dynamical microscopic and macro- micro approaches to

Dynamical Microscopic and Macro-Micro Approaches to Nuclear Fission

FIESTA2017

•  M.D.Usang,C.Ishizuka,Y.Iwata,S.Okumura,A.Etori,T.Nishikawa(TokyoTech)•  F.A.Ivanyuk(TokyoTech.,Kiev)•  J.A.Maruhn(FrankfurtU.)•  A.Ono(TohokuU.)•  M.Kimura(HokkaidoU.)•  T.Kawano(LANL)

SatoshiChibaProf.,TokyoInsQtuteofTechnology(TokyoTech）Director,AtomicEnergySocietyofJapan(AESJ)

Contents

•  IntroducQonandBackground• NuclearfissionbyLangevinequaQon:Macr-micro• NuclearfissionbyanQsymmetrizedmoleculardynamics(AMD)

• NuclearfricQonbyTDHF•  Summary

2

Introduc;on and Background

•  Eventhoughthehistoryofresearchesinnuclearfissionisalreadymorethan70years-long,manymysteriessQllexistactuallywhatisgoingonduringtheprocessofnuclearfission,especiallybeforescission

•  DuetothedifficulQes,accuracyofthefission-relatednucleardataissQllconsiderednothighenough

•  ToresolvethissituaQonandtounderstandmanydifferentaspectsofnuclearfissioninaconsistentway,wethinkaccuratedynamicaltreatmentofnuclearfissionisnecessary

• WearemakingresearchesinfissionbydynamicalapproacheswithLangevinequaQon,AMD(anQsymmetrizedmoleculardynamics)andTDHF

•  Hereweexplainthembriefly 3

Origin of extra spin that fission fragments have

Spinoffissi

on

fragment

148Pm

7ℏ

4

spinof148Pmfromfissionofn+235UsystemasafuncQonofexcitaQonenergyofcompoundnucleusAumannetal.,Phys.Rev.C16,254-265(1977).

5

タイトルタイトル

PRC21,204(1980)Maruhnetal.,Phys.Rev.C74,027601(2006).

Challenges

Dofissionfragmentsrotate?Ifso,whatistheoriginoftherotaQonalmoQonandhowmuchangularmomentatheyhave?

TwisQng

Bending

TwisQngmodeinfusionreacQons

TwisQng&Bendingmodesinfission

TDHForAMDapproach

6

AnomalyintheaverageTotalKineQcEnergyofFissionFragments

7

Our approaches (CN to scission)Macro-microapproach：Langevinmodel•  PESfrommacro-micro,transportcoefficientsfrommacroormicro+randomforce•  3DLangevinwithmacroscopictransportcoefficients(withF.Ivaniuk)•  3DLangevinwithmicroscopictransportcoefficients(bylinearresponsetheory)•  4DLangevinwithmacroscopictransportcoefficients•  4DLangevinwithmicroscopictransportcoefficients•  5DLangevin--talkbyA.SierkMicroscopicapproachesAnQsymmetrizedMolecularDynamics(AMD)•  SlaterdeterminantwithGaussianwavepackets(coherentstate)asbasisfuncQon•  meanfieldcalculatedwithSLy4interacQon•  StochasQcnucleon-nucleoncollisionstoexpressbranchingofSlaterdeterminant

anddeformaQonofGaussianwavepacket

Time-dependentHartreeFock(TDHF)•  NorestricQononsingle-parQclewavefuncQon(3DmeshcalculaQon)•  meanfieldbySV-BasinteracQon•  basedonOAK3DandSKY3D(nopairing)•  toobtainnuclearfricQoncoefficientfromdynamicalmodel

8

neutron

235U

SimulaQonofnuclearfission(235U+140MeVn)byJQMD

JQMD:JAERIQuantumMolecularDynamics=asemiclassicalmoleculardynamicsfornuclearreacQons(meanfield+NNcollision)

9

JQMD(分子動力学)による140MeV中性子＋238Uの核分裂の時間発展　　　

K.Niita, T. Maruyama, Y. Nara, S. Chiba and A. Iwamoto, JAERI-Data/Code 99-042(1999)

TimeevoluQonof235U+140MeVnreacQonbyJQMD

QMDsimulaQonofnuclearfission(235U+140MeVn)

Nuclear fission by Langevin equa;on

These2differentd.o.fhavedifferent2mescales:

BrowningmoQon

• nucleonmo2on:1to10fm/c• shapemo2on:～>10,000fm/c

NuclearshapeevoluQonisdrivenbyrandomkicksbynucleonsinthermalequilibrium(microscopicd.o.f.)giventothenuclearsurface(macroscopicd.o.f)frominsidethesurface

10

qi : deformation coordinates （nuclear shape） in two-center shell-model 　　 parametrization

pi : : momentum conjugate to : mass tensor : Hydrodynamical mass (Werner-Wheeler)

( )

( ) ( ) )(21 11

1

tRgpmppmqq

Vdtdp

pmdtdq

jijkjkijkjjkii

i

jiji

+−∂

∂−

∂

∂−=

=

−−

−

γ

ijjkk

ik

ijjii

Tgg

tttRtRtR

γ

δδ

=

−==

∑)(2)()( ,0)( 2121

( ) 21*int )(

21 aTqVppmEE jiij =−−= −

paramterdensity level:energycorrectionshellStrutinsky:)(

)()()(:n deformatioat potential : )V(re temperatu: energy, excitation : energy, intrinsic : *

int

aqV

qVqVqVqqTEE

sh

shLDM +=

多次元ランジュバン方程式　　Langevin Equations for nuclear fission

Einstein relation

orfromLinearResponseTheory(microscopic)

iq

ipijm

iq

ijγ : friction tensor : Wall and Window formula (one-body dissipation)

Intrinsic energy and Temperature :

Shell correction to the potential energy :

White noise

Friction　Random force

Random force :

11

Macroscopictransportcoefficients

{ } },,,{ 2104 αδδZZq D ={ } },,{ 2103 αδδ == ZZq D

Two-center oscillator model (Maruhn and Greiner, Z. Phys. 251(1972) 431)

　　Shape parametrization

: Radius of compound nucleus　　　　　　

Massasymmetry

ElongaQon

DeformaQonoffragmentsatouterQps2,1,2

)(3=

+

−= i

baba

ii

iiiδ

●

●

●

RzZZ 0

0 =

21

21

AAAA

+

−=α

R

12 11222

2

2 2 1 1

1

b1b1

b2b2 b1

35.0=ε neck parameter : fixed●

Rneck

Collective coordinates （4 dynamical variables)

12

4D : are independent, 3D : 21, δδ

{ } },,,{ 2104 αδδZZq D =

fragmentleft theof mass : fragmentright theof mass :

2

1

AA

3/12.1 CNA=

{ } },,{ 2103 αδδ == ZZq D

21 δδ =

Notethatneckradiusdependsonallthe5parameters

13


IniQalcondiQonofLangevincalculaQon

PotenQ

alene

rgy

ElongaQon

GroundState

Shapeisomer

Neutron-inducedfission

Tunnelingfission

Fissionfromisomer

Spontaneousfission

Westartfromhereorherewithzeroini2almomenta(thereisnodistribu2onintheini2alstate)

Manypeoplestartfromthisregion

{ } { }( ) { } { }( ) { }( )pqqpqP δδ 0, −=

{ }0q{ }0q

Exampleof4DLangevintrajectories(236U)

(MeV)V

RzZZ 0

0 =14

234U 238Np

240Pu

242Am

MassDistribuQonofFissionFragmentsatEx=20MeV

258Fm

15

258Fm

Distribu;ons of TKE (total kine;c energy of fission fragments)

236U

SuperLong

SuperShort

Q-value

16

Systema;cs of <TKE> by 3D Langevin

17

Extension to 4D Langevin

3DLangevin

4DLangevin

235U+nsystem

{ } },,,{ 2104 αδδZZq D =

18

19

(MeV

)

(MeV)

Present(Macro4D)

1stchance

σTKE by 4D Langevin

3DLangevin

4DLangevin

235U+nsystem

Nuclear fission by An;symmetrized Molecular Dynamics: AMD

•  EventhoughtheLangevin-modeldescripQongivessaQsfactoryresultsinmanyaspectsofnuclearfission,ithasonly3to4degrees-of-freedomtoexpressnuclearshape.Itshouldbebemertohavewaystodescribethenuclearfissionbymicroscopictheorieshavingd.o.f.ofallthenucleons

•  AMDisaverysuccessfulmicroscopicmodelofnuclearreacQonsandstructures(A.Onoetal,Prog.Theor.Phys.,87(5),(1992),1185-1206)

•  WavefuncQonofeachnucleonisexpressedbyaGaussianwavepacketregardedasacoherentstate

•  ThetotalwavefuncQonisgivenbyaSlaterdeterminantofthesesingle-parQclewavefuncQons→ unlikeQMD,thisisaproperlyanQsymmetriedquantum-mechanicaltheory

•  SkyrmeinteracQonisintroducedasaneffecQvenuclearforce•  Ground-stateofnucleiareobtainedbyfricQonalcoolingmethod,whichyieldsgoodBE/Aandnuclearradii

•  TheequaQonofmoQonofwavepacketsisgivenbyQme-dependentvariaQonalprinciple

•  StochasQcnucleon-nucleoncollision,whichisabsentinTDHFtheory,isintroducedwithPauliblockingeffectinthefinalstate,whichwillactasasourceof2-bodyfricQon. 20

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U,expU,Sly4

BE/A for U isotopes

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A

cros

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b)AMDexp

180MeVp+27Al

180MeVp+27Al

AMD-1 : Basic formula;on

Time-dependentvariaQonalprinciple:

Single-parQclewavefuncQon:Gaussiancoherentstate

21

< 𝒓 | 𝝓↓𝒁↓𝒊   > = ( 𝟐𝝂/𝝅 )↑𝟑/𝟒   𝐞𝐱𝐩[−𝝂(𝒓 − 𝒁↓𝒊  /√𝝂  )↑𝟐 ]

< 𝒓 | 𝝓↓𝒁↓𝒊   > = (𝟐𝝂/𝝅 )↑𝟑/𝟒 𝐞𝐱𝐩[−𝝂(𝒓 − 𝒁↓𝒊  /√𝝂  )↑𝟐 ]𝝌↓𝜶↓𝒊  

𝒁↓𝒊  =√𝝂 𝒓↓𝒊  + 𝒊/𝟐ℏ√𝝂  𝒑↓𝒊  

|𝜱(𝒓↓𝟏 , 𝒓↓𝟐 ,…, 𝒓↓𝑨 )> = 𝟏/√𝑨!  𝐝𝐞𝐭[𝝋↓𝒊 (𝒓↓𝒋 )]TotalwavefuncQon

𝝌↓𝜶↓𝒊  =𝒑↑,𝒑↓,𝒏↑,𝒏↓

𝜹∫𝒕𝟏↑𝒕𝟐▒𝒅𝒕𝚽𝒊ℏ 𝒅/𝒅𝒕 −𝑯 𝚽 /𝚽𝚽  =𝟎 

EquaQonofmoQonofthemeanfield:

𝒊ℏ∑𝒋𝝉↑▒𝑪↓𝒊𝝈,𝒋𝝉 𝒁↓𝒋𝝉  = 𝝏𝓗/𝝏𝒁↓𝒊𝝈↑∗    𝐧𝐝 𝐜.𝐜.

𝓗= <𝚽|𝑯|𝚽>/<𝚽|𝚽> 𝑪↓𝒊𝝈,𝒋𝝉 = 𝝏↑𝟐 /𝝏𝒁↓𝒊𝝈↑∗ 𝝏𝒁↓𝒋𝝉  𝐥𝐨𝐠<𝜱|𝜱>where

•  EffecQveN-NinteracQon(originofnuclearmean-field)•  SLy4interacQon

AMD-2 : NN interac;ons

• StochasQcNNcollisionasapartofresidualinteracQon(AtestparQclesfromWignerfuncQon)

mimicsrandomforceinLangevintheoryü In-mediumNNcrosssecQonü PauliBlockingaoerthecollisionü branchinganddeformaQonofwavefuncQons-->tunneling,distribuQon StochasQce.o.m.

𝑣↓𝑖𝑗 =𝑡↓0 (1+ 𝑥↓0 𝑃↓𝜎 )𝛿(𝒓)+ 1/2 𝑡↓1 (1+ 𝑥↓1 𝑃↓𝜎 )[𝛿(𝒓)𝒌↑2 +𝒓𝛿(𝒓)]+ 𝑡↓2 (1+ 𝑥↓2 𝑃↓𝜎 )𝒌・𝛿(𝒓)𝒌+ 𝑡↓3 (1+ 𝑥↓3 𝑃↓𝜎 )[𝜌(𝒓↓𝒊 )]↑𝛼 𝛿(𝒓)

22

jiijB ϕϕ=

( ) ( ) 1

1 1

222/32 −

= =

−−∑∑⎟⎠

⎞⎜⎝

⎛= jiij

A

i

A

jBBe ijRrr ν

πν

ρ ( )jiij ZZR += *

21ν

ji rrr −= ( )ji ppk −=!21where

Ini;al state of 236U (235U + n)AsafirststeptosimulaQonofnuclearfissionbyAMD,herewesimplygiveaboostmomentumtoeachnucleon,totheoppositedirecQonsfornucleonsstayingattheleoregionandrightregion

23

E+～1.3MeV/u

Time evolu;on by AMD, 236U (235U + n)

24

𝐸↑∗ =309.2361[MeV]

Examination of phenomenological assumptions

𝟐𝒍=𝟏𝟏𝒓�𝒍=𝟓.𝟓𝒓

AMD　：　𝒍～𝟓𝒓

Brosa, Grossmann & Müller, Phys.Rep.197, 167(1990)

•  Brosa's random neck rupture condition due to hydrodynamical Rayleigh instability

2l

r

25

ℓ2

ThisverifiesthatBrosa'sesQmaQonofneckrupturecondiQonissensible

26

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0.02

0.04

0.06

0.08

0.10

A mass number

FPY

FPY

Results : FPY (left) and rotational angular momenta (left)

•  Due to our specific way of boost, the mass distribution corresponds to symmetric fission, which is categorized as super-long mode in Brosa's terminology (low TKE component)

•  Each fragment has, on the average 5 to 10ℏ (on the average around 7ℏ) of rotational angular momentum, which is enough to account for the origin of extra spins of fission fragments as explained in the introduction

File = angular Date = 22:16 06-Sep-2017

100 110 120 130 1400

5

10

15

20

A mass numberL[

h̄] L

236U,boostfissionbyAMD

FragmentmassdistribuQon AngularmomentumdistribuQon

Ac/2

27

千葉先⽣⽤TKE (only exp)と2核⼦衝突ありの動画

80 100 120 140 160120

140

160

180

200

220

Mass, A(u)

TKE(

MEV

)Results : Mass-TKE correlation (left) and excitation energy of each fragment (left)

exp:236UatexcitaQonenergyof12MeV

•  Our result, which corresponds to the symmetric components, has the same TKE values deduced from experiment. This is in some sense quite satisfactory as well as surprising if we consider the difference of excitation energy of the data (20MeV) and our calculation (300 MeV).

80 100 120 140 160120

140

160

180

200

220

Mass, A(u)

TKE(

MEV

)

AMD

80 100 120 140 160120

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Mass, A(u)

TKE(

MEV

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A mass numberE*

[MeV

]

E*

•  This model has a capability to deduce excitation energy of each fragment, which is not the case in Langevin-type approaches. The results E*〜A denotes equi temperature Fermi-gas scheme

1.  ThereseemstobenooscillaQonintheelongaQondirecQonatalleventhoughthefactthatthewaywegavetheiniQalcondiQonisexpectedtoresultinstrongoscillaQon.However,theredoexistsurfaceoscillaQon

2.  ThisresultisincontradicQontoLangevintheorypredicQons,butinaccordwithTDHFandQMDresults

3. WenoQcethatneckisthickerthanweexpected,andparQcleexchangethroughtheneckregionseemstobequiteimportant,whichmayresultinrotaQonoffragments

4. Nuclearshapeisquitecomplicatedduringfission,whichisoutofreachifweuseonly3,4or5shapeparameters(suchasusedinLangevindescripQon).Inthisrespect,descripQonofnuclearfissionbymicroscopictheoriesisrecognizedtobeveryimportant

Results from AMD calcula;on for 236U (235U + n)

28

skipTDHF

Summary

Nuclear fric;on by TDHF

•  We calculate the relative distance of fragment centers

•  From its time evolution, we derive average friction coefficient as explained in the next slide

𝜌(𝒓)=∑𝑖↑𝑁▒|𝜑↓𝑖 |↑2   𝑅=|∫↑ ▒𝒓𝜌(𝒓)𝒅𝒓 |𝑖ℏ 𝑑/𝑑𝑡 𝜑↓𝑖 = ℎ ↓𝑖 𝜑↓𝑖 

410

86

29

How to obtain fric;on coefficients

Fromthemacroscopice.o.m.forthecenterdistanceR

∫𝐼↑𝐹▒𝜇𝑅 𝑑𝑅 +∫𝐼↑𝐹▒𝑑𝑉/𝑑𝑅 𝑑𝑅 +𝛾∫𝐼↑𝐹▒𝑅 𝑑𝑅 =0

Wecanintegrate

𝜸= (𝟏/𝟐 𝝁𝑹↓𝑩𝑭↑𝟐  + 𝑽↓𝑩𝑭 )−(𝟏/𝟐 𝝁𝑹↓𝑨𝑭↑𝟐  + 𝑽↓𝑨𝑭 )/∫𝑩𝑭↑𝑨𝑭▒𝑹 𝒅𝑹  

𝜇𝑅 + 𝑑𝑉/𝑑𝑅 +𝛾𝑅  =0

ThisistheaveragefricQoncoefficients,whichdependsoncollisionenergy

Numbersinredcharacgersinducate 𝛾∕𝐴 at 𝐸↓𝑐𝑚 ∕𝐴 =7MeV

𝐵𝐹:BeforeFusion𝐴𝐹:AoerFission 30

For 236U (=118Pd + 118Pd) system

Wall-and-Windowfric2onisa(sta2candsemiclassical)hydrodynamicalresult,whereasourresultsarepurelydynamicalandquantalones,buttheycoincidequitewell. 31

FricQonbyTDHF

Wallandwindow

32

Possibility of describing ternary and quarterly fission by TDHF

6MeV/A 7MeV/A

208Pb+208Pb

Summary•  Wehavedescribed3differentdynamicalapproachestonuclearfissioncurrentlystudiesatTokyoTech:

1.  DynamicaldescripQonoffissionbyLangevinequaQonisexplained•  CalculaQonsareperformedforacQnidestoFmregionwithboth3Dand4Dversions,macroandmicrotransportcoefficients

•  MassandTKEdistribuQonoffissionfragmentsarereproducedwithahighaccuracy,includingtheirsystemaQcaltrends

•  Withthisdegreeofaccuracy,thismethodcanbeconsideredtobeoneofthedataproducQontool

2.  OurfirstamempttodescribenuclearfissionbyAMDwasdescribed•  Quitepromising•  RotaQonalangularmomentumcouldbealreadyderivedtobefinite•  Itcanaccountfortheextraspinwhichfissionfragmenthasattheonset

3.  TDHFapproachwasalsodescribed•  todeducenuclearfricQonfromquantaldynamicalmodel•  possibilityofdescribingternaryorquarterlyfission

33

34

ThankyouforyouramenQon

35

AsymmetricandfinestructuresofmassdistribuQonofprimaryfissionproductsfromneutron-inducedfission

A=134A=100

post-neutronmassdistribuQon

36

タイトルタイトルn

n

n

γγ

e-e- n

γ

StaQsQcaltheory

DynamicalTheory Brosa-inspired

TDGCMTDHFAMD

RandomWalkLangevinequa2on

MulQ-modalanalyses

Weisskopf-EwingMadland-Nix(LAM)Hauser-Feshbach

GrossTheoryQRPAShellModel

β-decay+staQsQcaltheory

Scission•  IsotopedistribuQon•  SharingofE*between2fragments,•  JπdistribuQon

eν eν

Weisskopf-EwingHauser-Feshbach

StaQcTheory

FongWilkinsCHFRMF

NuclearTheorieswhichcandescribevariousaspectsoffissionCoupled-ChannelsMethod

37


n

n

即発中性子及びγPromptnandγn

γγ

e-

e-n

γ

核分裂片(FissionFragment)

（一次)核分裂生成物(PrimaryFissionProduct)

複合核の形成 Compoundnucleus

サドル点 Saddle,2ndminima

断裂点 Scission

β-decay

eν

独立収率(IndependentYield)

累積収率(CumulaQveYield)

1次収率(PrimaryYield)

NuclearData

Isotopedistribu2on,sharingofexcita2onenergy,spin-paritydistribu2onsdetermineactasini2alcondi2onsforthefollowingprocesses,butarenotknowwell

（二次)核分裂生成物(SecondaryFissionProduct)

断裂中性子Scissionneutron

Delayedn・γ

TimeevoluQonoffission

courtesyF.MinatoConstrainedHartree-Fock

38

neutron

235U

SimulaQonofnuclearfission(235U+140MeVn)byJQMD

JQMD:JAERIQuantumMolecularDynamics=asemiclassicalmoleculardynamicsfornuclearreacQons

39





neutron

235U

40




NuclearelongaQonseemstoevolvemonotonically(notoscillatory)insemi-classicalmoleculardynamics.ThisisalsoseeninTDHFcalculaQonbyBulgacetal.(Univ.Washington)FissionQmescale～ 10,000fm/c


41

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nerg

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Binding energy/A of medium and heavy nucleiby AMD

Sn isotopes U isotopes

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time[fm/c]

Skyr

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−5800

−5600

−5400

time[fm/c]

Skyr

me

Forc

e[M

eV]

Skyrme force

Time evolution of <V> before (left) and after (right) scission

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Essence of computa;onal method• Shape parametrization : Two-center model in 3 and 4

dimensions

• Potential energy -- macroscopic-microscopic method •  Macroscopic part : Krappe-Nix (double-folded Yukawa model)•  Microscopic part : two-center shell model or two-center

Woods-Saxon model + Strutinsky + BCS

•  Excitation energy dependence of the shell damping : Ignatyuk or Randrup-Moeller prescription

• Transport coefficients•  Werner-Wheeler method for mass tensor (3D and 4D)

•  Wall-and-Window model for friction tensor (3D and 4D)

•  Linear response theory for microscopic mass and friction tensors (3D only) (Ivaniuk et al., JNST DOI:

10.1080/00223131.2015.1070111)

• 

43

44

𝐩(𝟏𝟖𝟎𝐌𝐞𝐕)+^↑𝟐𝟕↓𝐀𝐥 𝐛𝐲 𝐀𝐌𝐃 

impact parameter:0<b<6force:SLy4

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Nuclear fission by An;symmetrized Molecular Dynamics: AMD

•  EventhoughtheLangevin-modeldescripQongivessaQsfactoryresultsinmanyaspectsofnuclearfission,ithasonly3to4degrees-of-freedomtoexpressnuclearshape.Itshouldbebemertohavewaystodescribethenuclearfissionbymicroscopictheorieshavingd.o.f.ofallthenucleons

•  AMDisaverysuccessfulmicroscopicmodelofnuclearreacQonsandstructures(A.Onoetal,Prog.Theor.Phys.,87(5),(1992),1185-1206)

•  WavefuncQonofeachnucleonisexpressedbyaGaussianwavepacketregardedasacoherentstate

•  ThetotalwavefuncQonisgivenbyaSlaterdeterminantofthesesingle-parQclewavefuncQons→ unlikeQMD,thisisaproperlyanQsymmetriedquantum-mechanicaltheory

•  SkyrmeinteracQonisintroducedasaneffecQvenuclearforce•  Ground-stateofnucleiareobtainedbyfricQonalcoolingmethod,whichyieldsgoodBE/Aandnuclearradii

•  TheequaQonofmoQonofwavepacketsisgivenbyQme-dependentvariaQonalprinciple

•  StochasQcnucleon-nucleoncollision,whichisabsentinTDHFtheory,isintroducedwithPauliblockingeffectinthefinalstate,whichwillactasasourceof2-bodyfricQon. 45

dynamical microscopic and macro- micro approaches to

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