-
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
0.75 1.00 1.25 1.50 1.750.50
0.75
1.00
Distance between Mass Centers r (Units of R0)
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:
Oslo:May2017 3JørgenRandrup
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
0.75 1.00 1.25 1.50 1.750.50
0.75
1.00
Distance between Mass Centers r (Units of R0)
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
0.75 1.00 1.25 1.50 1.750.50
0.75
1.00
Distance between Mass Centers r (Units of R0)
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
0.75 1.00 1.25 1.50 1.750.50
0.75
1.00
Distance between Mass Centers r (Units of R0)
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
0.75 1.00 1.25 1.50 1.750.50
0.75
1.00
Distance between Mass Centers r (Units of R0)
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
0.75 1.00 1.25 1.50 1.750.50
0.75
1.00
Distance between Mass Centers r (Units of R0)
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
0.75 1.00 1.25 1.50 1.750.50
0.75
1.00
Distance between Mass Centers r (Units of R0)
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
Finite-rangeliquiddrop:
U(Z,N,shape)=Umacro(Z,N,shape)+Umicro(Z,N,shape)
Single-par4clelevelsintheeffec4vefield
Shell&pairing:
Oslo:May2017 4JørgenRandrup
Swiatecki1963Stru4nsky1966
Umacro=Evol+Esurf+Ecoul+…
Umicro=Eshell+Epair
0.75 1.00 1.25 1.50 1.750.50
0.75
1.00
Distance between Mass Centers r (Units of R0)
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
0.75 1.00 1.25 1.50 1.750.50
0.75
1.00
Distance between Mass Centers r (Units of R0)
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
0.75 1.00 1.25 1.50 1.750.50
0.75
1.00
Distance between Mass Centers r (Units of R0)
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
-
NuclearshapeevoluJonasarandomwalkonthe5DpotenJalenergylandscape
Oslo:May2017 5JørgenRandrup
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
-
JørgenRandrup Oslo:May2017 6
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
Oslo:May2017 7JørgenRandrup
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
0.75 1.00 1.25 1.50 1.750.50
0.75
1.00
Distance between Mass Centers r (Units of R0)
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
0.75 1.00 1.25 1.50 1.750.50
0.75
1.00
Distance between Mass Centers r (Units of R0)
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
0.75 1.00 1.25 1.50 1.750.50
0.75
1.00
Distance between Mass Centers r (Units of R0)
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
0.75 1.00 1.25 1.50 1.750.50
0.75
1.00
Distance between Mass Centers r (Units of R0)
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
0.75 1.00 1.25 1.50 1.750.50
0.75
1.00
Distance between Mass Centers r (Units of R0)
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
0.75 1.00 1.25 1.50 1.750.50
0.75
1.00
Distance between Mass Centers r (Units of R0)
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
UE(Z,N,shape)=Umacro(Z,N,shape)+Umicro(Z,N,shape)
Oslo:May2017 8JørgenRandrup
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
Oslo:May2017 9JørgenRandrup
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
-
0.75 1.00 1.25 1.50 1.750.50
0.75
1.00
Distance between Mass Centers r (Units of R0)
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
JørgenRandrup Oslo:May2017 10
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)
JørgenRandrup Oslo:May2017 11
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
-
JørgenRandrup Oslo:May2017 12
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:
-
JørgenRandrup Oslo:May2017 13
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
-
JørgenRandrup Oslo:May2017 14
05
10152025
Yiel
d Y(Z f
) (%
)
I = 0
30 40 50 60 70Fragment proton number Zf
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:
Oslo:May2017 15JørgenRandrup
ThenuclearshapeevoluJonisakintoBrownianmoJonandcanbeapproximatelydescribedasarandomwalkonthemulJ-dimensionaldeformaJon-energysurface
TheenergydependenceoftheshapeevoluJonhadbeentreatedbymeansofasuppressionfuncJon;thoughquitesuccessful,thisapproachisnottheoreJcallysaJsfactory
Ageneral&consistentdescripJonwasobtainedbyusingthemicroscopicleveldensiJescalculatedforeachshapebymeansofarecentlydevelopedcombinatorialmethod;thegradualdisappearanceofshellandpairingeffectsisthenautomaJcallyensuredwithoutanynewparameters
✔
✔
✔
UE=Umacro+Umicro×S(E*)
Energydependenceofthenuclearshapeevolu4on
0.75 1.00 1.25 1.50 1.750.50
0.75
1.00
Distance between Mass Centers r (Units of R0)
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
0.75 1.00 1.25 1.50 1.750.50
0.75
1.00
Distance between Mass Centers r (Units of R0)
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
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)
30 40 50 60 70Fragment proton number Zf
0
5
10
15
20
Yie
ld Y
(Zf)
(%)
c) 234U (16 MeV)