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DOI:willbesetbythepublisher
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Superfluid fission dynamics with microscopic approaches
C.Simenel1,a,G.Scamps2,3,D.Lacroix4,andA.S.Umar5
1DepartmentofNuclearPhysics,ResearchSchoolofPhysicsandEngineering
AustralianNationalUniversity,Canberra,AustralianCapitalTerritory2601,Australia
2GANIL,CEA/DSMandCNRS/IN2P3,BoîtePostale55027,14076CaenCedex,France
3DepartmentofPhysics,TohokuUniversity,Sendai980-8578,Japan
6 4InstitutdePhysiqueNucléaire,IN2P3-CNRS,UniversitéParis-Sud,F-91406OrsayCedex,France
1 5DepartmentofPhysicsandAstronomy,VanderbiltUniversity,Nashville,Tennessee37235,USA
0
2
Abstract.Recentprogressesinthedescriptionofthelatterstageofnuclearfissionarereported. Dynamical
n effects during the descent of the potential towards scission and in the formation of the fission fragments are
a
studiedwiththetime-dependentHartree-FockapproachwithdynamicalpairingcorrelationsattheBCSlevel.
J
Inparticular, thisapproachisusedtocomputethefinalkineticenergyofthefissionfragments. Comparison
9
withexperimentaldataonthefissionof258Fmaremade.
1
]
h
t 1 Introduction isobtainedusingarecentcodesolvingthetime-dependent
-
l Hartree-FockequationwithBCSdynamicalpairingcorre-
c Despite important progresses since the discovery of nu- lations[37].
u
clear fission in 1939 [1, 2], it remains an important chal- Theimportanceofthedynamicaleffectsisinvestigated
n
[ lengefortheorists. Comparedwithothernucleardynam- in fission of 258,264Fm [6, 25]. It is shown that dynam-
ical processes such as fusion, the overall time-scale for ics has an important effect on the scission configuration
1
fission is relatively long. This suggests that the evolu- and on the kinetic and excitation energies of the frag-
v
2 tionacrossthepotentialenergysurfaceisaslowprocess. ments.Thevibrationalmodesofthefragmentsinthepost-
2 However, it is also expected that the systems could en- scissionevolutionarealsoanalyzed.Quantumshelleffects
1 counterrapidshapeevolutioninthelaterstagesnearscis- are shown to play a crucial role in the dynamics and for-
5 sion. Theoretical prediction of the fission fragments and mationofthefragments.
0
their characteristics are often based on the adiabatic ap-
.
1 proximation[3]. Itisassumedthattheinternaldegreesof
0 freedom are equilibrated while the system evolves along 2 Adiabatic calculations
6
the fission path. However, the shape evolution near scis-
1 Theadiabaticevolutionisdescribedinthetraditionalway
sion is expected to be non-adiabatic and the approxima-
v: tion may break down in the latter stage of fission [4, 5]. byminimizingtheHFenergyunderanexternalconstraint
Xi Thus, theinclusionofthedynamicaleffectsnearscission inducingelongationofthesystemuptothescissionpoint.
ThisconstraintcanbeonthedistanceRbetweenthecen-
are crucial. In particular, the dynamics has been shown
r
tresofmassofthefragments(definedassumingasharpcut
a togeneratemostoftheexcitationenergyinthefragments
attheneck),oronmultipolemoments. ThestaticHartree-
[6].
FockequationissolvedwithBCSpairingcorrelationsus-
Microscopic approachesare wellsuited toinvestigate
ing the ev8 code [38]. The SLy4d parametrization [39]
the dynamics of nuclear fission [5–21]. Microscopic nu-
oftheSkyrmefunctional[40]isusedwithasurfacepair-
clear dynamics is usually treated at the mean-field level
inginteraction[41]. Thecalculationsareperformedona
withthetime-dependentHartree-Focktheory(see[22,23]
Cartesiangridwithmeshsize0.8fm.
for reviews). In particular, Negele et al., have demon-
TheadiabaticpotentialhasbeencalculatedinRef.[6]
stratedthepossibilitytoaccountforimportantdynamical
for 264Fm and in Ref. [25] for 258Fm. The results are re-
effects in the fission process within the time dependent
portedonFigs.1and2for264Fmand258Fm,respectively.
Hartree-Fock approximation [24]. Similar approaches
In both cases, the resulting barrier is in good agreement
haverecentlybeenusedbyseveralgroupstostudyfission
withothercalculations[9,10,17,42,43].
[6, 25, 26] and quasi-fission [27–32]. Static or dynami-
An important question is which configuration should
cal pairing correlations are sometimes included in mean-
we choose as initial state of the time-dependent calcula-
fieldcalculations[33–37]. Here, thesuperfluiddynamics
tions. Apossiblechoicewouldbetostartwithaconfigu-
ae-mail:[email protected] rationwherethefragmentshaveestablishedtheiridentity.
2
The largest calculations (e.g., a collision of two actinides) involve ~500 wave-functions (one per nucleon) on a
100x100x32 grid with a mesh size of 0.8t ofmpro.p eUrlysudeaslcrliybe lfiesssiso nthdyannam 1ic0s0ac0ro0ss ttihme neu ciletaerratitoimness taepre1. 5ne1e0d�2e4ds. to describe the entire
⇥
chart. These correlations have been recently included in The CHF+BCS solutions with constraints on
collision process (~10-20 s). The interacretailoistnic tbimeet-wdepeeennde ntthmee ann-fiuelcdlecaolcnulsa tioinss [d21e–s25c]ribqeudad rbupyo leaQ S,koyctrumpoele Qde,nasnidtyh-edxaedpeceapnodleeQnt
20 30 40
effective interaction (zero-range with grwahdicihewnete xatenndd tospthiens-toudryboift fistseiornm. s). Several omthomeern tws aolornlgdt hleefiasdsioenrsax iisnle ttahpep eafriethlrdee hvaallveyes.
The purpose of this letter is to present a microscopic Two symmetric valleys (i.e., with a total Q = 0) lead
30
published results using this code (Otsukam, eMthoadrwuhhicnh,in Acorypiokra,t eLsbaocthrsouipxer,fl Muidadyrnuahmnic…san)d. tosymmetriccompactfragments(scf)andtosymmetric
non-adiabatice↵ectsinthelatterstageofthefissionpro- elongated fragments (sef), respectively. In the scf valley,
The recent developments of this code by the CI involve the inclusion of the particle number projection technique
cess. Fission in the 258Fm nucleus is considered as an the final fragments are almost spherical while they
(Simenel 2010) and of one-body fluctuaetxiaomnpsle aoft atphpleic aTtioDn.RTPhAis nlueclveuesl is(Skniomwnenexeple r-201e1xh)ib. itAanstorotnhgeprr orlaetceeshnapt eeinxttheenssefiovanll eyo.fA tthhiirsd
imentally to exhibit a bimodal fission [26] and consti- valley with Q = 0 leads to asymmetric elongated
code, made by my collaborator Prof. D. Lacroix and his student G. Scamps, includes pairi3n0 g6 correlations at the BCS
tute an ideal benchmark for theoretical studies of fission fragments (aef) with di↵erent masses and charges. Con-
level and is based on previous works on p[6a, 8ir,i9n, 2g7 ,d28y].namics by the CI and his collasbidoerriangtoanrsa d(iAabavteicze veoltu taioln 2in0t0he8a)s.y m metric valley
We use the standard approach in which it is assumed down to scission leads to A1X+A2Y fragments (replace
The TDHF3D code will be used extensively to study the dynamics near scission in the path to fission.
that the fission process is divided in two steps. In a with adiab results). Similar valleys were obtained
first step, the slow evolution near the fission barrier is by other groups for the same nucleus [8, 9, 28]. The
treatedadiabaticallybysolvingtheconstrainedHartree- potential energy along these three valleys is shown in
The microscopic structure as a signatuFroeck o(HfF t)heqeu aptirones-wfoithrmpairaintgioconrr eolaft iotnhsetr efarteadgamt enFitgs. 1. This first, adiabatic, stage of fission is crucial in
the BCS level (CHF+BCS). Several constraints are con- determining the outcome of the reaction. In particular,
The transition from one nucleus to two fragments is a key problem in the description of the fission process. Indeed,
sidered in order to find di↵erent valleys in the poten- we see that the scf and aef valleys have similar energies
the fragments are thought to be pre-formtieadl e nwergeylls ubrfeacfeo. rIen ascseicsonsdiostnep o, tchceunorns- a(dYiaboatuicnesu pantodr eGlatoivgelnyyla rg2e0d1ef1or)m. aTtiohniss( Qt2r0ans2i7t0iofmn2 )isin
⇠
descent of the potential towards scission is determined the descent to scission. The sef valley, however, is found
crucial as it is expected to trigger non-adiabatic effects in the further evolution. However, it is difficult to recognise
using the time-dependent HF equations with dynamical at higher energy.
the pre-formation of the fragments fromp atihrineg dcoerrnelsatiitoyns o(TfD HthFe+ BsCySs).tTehmep raonpedrt ioesnofet hheas toL etlouoskno wfoirn vaesntigoattheethre ssiegconndatsutargee oofft htehifiss-
fragments, in particular their mass, charge, and kinetic sion process, associated with the non-adiabatic descent
transition. As proposed by the CI (Simenel and Umar 2014), the best way to identify this transition is to look at the
energy, are then computed after scission and compared of the potential towards scission. TDHF+BCS calcu-
EPJWebofConferences
microscopic structure of the system. Thiwsi tihse xppoersimsiebntlael doatnalfyro min[2 6f]u. lly microscopic alaptpionros ahacvheebsee nsupecrfhor maesd twhieth HiniFtia lthcoenofigruyra.t iIonns
along theses valleys indicated by arrows in Fig. 1. More
particular, the distribution of single-particle energies is expected to converge toward the single-particle levels of the
compact configurations belong to the adiabatic phase as
280 HF (SLy4d) + BCS 258Fm single particle states cross the Fermi level and induce
final fragments. An example is shown in Figure 7 where the evolution of the proton single-particle energies is plotted
foV)r2 7t0h2e6 4Fsmame fission reaction as in Fig. 1V]MeV) a-n19d00 5. In thaies fcasseef, the 264Fm nucleusa[1 8ft,wiso3s-1b]i.oodnyTshd eisisdnipe natstwiitoynoed vu1oe3lu2ttSoionnths edinLoatunhdebaulny-oZn e-mnaedraiaegb↵aietccict
fnrrgy (Meuam22g5600mbeernst.s g T.sw.hietshe 5g0a ppsro, taonnds iena cpha.r tWicuel arer ctohE[Menergy (gant-1 i9as2te0 Zin= 5F0i,gs .ac rf7e tphree ssepnhte rwiAcesalyllm mbmeetarficgoirce gscaippFtsahsinsga t.isaoe2rso.anlsWer eoiwenrcestpihehaereietctsmehehnadatt se sondwaocnanictd-tauvhdcahri raasitborh gauaetesitcr t einempR↵aueer~scctti1ftlsoieo5rpnale aarbfyc mehamtwnm eai(oemsngdpeeitohecirne-
e E
FEnig24.0 5). This proves that the fragments are-1 9p4r0e-formed well beforSeym mscetiriscsion andfcraa lgctumhlaeantitto snnisnoptrnhede-iacatsdyaimahmebaevatyrticifrcav aglmelefeynfteasicntthtshe eTm1D42uHCsFst+r eBgbiCoeSn
included in the pre-scission dynamics. while in the adiabatic approximation we got A2Y (re-
230 0 100 200 300 400 500
placeandadaptthestatementifbothapproaches
MeV)MeV) 00 7 8 9 10 R 1(1fm)12 13 14 15Mnuamgibce rs FfuinFFgIicgGuut.rri1eoe:2 n(.7C Ao:ol dofEir atbovhantolieicnl epud)oQttPieisuonotattieadnanlrtn uiiaonpQclof2e2e5l e0n8 p[F ebmRrmrg]oy omfibsitenseoinotttnhn w(eb ws)etihtiehrnentegh evltaSehllLe-eyyp4s :dfarratglifisrecamalglidkemee tlenyoentttntosmhe beafreossgsrareidms eiptsesohtrn fiesbiri naub2gtl e6imot4fnhoFerensmhtt hosHw)e .fntFiaTsiinh lseaoitfhaoptsehnpyiemn reismoxentpeatteoorcrfiic hmFtm iewgnao.todsa3el.
y (eV)y (-1-100 50 132smStaSyklemyntqrrmumi.ca eedtTerrfliuouchpnnocgecoltaei mtopemnpdaaoolcfmtrtaaeenfgndrnmtasgtceumainrlatfecanulc(tlse aeet(fpes)dancif(r)easilnoo(rglbnigdlgauyserteh addefafrufislihnosnseceimdt)oionlan innatoedxh)fi,saet.hss yyeTmmswtoa--o-meIqtu iicsreaaldlscio↵ueinrlteaenrtteistotiminneg stiotso nrsoethaecothhwasctnitsh sieionnth. rFeTeihgeevs.oe lu5titm.io enWssareree-
gMg remficseostirogincnveilaosllneegya stteahdreefrr eapngrmeuseecnntltee(daa:efro)n (egmrleeeaandgindigoctteo dnslyiunmemm).ebtTriehcerfrsaar g--in the shell structure of the fragments.
ners (ner 28 ScaTmphsrcm,oae Cewlc nSugt,c slLoarar(tcredieroaosenispxhsn,o ePfnod hddyrsleti.a noaRecest)hvhh. aemCne so(dtRdda)oe r- t.n liinenIi spgntoroecdesoesa nnsifisynigmtuidemrsaietactirtoiacnhatoflerffatstghhm eetehdsnaytetsnua r(Fmadtoiiectotaenrldmi⇠m loe2dvezess,,l.r⇠ esp5.e4ctzivs,elayn.dT⇠he3s.e2vzasrifaotriotnhse ascrfe, laikefe,lyantdo sbeef
le eevelle e-2-200 20 dfiliegnnues)ri.attAyiod⇢na0sp/t2aend=dfr0fo.o0mr8t[fh2me5�]2.358aFrmeaglsrooupnlodt-tsetadtae.ttheseinitialcon- dsluoepetsoaandcodmi↵beirneanttioonneo-fbtowdoyfvaicsctoorssit:ydwi↵heicrhenitspexopteencttieadl
clc
tin ti This novel idea of using the internal structtuordeep teond iodnesnhetlilfey↵ etchtse. transition between
parotopar-3-300 8 noTnhT-isahemdmaiyaeabbne-afidetiffilidcci usalotnbtotdadi neaeteddrmwiaiitnhbetafhrtoeimScl ythr4edet[go29tia]mlSdkeeynsrs miwtey.ill bTeh egfiennaletrotaalliksineedti cteone rmgyo(rTeK Ec)oomf thpelfircagamteendts
gle Prgle syetHfshontoreeemwrpgmeeyavdisedrfire nr,nogismstictuihysthacfnpeuhonns secislnt.iigabollTnesea-h plteoaatrnCthsdiHecoelaFescw+loeehBnv esCertleSaisnnf[atr6cn-aG]gdl.umiTIndnetDneittHrhnsaeFcgat+cri eaoBsnpeCpirSnoea-firiidnsisgatin nogtcuhioesrhrirbmeeptlwoaretetaninottnhoesbs.fie srsviIaotbn lemwwodhielicslh. cIenannpaburebelluyesea ddiatao-
SinSin-4-400 2 coce2n6av4ls8Fcuim[sl3a,t0tte]ihonaennstpd raottrterodenhobafat3ntaddminn[e2eed9un]twrtcoi ontdhoseisfmn, gorbldeesio-fippeteadchrtti ivcvealeerlsydei,onianeassrsgbuoimeafstitnhaigrece absncaidtsisc ioannpopcornon-aficaghudersai,tatiohbneaTwtKhiicEch isipsuhisduaeansltlyiefiseesd tioomnfa tthefedispfrsootimeonnttiha,el
plottedinthemiddleandbottompanelsofFig.1,respec-
) 0 180 10 12 12 1414 whotinivcleylhyo. nhMeaapnslay nbleeevoeeflnssy clmraomscsektartiyns.hgoAr tlsldocisa tflacanucrlae tii(noRn sp<ar1ree0vpFiemor-)u. s emneircgyrosusrcfaocep,bicas esdtoundsioemse.c r iteria[6,8,19]. Thead-
V RR ((ffmm)) 82 formed on a Cartesian grid of 88 19.2 19.2 fm3 with a vantage of using the TDHF+BCS approach is not only
e mWeeshalssiozeo0b.s8erfvme.sTudhdeetnimjuemevposlui⇥ntiothneis⇥sionbgtlae-inpeadrtiwcliethena- to include non-adiabatic e↵ects in the formation of the
MV)-10 ergies due to changes of macroscopic shapes. At larger
T (eime evolution and final pro5p0erties of dtihstaen cfers,awgemobesenrvtes the appearance of energy gaps as-
yM
g sociated with the spherical magic numbers. As the 132Sn
Tr(h-e2 0main outcome of this project will be to describe the properties of fission and quasi-fission fragments. These
es fragments are doubly magic, there is no crossing of the
ienelnclude the distribution of the2i8r mass anFder mcihsuarrfagcee,a ftaers thwemealgl icagsa ptshaetZir= e5x0canidtaNti=on82 and kinetic energies and their angular
micle n levo-m30entum. Although TDHF 2i0s optimiseadre bTfohotehrs fyotmrhmmeee dte.riTxchfipissesoicocntcaumrtsoiadotenRo f∼v251a80Fl.mu5efims ao.lsfo dorniveen-bbyody observables (Balian and Vénéroni,
1artro98-440), nucleon number distributions of tshphee roicaultsgheollineffge ctfsr[a4g4]m. Iendnetesd, cthaenfr abgmee nctaslhcavuelaated using a particle number projection
te peutechnique (Simenel, 2010). T8his is becparoutosnem aTgiDcnHumFb eriZs =a5 0f.uThlleyev omlutiiocnroofspcrootopniacn dapproach which is not limited to the
dnglNes-5c0ription of the gross feature2s of the dynsyenmuatmrmoentrisiccinsgm leao-dspea .srthiAcolseiwnlevnthe elsi2na6r4 eFFmsihgocw.a sn1e,i nwaFenidogb. s36er,fvo ebr ttuhheet also gives access to the time evolution
i
oSf each single-particle wave function. Refocremnattiolyn,o fthZe= C50Ia nsdigNn=ifi8c2amnatglicyg aimpspafrteorvsoemde the theoretical description of fragment
-60
8 10 12 14 global elongation of the fissioning system (quantified by
mass and charge distributions by applying a prescription from Balian and Vénéroni, 1984 (Simenel, 2011). This
R (fm) the quadrupole moment) has been reached. However, in
prescription is also known as the time-detpheecnadseeonfnte RutrPonAs, th(eTleDveRlsnPeAart)h.e Fermisurface(rep-
Figure 1. Adiabatic calculation of 264Fm symmetric fission resentedwithadashedline)areonlypartiallyfilled.
These theoretical tools will be adapted to the fission studies to get a deep insight into the properties of fission
with the SLy4d Skyrme functional and surface pairing. The Thecaseof258Fmasymmetricfissionismorecompli-
dfisrtaangcembeetwnetesn.t heSfryagsmteenmts Ratisicde finceadlincuRelfa.t[6i]o.n(tsop )andc atecdoamsthpeafrraigsmoenntss arweniotthm ageixcpanedrtihmereefonretacaln bme ass and kinetic energy distributions
Adiabatic potential and isodensities at half the saturation den- deformed. Indeed, the associated single-particle energies
obtained at the Heavy-Ion Accelerator Facility will provide a fundamental understanding of the role of quantum
sity ρ /2 = 0.08 fm−3. The energy is defined with respect to
0 plotted in Fig. 4 show no energy gap neither for protons
tshehaesylmlspt ooticnfi ntahle13 2Sfno+r1m32Snasttiaoten. S iongfl et-phareti cfleinlevaells farreagmneornftosr.n eSuitrmonisl.aTrh ecroefmorep, aitriissnootna lbweaytswsteraeignh tefoxr-periment and theory will be carried out
plottedforprotonsandneutronsinthemiddleandbottompanels,
rteosp euctinvedlye. Prosstitaivnedan dthneega trivoelpear itoyfst adteissasreisphaowtnioinnre din thwea rddtyonidaenmtifiyctshe, pirne- fporamrattiiocnuolfathre ifnra gtmheen tsrseimppalrytition of the final excitation energies of
bylookingattheformationofsphericalmagicgaps.
andblue,respectively.Adaptedfrom[6].
the fragments.
To describe the non-adiabatic phase of fission, one has to start the time-dependent calculation with a static HF
3 TDHF+BCS calculations
calculation under one or several external constraints at the transition between adiabatic and non-adiabatic regimes
In general, it is necessary to consider a range of initial
(see previous section). A recent exemple coonfd istiuoncshin ao rcdearltcouinlvaetsitiogante itshe sdhynoawminca lienv oFluitgio.n 1.
along the fission path with time-dependent calculations.
Note that the resulting sequence of shapes around scission presents differences with the standard adiabatic description
Startingwithamorecompactconfigurationisdesirablein
(Fig. 5, see Simenel and Umar 2014) whoircdehr taorceap tcurreitmicosatlo fftoher dayn pamroicapleeffr edctestinetrhmeeivno-ation of the fragment characteristics. In
lution along the fission path. However, the TDHF+BCS
approach accounts only for part of the correlations and it
DP150102372 (In draft) Dr Cedric Simenel PDF Created: 13/03/2014 Page 10 of 43
NSRT15
SCF protons AEF protons
-3 -4
Protons Protons
-4
-5
-5
V) V) -6
Me -6 Me
E (s.p. -7 E (s.p.--87
-8
-9 -9
AEF neutrons
-10
-2 60 80 100 120Ne14u0tr1o60ns180 200 220 240 -2320 240 260 280
Q [b] Neutrons
20 Q [b]
20
-4
-4
)
V ) -5
e V
M e
E (s.p.-6 (Ms.p.-6
E -7
-8
-8
60 80 100 120 140 160 180 200 220 240
Q [b] 220 240 260 280
20 Q [b]
20
Figure3.Adiabaticcalculationof258Fmsymmetricfissionwith
Figure 4. Same as Fig. 4 for the asymmetric fission valley of
theSLy4dSkyrmefunctionalandsurfacepairing.Single-particle
258Fm.
levelsareplottedforprotonsandneutronsinthetopandbottom
panels,respectively.ThedashedlineshowstheFermilevel.
sult,thetimevariationoftheoccupationnumbersinduces
islikelythatsomecorrelationswillbemissingforthemost anevolutionofthepairingenergy[25].
compactconfigurations. Thecomputationaltimeisalsoa Accounting for dynamical pairing is also crucial for
practical limitation. However, starting with a configura- fissiontooccur. ThisisillustratedinFig.6whichshows
tiontooclosetoscission,wewouldmisssomeofthedy- theevolutionofthequadrupolemomentasaproxyforthe
namicaleffects. Inparticular, thefinalobservables(mass totalelongationofthesystem.Inthefrozenoccupationap-
and charge distributions, total kinetic energy...) may ex- proximation(FOA),i.e.,assumingthatthesingle-particle
hibitadependencewiththeinitialconfigurationwhenthe numbers are those of the initial configuration, the system
latteristooclosetoscission[25,26]. isnotalwaysabletofission. Inaddition,trajectorieslead-
Acompromisehastobefoundbyinvestigatingseveral ingtofissionintheFOAexhibitasignificantdependence
initialconditions. Asanexample,wefoundthatthefinal ofthefinalobservableswiththechoiceoftheinitialcon-
properties of the fragments in 258Fm asymmetric fission dition[26].
wereessentiallyindependentontheinitialconditionifthe Aninterestingfeatureofthedynamicalcalculationsis
latterwaschosentobeintherange210b< Q < 270b the possibility to predict the total kinetic energy. As dis-
20
(see Fig. 2). The sensitivity to the initial condition (in cussed above, the latter is essentially independent of the
TDHF calculations without dynamical pairing) has also initialconditionifitisnottooclosetoscissionandifpair-
recently been investigated in details by Goddard and col- ing correlations are included dynamically. The total ki-
laboratorsinref.[26]. netic energy of the fragments is computed from the sum
The importance of dynamical pairing correlations in of their Coulomb repulsion and kinetic energy. In partic-
the evolution was demonstrated in Ref. [25] at the BCS ular,thesequantitiesarewelldefinedafterscissionasthe
level. The time-dependent BCS equations, giving the nuclearinteractionbetweenthefragmentsvanishesdueto
evolutionofthesingle-particleoccupationnumbers,were its short range nature. Indeed, we see in Fig. 6 that the
solved simultaneously with the TDHF equation for the totalenergyisconstantafterscission. Itisequaltothefi-
evolution of the single-particle wave-functions. An ex- nal total kinetic energy (TKE) as the Coulomb energy is
ample of the resulting dynamics of the occupation num- transformedintokineticenergyatlargedistances.
bersisshowninFig.5forthesymmetricfissionmodeof Figure 7 shows a comparison of the predicted frag-
258Fm. An important rearrangement of the neutron occu- mentmass(top)andTKE(bottom)[25]withexperimen-
pationnumbersisobservedduringtheevolution. Asare- tal distributions [44] in 258Fm fission. The experimental
EPJWebofConferences
1500
)
b
(
t
n
e
m
1000
o
m
e
l
o
p 500
total mber Neutrons dru total mber Neutrons
nu ua nu
Coulomb pation Q 00 2 T4ime (zs6C) oulom8 b 1pation 0
kinetic u kinetic u
c c
c c
O O
Figure 6. Time-dependent Hartree-Fock with dynamical BCS
Single-particle energy (MeV) Single-particle energy (MeV)
pairing (TDBCS) and in the frozen occupation approximation
total mber Neutrons (aFsyOmAm)ectarilccufilastsiioonnswoifthththeetoStLayl4qduaSdkryurpmoleefumnocmtitooenntatallinanmber2d58sFumr- Neutrons
u u
n facepairinginteraction. n
Coulomb on Coulomb on
ati ati
p p
kinetic u kinetic u
c c
c c
O O
total mber Neutrons total mber Single-particle enNeerguyt r(oMnesV) Single-particle energy (MeV)
u u
n n
Coulomb on Coulomb on
ati ati
p p
kinetic u kinetic u
c c
c c
O O
Coulomtbotal on number Single-particle eNneerugtyro (nMseV) CCoouulloommttbboottaall on numberpation number Single-particle eNNneeeruugttyrroo (nnMsseV) Coulomtbotal pation number Neutrons
ati kinetic aticu kinetic cu
p pc c
kinetic u kinetic uO O
c c
c c
O O Single-particle energy (MeV) Single-particle energy (MeV)
Single-particle energy (MeV) total mber Single-particle eNneeurgtryo (nMseV) total mber Neutrons
u u
n n
Coulomb on Coulomb on
ati ati
p p
kinetic u kinetic u
c c
c c
O O
total mber Neutrons total mber Single-particle enNeerguyt r(oMnesV) Single-particle energy (MeV)
u u
n n
Coulomb on Coulomb on
ati ati
p p
kinetic u kinetic u
c c
c c
O O
Single-particle energy (MeV) Single-particle energy (MeV) Figure7. Experimentalmass(top)andTKE(bottom)distribu-
total mber Neutrons total mber Neutrons tpiroendsicintio25n8sFwmitfihssdiyonnafmroimcalRpeafi.r[i4n4g]c.oTrhreelaartiroonwss.showtheTDHF
u u
n n
Coulomb on Coulomb on
ati ati
p p
kinetic u kinetic u
cc cc datapresentatailinthemassdistributioninterpretedasan
O O
asymmetricmodeincompetitionwiththedominantsym-
Single-particle energy (MeV) Single-particle energy (MeV)
metricmode[44]. Thecalculationsalongtheasymmetric
valleypredictmassesinagreementwiththistail. Thecal-
Figure 5. TDHF with dynamical BCS pairing calculation of
culatedTKEforbothmodesalsoagreewiththetwomain
258FmsymmetricfissionwiththeSLy4dSkyrmefunctionaland
contributions in the experimental distribution, namely a
surface pairing interaction. The density profile, energy repar-
peak at high TKE and a tail at low energy. According to
tition of the fragments and single-particle neutron occupation
numberdistributionsareshownatfourdifferenttimes. thecalculations,thehighTKEmodeisassociatedwiththe
NSRT15
n magic gaps in compact symmetric modes leading to tin
io 2+1 Quadrupole fragments are formed well before scission, indicating a
t
c pre-formationofthefragments. Accountingforthetime-
n
u dependenceofthepairingcorrelations,leadingtoanevo-
f2
Qh lution of the occupation numbers, is crucial to allow the
t GQR
g system to fission and to reduce the dependence with the
n
e initial condition. The results are used to interpret the ex-
r
St perimental mass and kinetic energy distributions. It is
5 10 15 20 shown that at least part of the internal excitation energy
E (MeV) is stored into low-lying collective vibrations, while, as in
fusion,thedynamicsislesscoupledtohighenergymodes.
on 3-1 Octupole Quantum fluctuations beyond the independent
cti particle/quasi-particle picture need to be incorporated in
n
u the future in order to reproduce the experimental distri-
Q f3 butions. The mass and charge distributions in the final
h
t fragments of a TDHF evolution can be calculated with
g
en HEOR a particle number projection technique [63]. However,
r these fluctuations are underestimated at the mean-field
t
S
level [64]. A possible approach is to incorporate beyond
10 20 30
E (MeV) mean-field fluctuations with the TDRPA [65] which has
already been applied to heavy-ion collisions [66–68].
Applications to fission (neglecting pairing correlations)
Figure8.Quadrupole(top)andoctupole(bottom)strengthfunc-
are promising [25]. However, numerical solution of the
tionsinthe132Snfissionfragments(solidlines)comparedwith
TDRPAincludingpairingstillhastobedone.
RPAcalculationsofvibrationalmodesbuildonthe132Snground
state.AdaptedfromRef.[6].
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