fission dynamics with microscopic level...

Oslo:May2017 0JørgenRandrup

236U

6thWorkshoponNuclearLevelDensi6esandGammaStrength

Fissiondynamicswithmicroscopicleveldensi4es

DanielWard1,GillisCarlsson1,ThomasDøssing2,PeterMöller3,JørgenRandrup4,SvenÅberg1

1TekniskaHögskolan,UniversitetiLund,S-22100Lund,Sverige2NielsBohrInsJtutet,KøbenhavnsUniversitet,DK-2100KøbenhavnØ,Danmark3LosAlamosNaJonalLaboratory,LosAlamos,NewMexico87545,USA4LawrenceBerkeleyNaJonalLaboratory,Berkeley,California94720,USA

Oslo,8-12May2017

Energydependenceofthenuclearshapeevolu4on

Editors’Sugges4on:Phys.Rev.C95,024618(2017)

Oslo:May2017

N.Bohr&J.A.Wheeler,PhysRev56(1939)426:TheMechanismofNuclearFission

Nuclearfissionisaresultofshapedynamics

JørgenRandrup 1

JohnA.Wheeler(1911-2008) NielsBohr(1885-1962)

LiseMeitner(1878-1968)O`oR.Frisch(1904-1979)

L.Meitner&J.A.O.R.Frisch,Nature143(1939)239:Disintegra4onofUraniumbyNeutrons:ANewTypeofNuclearReac4on

JørgenRandrup Oslo:May2017 2

TheshapemoJonishighlydissipa4ve:

SmoluchowskiequaJon:0=Fcons+Fdiss

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1.00

Distance between Mass Centers r (Units of R0)


Frag

men

t Elo

ngat

ion σ

(Uni

ts o

f R0)

Frag

men

t Elo

ngat

ion σ

(Uni

ts o

f R0)

Family of shapes considered

LangevinequaJon:dp/dt=Fcons+Fdiss

Nuclearshapedynamics->randomwalk

TheJmeevoluJonofthenuclearshapeparameters,q(t):

PaulLangevin(1872-1946)

MarianSmoluchowski(1872–1917)

M(q) U(q) γ(q)

U(q) γ(q)

BrownianmoJon

IfP(Af)is≈insensi4vetoγ(q):RandomwalkonU(q)

236U

Metropoliswalk…

Startatground-state(orisomeric)minimum

Walkun4ltheneckhasbecomethin…

ElongaJon

Asym

metry

Pup=exp(-ΔU/T)Pdown=1

! ?

…onthepoten4al-energysurface:


P.Möller,Nucl.Phys.A192(1972)529

ΔU:ChangeinpotenJal

T:Localtemperature

Metropolis,Rosenbluth2,Teller2,J.Chem.Phys.21(1953)1087

NicholasC.Metropolis(1915-1999)

…thenbinthemassasymmetry

elongation

Utotal

Umacro

Poten4alenergy:Macroscopic-microscopicmethod

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Frag

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ngat

ion σ

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ts o

f R0)

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men

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ngat

ion σ

(Uni

ts o

f R0)


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ngat

ion σ

(Uni

ts o

f R0)

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men

t Elo

ngat

ion σ

(Uni

ts o

f R0)


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ion σ

(Uni

ts o

f R0)

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(Uni

ts o

f R0)


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ion σ

(Uni

ts o

f R0)

Frag

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ngat

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(Uni

ts o

f R0)


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Frag

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ion σ

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ts o

f R0)

Frag

men

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ngat

ion σ

(Uni

ts o

f R0)


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Frag

men

t Elo

ngat

ion σ

(Uni

ts o

f R0)

Frag

men

t Elo

ngat

ion σ

(Uni

ts o

f R0)


Finite-rangeliquiddrop:

U(Z,N,shape)=Umacro(Z,N,shape)+Umicro(Z,N,shape)

Single-par4clelevelsintheeffec4vefield

Shell&pairing:


Swiatecki1963Stru4nsky1966

Umacro=Evol+Esurf+Ecoul+…

Umicro=Eshell+Epair

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Frag

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men

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ts o

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ion σ

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ts o

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men

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Frag

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t Elo

ngat

ion σ

(Uni

ts o

f R0)

Frag

men

t Elo

ngat

ion σ

(Uni

ts o

f R0)


NuclearshapeevoluJonasarandomwalkonthe5DpotenJalenergylandscape


J.RandrupandP.Möller,Phys.Rev.Le`.106,132503(2011)

ElongaJonQ2NeckradiuscDefsεf1&εf2Asymmetry α

Q2

45 Q2 ~ Elongation (fission direction)

15 εf1 ~ Left fragment deformation

εf1 εf2

15 εf2 ~ Right fragment deformation

15

⊗

⊗

⊗

⊗d ~ Neck

d

35 αg ~ (M1-M2)/(M1+M2) Mass asymmetry

Five Essential Fission Shape Coordinates

M1 M2

⇒ 5 315 625 grid points − 306 300 unphysical points⇒ 5 009 325 physical grid points

RayNix1969

5Dshapefamily

U(q)=Umacro(q)+Umicro(q)

q

a b

c d

Exp. 233U(n,f) Calc. (6.54 MeV) 234U

30 40 50 60

0

5

10

15

20

25

Exp. 239Pu(n,f) Calc. (6.84 MeV) 240Pu

0

5

10

15

20

25

Yiel

d Y(

Z f) (

%)

Exp. 235U(n,f) Calc. (6.54 MeV) 236U

Exp. 234U(γ,f) Calc. (11.0 MeV) 234U

30 40 50 60 Fragment Charge Number Zf

>5Mshapespernucleus

>5knuclei


P.Möller&J.Randrup,PRC91,044316(2015)

Asymmetric Symmetric0.2 0.4 0.6 0.8

Fission-Fragment Symmetric-Yield to Peak-Yield Ratio

90 100 110 120 130 140 150Neutron Number N

70

80

90

Prot

on N

umbe

r Z

Asymmetric Symmetric0.2 0.4 0.6 0.8

Fission-Fragment Symmetric-Yield to Peak-Yield Ratio

90 100 110 120 130 140 150Neutron Number N

70

80

90

Prot

on N

umbe

r Z

A.N.Andreyevetal.,PRL105,252502(2010)

180Hg

a b

c d

Exp. 233U(n,f) Calc. (6.54 MeV) 234U

30 40 50 60

0

5

10

15

20

25

Exp. 239Pu(n,f) Calc. (6.84 MeV) 240Pu

0

5

10

15

20

25

Yiel

d Y(

Z f) (

%)

Exp. 235U(n,f) Calc. (6.54 MeV) 236U

Exp. 234U(γ,f) Calc. (11.0 MeV) 234U

30 40 50 60 Fragment Charge Number Zf

J.RandrupandP.Möller,Phys.Rev.Le`.106(2011)132503


EnergydependenceofthefissionshapeevoluJon

UseaneffecJveenergylandscapeobtainedbysuppressingthemicroscopicterms

UsemicroscopicleveldensiJestoguidetherandomwalk

J.RandrupandP.Möller,Phys.Rev.C88,064606(2013)

D.E.Ward,B.G.Carlsson,T.Døssing,P.Möller,J.Randrup,S.Åberg,Phys.Rev.C95,024618(2017)

elongation

Utotal

Umacro

Energy-dependenteffec4vepoten4alenergy

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Frag

men

t Elo

ngat

ion σ

(Uni

ts o

f R0)

Frag

men

t Elo

ngat

ion σ

(Uni

ts o

f R0)


0.75 1.00 1.25 1.50 1.750.50

0.75

1.00



Frag

men

t Elo

ngat

ion σ

(Uni

ts o

f R0)

Frag

men

t Elo

ngat

ion σ

(Uni

ts o

f R0)


0.75 1.00 1.25 1.50 1.750.50

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1.00



Frag

men

t Elo

ngat

ion σ

(Uni

ts o

f R0)

Frag

men

t Elo

ngat

ion σ

(Uni

ts o

f R0)


0.75 1.00 1.25 1.50 1.750.50

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Frag

men

t Elo

ngat

ion σ

(Uni

ts o

f R0)

Frag

men

t Elo

ngat

ion σ

(Uni

ts o

f R0)


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Frag

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t Elo

ngat

ion σ

(Uni

ts o

f R0)

Frag

men

t Elo

ngat

ion σ

(Uni

ts o

f R0)


0.75 1.00 1.25 1.50 1.750.50

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Frag

men

t Elo

ngat

ion σ

(Uni

ts o

f R0)

Frag

men

t Elo

ngat

ion σ

(Uni

ts o

f R0)


UE(Z,N,shape)=Umacro(Z,N,shape)+Umicro(Z,N,shape)


E

×S(E*(shape))

E*-dependentsuppressionfuncJon S(E*)

TheshellandpairingcorrecJonswerecalculatedforT=0;buttheygenerallydiminishwithincreasingtemperature=>Energy-dependenteffec4vepotenJalenergy:

E*(shape)=E-U(shape)

E*

J.RandrupandP.Möller,Phys.Rev.C88(2013)064606

Leveldensi4esindynamics


PotenJalenergyU(χ)

Shapecoordinateχ

TotalenergyE

LocalstaJsJcalexcitaJonE*(χ)=E–U(χ)

LocalleveldensityρE(χ)

Detailedbalance:

Temperature:

Drivingforce:≈exp(-ΔU/T)

CollaboraJonforthepurposeofobtainingleveldensiJesforallrelevantfissionshapes:GillisCarlsson,ThomasDøssing,PeterMöller,JørgenRandrup,DavidWard,SvenÅberg

(=>Metropolis)

DETAILEDBALANCE

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H.Uhrenholt,S.Åberg,A.Dobrowolski,Th.Døssing,T.Ichikawa,P.Möller:NPA913(2013)127

groundstate 2p-2hstate

Combinatorialmethodforthenuclearleveldensity

=>

ConsiderallmulJplep-hexcitaJonsforprotonsandneutronsseparately

CalculateBCSpairingforeachone

E=Ep+En+Erot

Expectedtobeunimportant=>ignored

Erot(I,K)=[I(I+1)-K2]/2Iperp(χ,Δp,Δn)RotaJonalbandbuiltoneachintrinsicstate:

Intrinsicstates:

Rota4onalenhancement

Vibra4onalenhancement

OBS:HigherI=>LowerEintr

1p-1hstatePairing

A.Schiller,etal.,PRC63(2001)021306(R)


H.Uhrenholt,S.Åberg,A.Dobrowolski,Th.Døssing,T.Ichikawa,P.Möller:NPA913(2013)127

Combinatorialmodelforthenuclearleveldensity

=>

E.Melby,etal.,PRC63(2001)044309

S.Siem,etal.PRC65(2002)044318

M.Gu`ormsen,etal.,PRC68(2003)064306


Idea/plan:

Usethecombinatorialmethodtocalculatethemicroscopicleveldensityforall(>5M)3QSshapesforwhichthepotenJalhasbeentabulated:

ρZA(E,I,shape)foreachindividualfissioningnucleusAZ(UZA(shape)existsfor>5kAZ)

Usethoseasthebasisfortherandomwalk:Pdown:P(U’≤U)=1-->P(ρ’≥ρ)=1Pup:P(U’≥U)=exp(-ΔU/T)-->P(ρ’≤ρ)=ρ’/ρ

ThenthegradualdisappearanceofpairingandshelleffectswithexcitaJonisautoma4callyincludedintheshapeevoluJon

=>PhDthesisprojectforDanielWard(Lund)

AsymmetricshapesReplace{εn}by3QSGetalls.p.levels(PM)

TrivialcodemodificaJon

Fullyconsistent:sames.p.levelsusedforUandρ(noparameters)

Friday13Jan2017

22ndASRCInt’lWorkshopTokai,3-5December2014:SvenÅberg(Lund,Sweden)

KrapperupCastle

PlanningmeeJng:


0

5

10

15

20

25

Yiel

d Y(Z f

) (%

)

a) 234U (6.84 MeV) ExpUEρmic

0

5

10

15

20

Yiel

d Y(Z f

) (%

)

b) 234U (11 MeV)

30 40 50 60 70Fragment proton number Zf

0

5

10

15

20

Yiel

d Y(Z f

) (%

)

c) 234U (16 MeV)

0

5

10

15

20

25

Yiel

d Y(Z f

) (%

)

0

5

10

15

20

Yiel

d Y(Z f

) (%

)

ExpρmicUE

20 30 40 50 60 70Fragment proton number Zf

0

5

10

15

20

Yiel

d Y(Z f

) (%

)

b) 236U (6.55 MeV)

c) 240Pu (6.53 MeV)

a) 226Th (11.20 MeV)

LeveldensiJesversussuppressionfactor


05

10152025

Yiel

d Y(Z f

) (%

)

I = 0


05

101520

Yiel

d Y(Z f

) (%

)

Even IOdd I

a) 234U (6.84 MeV)

b) 234U (11 MeV)

DependenceonangularmomentumI

I=0:smallestI=>largestT

Erot(I,K)=[I(I+1)-K2]/2Iperp(χ,Δp,Δn) OBS:HigherI=>LowerEintr

I=0,2,4,6,8

I=1,3,5,7,9

Fissiondynamicswithmicroscopicleveldensi4es:


ThenuclearshapeevoluJonisakintoBrownianmoJonandcanbeapproximatelydescribedasarandomwalkonthemulJ-dimensionaldeformaJon-energysurface

TheenergydependenceoftheshapeevoluJonhadbeentreatedbymeansofasuppressionfuncJon;thoughquitesuccessful,thisapproachisnottheoreJcallysaJsfactory

Ageneral&consistentdescripJonwasobtainedbyusingthemicroscopicleveldensiJescalculatedforeachshapebymeansofarecentlydevelopedcombinatorialmethod;thegradualdisappearanceofshellandpairingeffectsisthenautomaJcallyensuredwithoutanynewparameters

✔

✔

✔

UE=Umacro+Umicro×S(E*)

Energydependenceofthenuclearshapeevolu4on

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Frag

men

t Elo

ngat

ion σ

(Uni

ts o

f R0)

Frag

men

t Elo

ngat

ion σ

(Uni

ts o

f R0)


0.75 1.00 1.25 1.50 1.750.50

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1.00



Frag

men

t Elo

ngat

ion σ

(Uni

ts o

f R0)

Frag

men

t Elo

ngat

ion σ

(Uni

ts o

f R0)


0

5

10

15

20

25

Yie

ld Y

(Zf)

(%)

a) 234U (6.84 MeV) ExpUEρmic

0

5

10

15

20

Yie

ld Y

(Zf)

(%)

b) 234U (11 MeV)


0

5

10

15

20

Yie

ld Y

(Zf)

(%)

c) 234U (16 MeV)

fission dynamics with microscopic level...

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