALREVIEWC84,044608(2011)
experimentsdidnotprovideevidenceforCRinthisregionand approachafissionbarrierwithaspecificmassdivisionmustbe
quantitatively they only gave lower limits for the branching determined.Thelocallymaximalbarriertransitionprobability
ratiosfor12Cemission. for the specific fragments with large mass asymmetry points
Cluster radioactivity is an exotic process. The partial to the possibility of fission with a cluster as one of the
half-livesareverylongandvaryinawiderangefrom1011 to fragments. Usually the potential energy surface (PES) has
1026s.Branchingratiostothedominantαdecayinthesenuclei to be determined as a function of the relevant deformation
are very small and fall between 10−9 and 10−16. Moreover, parameters, including elongation and reflection asymmetry
spontaneousfissionisalsoacompetingdecaychannelinsome coordinates. The path in this multidimensional deformation
heavyclusteremitters[27].Thesereasonsclearlyjustifywhy surfaceleadingtofissionwithlargefragmentmassasymmetry
CRwasexperimentallydiscoveredaslateas45yearsafterthe has to be found and, finally, the fission barrier must be
firstfissionevents,whichwerereportedbackin1939[28].In specified. In this approach it is reasonable to use two-center
thelastfewdecadesandthankstoboththeinterestraisedbythe modelsinthedescriptionofthenuclearpotential[67,68].
phenomenonandtheimpressiveimprovementofexperimental WewanttoshowthatCRcanbefullydescribedmicroscop-
techniques,manyexamplesofCRhavebeenfoundinseveral icallyasaveryasymmetricfissionprocess.Weapplystandard
actinide nuclei. Various experimental methods have been methodsusedinthetheoreticaldescriptionofnuclearfission
appliedtodetecttheproductsofclusteremission[22,23,29]. which are well established in the literature [69–77]. We use
Initialobservationswerebasedontechniquesborrowedfrom themean-fieldapproximationintheHartree-Fock-Bogoliubov
αdecaystudies.A(cid:3)E-Etelescopemadeofsilicondetectors (HFB) scheme with the finite-range Gogny force [78] to
was used by Rose and Jones in the first experiment [1]. This compute the nuclear wave functions. Axial symmetry of
method was inconvenient due to the huge α radioactivity the nuclear system is assumed throughout the calculations.
background,whichcouldevendestroytheexperimentalsetup. Constraints on the quadrupole and octupole moments allow
Later,amagneticfieldwasappliedtoremovethebackground us to control simultaneously the elongation and reflection
of charged α particles. Another method used in experiments asymmetry of the system as it evolves to the scission point.
was the detection of γ rays emitted from exited clusters. The wave functions obtained in this way can be used to
Numerousclusterswereidentifiedinsolid-statenucleartrack calculatethenecessaryquantities(energies,collectivemasses,
detectors. In this technique plastic or glass layers absorb etc.) for a physical description of the process. Moreover,
the ionized cluster emitted from the radioactive probe. The an extra constraint on the number of nucleons in the neck
material of the layer cannot be sensitive to α radiation and (the neck thickness) has been used to control the density
plasticorglassmaterialswithproperionizationthresholdsare distribution around the scission point. A description of CR
the standard choices. After irradiation, the layer is etched to is possible owing to the identification of a new valley in the
enlargethetrackcreatedbytheemittedclusterandmakeitvis- PES leading to hyper-asymmetric fission. Charge and mass
ibleandwelldefinedunderthemicroscope.Theanalysisofthe numbersofthelightfragmentcreatedatthehyper-asymmetric
geometryofthetrackallowsonetoidentifytheemittedcluster. scissionpointcorrespondtowhatisobservedexperimentally
On the theoretical side, the first successful theoretical for a given nucleus. In contrast to the standard fission path
description of cluster decay was made by Sandulescu et al. wheretheleadingcoordinateisthequadrupolemoment,inthe
[30], four years before the experimental discovery of this hyper-asymmetric fission path the relevant coordinate turns
reaction. Since the pioneering work of Sandulescu, nu- outtobetheoctupolemoment.Therefore,inourdescription
merous theoretical papers devoted to this end have been of CR, all physical quantities will be given in terms of the
published [31–59]. A thorough overview of most of the octupolemoment.
theoretical(mostlysemimicroscopic)methodscanbefoundin The first results obtained in this approach have been
Refs.[23,60–63]. publishedinthepreviouspapers[31–34].Clusterradioactivity
AsCRisadecaymode“inbetween”αemissionandnuclear of selected nuclei have been discussed there with some
fission,methodsalreadyknowntobothofthemcanbeusedto approximationsofthemodel.
describeclusterradioactivity.Forinstance,theGamowmodel In this paper we want to investigate from a microscopic
ofα emissioncanbeextrapolatedtodescribetheemissionof perspectivealleven-evenactinidenucleiwhereclusterradioac-
heavier clusters. It requires the assumption that a cluster of tivityhasbeenexperimentallydetected.Therearetwelvesuch
nucleons is preformed in the interior of parent nucleus and isotopes:222,224,226Ra,228,230Th,230,232,234,236U,236,238Pu,and
then it tunnels through the barrier of nuclear and Coulomb 242Cm.Moreover,threeothernuclei(226,232Th,240Pu),where
potential.Inthiswayakindofuniversaldecaylawsimilarto experimentshaveonlyprovidedlowerlimitsforhalf-livesof
theGeiger-Nuttallformulaforα emissioncanbeformulated CR,havebeenexamined.
[23,52,57]. The main drawback of this approach is that the The structure of the paper is as follows: In Sec. II the
preformationoftheclusterinsidetheparentnucleusisapoorly theoretical model used in this investigation is described in
knownandhardtocharacterizeprocess[64–66].Nevertheless, detail. Two typical and representative examples of cluster
the half-lives predicted by this method agree very well with radioactivity corresponding to the parent nuclei 224Ra and
theexperimentaldata. 238Pu are thoroughly discussed in Sec. III to establish the
The other method treats cluster radioactivity as a very relevantphysicsdrivingtheclusteremissionprocess.Results
asymmetric fission reaction (see, e.g., Refs. [30,60,62,63]). forallfifteennucleiconsideredinthispaperarepresentedin
The formation of the cluster is a direct consequence of a Sec.IV.Weconclude inSec.Vwiththemainconsequences
specific kind of deformation of the parent nucleus. In this extractedfromourtheoreticaldescription.
044608-2
MICROSCOPICDESCRIPTIONOFCLUSTER... PHYSICALREVIEWC84,044608(2011)
II. THEORY projected calculation (see also [90] for a comparison in
light nuclei). The conclusion is that, for strongly deformed
As a first step in our theoretical description of cluster
configurations,theexactRECisroughlyafactor0.7smaller
emission we solve the mean-field HFB equation [79] with
thantheonecomputedwiththe“cranking”approximationto
theusualconstraintsontheaveragenumberofparticlesand,
theYoccozmomentofinertia.Ithasalsotobementionedthata
in the present case, with a constraint on the value of the
similarbehaviorhasbeenobservedforthedifferencesbetween
massmultipolemoments(cid:3)Qˆ (cid:4)=Q toanalyzethephysical
λ0 λ theThouless-Valatinmomentofinertiacomputedexactlyand
contentsoftheprocess.Theaxialquadrupole(Q ),octupole
2 thatinthe“cranking” approximation [93,94].Wehave taken
(Q ),andhexadecapole(Q )momentsaredefinedthroughthe
3 4 this phenomenological factor into account in our calculation
standardLegendrepolynomials
oftheREC.
Qˆ =rλP [cos(θ)]. (1) In Sec. IV we will discuss half-lives corresponding to
λ0 λ
cluster emission and compare them with experimental data.
The nonlinear HFB equation is solved using the gradient Thehalf-livesforclusteremissionarecomputed(inseconds)
method [80] and taking into account approximately second- usingthestandardWKBframework[95]
ordercurvatureeffects[69,81].TheHFBquasiparticlecreation
t =2.86 ×10−21[1+exp(2S)]. (2)
andannihilationoperatorsareexpandedinaharmonicoscilla- 1/2
tor(HO)basisandspecialattentionispaidtotheconvergence
ThequantitySenteringthisexpressionistheactionalongthe
of the results with the basis size (see Appendix A for
Q constrainedpath:
3
furtherdetails).Theinteractionusedisthefinite-rangeGogny (cid:2)
b (cid:3)
forcewiththeD1Sparametrization[73].Thisinteractionhas
S = dQ 2B(Q )[V(Q )−E ]. (3)
been proven to successfully describe the fission process in 3 3 3 0
a
heavy nuclei [69,74,76,77,82–87]. The other Gogny forces
For the collective inertia B(Q ) we have used the adiabatic
developedrecently—D1N[88]andD1M[89]—arediscussed 3
time-dependent HFB (ATDHFB) expression computed again
in Appendix B. Other details of the HFB calculations are
inthe“cranking”approximationandgivenby[94]
as follows: the two-body kinetic energy correction (2bKEC)
hasbeenincludedintheminimizationprocess.Theexchange B (Q )= M−3(Q3) (4)
CoulombcontributionisevaluatedintheSlaterapproximation. ATDHFB 3 M2 (Q )
−1 3
All calculations have been performed in the axially sym-
metricregime.Thisseemstobearationalchoiceasthesystems withthemomentsM−ngivenby
(cid:5)(cid:6) (cid:7) (cid:5)
studiedtendtobebuiltfromalargesphericalpartreproducing (cid:4) (cid:5) Q20 (cid:5)2
pTrhoepelirgtihetseorffdraogumbleyn-tmiasgoicftneunclsepihweritihcaalsimnatlhleadgdriotiuonndalsptaatret.. M−n(Q3)= μν (Eμ3+0 Eμνν)n. (5)
Therefore,theinfluenceofnonaxialeffectsisexpectedtobe
rathersmall,ifany,andmayonlyaffecttheshapeofthebarrier Inthisexpression,(Q2300)μν isthetwo-quasiparticle–zero-hole
justbeforescission,slightlyreducingitsheight. component of the octupole operator Qˆ in the quasiparticle
30
To evaluate the PES we take into account correlation representation[79]andE aretheone-quasiparticleexcitation
μ
energiesbeyondthemeanfield.Tothisendwesubtractfrom energiesobtainedastheeigenvaluesoftheHFBHamiltonian
the HFB energy the rotational energy corrections (RECs) matrix.
stemming from the restoration of rotational symmetry. This In the expression for the action V(Q )=E (Q )−
3 HFB 3
correction has a considerable influence on the energy land- REC(Q )−(cid:8) (Q ) is given by the HFB energy minus the
3 0 3
scape (and therefore on the height of fission barriers) as it is REC and the zero-point-energy (ZPE) correction (cid:8) (Q )
0 3
proportionaltothedegreeofrotationalsymmetrybreaking.A associated with the octupole motion. This ZPE correction is
fullcalculationoftheRECwouldimplytheevaluationofthe givenby
angular momentum projected energy [90,91]. Unfortunately,
(cid:8) (Q )= 1G(Q )B−1 (Q ), (6)
thiskindofbeyond-mean-fieldcalculationisonlyfeasiblefor 0 3 2 3 ATDHFB 3
light nuclei with present-day computer capabilities. In order where
to estimate the REC we have followed the usual recipe [79]
(whichiswelljustifiedforstronglydeformedconfigurations) G(Q )= M−2(Q3) . (7)
ofsubtractingfromtheHFBenergythequantity(cid:3)(cid:3)J(cid:5)2(cid:4)/(2J ), 3 2M−21(Q3)
Y
where (cid:3)(cid:3)J(cid:5)2(cid:4) is the fluctuation on angular momentum of IntheexpressionfortheactionanadditionalparameterE
0
the HFB wave function and J is the Yoccoz moment of isintroduced.ThisparametercanbetakenastheHFBenergy
Y
inertia[92].Thismomentofinertiahasbeencomputedusing ofthe(metastable)groundstate.However,itisarguedthatin
the“cranking”approximationinwhichthefulllinearresponse aquantaltreatmentoftheproblemtheground-stateenergyis
matrixappearinginitsexpressionisreplacedbythezero-order given by the HFB energy plus the ZPE associated with the
approximation(thatis,thesumoftwoquasiparticleenergies). collectivemotion.Toaccountforthisfact,theusualrecipeis
TheimpactofthisapproximationonthevalueoftheYoccoz to add to the HFB energy an estimation of the ZPE for the
moment of inertia was analyzed with the Gogny interaction groundstateinordertoobtainE .Inourcalculationswehave
0
forheavynucleiin[86]bycomparingtheapproximatevalue considered this ZPE as a phenomenological parameter and
with the one extracted from a complete angular momentum givenareasonablevalueof0.5MeVforallisotopesconsidered
044608-3
M.WARDAANDL.M.ROBLEDO PHYSICALREVIEWC84,044608(2011)
[96].Noticethatthisconstantquantityisnotcancelingoutthe 666666000000 50
impactof(cid:8) (Q )inV(Q ).Finally,theturningpointsa and ((((((aaaaaa))))))
0 3 3
b in the limits of the integral of Eq. (3) are the intersection
5555500000
40
pointsbetweenthehorizontallineatE andtheV(Q )curve. 222222222222444444
0 3 RRRRRRaaaaaa
444444000000
EEEEEE [[[[[[MMMMMMeeeeeeVVVVVV]]]]]]
30
III. CLUSTERRADIOACTIVITYIN224RaAND238Pu 3/23/23/23/23/23/2b]b]b]b]b]b] 333333000000
Q [Q [Q [Q [Q [Q [333333
TheanalysisofCRrequiresthedeterminationofthePES 20
for each nuclei considered in this paper. After performing 222222000000
thesecalculationswehavefoundthattherearenosubstantial
10
qualitative differences among the various actinide isotopes 111111000000
considered.InallcasesthePESissimilarandonlyquantitative
variationsarefound.Thereforewewillnotdescribeindetail 000000 0
thePESofallactinides.Inthissectionwewillconcentrateonly 000000 111111000000 222222000000 333333000000 444444000000 555555000000 666666000000
ontheCRoftworepresentativenuclei,namely,thelightcluster QQQQQQ222222 [[[[[[bbbbbb]]]]]]
emitter224Ra,inwhichemissionof14Cisobserved[2,15],and 77770000 40
one of the heaviest emitters 238Pu, that decays by producing ((((bbbb))))
the relatively large clusters 28,30Mg and 32Si [7]. A detailed 66660000 35
222233338888
account of our previous calculations in some other isotopes PPPPuuuu
30
55550000
canbefoundin[31–34].
EEEE [[[[MMMMeeeeVVVV]]]]
25
A. ThePESandshapesoffissioningnuclei 3/23/23/23/2Q [b]Q [b]Q [b]Q [b]3333 3434343400000000 20

15
In Fig. 1 we show the PES of 224Ra and 238Pu in the
22220000
deformation space of the quadrupole Q and octupole Q 10
2 3
moments. This figure shows how the energy of the system
11110000
5
changes with the simultaneous changes of elongation (con-
trolled by Q2) and reflection asymmetry (governed by Q3). 0000 0
Calculations have been performed on a grid, with a spacing 0000 11110000 22220000 33330000 44440000 55550000 66660000 77770000
of 5 b in the Q2 direction and of 5 b3/2 in the Q3 direction. QQQQ2222 [[[[bbbb]]]]
The oscillator lengths characterizing the single-particle basis
FIG.1. (Coloronline)MapsofthePESof(a)224Raand(b)238Pu
havebeenoptimizedineverymeshpointtominimizethetotal
asafunctionofquadrupolemomentQ andoctupolemomentQ .
HFB energy. All the values of potential energies presented 2 3
Linesofconstantenergyareplottedevery2MeV.Bolddot-dashed
in this paper are the corresponding HFB energies corrected
linesareplottedalongfissionpaths.
by the correlation energies of 2bKEC and REC as described
in Sec. II. Both quantities represent correlation energies
gained by restoring (in an approximate way) the rotational Q =55–60bwithQ =15b3/2.Thisisthetypicalscenario
2 3
and translational symmetries spontaneously broken by the offissioninmanyheavynucleileadingtoasymmetricfission.
mean-fieldapproximation.Moreover,tofacilitatetheanalysis Such a valley is usually called an “elongated fission valley”
ofthebarriersheights,wehavenormalizedenergiestozeroin [69,70] as the shapes of the nucleus along it are relatively
thegroundstate. stretched with a long neck coupling a typically spherical
In both nuclei the ground state is well deformed. Its fragmentand atypically prolate-deformed nascent fragment.
quadrupole moment is Q =8.3 b (β =0.18) for 224Ra The value of the fission barrier height is around 10 MeV,
2 2
and Q =14.1 b (β =0.27) for 238Pu. A small octupole whichisavaluealittlebitlargerthanthoseusuallycalculated
2 2
deformation Q =4.2 b3/2 (β =0.14) can also be found in intheheavyactinides.Incontrasttothesenuclei[69–71]the
3 3
thegroundstateof224Ra.Fissionvalleysarecharacterizedby barriers are extremely wide in light actinides. In 224Ra the
alocaldecreaseoftheslopeinthePESfromthegroundstate potential energy oscillates around 10 MeV with increasing
toward scission. Fission paths can be found in the bottom of elongationandwehavenotbeenabletofindasecondturning
these valleys to determine locally the lowest energy barriers. point even for very large Q values. The fission barrier of
2
The direction corresponding to the slowest increase of the 238Pu finishes beyond Q =100 b. A very extended barrier
2
potential energy with deformation can be easily found along causeslongfissionhalf-livesinallconsiderednuclei.Alsothe
the reflection symmetric axis. This barrier is also plotted in experimentalbranchingratioofspontaneousfissiontoαdecay
Fig. 2 with a green short-dashed line. At Q =20–25 b the is very small in all of them [27] and they are stable against
2
barrier reaches a saddle point and then it slowly descends. fission.
At larger elongation, from Q =50 b, the potential energy InFig. 1 one can alsosee a second valley in the PES that
2
increases again, producing a second hump of the barrier. At goesfromthegroundstatethroughthereflection-asymmetric
this stage, the fission valley turns into reflection-asymmetric shapes with nonzero octupole moment. The huge octupole
shapes and a second saddle point can be found around moment values obtained for small elongation suggests a
044608-4
MICROSCOPICDESCRIPTIONOFCLUSTER... PHYSICALREVIEWC84,044608(2011)
120 valley.” Along the fission path at the bottom of this valley
(a) (b)
the elongation of the nucleus rises along with reflection
2Q [b]4 4800 aFsiygm.1m,aestrtyh.eMgroorweotvheorf,tthheefiqussaidornuppaotlhecmreoamteesnatisstrpariogphotrltiinoenianl
0 totheincreaseoftheoctupolemoment.
(c) (d) compound
40 separated InFig.2thehyper-asymmetricfissionpathisalsoplotted
typical fission
withasolidbluelineasafunctionofthequadrupolemoment.
30 Fromthisfigure,itisclearthatthehyper-asymmetricbarrieris
V] 224Ra 238Pu muchhigherthantheclassicalone.Thepotentialenergygrows
Me
E [ 20 very quickly with deformation in the CR path up to around
25MeV.Itsheightisextremelylargeincomparisonwiththe
10 classical fission barrier. This implies very long half-lives for
the decay along this channel (over 1010 s) and explains why
0 theCRpathwasignoredsofarasthepossiblefissionpath.The
0 20 40 60 80 0 20 40 60 80 experimentalevidenceofCR,whichischaracterizedbyhalf-
Q2 [b] Q2 [b] livesofthesameorderofmagnitude,forcesustoconsiderthe
FIG.2. (Color online) Fission barriers in 224Ra (left) and 238Pu hyper-asymmetricpathasthepossibleexoticdecaychannel.
(right) plotted as a function of the quadrupole moment Q (lower The evolution of the shapes of nuclei along the CR path
2
panels). The values of the hexadecapole moment Q of the nuclei fromgroundstatetothesaddlepointisshowninFigs.3(a)–
4
alongthefissionpathsareplottedasafunctionofquadrupolemoment 3(e) for 224Ra and in Figs. 4(a)–4(f) for 238Pu. One can see
Q intheupperpanels. thataclusterofnucleonsisbuddingfromtheparentnucleus
2
as elongation and asymmetry grow and already at a modest
large asymmetry in the mass distribution. As the saddle octupoledeformationofQ =20–30b3/2 aneckstartstobe
3
point is reached, the matter density distribution starts to clearlyvisibleinbothcases.
resemble a molecular shape with a small sphere touching Around Q =20 b a bifurcation can be found in the
2
a larger one. The large spherical fragment has numbers CR path of 224Ra. One of the branches goes toward large
of protons and neutrons that are consistent with those of deformation parameters (Q =45 b, Q =55 b3/2) with the
2 3
208Pb. This observation points toward a clear relationship energyreachingvaluesover40MeVabovethegroundstate.
between this valley and the phenomenon of CR. We will This path cannot lead to fission, as the nucleus takes on a
refer to this valley as “hyper-asymmetric” or as the “CR conelikeshape[seeFig.3(d)]withoutawell-definedneck.The
20 3/2 3/2 3/2 3/2 3/2
Q = 0 b Q = 10 b Q = 20 b Q = 30 b Q = 25 b
3 3 3 3 3
224
Ra
10
]
m
f
[
z
0
(a) (b) (c) (d) (e)
-10
20 Q3 = 30 b3/2 Q3 = 40 b3/2
3/2 3/2 3/2
10 Q = 50 b Q = 60 b Q = 70 b
3 3 3
]
m
f
[
z
0
(f) (g) (h) (i) (j)
-10
-5 0 5 -5 0 5 -5 0 5 -5 0 5 -5 0 5
r [fm] r [fm] r [fm] r [fm] r [fm]
perp perp perp perp perp
FIG.3. (Coloronline)Shapeevolutionof224RawithincreasingoctupolemomentQ .Panels(a)–(d)correspondtotheup-goingpartof
3
thefissionpath,panel(e)correspondstotheshortbrancharoundthesaddlepoint,andpanels(f)–(j)correspondtothedecreasingpartofthe
fissionpath.
044608-5
M.WARDAANDL.M.ROBLEDO PHYSICALREVIEWC84,044608(2011)
20 3/2 3/2 3/2 3/2 3/2
Q = 0 b Q = 10 b Q = 20 b Q = 30 b Q = 40 b
3 3 3 3 3
238
Pu
10
]
m
f
[
z
0
(a) (b) (c) (d) (e)
-10
20 Q3 = 50 b3/2 Q3 = 50 b3/2 Q3 = 60 b3/2 Q3 = 70 b3/2 Q3 = 80 b3/2
10
]
m
f
[
z
0
(f) (g) (h) (i) (j)
-10
-5 0 5 -5 0 5 -5 0 5 -5 0 5 -5 0 5
r [fm] r [fm] r [fm] r [fm] r [fm]
perp perp perp perp perp
FIG.4. (Coloronline)ThesameasinFig.3butfor238Pu.Panels(a)–(f)correspondtotheup-goingpartofthefissionpath,andpanels
(g)–(j)correspondtothedecreasingpartofthefissionpath.
densityprofilecorrespondingtotheshorterbranch,presented before scission; cf. also Sec. IIIC.] The lighter fragment
inFig.3(e),showstwonearlysphericalfragmentsseparatedby may be slightly deformed (prolate or oblate). Its shape is
aneck.Thisconfigurationischaracterizedbyahexadecapole mostly determined by the shape of the ground state of the
momentwhichissubstantiallylarger,asseeninFig.2(a).The corresponding nucleus as the Coulomb interaction with the
samefissionpathbifurcationcanbefoundalsoin222Ra.Ina heavier fragment is not strong enough. In the case of 30Mg
subsequentanalysiswewillconsideronlythesecond,shorter emittedfrom238Pu,thegroundstateisoblate(β =−0.222)
2
branchastheonlyrelevantoneforthedescriptionofCR. [98].Intheothernucleus,thespherical14Cisotopeconstitutes
IntheupperrightcornersofbothpanelsinFig.1distinct thelighterfragmentoftheCRfrom224Ra.
region of the PES is found, with the energy decreasing with Oncethesystemhassplitintwo,theshapesofthefragments
increasing deformation. In this part of the PES the system do not change significantly as they move apart and the
of nucleons is split into two fragments. It may be called: increase of the total quadrupole and octupole momenta is a
“fusion”valleyincontrasttothefirstpartwhichiscommonly consequenceoftheincreasingdistancebetweenthefragments.
called“fission”valley.The“fusion”and“fission”valleysare Therefore the change in the potential energy after scission is
strongly correlated with each other as they are linked in the mainlyduetothedecreasingCoulombrepulsionanditshould
same region of the deformation space and the mass splitting declinehyperbolicallywiththedistancebetweenthecentersof
betweenfragmentsissimilarinbothcases.Theminimumof fragments,whichisroughlyproportionaltoQ .Suchbehavior
2
theenergyinthisvalleycreatesthedescendingbranchofthe can be seen in Fig. 2 close to the saddle point. However, for
CR fission barrier, which is plotted with a red dashed line in largerdeformationsweobserveadeparturefromtheexpected
Fig.2.IntheupperpanelsofFig.2weobservethecoincidence behaviorthatcallsforalargerbasis.Unfortunately,theuseofa
ofthehexadecapolemomentsofbothbranchesofthebarrier largerbasiscanbeproblematicasaconsequenceofnumerical
atthesaddlepoint. instabilitiesintheevaluationofmatrixelementsduetofinite
The density distributions in the “fusion” path are given computer accuracy. Those instabilities lead in some cases to
in Figs. 3(f)–3(j) for 224Ra and in Figs. 4(g)–4(j) for 238Pu. strange behaviors in the energy, preventing the use of a very
Someimportantinformationcanbededucedfromtheseplots. large basis (see also Appendix A where the convergence of
First,thesystemisbuiltfromtwoalmostsphericalfragments. theenergyisdiscussed).Toavoidthesedifficulties,whichare
The space between their surfaces is wide for at least a few criticalforthedeterminationofhalf-livesintheWKBscheme,
femtometers and increases with Q and Q . The heavier we have adopted an approximate strategy to be discussed
2 3
fragment is the doubly-magic spherical 208Pb after scission in Sec. IIIB below. The insufficient size of the basis also
of 224Ra or 210Pb in the case of 238Pu. [Those values are manifests in the matter distributions of the lighter fragment
in good agreement with the mass and charge of the heavy seeninpanels(i)and(j)ofFig.3,whereanunnaturalstretching
pre-fragments of configurations (e) in 224Ra and (f) in 238Pu towardlargezvaluescanbenoticed.
044608-6
MICROSCOPICDESCRIPTIONOFCLUSTER... PHYSICALREVIEWC84,044608(2011)
The solution of the HFB equation often depends on the
(a) (b) (c) (d)
nuclearmatterdistributionsoftheinitialwavefunctionusedin 30
Q = 5 b3/2 Q = 10 b3/2 Q = 15 b3/2 Q = 20 b3/2
theiterativeprocedure.InmanyregionsofthePES,especially 3 3 3 3
V] 20
close to the scission line, two solutions may be obtained for Me
thesameconstraints.Ifthecalculationbeginswithacompact E [ 10 238
shape of the nucleus, the final solution will have similar Pu
properties. If a configuration with two separated fragments 0
ischosenastheinitialcondition,againasolutionwithsimilar (e) (f) (g) (h)
30
properties will be found. When the same constraints are put
on the system the two results will have the same quadrupole V] 20
and octupole momenta but they may have different higher Me
multipolaritiesaswellasenergy.Becauseofthis,the“fusion” E [ 10 Q = 25 b3/2 Q = 30 b3/2 Q = 35 b3/2 Q = 40 b3/2
valleyextendstowardgroundstatemuchfurtherthanisshown 3 3 3 3
in Fig. 1. It covers the area around the “fusion” path in 0
the Q -Q deformation space. Its part is hidden below the 0 20 40 0 20 40 0 20 40 0 20 40
2 3
“fission”valleyshowninFig.1.Inthisfigurewehavemarked Q2 [b] Q2 [b] Q2 [b] Q2 [b]
both the fission paths and “fission” valley but not the whole
FIG.6. (Coloronline)Potentialenergiesof238Puasafunctionof
“fusion”valley.
thequadrupolemomentQ forseveralvalues(shownineachpanel)
2
oftheoctupolemomentQ .
3
B. Trackingfissionpathsasafunctionofoctupolemoment ThepotentialenergyforfixedQ3asafunctionofQ2isplotted
inFig.6.Herethehyper-asymmetricvalleyisclearlyvisible
Tracking the hyper-asymmetric fission path in the PES
ateverypointanditistrivialtodeterminelocalminimathere
is a difficult task from a numerical standpoint. Usually, the
and track the fission path. The octupole moment can also
fission path is determined by searching for the local minima
be used as the driving coordinate to determine half-lives in
of the energy along cuts of constant Q . This method could
2 theWKBapproximation,asalreadydescribedinSec.II.We
also be applied to the CR path as is shown in Fig. 5, where
conclude that the octupole moment is better suited than the
the potential energies of the 238Pu nucleus are plotted as a
quadrupolemomenttodescribetheCRpathsinthePESand
function of Q for fixed values of Q . It is clear that a local
3 2 thereforewewilluseitasaleadingcoordinateinthefollowing
minimumcorrespondingtothehyper-asymmetricfissioncan
discussion.
bedeterminedinmostofthecases,usuallyathigherenergies
thantheminimumoftheclassicalfissionobservedatQ =0 The profiles of the CR path in 224Ra and 238Pu, presented
3 already as a a function of quadrupole moment in Fig. 2, are
b3/2. However, in many nuclei there are certain Q values
2 plottednowasafunctionofoctupolemomentinFigs.7(e)and
whereaplateauisobservedinsteadofawell-definedminimum
[e.g.,forQ =30bin238PuinFig.5(d)].Thisproblemcanbe 7(f). Initially, the energy increases with increasing octupole
2
solvedbyusinganalternativechoiceofcoordinatetodescribe
20
theformationofthedaughternuclei.Asmentionedbefore,in (a) Ncluster (b)
othcetuCpRolpeamthoQm2enistrcoaunghallsyopbroepuosretidonaasltthoeQle3aadnindgthceoroerfdoirneatthee. nucleons 1105 Zcluster
-6m] 5 (c) (d)
(a) (b) (c) (d) -1V f 10
V] 2300 238Pu -8B [10 Me 05
Me (e) compound (f)
E [ 10 Q = 15 b Q = 20 b Q = 25 b Q = 30 b 20 secplaarsastiecdal
2 2 2 2 V]
Me
0 E [ 10 224Ra 238Pu
(e) (f) (g) (h)
30
0
V] 20 0 20 40 60 80 0 20 40 60 80
E [Me 10 Q3 [b3/2] Q3 [b3/2]
Q2 = 35 b Q2 = 40 b Q2 = 45 b Q2 = 50 b FIG.7. (Coloronline)Hyper-asymmetricfissionbarriersin224Ra
0 (left) and 238Pu (right) as a function of the octupole moment Q3
(lower panels). Approximate Coulomb repulsion energies [Eq. (8)]
0 20 40 0 20 40 0 20 40 0 20 40
Q [b3/2] Q [b3/2] Q [b3/2] Q [b3/2] forcorrespondingclustersarealsoplotted.Inthemiddlepanels,the
3 3 3 3
massparameterB(Q )calculatedinamicroscopicwayisplotted.In
3
FIG.5. (Coloronline)Potentialenergiesof238Puasafunctionof addition,theclassicalvalue[Eq.(11)]correspondingtotwoseparate
theoctupolemomentQ forseveralvalues(shownineachpanel)of fragmentsisalsogiven.Intheupperpanel,thenumberofnucleons
3
thequadrupolemomentQ . inclustersisgivenasafunctionoftheoctupolemomentQ .
2 3
044608-7
M.WARDAANDL.M.ROBLEDO PHYSICALREVIEWC84,044608(2011)
momentinanalmostquadraticfashionfromthegroundstate, therefore the contribution to the action [Eq (3)] from the
which may be refection symmetric (238Pu) or asymmetric postscission region will be very similar in our calculations
(224Ra). The slope of energy decreases when approaching to the ones of the analytical superasymmetric fission model
the top of the barrier. At some point the branch with two (ASAFM)ofRef.[38].
fragmentsbecomesthelowestenergysolutionwithanenergy InFigs.7(a)and7(b)wehaveshownthenumbersofprotons
which is essentially the Coulomb repulsion of the fragments and neutrons of the lighter fragment after scission in the CR
expressed as a function of the octupole moment of the two path.Inthiswaytheclusteremittedinthehyper-asymmetric
fragments. The Coulomb repulsion energy can be very well fission can be identified. In 224Ra it corresponds exactly to
be approximated by the classical value corresponding to two the experimentally observed cluster 14C. The PES of 238Pu
uniformlychargedspheres: indicates30Mgasapotentialcluster.Thisisoneoftheclusters
observedinthedecayofthisnuclide(28,30Mgand32Si).
Z Z
V(Q )=V −Q=e2 1 2 −Q, (8) Wewouldliketopointoutanimportantaspectoftracking
3 Coul
R
thefissionpathafterthescissionpoint.Inthelaboratoryitisnot
whereR representsthedistancebetweenthecentersofmass possibletotransfernucleonsbetweenthedaughternucleionce
of the fragments. Asymptotically, the total energy tends to thefragmentsarecreated.Wehavecheckedthatthenumbers
the Q value of the reaction that can be extracted from the ofneutronsandprotonsareusuallyconstantintheminimum
experimental binding energies [97]. The connection between ofthe“fusion”valley,althoughtheymaydifferslightlyfrom
the variable R and the octupole moment Q is obtained in a integer numbers. Imposing given fragment masses will lead
3
simplegeometricalwaywhenthetwofragmentsarespherical toconfigurationswithhigherenergies.The“fusion”pathsfor
orwhentwopointmassesareconsidered: thosesystemswithmassasymmetrydifferingbyafewnucle-
onsfromtheonecorrespondingtotheminimumenergyconfig-
Q =f R3, (9)
3 3 urationrunparalleltotheminimumenergypathandtheyreach
where the scission point at almost the same position in the Q2-Q3
A A (A −A ) deformation space, i.e., in the saddle, with similar energy.
f = 1 2 1 2 (10) Atinyinstabilityaroundthesaddlemayleadtoanalternative
3
A A
choiceofclusterconfiguration.Thelengthofthefissionbarrier
is given in terms of the total mass number A, and the mass correspondingtoeachpossiblenascentfragmentwoulddeter-
numbers of each of the fragments, A1 and A2. In Figs. 7(e) mine which one will be observed in the experiment. Further
and7(f)weobservethataroundthesaddlepointboththeHFB detailed investigations should be performed using additional
and the approximate Coulomb repulsion energy of Eq. (8) constraintsonthenumberofnucleonsofeachfragment.
coincide with a noticeable agreement of the order of 2 MeV.
Small differences can be mainly attributed to the excitation
of the lighter fragment in the presence of the Coulomb field C. Scissionpointtransitionfromacompoundnucleustotwo
oftheheavy-massresidueaswellastothedeformationofthe separatedfragments
emittedclusterthatcanbedifferentfromtheoneofitsground Two independent branches are clearly visible in the CR
state.AsalreadymentionedinSec.IIIA,atlargerQ3 values fissionbarriersofFigs.2and7.AsdescribedinSec.IIIBthey
the HFB energy results are more affected by the finite size differsubstantiallyintheshapesofthenucleuscorresponding
of the basis used and therefore they lie at an energy higher to each of them. In the first, the up-going part of the barrier,
thantheoneofaninfinitebasiscalculation.Becausethisrange called the “fission” path, the shape corresponds to that of a
ofoctupolemomentsisveryrelevantforthedeterminationof compound nucleus [Figs. 3(a)–3(e) and 4(a)–4(f)]. For the
half-livesintheWKBframework,wewillusetheapproximate deformationsaroundthegroundstatethecorrespondingshape
expressionofEq.(8)inthecalculationofhalf-livesinsteadof isnottoodistantfromtheellipsoidandthereforewecansay
theHFBenergy. that the nucleus takes a compact shape. However, for large
ThecollectivemassB(Q3)linkedtotheoctupolemoment deformationsaneckcanbeclearlydistinguishedinthisbranch
is also plotted in Figs. 7(c) and 7(d). The collective mass of andthedensitydistributionofthenucleusisofmoleculartype.
thecompoundsystemcomputedmicroscopicallysubstantially Acompletelydifferenttypeofshapeisobtainedonthedown-
differsfromthesemiclassicalvaluegivenbythereducedmass slopesideofthebarrier,calledthe“fusion”path[Figs.3(f)–
of the two fragments, μ=mnA1A2/(A1+A2) (a quantity 3(j)and4(g)–4(j)].Twowell-separatednucleicanbeobserved
connected to the kinetic energy for the coordinate R) but thereasthematterdensityintheregionbetweenthemgoesto
writtenintermsofQ3: zeroandtheshortestdistancebetweenthenuclearsurfacesof
μ thetwofragmentsisatleastafewfemtometers.
B(Q )= . (11)
3 9Q4/3f2/3 Inspiteofthedifferentshapesofthedensityprofilesalong
3 3 thetwobranches,theysharemanysimilarnuclearproperties
This quantity derived from the ATDHFB model in Eq. (4) at the top of the barrier. For instance, comparable values of
variesconsiderablywhenthenucleusisstretchedout.Thisis thequadrupole,octupole,andhexadecapolemomentscanbe
aconsequenceofthestrongdependenceofcollectivemasson found there. The density distribution before scission is close
thesingle-particleeffectsthatshowupduringthedevelopment totheoneafterseparationandtheonlyimportantdifferences
of the neck. After scission the microscopic collective mass canbefoundintheneckregion.Moreover,theenergiesalso
B(Q ) is very close to the classical value, as expected, and havesimilarvaluesandwecaneasilyfindinFig.7acrossing
3
044608-8
MICROSCOPICDESCRIPTIONOFCLUSTER... PHYSICALREVIEWC84,044608(2011)
point where the potential energy on the two branches is the centeredatthepositionz andofwidtha.Inthepresentcase
0
sameforsomevalueofQ .Afirstandroughapproximation wehavechosena =0.1fm,whichgivesusasufficientlythin
3
couldbetoconsiderthisasthescissionpoint.Thisassumption slice,andz =7.5fm,whichcorrespondstothepositionofthe
0
canbeusedtogetaquitereasonableestimationofthesizeof neck.Theneckparameteriscorrelatedwiththehexadecapole
thebarrierandfissionhalf-lives.Nevertheless,amoreprecise moment, a quantity that has been used routinely in fission
analysisshowsthattheshapeofthesystemisclearlydifferent calculations [74] to study the scission process, but the neck
in the two branches and none of them can be considered as parameterismoresuitedtodrivethesystemthroughscission
twotouchingfragments. when z and a are chosen conveniently. The quantity Q
0 N
Thepassagefromacompactshapetoatwo-fragmentone never goes to zero in any physical situation because of the
cannotbetreatedasaninstanttransitionatthecrossingpoint. nonvanishingtailofthenucleardensitydistributionbutitcan
Some energy barrier, not seen clearly in the PES spanned in bearbitrarilysmallifthesliceisproperlylocatedintheregion
theQ -Q space,existsbetweenthe“fission”andthe“fusion” betweenthetwoseparatedfragments.
2 3
path. These two constraints are not sufficient to describe the InFigs.8(a)and9(a)thePESof224Raand238Puareplotted,
continuouspathconnectingthetwobranches.Insuchapath, respectively, as a function of the octupole moment Q and
3
the nuclear density in the neck would decrease gradually to the neck parameter Q . In these plots, we only show the
N
zero and then the two fragments would be disengaged. The relevantregionaroundthetopofthebarriers.Theminimaof
relevantparameteralongthispathistheneckparameter[69,74] thevalleysonthissurfacearemarkedbygreendashedlines.
definedthroughthemeanvalueoftheoperator The “fission” path goes from Q =20 b3/2, Q =0.65 to
3 N
(cid:8) (cid:9) Q =28 b3/2, Q =0.18 in 224Ra and from Q =45 b3/2,
3 N 3
Qˆ =exp (z−z0)2 . (12) QN =0.45 to Q3 =55 b3/2, QN =0.20 in 238Pu. The
N a2 “fusion” path is marked by an almost horizontal line with
neck parameter in the range from Q =0.02 to Q =0.05
N N
The value of the neck parameter roughly corresponds to in both nuclei. In Fig. 8(a) a horizontal line at Q =0.65 is
N
thenumberofnucleonsinasliceperpendiculartothezaxis,
0.8 34
((((aaaa)))) (c)
0.7 AAAAA 32
A B C
0.6
30
7.5 fm 7.5 fm 7.5 fm
0.5 BBBBB 222222224444RRRRaaaa
28
0.4 EEEE [[[[MMMMeeeeVVVV]]]] 0 0 0
26
0.3
CCCCC
24
0.2
DDDDD
m) 0.1 sssssccccciiiiissssssssssiiiiiooooonnnnn EEEEE 22
f FFFFF
0.1 20
=
a
m,
f 0.8 29
5 ((((bbbb))))
7. 0.7 AAAAA 28
=
0 27 D E F
z 0.6
(N 26 7.5 fm 7.5 fm 7.5 fm
Q 0.5 BBBBB 222222224444RRRRaaaa 25
0.4 QQQQ [[[[bbbb]]]] 24
2222 0 0 0
23
0.3
CCCCC
22
0.2
DDDDD 21
0.1 sssssccccciiiiissssssssssiiiiiooooonnnnn EEEEE 20
FFFFF
19
20 22 24 26 28 30
3/2
Q [b ]
3
FIG.8. (Coloronline)(a)SaddleregionofthePESand(b)quadrupolemomentQ plottedasafunctionofoctupolemomentQ andneck
2 3
parameterQ in224Ra.Greendashedlinesarefissionpathsandtheredshort-dashedlineisthescissionline.(c)Contourplotsofdensity
N
distributionsforsomedeformationsmarkedinpanel(a).Equidensitylinesareplottedevery0.02fm−3.
044608-9
M.WARDAANDL.M.ROBLEDO PHYSICALREVIEWC84,044608(2011)
0.7 36
((((aaaa)))) (c)
0.6 222233338888PPPPuuuu 34
A B C
32
0.5 EEEE [[[[MMMMeeeeVVVV]]]]
AAAAA 7.5 fm 7.5 fm 7.5 fm
30
0.4 BBBBB
28
0 0 0
CCCCC
0.3
26
DDDDD
0.2
24
sssssccccciiiiissssssssssiiiiiooooonnnnn EEEEE
m) 0.1 22
f FFFFF
0.1 20
=
a
m,
f 0.7 58
=7.5 0.6 222233338888PPPPuuuu ((((bbbb)))) 56
0 D E F
z 54
(N 0.5 AAAAA QQQQ2222 [[[[bbbb]]]] 7.5 fm 7.5 fm 7.5 fm
Q
52
BBBBB
0.4
50
0 0 0
CCCCC
0.3
48
DDDDD
0.2
46
sssssccccciiiiissssssssssiiiiiooooonnnnn EEEEE
0.1 44
FFFFF
42
46 48 50 52 54 56 58 60
3/2
Q [b ]
3
FIG.9. (Coloronline)ThesameasinFig.8butforthe238Punucleus.
also shown. It corresponds to the branch of the fission path andDareductionoftheneckparametercanbenoticed.The
in224Rathatgoesupinenergyandthatshowsshapesthatdo neckbecomesthinnerandadecreaseofthenucleardensityup
notdevelopasizableneck.Theredshort-dashedlinearound tohalf ofthebulk value inconfiguration Dcan beobserved.
Q =0.10isthescissionlinedescribingtheseconfigurations BetweenconfigurationsD,E,andFthescissionprocesstakes
N
wherethedensityintheneckregiongoesbelow0.4fm−3.It placeandtheshapeofthelighterfragmentevolvesfromprolate
liesalongtheridgeontheenergysurfaceseparating“fission” inDtosphericaloroblateinF.Itisalsointerestingtonotice
and“fusion”valleys.Owingtotheuseoftheneckparameter that shapes D, E, and F have essentially the same octupole
both valleys are linked in a continuous way along the whole momentandaverysimilarquadrupolemoment.
scissionlineandthereisnosuddenenergychange. WealsoobserveinFigs.8(a)and9(a)thatthecrossingpoint
InFigs.8(b)and9(b)thequadrupolemomentsofthe224Ra ofthetwobranches(“fission”and“fusion”)atQ =50b3/2
3
and238Puareplottedinthesamespaceofdeformationsasin for 238Pu and at Q =25 b3/2 for 224Ra are well separated
3
Figs.8(a)and9(a).Weobservehowthequadrupolemoment as they correspond to different values of Q . It is now clear
N
increases monotonically with increasing octupole moment. that to pass directly from one configuration to the other it is
The variations with the neck parameter are much smaller necessary to climb the “neck barrier,” which is over 1 MeV
though. This explains why the quadrupole moment does not high, although the energy of both the “fission” and “fusion”
provideaquantitysensitiveenoughforthedetaileddescription paths is the same. From these plots it becomes clear that
oftheruptureofthenucleusintotwopieces.Thehexadecapole it is energetically preferable to follow the “fission” path to
moment is also insensitive to changes of the neck parameter the very end, where the neck is very thin (see the shape
and it varies by not more than 4 b2 for the fixed octupole ofthenucleusatpointD)andthereisnobarrierseparatingthe
moment in the configurations considered in the figures. A nucleusfromthescissionline,thantoclimbthe“neckbarrier.”
largermonotonicincreaseofQ withQ canbeobserved. The subsequent evolution of the shape of the nucleus should
4 3
Finally,inFigs.8(c)and9(c)thematterdensitydistribution follow the direction corresponding to the maximal decrease
at the different stages of the scission process [marked by the in energy (the gradient direction). In fact, this means that
letters A, B, C, D, E, and F in panels (a) and (b)] is shown. the neck parameter should decrease rapidly almost without
Followingthepointsatthe“fission”pathmarkedasA,B,C, change of the octupole moment until it reaches the bottom
044608-10